The Network Packing Problem in Terrestrial Broadcasting
C. Mannino
¤
F. Rossi, S. Smriglio
y
Abstract The introduction of digital terrestrial broadcasting all over Europe requires a complete and challenging re-planning of in-place analog systems. But an abrupt migration of resources (transmitters and frequencies) from analog to digital networks
cannot be accomplished, since the analog services must be preserved temporar-
ily. Hence, a multi-network, multi-objective, problem arises, referred to as the
Network Packing Problem, in which several networks, both analog and digital, sharing a common set of resources, have to be designed. In Italy this problem is particularly challenging, because of a large number of transmitters, orographical
features and strict requirements imposed by Italian law. In this paper we report
our experience developing solution methods at the major Italian broadcaster Ra-
diotelevisione Italiana (RAI S.p.A.). We propose a two-stage heuristic. In the ¯rst stage emission powers are assigned to each network separately. In the second stage frequencies are assigned to all networks so as to minimize the loss from mu-
tual interference. A software tool incorporating our methodology is currently in use at RAI to help discover and select high-quality alternatives for the deployment of digital equipment.
Subject classi¯cations: Communications: Frequency and Power Assignment. Programming, integer, heuristic: neighbourhood search. ¤
Dipartimento di Informatica e Sistemistica, Universitµ a di Roma \La Sapienza", e-mail:
[email protected] y Dipartimento di Informatica, Universitµ a di L'Aquila, e-mail: rossi[smriglio]@di.univaq.it
1
1
Introduction
In the context of terrestrial video broadcasting, digital technology most likely will re-
place analog technology. This transition has been recently claimed (Paris, September 2003) by the French and German premiers as one of the main challenges of the Eu-
ropean Union development programs [4]. The Terrestrial Digital Video Broadcasting standard (DVB-T) has been introduced by European Telecommunication Standard Institute (ETSI) in 1997. Details on the development of DVB-T can be found in the web sites of the major public bodies involved, namely, ETSI [12] and the International Telecommunication Union (ITU) [18], as well as in the industry-led consortium Digital
Video Broadcasting [9]. DVB-T has several advantages. First, it utilizes the available
bandwidth more e±ciently than analog systems: in fact, it allows to broadcast a multiplex of programs (from two to ¯ve, depending on the required quality of service) over a 8 Mhz width portion of the frequency spectrum (referred to as channel) instead of a single
analog program. Second, DVB-T embeds a Multimedia Home Platform representing a
¯rst step towards the integration of video broadcasting with both Internet and Global System for Mobile Communications (GSM) [21], making additional (interactive) services, such as e-commerce and home-banking, accessible through television [3]. Finally,
receiving DVB-T programs only requires the installation of a decoder with actual roof antenna, a modest overload for users. DVB-T implementation is under investigation in several European countries, such as Great Britain, Spain, Germany, Norway, Finland
and Denmark [3]: the current status for each country can be found in [11]. Worldwide launch dates for DVB-T and analog switch-o® dates are reported in [9].
In this paper we address new planning issues arising in the Italian broadcasting sys-
tem because of the analog to digital transition. Due to the large number of broadcasters and to the scarce penetration of cable and satellite networks, the Italian context appears to be one of the most complex in Europe (details are reported in [17]). For instance,
23,506 transmitters operate versus 12,455 in France and 10,099 in Germany (European
Radiocommunications O±ce, May 2002 [11]). A recent study on the introduction of digital broadcasting has been carried out by the Italian Communications Authority (AGCOM). The results, reported in the White Book [3], provide the ground for the
Digital National Plan for Frequency Assignment (D-PNAF) [3], devised by AGCOM
in the year 2003. The purpose of D-PNAF is to rationalize the resource utilization so 2
as to guarantee the open competition in the video market. In particular, D-PNAF de-
¯nes a target con¯guration for the future Italian broadcasting system, consisting of 18 equivalent reference networks. Each network utilizes three channels and 300 transmit-
ters (geographical sites) in order to broadcast a multiplex of ¯ve programs to at least 90% of the Italian population. The implementation of D-PNAF looks quite complex and will take place through several steps. At the time of this study, the Italian law 66/2001 ¯xed two important milestones: (i) D-PNAF has to be implemented within
the end of year 2006, when all analog networks will be switched-o®; (ii) the public Italian broadcaster Radiotelevisione Italiana (RAI), has to activate two digital networks reaching at least 50% of the population by January, 1, 2004, keeping safe the actual
coverage of analog networks. RAI, a state-owned Company with an executive board
designated by the Parliament, is also the major Italian broadcaster. The present study
was promoted by RAI Way S.p.A, the RAI network provider, to understand whether requirement (ii) could be accomplished with the available (and currently used entirely by analog networks) resources.
Even when not required by law, other broadcasters also must face the same problem.
In fact, the available system capacity (which depends on available transmitters and
channels) is not su±cient to grow one or more digital networks without reducing the
coverage of analog networks. On the other hand, the resources from analog networks
cannot be released sharply since the actual service must be provided until the new networks guarantee a su±cient coverage. Under this respect, the high level reached
by analog networks (i.e., around 95% of population) represents a severe barrier for the set-up of digital networks which appear, at the beginning, not competitive from a
commercial point of view. Managing this transition is actually the key challenge for major broadcasters.
In conclusion, either the implementation of D-PNAF or, simply, the introduction
of new digital networks requires an expensive re-planning activity in which a set of resources is shared among di®erent (analog or digital) networks. Since each network
has its own technical features and commercial goals, the problem consists in allocating resources to networks so as to optimize a multi-objective performance indicator (e.g.,
the population coverage for each network). We designate this problem as Network Packing Problem (NPack, x3). In Section 5 we illustrate a two-phase heuristic for the solution of NPack. This is based on two major ingredients: (i) an algorithm for the 3
optimal assignment of power and frequency to transmitters of a single network (Single Network Planning problem, NPlan, x4) and (ii) an algorithm for the solution of a variant of the classical Frequency Assignment Problem (FAP, x5.2).
This paper introduces three major novelties in the models addressed so far by the
OR community involved in frequency assignment. First, the coverage assessment is based on the de¯nition of testpoints and is carried out by a statistical procedure recommended for implementation purposes [3, 8]. Second, unlike problems investigated in the literature, NPlan includes both emission powers and frequencies as decision
variables and simultaneously optimizes over these variables. Third, several networks are optimized on a common set of resources.
The resulting software tool has been tested together with RAI Way engineers and
is designed to cope with their assessed planning procedures. In particular, the tool not only returns good solutions for very large scale instances, but also allows engineers to evaluate several high-quality alternatives.
Computational results from several key experiments proposed by RAI Way engi-
neers are reported in x7 and their in°uence on strategic decisions is discussed. RAI
investments in DVB-T implementation concern technological infrastructures and pur-
chase of frequencies. As for the former, RAI management approved a 79 million Euro budget in August 2003 (from the major Italian ¯nancial newspaper: Il Sole 24 Ore, August, 7, 2003).
2
System elements
A broadcasting system consists of a set R = f1; : : : ; jRjg of (either analog or digital)
networks. Network k distributes its own program from a set T k of transmitters. T = SjRj
r=1
T r denotes the whole transmitter set, where T k \ T h = ;, for all networks h 6 = k.
Each transmitter is located in a geographical site (typically a site accommodates sev-
eral transmitters of di®erent networks) and its con¯guration depends on: transmission
frequency, emission power, antenna height, polarization (horizontal/vertical), antenna diagram (directivity) and time o®set. In this paper we deal with deciding frequencies and emission powers of transmitters, while all other quantities are ¯xed and will be referred to as transmitter parameters (see x7).
The frequency spectrum is subdivided into a set F = f1; : : : ; jF jg of equally sized 4
intervals called channels (or frequencies). The set of feasible frequencies for transmitter i 2 T is denoted by Fi µ F. It may happen that Fi ½ F because of technical or commercial constraints (international agreements at boundaries, licensing, etc.).
The transmission frequency assigned to transmitter i 2 T is denoted by fi , whilst
Pi denotes its emission power, ranging in the interval [0; PiU ]. For convenience, we also introduce the power fading pi 2 [0; 1], that measures power attenuation w.r.t. the maximum power PiU (i.e., Pi = pi ¢ PiU ).
Network k is designed to distribute video programs within a given territory por-
tion called target area. This is decomposed into a set Z k of \small", approximatively squared, areas (about 2 £ 2 Km) called testpoints (T P s). Z = [r=1;:::;R Z r denotes the whole testpoint set.
Each testpoint, identi¯ed by its coordinates, represents the behavior of all receivers
(i.e., roof antennae with TV sets) within it. In practice, distinct testpoints belonging to di®erent networks target areas often coincide on the map. All antennae in a TP have the same, ¯xed, directivity (see [8] for details). A revenue uj 2 Z+ is de¯ned for each TP j. Here, uj equals the number of inhabitants of TP j (for S µ Z, u(S) =
P
j2S
uj ).
The signal emitted by a transmitter propagates according to transmitter directivity
and orography. The power density Pij (watt=m2 ) received in TP j from transmitter i is proportional to the emission power. In particular, Pij = Aij ¢ Pi , where Aij 2 [0; 1]
is de¯ned through a propagation model (detailed in [3]) and is given for each pair i 2 T; j 2 Z. A general treatment of propagation models can be found in [26]. We refer to the matrix [A] = [Aij ]i2T;j2Z as fading matrix. Finally, given a TP j, T (j) = fi 2 T : Aij 6 = 0g µ T denotes the set of signals received in j.
Whenever a network k is received "clearly" in a TP j, this is said to be covered by
k. The coverage area of network k is the subset C k µ Z k of testpoints covered under the current assignment of emission powers (vector pk ) and frequencies (vector f k ) to
transmitters. Coverage depends on system type (i.e., analog or digital) and receiver behavior, as described in the next section.
2.1
Coverage evaluation
We resume the receiver behavior (more details can be found in [6]) and coverage eval-
uation models adopted for practical implementation purposes [8, 10]. For the sake of 5
presentation, we start with a single network in which all transmitters have the same frequency (i.e., jF j = 1).
It is well known that the quality of service increases with the received power level
of wanted signals and degrades with interfering signals. In analog systems, di®erent signals arriving on a receiver with the same frequency always interfere (co-channel in-
terference). On the contrary, the Orthogonal Frequency Division Multiplexing (OFDM) scheme, adopted in the DVB-T standard, permits receivers to combine co-channel sig-
nals so as to obtain a stronger wanted signal. To explain this, recall that, in DVB-T
systems, transmitters (in the same network) broadcast simultaneously the same data symbol; symbols arrive in a receiver (in TP j) according to the travel times over the cor-
responding transmitter-TP distances. The receiver in TP j locates a pass band ¯lter at time ¿j (detection window), represented in Figure 1. A signal (symbol) from transmitter i arriving in TP j at time ¿ij fully contributes to the wanted signal if ¿j · ¿ij · ¿j + Tg
while it is fully interfering if ¿ij > ¿j + 2Tg . In the technical literature, Tg is known as Guard Interval and a typical value for commercial receivers is Tg = 224 ¹sec. Formally,
the power contribution from the i¡th transmitter to the wanted signals is wij Pij and
the contribution to the interfering signal is (1 ¡ wij )Pij , where wij is the weighting function [10]:
8 1 >
(1:5 ¡ : 0
m
l
if ¿j · ¿ij · ¿j + Tg
if ¿j + Tg · ¿ij · ¿j + 2Tg
¿ij ¡¿j 2 ) 2Tg
otherwise
k j
(1)
h h
tj
l
tj+Tg
k
tj+2Tg
m
Figure 1: TP j synchronized with signal h: h and l are fully wanted, m fully interfering, k is both wanted and interfering
Although the number of window positions is in¯nite, it su±ces to consider only
values of ¿j corresponding to signal arriving times, i.e., ¿j = ¿hj , h 2 T (j) (see [27]). 6
For this reason, a window position in TP j is denoted by the corresponding signal
h 2 T (j). Once a position h is chosen, the sets W (j; h) = fh : whj Phj > 0g of wanted contributions and I(j; h) = fh : (1 ¡ whj )Phj > 0g of interfering contributions
are identi¯ed. After such a classi¯cation, the coverage quality of TP j is assessed by
statistical methods. According to [3], we use the k-LNM method [5] [10]. Given a position h of the detection window, k-LNM combines signals W (j; h) and I(j; h) so as
W 2 to obtain two Log-normal distributions with mean value Ptot (h) and variance ¾W;tot (h) I 2 for the wanted contributions (Ptot (h) and ¾I;tot (h) for the interfering contributions). If
we let
¢P (h) = q
W I Ptot (h) ¡ Ptot (h)
(2)
2 2 ¾W;tot (h) + ¾I;tot (h)
then TP j is regarded as being covered under window h if Erf (¢P (h)) ¸ 0:95
(3)
where Erf (¢) is the Gaussian error function. Summarizing, deciding whether TP j is covered (under the current setting) or not
consists in ¯nding a detection window h satisfying (3) or proving that none exists.
Therefore, coverage evaluation in TP j is carried out by enumerating all jT (j)j val-
ues Erf (¢P (h)). If j is covered, the receiver chooses signal h = s(j) maximizing Erf (¢P (h)), called reference signal.
In analog networks, the procedure is simpler: only the strongest signal is wanted;
all other co-channel signals are interfering and combined by Power Sum method [10].
In case of multiple frequencies (jF j > 1), TP j is regarded as being covered if it is
covered for at least one frequency in F . Hence, TP coverage evaluation is repeated for each set of co-channel signals. In practice, this corresponds to tuning the receiver on each frequency of the network.
When multiple networks are concerned, one must regard as interfering all co-channel
signals received in TP j 2 Z k from any network h 6 = k.
In conclusion, the coverage C k of network k is computed by evaluating the coverage
of each testpoint in Z k : it depends on emission powers and frequencies assigned to all
7
F = f1; : : : ; jFjg (Fi ) R = f1; : : : ; jRjg T (T k ) Z (Z k ) T (j), j 2 Z fi Pi (PiU ) pi Aij Pij s(j) (pk ; f k ; sk ) W (j; h), h 2 T (j) I(j; h), h 2 T (j) uj C k (pk ; f k ; sk )
set of available frequencies (of transmitter i) set of networks set of transmitters (of network k) set of testpoints (target area of network k) set of transmitters received in j transmission frequency of transmitter i emission power (upper bound for emission power) of transmitter i power fading at transmitter i (w.r.t. PiU ) fading out from propagation between transmitter i and TP j power density [W=m2 ] measured in TP j from transmitter i reference signal for TP j network con¯guration for network k. set of (co-channel) wanted contributions in TP j under window h set of (co-channel) interfering contributions in TP j under window h revenue (typically, number of inhabitants) of testpoint j coverage of network k
Table 1: Summary of notation transmitters in T (not only to those in T k ) and on chosen reference signals. These are represented by a (characteristic) vector sk 2 f0; 1gjZ skjh =
(
k j£jT k j
, where
1 if h 2 T k is reference signal of j 2 Z k 0 otherwise
(4)
The vector triple (pk ; f k ; sk ) is called con¯guration of network k. Observe that, once
pk and f k have been ¯xed for all networks, the coverage evaluation procedure returns the vector of reference signals sk as well as the resulting coverage area C k (pk ; f k ; sk ), for all k 2 R.
3
Problem de¯nition
Problem 3.1 (NPack) Given a set R = f(T 1 ; Z 1 ); : : : ; (T R , Z R )g of networks, the fading matrix [A]i2T;j2Z , the power upper bound vector PU 2 IRjT j and the feasible frequency domain Fi 2 F for all i 2 T , the Network Packing Problem is the one of ¯nding network con¯gurations f(p1 ; f 1 ; s1 ); (p2 ; f 2 ; s2 ); : : : ; (pR ; f R ; sR )g such that the vector function fu(C 1 (p1 ; f 1 ; s1 )); u(C 2 (p2 ; f 2 ; s2 )); : : : ; u(C R (pR ; f R ; sR ))g is maximized. 8
AGCOM AS DVB-T FAP MFAP NPack NPlan OFDM RAI TP
the Italian Authority for Communications Antenna Siting problem Terrestrial Digital Video Broadcasting Frequency Assignment problem Multi-network Frequency Assignment problem Network Packing problem Network Planning problem: optimization of coverage revenue of a a single network Orthogonal Frequency Division Multiplexing Radiotelevisione Italiana, the Italian State owned broadcasting Company Testpoint
Table 2: Acronyms The presence of several networks yields a multi-objective optimization problem. In
Appendix A, NPack is shown to be NP-hard. A hierarchy of problems is derived from NPack for algorithmic purposes. In particular, relevant subproblems are obtained
combining three restrictions: (i) single network (R = 1); (ii) ¯xed frequencies; (iii) ¯xed emission powers.
R = 1 ) NPlan. When the system consists of a single network (i.e., R = 1), NPack
reduces to a single objective problem. For the sake of presentation, we refer to this problem as the Network Planning problem (NPlan). It has been introduced in the White Book of AGCOM [3].
Fixed powers ) MFAP. The emission power of all transmitters (of all networks) is ¯xed and frequencies have to be assigned. We refer to it as Multi-network Frequency Assignment problem (MFAP). R = 1, ¯xed powers.
NPack reduces to the Frequency Assignment problem, widely
studied in the literature [1].
R = 1, ¯xed frequencies. The frequency of all transmitters is ¯xed and NPack reduces to a problem often referred to as Antenna Siting problem (AS) [20, 27].
The problems hierarchy is summarized in Figure 2. So far, only the most constrained
problems (white boxes) have been addressed in the literature. In x5 we illustrate a 9
heuristic for NPack incorporating algorithms for NPlan and MFAP as subroutines. Multi-network (multi-objective) problems Multi-network Frequency Assigment (MFAP)
Fixed Powers
Network Packing (NPACK)
R =1
R =1 Frequency Assignment (FAP)
Fixed Powers
Network Planning (NPLAN)
Fixed Frequencies
Antenna Siting (AS)
Single network (single objective) problems
Figure 2: Problems hierarchy
Formulation, Bound and Heuristic for NPlan
4 4.1
Mixed Integer Linear Programming Formulation
A Mixed Integer Linear Programming (MILP) formulation for NPlan is here introduced. We start with a digital single frequency network (T; Z) and then discuss the multi-frequency case. The analog model turns out to be a special case of the digital
one. The key step concerns the coverage condition (3). We show that, under a suitable assumption, it becomes a linear inequality of the power fading vector p.
Let us consider a TP j 2 Z and a detection window h 2 T (j). From (3), we get: q
W I 2 2 Ptot (h) ¡ Ptot (h) ¸ Erf ¡1 (0:95) ¾W;tot (h) + ¾I;tot (h) = SIR
(5)
where Erf ¡1 (¢) is the inverse of the Gaussian error function and SIR [dBW=m2 ] is the 2 2 required Signal to Interference Ratio. The ¯gures ¾W;tot (h) and ¾I;tot (h) are evaluated
by the k-LNM method as functions of emission powers [5, 10]. Nevertheless, we let 10
2 2 ¾W;tot (h) and ¾I;tot (h) be equal to the values returned by the k-LNM method when all
emission powers are ¯xed to their upper bounds (i.e., Pi = PiU , for all i 2 T ). Preliminary computations showed that this assumption introduces a negligible approximation in coverage evaluation, while making SIR independent from emission powers. Then, recalling from [10] that Ptot = log(kP log P
P
P
i Pij )
¡
2 ¾tot , 2
we obtain:
2 ¾2 ¡ ¾I;tot Pij ¸ W;tot + SIR = £ 2 i2I(j;h) Pij + PC=N i2W (j;h)
(6)
where PC=N , representing the system noise, is treated as an interfering signal (inde-
pendent from p) with zero variance and mean value PC=N (values of PC=N for di®erent channel types are reported in [8]). After some algebra one has: X
i2W (j;h)
X
Pij ¸ 10£ (
i2I(j;h)
Pij + PC=N )
(7)
Expressing the power received in j (from i) as Pij = pi ¢Aij ¢PiU and letting aij = Aij PiU ,
for i 2 W (j; h); bij = 10£ Aij PiU for i 2 I(j; h)); dj = 10£ ¢ PC=N , the following linear inequality is obtained: X
i2W (j;h)
aij pi ¡
X
i2I(j;h)
bij pi ¸ dj
(8)
If inequality (8) is satis¯ed, then TP j is covered and h is a potential reference signal
for j. On the other hand, j is covered if there exists at least one potential reference signal h 2 T (j). This requirement can be expressed by the following set of constraints: µ
X
i2W (j;h)
aij pi ¡
X
i2I(j;h)
¶
bij pi ¡ dj wjh ¸ 0 zj ·
X
h2T (j)
whj
8h 2 T (j)
(9)
(10)
where zj 2 f0; 1g is equal to 1 if TP j is covered (0 otherwise) and wjh 2 f0; 1g is equal to 1 if signal h is a potential reference signal for j. Constraint (9) can be linearized,
as shown below. The objective function for Nplan amounts to maximizing coverage revenue. Thus, an MILP model for NPlan (single frequency digital network) reads: max
X
j2Z
uj z j
subject to 11
X i2W (j;h)
aij pi ¡
X i2I(j;h)
bij pi ¡ Mwjh ¸ dj ¡ M zj ·
X
h2T (j)
whj
0 · pi · 1 8i 2 T
8j 2 Z; 8h 2 T (j) (11) 8j 2 Z
zj 2 f0; 1g 8j 2 Z
where M is a constant larger than
wjh 2 f0; 1g 8j 2 Z; 8h 2 T (j)
P
i2I(j;h) bij
+ dj .
The model also applies to analog networks where only one positive contribution
(reference signal) appears in constraint (11).
Observe that an optimal solution (p¤ ; w¤ ; z¤ ) may not identify a unique (single
frequency) network con¯guration (p¤ ; s¤ ). In fact, for each TP j, one can have more ¤ than one variable wjh equal to one. Hence, reference signals s¤ are chosen as in x2.1.
Practical applicability The above formulation can be easily extended to multifrequency network. To this purpose, new binary variables xif , for i 2 T and f 2 F, are
introduced, with the following meaning: xif is equal to 1 if frequency f is assigned to transmitter i (0 otherwise). Furthermore, each variable wjh is replaced with a variable
wjhf which is equal to 1 if signal h at frequency f is the reference signal of TP j (0 otherwise). Finally, each constraint (11) is substituted by jF j constraints, one for each f 2 F, of the following form: X i2W (j;h)
aij xif pi ¡
X i2I(j;h)
bij xif pi ¡ M wjhf ¸ dj ¡ M
(12)
Even this constraint can be easily linearized (details are omitted).
The MILP model may be of practical usage only for small sized instances of NPlan
with jFj · 2. On the contrary, for large instances the MILP model cannot be solved to optimality. For instance, a small test problem (jT j = 10; jZj = 1; 000) with jF j = 3 leads to a MILP with 30,000 variables w and 1,000 coverage constraints. Real-life instances are typically 50 times larger in size. The MILP size grows even more when the model is extended to NPack, requiring an additional dimension representing networks. Hence, in order to tackle such large scale problems, we devise the heuristic approach of x5. 12
Minimizing total emitted power Consider a network con¯guration (p; f ; s) and assume both f and s are ¯xed. Then, all binary variables xif and wjhf in (12) are ¯xed,
leading to a set of linear constraints Ap ¸ d. Letting P = fp 2 IRjT j : Ap ¸ d; p ·
11; p ¸ 0g, the linear program minp2P 110 p minimizes the total emitted power (under current reference signals). This will be of some practical usage (see x5.3).
4.2
Upper bound
An upper bound on the optimal value u(C ¤ ) = u(C(p¤ ; f ¤ ; s¤ )) of NPlan is obtained
by the following relaxation. Let us consider a network (T; Z) and a partition of Z into
k clusters Z1 , : : :, Zk . For i = 1; : : : ; k, compute the optimal values u(Ci¤ ) of (small) NPlan-s induced by (T; Zi ). An upper bound on u(C ¤ ) is given by
Pk
i=1
u(Ci¤ ).
Property 4.1 Consider the coe±cient matrix of constraints (12) in which rows in the same cluster are consecutive. If the submatrix restricted to variables p is block diagonal, then
Pk
i=1
u(Ci¤ ) equals the optimal value of NPlan.
In fact, when no cross interference among clusters occurs, the problem naturally
decomposes. The rationale of the approach relies on the existence of a matrix decomposition which is close to ful¯lling the above property. There is a trade-o® between
the quality of the bound and the size of the clusters: small clusters produce poor bounds, while large clusters may result in instances of Nplan which cannot be solved to optimality.
In order to ¯nd a decomposition, de¯ne the complete bipartite graph G = (T [Z; E),
with edge weights cij = Aij PiU , for ij 2 E. Consider a partition of vertices of G into k ¸ 2 non-empty subsets V1 ; : : : ; Vk : a k-cut of G with shores V1 ; : : : ; Vk is the set of edges with endpoints in di®erent shores (crossing edges). The weight of a k-cut is the
sum of its edge weights. A k-cut with value B in G corresponds to a partition of (T; Z) into k disjoint sub-networks T1 [ Z1 ; : : : ; Tk [ Zk with overall cross-interference equal to B.
Remark 4.2 A k-cut of value 0 identi¯es k blocks of the coe±cient matrix (12) and Property 4.1 holds.
13
In practice, a k-cut of value 0 does not exist. However, a suitable decomposition is
constructed by computing the minimum weight cut in G. Such a min k-cut problem is
solved using the METIS solver by Karypis and Kumar [25] which returns the (heuristic) partition T1 [ Z1 ; : : : ; Tk [ Zk . The required testpoint clusters are precisely Z1 ; : : : ; Zk .
The tightest bound has been obtained for k = 20 and the results are reported in
x7. In addition, we observed (see Figure 3) that the corresponding clustering closely reproduces the Italian administrative decomposition, i.e., clusters approximate regional districts (some of the clusters are too small to perceive.)
Figure 3: Partition of Italy vs. regional districts This is coherent with the fact that the actual networks have been designed and imple-
mented by controlling cross interference among di®erent districts. In several cases this occurs because the boundary between two regions is located along mountains (mainly in
the center of Italy); in other cases this is obtained by properly shaping the transmitter directivity along the regional boundaries. The existence of such a decomposition with low cross interference yields tight bounds on the coverage by the proposed relaxation.
4.3
A GRASP algorithm for NPlan
In this section we illustrate a Greedy Randomized Adaptive Search Procedure (GRASP) for NPlan. GRASP is a metaheuristic for combinatorial optimization problems which
has been successfully applied in several application contexts. The general GRASP
scheme consists of repeated applications of a randomized greedy algorithm. The latter di®ers from a standard greedy algorithm since the (exact) greedy choice is replaced by a random choice in a set of candidate solutions. The candidate set contains, besides 14
the greedy solution, those whose value is not "too far" from it. At the end of the
construction phase a local search is performed in order to provide a local optimum. For an exhaustive treatment of GRASP methods we refer the reader to [15].
Two tasks a®ect the algorithm performance: (i) optimizing over the greedy neigh-
borhood and (ii) evaluating the ¯tness function. Point (ii) requires the execution of the
coverage evaluation procedure (see x2.1): therefore, we developed an e±cient algorithm to carry out this task (see Appendix B).
As for point (i), the greedy neighborhood adopted for NPlan contains all con-
¯gurations obtained by the activation of one switched-o® transmitter. Most of the neighborhoods proposed in related literature have polynomial size (e.g. [20, 29]). On
the contrary, we de¯ne an exponential neighborhood searchable in polynomial time (exponential neighborhood search [2]). For the sake of simplicity, let us consider a network (T; Z) where all TP-s have unit revenue (i.e., u(C) = jCj). In this section (¹ p; ¹ f ;¹ s) denotes the initial con¯guration.
Neighborhood structure Let T¹ = ft 2 T : p¹t > 0g (set of active transmitters). For any i 2 T n T¹, let Ni (¹ p; ¹ f; ¹ s) be the set of con¯gurations (~ p; ~ f; ~ s) such that p~t = p¹t and f~t = f¹t , for all t 2 T , t 6 = i. Then, the neighborhood of (¹ p; ¹ f; ¹ s) is de¯ned as N (¹ p; ¹ f; ¹ s) = S
i2T nT¹
Ni (¹ p; ¹ f; ¹ s). Observe that the number of di®erent feasible vectors s 2 Ni grows
exponential in jZj and jT j, being in correspondence with the feasible assignments of TPs to reference signals. In fact, after the activation of a new transmitter i, the reference signal of every TP must be re-computed.
Neighborhood search Exploring the neighborhood N consists in searching each Ni , 8i 2 T n T¹ , and then choosing the best con¯guration obtained. Searching Ni is equivalent to ¯nding the con¯guration (p¤ ; f ¤ ; s¤ )i 2 Ni (¹ p; ¹ f ;¹ s) so that jC((p¤ ; f ¤ ; s¤ )i )j
is maximized. Given TP j, signal i 2 T (j) and window h 2 T (j), the total power ¢P (h) (expression (2)) is a monotone increasing function of pi if i 2 W (j; h), while it is a monotone decreasing function of pi if i 2 I(j; h). Now, assume j is covered under window h when pi = p~. Since Erf (¢P ) is increasing with ¢P , two cases are possible:
(1) i 2 W (j; h). Then, for any pi 2 [~ p; 1], j remains covered; (2) i 2 I(j; h). Then, for any pi 2 [0; p~], j remains covered. Therefore, the following property holds
Property 4.3 The power fading values for i 2 T n T¹ such that j is covered correspond 15
to two (possibly empty) intervals of the real line of the form °lj = [0; uji ] and °uj = [lij ; 1].
B A C
B
0
A
B
C
E
B A E
A
1
B
B
A
A
pi
C
E
Figure 4: Power intervals and intersection graph This property allows for an e±cient algorithm to search Ni . Assign to i a feasible
frequency fi . For all testpoints j 2 Z, compute intervals °lj and °uj and associate with
each interval a reference signal: this is e±ciently carried out by exploiting the coverage evaluation algorithm (see Appendix B). Three cases are possible for TP j: (i) °lj and
°uj are both empty: j will stay uncovered for any value of pi ; (ii) °lj [ °uj = [0; 1]: j is
covered for any value of pi ; (iii) °lj [ °uj 6 = ; and °lj \ °uj = ;. All testpoints satisfying (i) and (ii) can be neglected. In fact, our goal is to ¯nd one power value p¤i contained in the maximum number of intervals, that is, maximizing the coverage.
To this purpose, build the intersection graph G = (V; E) in which V is the set
of intervals associated with testpoints satisfying (iii) and edges correspond to pairs of overlapping intervals. Intervals in a clique Q = f°1 ; : : : ; °jQj g of G identify a set Z(Q) of distinct testpoints which can be covered simultaneously (along with the corresponding reference signals). In fact, \j2Q °j identi¯es, by Property 4.3, the power values covering all the testpoints in Z(Q). Moreover, since the two intervals associated to the same testpoint are disjoint, then jQj = jZ(Q)j.
The power fading value pi covering the maximum number of testpoints in Z can be
therefore computed by ¯nding a clique of maximum size in an interval graph G. This can be done in O(jZjlogjZj) ([16]).
Thus, searching Ni consists in repeating the above computation for any fi 2 Fi .
We refer to it as Search(Ni ; Fi ). Finally, the neighborhood search consists in repeating Search(Ni ; Fi ) for any i 2 T n T¹ and choosing the best overall solution. We have therefore proved the following:
16
Theorem 4.4 The neighborhood search can be done in O(jT j ¢ jF j ¢ (jZjlogjZj)).
This low complexity is crucial for increasing the number of iterations of GRASP in
very large scale problems.
The GRASP algorithm The algorithm is referred to as Solve NPlan and summarized in Table 3. At each iteration of the activation loop, the neighborhood of the current solution is explored, and one solution in the candidate set is chosen at random.
The candidate set contains all solutions whose value is at least ®cmax , where ® 2 (0; 1) and cmax is the optimal value. Parameter ®, initialized at 0.9, is decreased at each iteration. Remark that (initial) larger values of ® enhance the probability of selecting (one of) the most covering transmitters; smaller values increase diversi¯cation. The loop
ends when no improving solutions are found. Then, a local search is performed: following the activation order, one transmitter at time is switched o® and the neighborhood is explored. The algorithm is repeated num iter times.
5
Heuristic for NPack
In a preliminary study, we investigated di®erent extensions of the above GRASP ap-
proach to solve NPack, obtaining poor results. Indeed, it is common experience that large scale problems with multiple sets of decision variables are e®ectively tackled by some type of decomposition. Within this framework, two-phase heuristics decompose the original problem into two more tractable subproblems (recent examples in [19, 30]), typically corresponding to di®erent sets of decision variables.
We propose a two-phase heuristic for NPack: in the ¯rst phase emission powers are
determined for each stand-alone network (all other networks switched o®); in the second phase frequencies are simultaneously assigned to all networks. The heuristic is based
on the goal programming approach to multi-objective optimization: the decision maker speci¯es (optimistic) aspiration levels for the objective functions and the (weighted) sum of the deviations from these aspiration levels is minimized (weighted approach for goal programming [24]). The goal programming version of NPack is obtained by (i) de¯ning suitable threshold values u¹k for the coverage revenue of each network k 2 R,
and (ii) minimizing the weighted sum of the deviations u¹k ¡ u(C k ). This function has 17
||{|||||||||||||||||||||||||||||||||||||| Solve NPlan Input: network (T; Z), power upper bound vector PU 2 IRjT j , fading matrix [A]i2T;j2Z , revenue vector u 2 IRjZj , frequency spectrums Fi , for all i 2 T . Output: A network con¯guration (p; f; s) repeat num iter times p = 0jT j ; ® = 0:9; queue = ;; activation loop: for all i 2 T n T¹ apply Search(Ni ; Fi ) to compute optimal con¯guration (p¤ ; f ¤ ; s¤ )i ; cmax = maxi2T nT¹ fu(C((p¤ ; f ¤ ; s¤ )i ))g; if cmax ¸ u(C((p; f ; s))) then choose randomly t 2 T n T¹, satisfying ®cmax · u(C((p¤ ; f ¤ ; s¤ )t )) · cmax ; (p; f ; s) = (p¤ ; f ¤ ; s¤ )t ; push t in queue; ® = maxf0:1; ® ¡ 0:01g; goto activation loop; local search: while queue is not empty pop t from queue; (¹ p; ¹ f;¹ s) = (p; f ; s) ; p¹t = 0 ; for all i 2 T n T¹
apply Search(Ni ; Fi ) to compute optimal con¯guration (p¤ ; f ¤ ; s¤ )i ; cmax = maxi2T nT¹ fu(C((p¤ ; f ¤ ; s¤ )i ))g = u(C((p¤ ; f ¤ ; s¤ )q )); if cmax ¸ u(C(p; f ; s)) then (p; f ; s) = (p¤ ; f ¤ ; s¤ )q ; endrepeat; ||||||||||||||||||||||||||||||||||||||||{
Table 3: GRASP Algorithm for NPlan
18
been discussed with RAI Way engineers and validated in the testing phase (see x7). From now on we denote the complete two-phase heuristic by Solve NPack.
5.1
Phase 1: (stand-alone) Network Planning
The input of this phase is an instance of NPack and one aspiration revenue u¹k for each
network k 2 R. The output is a con¯guration (pk ; f k ; sk ) such that u(C k ) ¸ u¹k , for k 2 R.
Phase 1 treats each network k as a stand-alone network, by an iterative procedure:
starting from the initial value jF j = 1, a slight modi¯ed version of Solve NPlan (x4.3) is executed in which the algorithm stops (precisely, exits the activation loop) as soon as one solution with u(C k ) ¸ u¹k is found. If no such solution is found, then the spectrum size is increased by one and a new search is performed.
This algorithm intends to minimize the number of frequencies used to meet the
threshold. These are temporary frequencies, since they are re-assigned in Phase 2, and
do not correspond to the actual available frequencies. Thus, the temporary spectrum size is a parameter, whose e®ect on overall networks coverage is investigated in x7.
Finally, observe that it is not required that all pairs of (intra-net) interfering trans-
mitters receive distinct temporary frequencies. Indeed, interfering transmitters often receive the same temporary frequency in the ¯nal solution returned by Solve NPlan.
5.2
Phase 2: Multi-Network Frequency Assignment
The input of Phase 2 consists of a stand-alone con¯guration (pk ; f k ; sk ) and a threshold u¹k for each network k 2 R. The output is a (new) con¯guration (pk ; f k ; sk ) for each k 2 R.
The purpose of this phase is to assign an available frequency to every transmitter, so
as to approximate the revenue levels obtained in Phase 1 (satisfactory for the planner) when the networks are simultaneously active. This is an instance of the Multi-network Frequency Assignment Problem (MFAP), which is a generalization of well known FAP
(see x3). The latter is usually solved by a graph model referred to as the Minimum Interference Frequency Assignment Problem (MI-FAP) ([1]).
An instance of the MI-FAP is described by an undirected graph G = (V; E) (inter-
ference graph) with non-negative weights cuv for each (u; v) 2 E, where V is the set 19
of transmitters, E is the set of pairwise interfering transmitters and cuv represents the revenue loss when u and v receive the same frequency (computed as in [1]). Then the MI-FAP is the problem of assigning frequencies from F to the vertices of V so that the
sum of the revenue losses is minimized. When only co-channel interference is considered, the MI-FAP reduces to the well known max k-cut problem (where k = jFj) on weighted graphs, namely:
Problem 5.1 (max k-cut) Find a partition of V into k classes so that the sum of the weights of crossing edges (i.e. edges with endpoints in di®erent classes) is maximized.
The same graph model applies to MFAP, where edges connect transmitters either
from the same network or from di®erent networks and the revenue losses are computed according to power levels established in Phase 1. We denote such a graph by G = (T; E).
Using this de¯nition of multi-network graph we often experienced large gaps between
value of the max k-cut solutions and corresponding coverage value based on testpoints (x2.1). Precisely, we observed large deviations u¹k ¡ u(C k ) for the single networks coverage because of increased intra-net interference. This turns out to be a consequence of the modelling approximation introduced by the graph model.
On the contrary, we experienced that this bad e®ect is remarkably reduced by the following amendment to the standard model. Let E¹ k be the (possibly empty) set of transmitter pairs of network k in which the two transmitters are interfering but received
the same (temporary) frequency in Phase 1, yielding a coverage loss. The new multi¹ = (T; E), ¹ where E ¹ = E n [k2R E ¹ k . In practice, this network interference graph is G has two major e®ects: (i) forces the max k-cut solution to approximate the stand-alone
frequency assignment of Phase 1, ¯tting with the optimized power values; (ii) drives the frequency assignment toward coverage losses accepted by the aspiration thresholds.
Several solution algorithms have been proposed for MI-FAP. For large real-life in-
stances (like those considered in this paper) exact methods fail and one resorts to heuristics. Among these, the most promising appear to be Monte Carlo methods, such as Simulated Annealing (SA) and Threshold Acceptance [1]. In our tool MI-FAP is solved using the variant of SA devised in [13], which outperforms other known approaches on several real-life large scale instances [14, 23].
20
5.3
Post-optimization
At the end of Phase 2 a post-processing step concludes the heuristic for NPack. It consists in minimizing the overall emitted power with vectors f and s ¯xed by Phase
2. This is carried out as explained in 4.1. We experienced that a few uncovered TP-s
may gain a reference signal because of reduced interference from other signals, yielding a slight coverage increase.
5.4
Advances to current practice
The actual approach, referred to as segregation, adopted by the AGCOM in the elaboration of the national frequency plans ([3]) and also in implementation of multi-operators GSM systems works as follows. In order to con¯gure jRj distinct networks, the available bandwidth is subdivided into jRj mutually non-interfering sets of frequencies, each one being the set of feasible frequencies for exactly one network. In other words, each
network operates in a segregated frequency domain and no-interference can occur be-
tween transmitters belonging to distinct networks (inter-net interference). Then, every
single network can be separately designed. This is typically carried out by a further decomposition, namely, emission powers are established independently from frequency allocation. The latter decomposition represents a ¯rst drawback of the current practice: we experienced that this yields solutions to NPlan of poor quality. On the contrary,
Phase 1 of Solve NPack optimizes both emission powers and frequencies, providing better solutions.
Two other major drawbacks can be identi¯ed in the segregation approach. The
¯rst drawback concerns the °exibility of the approach. In practice, splitting frequencies among non-identical networks (i.e., deciding the segregation scheme) is a hard task,
requiring a trial-and-error process. Even more di±cult can be applying the segregation
approach under user-de¯ned coverage aspiration thresholds. In this case, infeasibility problems may occur, whenever all feasible segregation schemes do not satisfy the thresholds. The methodology introduced in this paper ¯lls in this lack of °exibility.
The second drawback is illustrated in Figure 5. In this example we consider two
identical networks N 1 ; N 2 (identical networks occur when the same network broadcasts di®erent programs, a common event), with T 1 = fa; b; c; d; eg and T 2 = fA; B; C; D; Eg.
The interference graph associated with each single network is depicted in Figure 5.a (we 21
A fA=2
fa=1
a
fe=3 e
fE=5
b fb=2 d
fd=2
a
e f =4 e
E
fa=1
fd=2
c
d
fc=1
fD=3
(a) segregated solution
D
fb=3 b
B
fB=4
fc=1
c
C
fC=5
(b ) multi-inteference graph
Figure 5: (a) segregation approach vs: (b) multi-interference graph can suppose that, for every edge, the co-channel revenue loss is equal to a large constant M ). The corresponding multi-network interference graph is represented in Figure 5.b.
Observe that the minimum number of frequencies required by a single network (say N 1 ) to avoid losses is 3. A 0-loss frequency assignment for a single network is shown in Figure 5.a, namely fa = fc = 1, fb = fd = 2 and fe = 3. Thus, in the segregation mode,
6 distinct frequencies are required for two networks. On the other hand, one can obtain
a 0-loss frequency assignment for the multi-network graph of Figure 5.b by assigning fa = fc = 1, fd = fA = 2, fb = fD = 3, fe = fB = 4 and fC = fE = 5. Therefore, if we let both networks draw frequencies from a common set, we can reduce the bandwidth
requirement to 5 frequencies. Observe that network N 1 receives frequencies from the set f1; 2; 3; 4g, while network N 2 from the set f2; 3; 4; 5g and the two sets have large overlap. Phase 2 of Solve NPack is designed to overcome the drawback highlighted in the example through an improved spectrum usage.
6
Implementation issues
The methodology described in this paper has been coded in a software tool which is
currently in use at RAI Way [28]. In particular, the tool was developed under constant coordination with company engineers during years 2000 and 2001. This process focused on some key points illustrated below.
22
Simulation vs Optimization Tools available at RAI Way at the beginning of this study perform ¯eld prediction and coverage evaluation, i.e., simulation of network con¯gurations. In practice, such tools allow the engineers to compute coverage of hand-
made network con¯gurations. Therefore, the planning activity was carried out by a trial-and-error process: (i) identi¯cation of promising modi¯cations of actual networks,
(ii) validation of new candidate con¯gurations through simulation and (iii) acceptance/rejection of new con¯gurations. New regulations from AGCOM and the DVB
launch called for a signi¯cant re-planning of the broadcasting system and the previous planning process became quickly ine®ective. On the contrary, our methodology has
been designed to cope with the new needs. In particular, the possibility of optimizing network con¯gurations and the automatic handling of the design process is of great help
for the company engineers. In fact, when system resources become scarce, the quality of hand-made con¯gurations is simply unacceptable. As far as we know, RAI Way is the ¯rst European broadcaster using automatic optimization, whereas most of the research e®ort at private broadcasters is focused on propagation and coverage assessment.
Tool °exibility In order to be compliant with the planning activity, the tool has been designed to satisfy several requirements of RAI Way engineers:
(i) when planning a multi-network system, one wishes to de¯ne a coverage requirement for every network, according to commercial strategies. The choice of goal
programming in Solve NPack permits obtaining di®erent optimized con¯gura-
tions for di®erent choices of thresholds u¹k . This also helps evaluating the trade-o® between coverage and resource usage.
(ii) some transmitter may have pre-assigned emission power and/or frequency. In this experience, two situations occur: one of two networks is completely ¯xed and
cannot be re-optimized (see x7); the frequency of transmitters near Italy's border is ¯xed by international agreements. The tool easily manages this requirement by variable ¯xing.
(iii) some TPs are regarded as critical for some networks. Such TPs typically coincide with important town centers. In this case, the revenue uj of the TPs can be increased according to the importance awarded to them. 23
Parameters All transmitter parameters (see x2) can be easily modi¯ed through input ¯les. Furthermore, several system parameters are available in order to build speci¯c scenarios. Among them, the following emerge as the important ones:
² the antenna gain and directivity can represent several receiving conditions: roof antenna, omnidirectional indoor antenna, etc. [8];
² the tool can adopt any propagation model [3] and work with all frequency bands, from VHF to L-band.
² the guard interval Tg (see x2.1) and system noise PC=N (see x4.1) can be easily modi¯ed. This allows the user to evaluate di®erent modulation schemes and di®erent channel models [8].
7
Experimental ¯ndings and practical achievements
The computational experience has three main purposes: (i) evaluating the quality of solutions obtained by Solve NPlan (x7.1); (ii) evaluating the quality of solutions
obtained by Solve NPack (x7.2); (iii) illustrating practical achievements concerning the RAI analog-to-digital (A to D) transition (x7.3). As for (iii), we discuss results obtained in reference scenarios based on the Italian law, which have been the technical
support for initial A to D transition strategies. Some details had to be omitted for not being authorized by the Company for publication (e.g., details about the networks used in the experiments).
All experiments regard subsets of 480 di®erent geographical transmitter sites. These
have been identi¯ed by the Italian Communications Authority (law 66/2001) as those accommodating the transmitters of the future television networks. The digital terrain
database provided to us by RAI is based on a grid with 2.5Km step, amounting to 55,012 testpoints (15,442 inhabited, u(Z) = 56; 804; 187). Three types of instances are
considered, according to their geographical extension: (a) national networks: the target area Z consists of all Italian inhabited TP-s; (b) regional networks: Z consists of all
inhabited TP-s of one Italian regional district; (c) multi-regional networks: Z consists
of all inhabited TP-s of several regional districts. The instances of type (a) represent the focus of this study and are of very large scale (compare, for instance, with the 800 24
testpoints instances in [20]). Instances of other types can be of practical interest too, but are considered here only to assess performance of algorithms.
A complete description of all system parameters, including transmission bands, an-
tenna diagrams and reception characteristics can be found in [8] and [3]. The algorithms Solve NPlan and Solve NPack have been coded in C++ and run on a Pentium IV 1.4 Ghz machine, with 512 MB of RAM.
Throughout the computational section, the objective function value is often dis-
played as the (percentage) ratio u(C)=u(Z) (i.e. covered over total population). All other percentage values derive consequently.
7.1
Evaluating Solve NPlan
The ¯rst experiment concerns regional networks and two spectrum sizes jF j 2 f1; 2g.
These instances can be solved to optimality by MILP formulations (x4.1), allowing the evaluation of Solve NPlan solutions. The optimal solutions of MILP-s are computed
by the commercial solver ILOG Cplex 8.0. After some testing, we established that the
best setting for Cplex is the default with one major modi¯cation: nodesel = 3 (strong branching). In some instances a decisive improvement is obtained by MIPemphasis = 3
and/or probing strategy = 3. For instances with jZj · 800 some advantage can be obtained with flowcovers generation level aggressive and heuristicfreq · 10.
Results are collected in Table 4. The ¯rst four columns display instance features:
district identi¯er, number of inhabited testpoints jZj, target population u(Z) and num-
ber of candidate transmitters jT j. The remaining columns report, for each value of the spectrum size, the following data: solution by Solve Nplan (percentage value
%u(C) = u(C)=u(Z) ¤ 100), optimal solution by MILP (%u(C ¤ )), number of transmitters activated by the two algorithms (#T and #T ¤ ), CPU time (in seconds) and relative percentage gap between the two solutions.
One can observe that for all instances a very small gap is measured (less than 0.8%
in the case jF j = 2), decreasing on average as the spectrum size increases. Furthermore,
the CPU time required by Solve NPlan is limited and does not increase signi¯cantly as the spectrum size augments (the parameter num iter is set to 50 for all instances). On the contrary, a large amount of CPU time is typically required by Cplex. Finally,
instances marked with an (¤ ) in the CPU time column (with jFj = 2), required a 25
speci¯c (instance dependent) ¯xing of variables in order to certify optimality. Observe
that optimal solutions accommodate a number of transmitters larger than the one from heuristic solutions.
When national networks are concerned, MILP formulations (x4.1) cannot be solved
to optimality. Therefore, Solve NPlan solutions are evaluated resorting to the upper bound (UB) of x4.2. Table 5 reports on national networks with two possible sets of candidate sites. The ¯rst set contains all of the 480 sites while the second corresponds
to a subset of 367 sites selected by the Company. For the second set, also an analog network is considered. In all the experiments the Solve NPlan parameter num iter is set to 100.
When jF j ¸ 2, a fairly small percentage gap (from 1.06% to 2.17%) is measured,
proving the successful behavior of the heuristic as well as the e®ectiveness of the bound
from decomposition. On the contrary, networks with jF j = 1 show a larger gap (from 3.58% to 6.75%). However, this is due to a weaker bound rather than a poor heuristic solution, because of large cross-interference among clusters.
On the practical side, large Single Frequency Networks are of great interest in digital
systems, while they do not play any role in analog ones. Before the present study,
coverage of Single Frequency Networks was estimated applying the digital coverage
evaluation procedure to the actual (analog) transmitter con¯guration (or to some handmade manipulation of it). Solve NPlan solutions largely outperform such estimates, revealing new insights to RAI Way engineers.
7.2
Evaluating Solve NPack
Two experiments are proposed to compare Solve NPack to the segregation approach (x5.4). In particular, the e®ect of Phase 2 is evaluated when applied over a ¯xed frequency allocation (segregation scheme). The comparison is performed in four steps:
(1) choose a segregation scheme; (2) compute by Solve NPlan a con¯guration for each network k 2 R with revenue u(C¹ k ); (3) set aspiration thresholds u¹k = u((C¹ k )); (4) run Solve NPack.
Under this choice of aspiration thresholds, the segregation approach coincides with
Phase 1 of Solve NPack: for each network, the minimum temporary spectrum size required by Solve NPlan to meet the threshold equals, by construction, the segregated 26
27
Network Attributes District |Z| u(Z) 1 507 1,343,650 2 53 75,977 3 602 3,451,121 4 1,258 3,854,439 5 475 799,703 6 566 1,630,641 7 189 328,848 8 793 2,093,580 9 1,092 3,515,468 10 385 1,979,830 11 2,227 9,368,780 12 947 5,614,973 13 918 5,237,954 14 617 1,140,811 15 333 703,387 16 223 641,388 17 905 5,139,317 18 1,421 4,555,377 19 588 1,289,670 20 1,561 4,025,781
Solve NPlan %u(C) #T CPU 98.25 15 32 95.86 8 4 82.24 7 100 89.69 8 104 92.24 40 20 94.44 17 28 89.90 8 20 93.55 27 68 94.98 20 124 93.28 11 28 93.70 22 232 98.48 24 108 94.10 21 56 93.45 23 72 99.55 13 40 96.57 17 68 96.25 19 76 90.53 11 132 98.34 20 36 95.56 12 108
MILP %u(C ∗ ) #T ∗ CPU 98.76 16 285 95.86 8 10 82.59 9 56,782 90.15 9 63,027 92.51 40 180 94.44 17 161 90.25 8 78 94.11 28 59,099 95.94 22 38,907 94.04 12 115 94.56 23 79,087 98.68 24 52,027 94.29 22 23,456 93.54 23 15,802 99.6 13 277 97.55 19 26,891 97.12 20 54,228 91.26 13 60,354 99.06 21 302 96.55 15 64,678 Gap 0.51 0 0.43 0.51 0.30 0 0.39 0.60 1.01 0.82 0.91 0.20 0.20 0.10 0.05 1.01 0.90 0.81 0.73 1.04
Solve NPlan %u(C) #T CPU 98.64 16 40 95.86 8 5 97.64 15 140 98.29 20 248 92.35 40 28 95.69 18 32 90.11 8 28 96.69 30 76 96.74 23 188 97.41 18 44 97.10 26 420 98.89 25 136 96.16 28 64 97.16 25 84 99.55 12 52 97.24 17 88 96.64 19 104 96.72 22 248 98.64 19 60 97.41 17 188
Table 4: Solve NPlan vs. MILP on Italian districts, |F| ∈ {1, 2}
|T | 19 18 16 45 32 18 17 34 31 29 56 32 31 37 17 26 22 36 29 35
|F| = 1 MILP %u(C ∗ ) #T ∗ CPU 98.94 17 6,042 95.86 8 24 97.91 17 72,890∗ 98.46 21 91,432∗ 92.51 40 1,928 95.69 18 5,039 90.25 9 866 97.03 31 75,490∗ 97.51 25 67,834∗ 97.71 20 1,507 97.79 28 96,453∗ 99.07 26 79,926∗ 96.21 28 59,735 97.20 25 75,845 99.60 12 3,352 97.55 18 75,238 97.12 21 84,820∗ 97.20 23 91,502∗ 99.16 21 5,194 98.17 19 92,415∗
|F| = 2 Gap 0.30 0 0.27 0.17 0.18 0 0.15 0.35 0.79 0.31 0.71 0.18 0.05 0.04 0.05 0.32 0.50 0.49 0.52 0.77
jF j Technology jT j u(C) %u(C) #T CPU UB %UB Gap 1 Digital 480 51,520,319 90.70 274 25,596 53,554,988 94.28 3.58 2 Digital 480 54,168,472 95.36 279 43,992 55,401,124 97.53 2.17 1 Digital 367 44,006,204 77.47 302 24,516 47,840,486 84.22 6.75 2 Digital 367 51,498,676 90.66 346 36,828 52,731,327 92.83 2.17 3 Digital 367 53,128,956 93.53 367 48,276 53,731,080 94.59 1.06 3 Analog 367 47,584,867 83.77 367 12,883 48,556,219 85.48 1.71 4 Analog 367 49,896,798 87.84 367 17,208 51,055,603 89.88 2.04 Table 5: Solve NPlan solutions vs. upper bounds for national networks spectrum size. In this setting, the experiment aims to demonstrate that Phase 2 is
e®ective to counterbalance the increase of inter-net interference by reducing intra-net interference.
Both experiments concern two networks N 1 and N 2 and represent two di®erent
situations, i.e., complete and partial overlapping target areas. The results are or-
ganized in tables with the following format. Each row corresponds to a segregation scheme (jF 1 j; jF 2 j) and displays: best percentage coverage %u(C¹ 1 ) (%u(C¹ 2 )) of N 1 (of
N 2 ) found by Solve NPlan (segregated solution), best percentage coverage %u((C 1 )¤ ) (%u((C 2 )¤ )) of N 1 (of N 2 ) found by Solve NPack, percentage sum of deviations %¢N 1 + %¢N 2 . The deviation is de¯ned as ¢N k = u¹k ¡ u((C k )¤ ), k = 1; 2 (x5).
Scenario 1. One analog national network N 1 = A and one digital national network N 2 = D (A and D have completely overlapping target areas). The candidate site is the one with 367 sites used in x7.1: therefore, best stand-alone heuristic con¯gurations and relative upper bounds can be found in Table 5.
Results are reported in Table 6, in which three spectrum sizes jFj 2 f4; 5; 6g and ¯ve
di®erent segregation schemes are considered. In this scenario (Phase 2 of) Solve NPack improves the segregation approach. In particular, for jFj 2 f4; 5g a small (i.e., within
0.5%) loss for network A is pro¯tably counterbalanced by a relevant (i.e., up to 6.6%) gain for network D. This leads to a negative sum of deviations. For jFj = 6, the solutions from Solve NPack dominate those from the segregation approach. The practical
relevance of this gain can be appreciated recalling that, in this scenario, an improvement of 1% corresponds to about 500,000 inhabitants. 28
jF j 4 5 5 6 6
Scheme Segregation approach Solve NPack Sum of deviations A D A D A ¤ D ¤ ¹ ¹ jF j jF j %u(C ) %u(C ) %u((C ) ) %u((C ) ) %¢A + %¢D 3 1 83.77 77.47 83.32 79.34 -1.42 4 1 87.84 77.47 87.37 84.05 -6.11 3 2 83.77 90.66 85.94 89.95 -1.46 3 3 83.77 93.53 85.33 93.99 -2.02 4 2 87.84 90.66 87.85 91.73 -1.08 Table 6: Segregation vs. Solve NPack: Scenario 1
Scenario 2. Two multi-regional networks N 1 and N 2 with partially overlapping target areas. Namely, Z 1 (Z 2 ) is the union of regional districts Abruzzo and Marche (Lazio and Abruzzo). Then, jZ 1 j = 3; 882, u(Z 1 ) = 2; 503; 813, and jZ 2 j = 4; 572,
u(Z 2 ) = 5; 924; 318. Two spectrum sizes are considered (jFj 2 f3; 5g) and four segregation schemes. Results are reported in Table 7, including also the number of active transmitters (#TX) . jF j 3 3 5 5
Scheme Segregation Approach Solve NPack Sum of deviations 1 2 1 1 2 2 1 ¤ 2 ¤ jF j jF j %u(C¹ ) #T %u(C¹ ) #T %u((C ) ) %u((C ) ) %N 1 + %N 2 1 2 80.77 45 92.34 56 90.86 93.18 -10.93 2 1 90.89 49 87.49 52 91.01 90.24 -2.87 2 3 90.89 49 94.75 57 92.55 94.71 -1.62 3 2 92.73 49 92.34 56 92.81 93.52 -1.26 Table 7: Segregation vs. Solve NPack: Scenario 2
When jF j = 5, the solutions from Solve NPack "almost" dominate the segregated ones. In fact, the segregation schemes (2,3) and (3,2) slightly improve (i.e. less than
0.1%) the coverage of the protected network (i.e., the one with 3 frequencies), but penalizes the coverage of the other network by about 2%.
When jF j = 3, Solve NPack largely outperforms the segregation approach. In fact,
solutions from both segregation schemes (1,2) and (2,1) are dominated by those from
Solve NPack. Notice that the gain for single frequency networks ranges from 3% to 10%. A similar enhancement was observed in tests involving a larger number of networks. Partially overlapping target areas yield remarkable advantages for Solve NPack. In summary, in all cases of both scenarios a negative sum of deviation is measured. 29
7.3
A to D transition
The key issue of digital start-up is growing digital networks sharing resources with operating analog networks. The experiments of this section include one RAI national analog network A, consisting of 367 active transmitters.
The two experiments motivating the present study are here illustrated. In the ¯rst
experiment one digital network is grown over network A under its current (¯xed) setting.
In the second experiment two digital networks are grown over A (which can now be replanned) according to the new requirements ¯xed by Italian law. In both experiments the available spectrum is jF j = 7.
Experiment 1: growing a national digital network D over A (¯xed). This scenario represents a classical start-up for a new service: managers wish to keep safe the old service until the new one guarantees at least the same revenue. Emission powers and frequencies of A are ¯xed. Under this setting, the population coverage of A is 91:21%,
used as reference threshold u¹A . Forbidding power and frequency re-con¯guration for A prevents users from re-tuning channels or turning antennas. Therefore this is considered an interesting transition scenario.
Two major questions were formulated by RAI Way engineers:
(i) set-up of D: is it possible to activate network D with at least 40% population coverage without a®ecting network A (namely, keeping A over 90% population coverage)?
(ii) development of D: what is the impairment on network A as the coverage of D increases?
Neither the simulation tools nor the segregation approach can be of any help for
answering the above questions. On the contrary, these have been addressed by running Solve NPack for increasing values of the threshold u¹D . The initial percentage threshold is 40%, considered by broadcasters a suitable start-up value.
Table 8 displays, for each value of u¹D , the following data: (temporary) spectrum size
in Phase 1 (jF T D j), number of active digital transmitters (#T D ), Phase 1 (stand-alone) digital population coverage (%u1 (C D )), digital population coverage (%u((C D )¤ )) found
by Solve NPack, digital percentage deviation (%¢D = %(u1 (C D )¡u((C D )¤ ))), analog 30
percentage deviation found by Solve NPack (%¢A = %(¹ uA ¡ u((C A )¤ ))) and sum of percentage deviations (%¢D + %¢A). The coverage requirement u¹D is considered ful¯lled with a 0.2% tolerance.
Notice that several spectrum sizes (jF T D j) are reported for each threshold u¹D : the
¯rst of such values is the one computed by Phase 1, whilst the others have been reported
for sensitivity analysis. Values f¹ uD = 77%; 90%; 93%g represent maximum revenues obtained for (jF T D j) = f1; 2; 3g, respectively. u¹D jF T D j 40 1 2 3 50 1 2 3 60 1 2 3 65 1 2 3 70 1 2 3 75 2 3 77 1 80 2 3 85 2 3 90 2 3 93 3
#T D %u1 (C D ) %u((C D )¤ ) 9 39.88 35.11 9 41.99 35.11 7 41.50 33.47 14 50.41 43.99 13 50.52 43.98 13 51.17 43.98 21 58.79 55.75 22 59.46 52.02 23 60.93 52.49 40 65.08 64.18 31 65.52 60.48 25 64.67 56.98 76 70.38 72.33 45 69.94 65.91 38 69.41 62.50 63 74.86 70.28 62 75.02 68.54 302 77.47 81.63 86 79.63 75.49 82 80.20 73.55 134 84.31 80.58 117 84.53 78.62 346 90.66 87.56 191 89.50 83.46 367 93.53 87.93
%¢D %u((C A )¤ ) 4.77 90.41 6.88 90.01 8.03 89.99 6.42 90.15 6.54 90.02 7.19 89.95 3.04 90.07 7.44 89.95 8.44 89.69 0.90 89.96 5.04 89.67 7.69 89.64 -1.95 89.78 4.03 89.53 6.91 89.52 4.58 89.15 6.48 89.14 -4.16 89.44 4.14 88.91 6.65 88.89 3.73 88.24 5.91 88.17 3.10 87.91 6.04 87.59 5.60 87.12
%¢A (%¢D + %¢A) 0.80 5.57 1.20 8.08 1.22 9.25 1.06 7.48 1.19 7.73 1.26 8.45 1.14 4.18 1.26 8.70 1.52 9.96 1.25 2.15 1.54 6.58 1.57 9.26 1.43 -0.52 1.68 5.71 1.69 8.60 2.06 6.64 2.07 8.55 1.77 -2.39 2.30 6.44 2.32 8.97 2.97 6.70 3.04 8.95 3.30 6.40 3.62 9.66 4.09 9.69
Table 8: Solve Npack results: one digital network over a ¯xed analog network 31
4.5 4 3.5 3 2.5 2 1.5 1 0.5 0
Analog deviation
Digital deviation
10 8 6 4 2 0 -2
threshold 40 50 60 65 70 75 80 85 90 95
total deviation
12 10 8 6 4 2 0 -2
threshold 40 50 60 65 70 75 80 85 90 100
threshold
1F 2F 3F
40 50 60 65 70 75 80 85 90
Figure 6: Digital, analog and total deviations vs. design threshold and number of frequencies
The con¯gurations obtained by jF T D j = 1 (covering from 43 to 72 %) provide
positive answers to question (i). Before the present study, RAI management could only guess the feasibility of digital start-up based on network engineers estimates.
As for question (ii), results of Table 8 represent attractive con¯gurations, covering a
wide variety of possible extensions of network D between two extreme con¯gurations: on the one hand, a concentrated network reaching 40% by covering only major towns with
few transmitters (7 or 9); on the other hand, an extended network reaching 88% using most of the available transmitters. In addition, the above results also allowed engineers to perform cost bene¯t analysis of networks, investigating the trade-o® between coverage and required number of transmitters.
Sensitivity analysis As a byproduct, results in Table 8 allows for a sensitivity analysis: the stopping criterion in Phase 1 of Solve NPack is removed and Phase 2 has been executed for all values of jF T D j satisfying thresholds (not only for the smallest one).
The results highlight that, for all values of u¹D , the greater the spectrum size jF T D j the
larger the sum of deviations. This e®ect is clear from Figure 6, in which digital, analog and total deviations are plotted as functions of design thresholds and spectrum size. 32
These results validate the stopping criterion of Phase 1, which terminates as soon as the threshold is met.
Finally, notice that jF T D j = 1 yields a negative digital deviation and the solution
(jF T D j = 1; u¹D = 77%) dominates the solutions (jF T D j = 2; u¹D = 80%) and (jF T D j = 3; u¹D = 80%) (see Table 8). The negative deviations arise since the increase of inter-
net interference caused by network packing is overcome by the reduction of intra-net interference obtained exploiting the shared spectrum.
Experiment 2 Planning one analog network A with 2 digital networks D1; D2.
This experiment consists of investigating system con¯gurations implementing the new regulation. This imposed on RAI Way two requirements: (i) implement by January 1,
2004, two digital networks covering at least 50% of the population (ii) implement by January 1, 2005, two digital networks covering at least 70% of the population. Purpose
of RAI Way is to exploit the available spectrum jF j = 7 to ful¯ll such requirements while minimizing the deviation of analog network A from actual value u¹A = 91:21.
The following experiments aim to show the achievements obtained by Solve NPack
versus those from the segregation approach (precisely, an improved segregation approach in which stand-alone networks are optimized by Solve NPlan).
Problems (i) and (ii) admit the same optimal segregation scheme, namely jF A j =
5; jF D1 j = 1; jF D2 j = 1. Corresponding percentage revenues computed by Solve NPlan are: u(C A ) = 82:83%, u(C D1 ) = u(C D2 ) = 70:38%. This solution represents a good
con¯guration for scenario (ii). On the contrary, requirements of scenario (i) are largely satis¯ed, yielding an unforced penalty on C A . The result deviation ¢A = 8:38 is indeed considered unacceptable for digital start-up. u¹
This drawback can be e®ectively managed by Solve Npack. In fact, choosing
D1
= u¹D2 = 55%, Solve Npack returns u(C A ) = 89:02%, u(C D1 ) = 51:40; u(C D2 ) =
49:82%. In this case ¢A = 2:19%, which is considered satisfactory. As for scenario (ii) Solve NPack returns u(C A ) = 82:91%, u(C D1 ) = 70:51; u(C D2 ) = 70:01%, obtained with thresholds u¹D1 = u¹D2 = 68%, representing a small gain over the segregated solution.
RAI managers considered such deviation for network A satisfactory for digital start-
up. This new achievement has been regarded as a conclusive argument to start the implementation design.
33
8
Conclusions
We presented a method for optimizing emission powers and frequencies in multi-network broadcasting systems in which di®erent networks share transmitter sites and frequencies, and each network has its own goal (Network Packing Problem).
The contribution of this paper is threefold: on models, algorithms and practice.
On models, we introduced a hierarchy of new and relevant problems generalizing the models studied in the literature so far. Concerning solution algorithms, a new local
search is introduced for simultaneously optimizing emission powers and frequencies on
a single network. The algorithm is able to ¯nd near optimal solutions for large scale instances. Also, an e®ective relaxation for computing upper bounds is described and
exploited to measure the quality of heuristic solutions; ¯nally a two-phase heuristic is devised able to approximate the Pareto optimal boundary of the multi-network (i.e., multi-objective) problem.
On the practical side, the problem investigated here represents a key problem in the
current transition from analog to digital television. This is the subject of hot political
controversy in Italy and has been receiving quotidian attention from the popular press
in the last three to four years. We reported on the development of solution methods for the Network Packing Problem at RAI Way, the network provider of the major Italian
broadcaster. The proposed approach guarantees high °exibility, allowing managers to obtain several high quality con¯guration alternatives for the Italian system under di®er-
ent exploitation patterns of the system resources. The results illustrated in this paper contributed to start the RAI digital service according to the schedule and requirements ¯xed by Italian law.
Acknowledgements. This work was partially supported by RAI Way s.p.a.. We
are grateful to Arturo Baglioni, Giuseppe Carere and Roberto Sera¯ni from RAI Way for promoting this study. We wish to thank Antonio Sassano for his invaluable suggestions and Andreas EisenblÄatter for his detailed observations. Finally, we are indebted to the anonymous referees, whose comments led to a signi¯cant improvement of the paper.
References [1] K. I. Aardal, S.P.M. van Hoesel, A.M.C.A. Koster, C. Mannino and A. Sassano, 34
Models and solution techniques for the frequency assignment problem,4OR, 1, 4, 2003, 261-317
Ä Orlin, J.B., Punnen, A.P., A survey of very large-scale [2] Ahuja, K.R., Ergun, E., neighborhood search techniques, Discrete Applied Mathematics, 123, 1-3, 2002, 75{102.
[3] Autoritµa Nazionale per le Garanzie nelle Comunicazioni, http://www.agcom.it. [4] BBC
news,
UK
edition,
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http://news.bbc.co.uk/1/hi/business/3121084.stm.
18
September,
2003,
[5] Beutler, R., Frequency Assignment and Network Planning for Digital Terrestrial Broadcasting Systems, Kluwer Academic Publishers, ISBN 1-4020-7872-2, 2004.
[6] R. Brugger and D. Hemingway, OFDM Receivers, EBU Technical Review. [7] CEPT/EBU, Planning and Introduction of Terrestrial Digital Television (DVBT) in Europe, Doc. FM(97) 160, 1997.
[8] The Chester 1997 Multilateral Coordination Agreement, Technical Criteria, Coordination Principles and Procedures for the introduction of Terrestrial Digital Video Broadcasting, 25 July 1997.
[9] The Digital Video Broadcasting project, http://www.dvb.org [10] European Broadcasting Union, Terrestrial Digital Television Planning and Implementation Considerations, BPN 005, second issue, July 1997.
[11] European Radiocommunications O±ce, http://www.ero.dk/ [12] European Telecommunication Standard Institute, http://www.etsi.org [13] Successful Applications to GSM Networks, http://www.dis.uniroma1.it/ mannino/fap.htm
[14] A. EisenblÄatter and A. Koster, FAP web - A website about Frequency Assignment Problems, http://fap.zib.de/
35
[15] P. Festa and M.G.C. Resende, GRASP: an annotated bibliography, in C.C.
Ribeiro and P. Hansen, Essays and Surveys on Metaheuristics, Kluwer Academic Publishers, 325{367, 2001.
[16] U.I. Gupta, D.T. Lee and J.Y.T Leung, E±cient Algorithms for Interval Graphs and Circular-Arc Graphs, Networks, 12, 1982, 459-467.
[17] Italian site of broadcasting, http://www.broadcasting.it [18] International Telecommunication Union, http://www.itu.int/home/ [19] Kim, S. and S.L. Kim, A Two-Phase Algorithm for Frequency Assignment in Cellular Mobile Systems, IEEE Transactions on Vehicular Technology, 43, 1994, 542{548.
[20] Ligeti, A. and J. Zander, Minimal Cost Coverage Planning for Single Frequency Networks, IEEE Transactions on Broadcasting, 45, 1, 1999, 78{87.
[21] Ligeti, A. and S.L. Kim, Optimization of DAB/DVB Networks for Wireless Personal Services with Limited Frequency Band, 10th IEEE International Symposium on Personal, Indoor, and Mobile Radio Communications, Osaka, Japan, 1999. [22] L. Lov¶asz and M.D. Plummer. Matching Theory. North-Holland, 1986. [23] C. Mannino, G. Oriolo and F.Ricci, Solving Stability Problems on a Superclass of Interval Graphs, Tech. Rep. N. 26-02, DIS-Universitµa di Roma "La Sapienza".
[24] K. Miettinen, Nonlinear Multiobjective Optimization, International Series in Operations Research and Management Science, Kluwer, 1999
[25] K. Karipys and V. Kumar, Family of multilevel partitioning algorithms, http://www-users.cs.umn.edu/ karypis/metis/
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36
[28] RAI Way o±cial web site, http://www.raiway.rai.it/ [29] Vasquez, M. and J.K. Hao, A heuristic approach for Antenna positioning in cellular networks, Journal of Heuristics, 7(5), 2001, 443{472.
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Appendix A
Complexity
We show that NPack is NP-hard, even when restricted to planning a single network (i.e. jRj = 1). In particular, we show that Nplan for one analog network is NPhard in strong sense. Recall that Pij = Aij Pi is the power density in testpoint j from
transmitter i 2 T , and I(j; i) is the set of co-channel (with i) transmitters r 2 T n fig with Prj > 0. Testpoint j is regarded as covered by i i® Pij ¸ K1
(13)
and, whenever I(j; i) 6 =; P
Pij
k2I(j;i)
Pkj
¸ K2
(14)
where K1 is the system noise and K2 > 1 is an adimensional quantity (protection ratio) [3].
Theorem A.1 NPlan is NP-hard in the strong sense. Proof. By reduction from graph coloring: Given a biconnected undirected graph G(V; E), does there exist a coloring of V with at most k colors?
Associate with G an instance I of NPlan de¯ned by a network (T; Z), a power
upper bound vector PU 2 IRjT j , a fading matrix [A]i2T;j2Z , a revenue vector u 2 IRjZj
and a frequency spectrum F . Speci¯cally, let T = V = fv1 ; : : : ; vn g, Z = E and
PiU = K1 , for all i 2 T . Since G is biconnected, E can be partitioned into n non-
empty subsets E 1 ; : : : ; E n such that, for j = 1 : : : ; n, vertex vj is endpoint of every edge in E j (i.e. E j µ ±(vj )). Such partition can easily be obtained by exploiting an ear decomposition of G, which certainly exists since G is biconnected [22]. For each 37
transmitter vi 2 T and for each testpoint ej (vi ; vr ) 2 E i , let Aij = 1, 1=K2 < Arj < 1 and Aqj = 0 for q 6 = i; r. Finally, let F = f1; : : : ; kg and u = 11.
Claim. G has a k-coloring if and only if I has a solution (p; f ; s) such that C(p; f; s) = Z (i.e. all testpoints are covered).
v5 e5
v1
e6
e1
e3 v4
e4
E1 = {e1} E2 = {e2,e3} E3= {e4} E4= {e5} E5= {e6}
v2 e2
v3
Figure 7: A partition of edges as in the proof of Theorem A.1 Only if. Let c : V ! f1; : : : ; kg be a k-coloring of G. For all i, assign to vi 2 T
frequency fi = c(vi ), power fading pi = 1 (i.e. Pi = pi ¢ PiU = K1 ) and let sji = 1 if ej 2 E i , sji = 0 otherwise. Then, (p; f ; s) is feasible and every testpoint in Z is
covered. Suppose not, and let ej (vi ; vr ) 2 E i be an uncovered testpoint. Since Aij = 1 and Pi = K1 , Pij = Aij Pi = K1 and (13) is satis¯ed. This implies that (14) is violated
and thus I(j; i) 6 = ;. Since Aqj = 0 for all q 6 = i; r, we have I(j; i) = frg. Since
(vi ; vr ) 2 E, and c is a coloring, it is c(i) 6 = c(r). But then fi 6 = fr and r 2 = I(j; i), a contradiction.
If. Let (p; f; s) be a solution of NPlan covering all testpoints in Z = E. Then, for all j 2 Z, i 2 T , such that ej 2 E i , it must be sji = 1 (i.e. vi is the (only) reference signal
of testpoint ej ). In fact, for all q 2 T nfig, it is Aqj < 1, implying Pqj = Aqj Pq < K1 and
(13) is not ful¯lled: this implies that sjq = 0 for all q 2 T n fig and then sji = 1 (since
ej is covered). Since E q 6 = ; for all q = 1; : : : ; n, each transmitter is reference signal for at least one testpoint, which, by (13), implies Pq = PqU = K1 , and thus pq = 1, for all q 2 T.
For all i 2 V , let c(vi ) = fi . We show that c is a coloring of the vertices of G. Suppose
not, then there exists an edge ej = (vi ; vr ) 2 E such that c(vi ) = c(vr ). This implies that transmitter vi and transmitter vr are assigned with the same frequency. Suppose w.l.o.g. that ej 2 E i . Then I(j; i) = frg and we have
P
q2I(j;i)
Pqj = Prj = Arj Pr > K1 =K2 ,
which implies that (14) is violated and ej is not covered by vi , a contradiction. 38
¦
Extension of the above proof to digital network planning is straightforward.
Appendix B
A computing digital coverage in linear amortized time
We describe a linear time implementation of the coverage evaluation procedure for a testpoint, introduced in x2.1. The coverage probability is calculated by the error funcW I tion (3), which depends upon mean wanted (interfering) signal Ptot (Ptot ) and variances 2 2 ¾W;tot and ¾I;tot . Recall that all of these values depend on the position of the detec-
tion window. We show how to e±ciently select the position of the detection window which maximises the coverage probability, by this simulating the behaviour of actual
commercial devices. Observe that if n signals arrive in a testpoint, then we have n
di®erent candidate windows, each containing at most n signals. So, a naive imple-
mentation of this computation would require O(n2 ) operations. We show here how to W I 2 2 compute total wanted power Ptot in linear time; the extension to Ptot , ¾W;tot and ¾I;tot
is straightforward.
A generic signal h 2 T arriving in a testpoint at time ¿h may contribute to the wanted
W I signal Ptot , to the interfering signal Ptot or to both. For sake of clarity, since coping
with a single testpoint, we remove all subscripts referring to testpoints, e.g. Phj = Ph.
W The contribution of signal h to Ptot is Ph ¢ w(¿h), where w(¿h ) is the weighting function
(1) which can be rewritten as: 8 1 >
t12 (tu + Tg + ¿ ¡ ¿h )2 :
u
0
if 0 · ¿h ¡ ¿ · Tg
if Tg < ¿h ¡ ¿ · 2Tg otherwise
(15)
where ¿ denotes the starting time of the detection window and tu is a constant (useful symbol period). Let T = f1; : : : ; ng be the n distinct signals in the testpoint. W.l.o.g.
we can assume ¿1 · ¿2 · : : : · ¿n . We have seen that ¿ 2 f¿1 ; : : : ; ¿n g, i.e., the
detection window starting time coincides with the arrival time of one of the incoming signals.
When the receiver is synchronized with the i-th signal (i.e. ¿ = ¿i ), let us denote by P W (i) the value of the total wanted signal, by U (i) = fh 2 T : ¿ = ¿i ; w(¿h ) = 1g the 39
set of fully contributing signals, and by M (i) = fh 2 T : ¿ = ¿i ; 0 < w(¿h ) < 1g the set of partially contributing signals, for i = 1; : : : ; n.
Since ¿1 · ¿2 · : : : · ¿n , then U (i) and M(i) are sets of consecutive integers and
we can write U (i) = fi; i + 1; : : : ; qi g and M (i) = fsi ; : : : ; ti g, for i1; : : : ; n (see Fig. 8). Trivially, i · qi < si · ti .
i
q i s i ti
i+1
Mi
Ui
qi+1 si+1 ti+1
Figure 8: Shifting the detection window from ¿i to ¿i+1 Denote by P U (i) =
and by P M (i) =
Pti
Pqi
j=i
Pj the overall contribution to P W (i) of the signals in U(i),
j=si Pj ¢ w(¿j ) M
the overall contribution to P W (i) of the signals in M(i)
(P W (i) = P U (i) + P (i) for all i). We show ¯rst how to e±ciently evaluate P U (i + 1) once P U (i) = Pi + : : : + Pqi is known. When the detection window shifts from position
¿i to the next position ¿i+1 we have that signal i is no more a wanted signal whilst signals qi + 1; : : : ; qi+1 are now fully wanted signals (see Fig. 8), i.e. P U (i + 1) = P U (i) ¡ Pi +
X j=qi +1;:::;qi+1
Pj :
Therefore, P U (i + 1) can be obtained from P U (i) by performing qi+1 ¡ qi + 1 sim-
ple arithmetic operations. Let us indicate as ri the number of elementary operations required to compute P U (i), i = 1; : : : ; n. The following recursive expression holds: ri+1 = ri + qi+1 ¡ qi + 1:
Assuming r0 = q0 = 0, the total number of elementary operations required to compute P U (n) is simply rn = rn¡1 + qn ¡ qn¡1 + 1 = : : : = qn + n = 2n. This shows that the contribution of the fully wanted signals can be computed in optimum time O(n).
40
We evaluate now the complexity of calculating the contribution to P W (i + 1) of the
partially contributing signals P M (i + 1) when P M (i) is known. By (15) we have (letting Aj = tu + Tg ¡ ¿j ): P M (i) =
ti ti ti ti X X 1 X 1 X 2 2 2 P (A + ¿ ) = [ P A + ¿ P + 2¿ Pj Aj ] j j i j j j i i t2u j=si t2u j=si j=si j=si
(16)
Observe ¯rst that computing the quantities Ak , Pk Ak , Pk A2k , ¿k2 and 2¿k , for k =
1; : : : ; n, requires O(n) elementary operations. As before, let us shift the window to the
next position ¿i+1 and show how the three summations in the right hand side of (16) Pti+1
can be e±ciently re-computed. Consider ¯rst the term ti+1
X
j=si+1
So the term
Pti+1
j=si+1
Pj A2j
=
ti X j=si
Pj A2j
¡
si+1
X
j=si
Pj A2j can be obtained from
Pj A2j Pti
j=si+1
+
j=si
ti+1
X
j=ti
Pj A2j . If si+1 · ti , then
Pj A2j :
Pj A2j by performing (si+1 ¡si )+
(ti+1 ¡ ti ) + 1 additions and subtractions. When si+1 > ti then the direct computation of
Pti+1
j=si+1
Pj A2j involves ti+1 ¡st+1 < ti+1 ¡ti additions. So, in any case we can compute
the new term in at most si+1 ¡ si + ti+1 ¡ ti elementary operations. 2 Similarly one can show that both the second summation ¿i+1
summation 2¿i+1
Pti+1
j=si+1
elementary operations.
Pti+1
j=si+1
Pj and the third
Pj Aj can be computed by at most (si+1 ¡ si ) + (ti+1 ¡ ti ) + 1
Finally, we can conclude that P M (i + 1) can be calculated from P M (i) by performing
at most 3(si+1 ¡ si + ti+1 ¡ ti ) + 2 elementary operations. By solving the corresponding
recursive equation, one can show that P M (1); : : : ; P M (n) can be computed in at most 8n elementary operations (plus O(n) required to compute Ak , Pk Ak , Pk A2k , ¿k2 and 2¿k , for k = 1; : : : ; n).
So, the overall wanted signal for each possible detection window can be computed
in linear (amortized) time. Similar arguments apply to the computation of the overall
interfering mean value, and the total wanted and interfering variances. This implies that the optimum detection window - and thus the coverage probability - can be calculated in O(n).
41
