The Range Assignment Problem in Non-Homogeneous Static Ad-Hoc Networks Christoph Amb¨uhly Instituto Dalle Molle di Studi sull’Intelligenza Artificiale Galleria 2, 6928 Manno - SWITZERLAND
[email protected] Andrea E.F. Clementi Miriam Di Ianni Gianluca Rossiz Dipartimento di Matematica, Universit`a degli Studi di Roma “Tor Vergata”. Via della Ricerca Scientifica 1, 00133 Roma - ITALY fclementi,diianni,
[email protected] Angelo Monti Riccardo Silvestri Dipartimento di Informatica, Universit`a degli Studi di Roma “La Sapienza” Via Salaria 113, 00198 Roma - ITALY fmonti,
[email protected] Abstract
1 Introduction Static Ad-hoc wireless networks are constituted by a system of stations connected by wireless links. In ad-hoc networks, a transmission range is assigned to every station. One station s can directly transmit to another station t if and only if t is within the transmission range of s. The ranges determine a (directed) transmission graph in which nodes correspond to the stations and edges correspond to the transmission links. If a station needs to transmit outside its range it can use intermediate stations (multi-hop communication). The range transmission of a station depends, in turn, on the energy power supplied to the station: In particular, the power Ps required by a station s to correctly transmit data to another station t must satisfy the inequality [20]
Range assignment problems in Ad-Hoc wireless networks have been the subject of several recent studies. All these studies deal with the homogeneous case, i.e., all stations share the same energy cost function. However, this assumption does not well model realistic scenarios in which the energy cost of a station varies dramatically depending on the particular enviroment conditions of its location. We introduce the weighted version of the range assignment problem in which the cost a station s pays to transmit to another station depends on the distance between the stations and on the energy cost of station s. Most of the algorithm results for the unweighted range assignment problem can not be applied to the weighted version. We thus provide a set of algorithmic results for this version and discuss some interesting related open questions.
Ps dist(s; t) where dist(s; t) is the Euclidean distance between s and t, 1 is the distance-power gradient, and > 0 is the transmission-quality parameter. In an ideal environment it holds that = 2, but it may vary from 1 up to 6 depending on the environment conditions of the place the network is located. If r(s) is the transmission range of s, the transmission power of s is Ps = r(s) . The cost of this transmission is
This work is partially funded by the Information Society Technologies programm of the European Commission, Future and Emerging Technologies, under the IST-2001-33135 project CRESCCO and by the Italian MIUR project ReAlWiNe. y Supported by the Swiss National Science Foundation project 200021-100539/1, “Approximation Algorithms for Machine Scheduling Through Theory and Experiments”. Results obtained while the author stayed at Universit`a di Roma “ Tor Vergata” in Italy supported by the European Union under the RTN Project ARACNE z Supported by the European Union under the IST FET Project CRESCCO.
cost(s; r(s)) = (s) Ps 1
= (s) r(s)
(1)
where (s) is the energy unit cost of station s. From now on we assume, without loss of generality, = 1. Notice that, hosts may be portable devices that have only limited power resource available. One of the main benefits of ad-hoc networks is the ability of the hosts to vary the power used in the transmission (and, therefore, the transmission range) in order to avoid interference problems and reduce the power consumption. Deciding the transmission range of the single hosts in order to (i) guarantee a “good” communication between hosts, and (ii) minimize the overall costs of the network, gives rise to interesting algorithmic questions. In particular, these two aspects yield a class of fundamental optimization problems denoted as minimum weighted range assignment problems.
All the above properties have a bounded-diameter version: An additional parameter h is specified and the communication graph must have a diameter at most h. In this paper we also consider the following generalization of the broadcast:
From now on, we do not specify the parameters and d when they are clear from the contest.
Definition 1 (WMinRange d ())
Solution: A range assignment r : S ! R+ such that the transmission graph induced by r satisfies the property . Measure: The total cost of the range assignment r that is:
u2S
(u) r(u) :
(2)
Several complexity results and algorithmic solutions have been obtained concerning range assignment problems in the case (s) = 1 for any s [17, 15, 7, 10], that is, when the energy unit cost does not depend on the stations. We refer to this problem as Minimum Range Assignment (MinRange d ()). This is a good model in the case in which the stations and the environment are homogeneous. However, different technologies or different external conditions may result in different energy unit costs for the stations. In the homogeneous model two main properties are studied, the same that we also consider in this paper.
The strong connectivity ( = SC): The induced communication graph must be strongly connected. This guarantees the all-to-all communication.
The broadcast ( = B RO ): The induced communication graph must contain a directed spanning tree rooted at a given source station s. This guarantees that s can broadcast a message to the network.
Previous results. The MinRange(SC) problem was introduced in [17] where it has been proven to be N P -hard in a three dimensional Euclidean space for any value of . The same paper provides a 2approximation algorithm for the general case and an exact polynomial time algorithm for the one dimensional case. The N P-hardness in the plane for every was given in [14]. Finally, an 1:5-approximation algorithm for the case = 1 has been providedin [1]. The MinRange(B RO) problem was introduced in [15]. In [9] it was proven that the problem is N Phard if the stations lie in Euclidean plane and 2. Moreover, it is approximable within some constant [10, 7]. In the one dimensional Euclidean space the problem is in P for any [6, 13]. Other variants of these problems are studied in several papers (see for example [2, 4, 11]). For a more comprehensive collection of results see [12]. Finally, in [3, 5] it is studied a more general version of the range assignment problem in which the network is modeled with a directed edge-weighted graph: The weight of the direct edge (u; v ) represents the cost of the communication from u to v . The authors independently provide a log n approximation for the broadcast and strong connectivity. This upper bound immediately apply to WMinRange(B RO) and WMinRange(SC). In all the above mentioned papers it is implicitly assumed the existence of a central authority that has a full control over all the stations. This is not a good model in the case of autonomous stations where every station is governed by users which are independent from the central authority. In order to compute a feasible solution, the central authority needs to know all the relevant information from the stations. On the other hand such information may be private. When the central authority asks the stations for this information, they may report false
Instance: A set S of stations placed in a ddimensional Euclidean space and a energy unit cost function : S ! R+ ;
X
The multi-source broadcast ( = MSB): Given two sets of stations R (source set) and T (target set) such that R; T S and R \ T = ;, the induced communication graph must have the property that for each v 2 T there exists a path from some station in R to v .
2
values with the hope of being assigned a smaller transmission range. Here the selfish behavior of the stations plays a crucial role. This can be avoided by designing a so called mechanism (see [18, 19]), that is, a set of payment functions which, combined with a suitable algorithm that constructs a solution (i.e., a range assignment), rewards each station for its expenses in implementing the solution (i.e., the power corresponding to its transmission range). In [1] it is studied the model in which only the stations know the information regarding their relative positions with respect to the other stations in the network. The central authority asks directly to the stations for their relative distances and, on the basis of their declarations, it computes the range assignment. The paper shows that, by using the well known VCG mechanisms [8, 16, 21] and some algorithmic results, it is possible to incentive the stations to declare true costs. This is done for the MinRange(SC) problem for = 1. In this case the VCG mechanism provides a constant approximation.
As for the strong connectivity property the problem is N P-hard for any 1; similarly for the broadcast property the problem is N P-hard for any > 1. This directly follows from the N P-hardness for the homogeneous case. We show that for = 1 both problems are approximable within a constant. For > 1 we have a constant approximation algorithm for interesting class of instances of the WMinRange(SC) problem. Finally we present a family of instances showing that the 2-approximation algorithm for the MinRange(SC) problem does not work in non-homogeneous case.
2 The Linear Case In this section we will see that the WMinRange() problem on the line admits polynomial time algorithms for strong connectivity, broadcast and multi-source broadcast. Strong connectivity. The algorithm proposed in [17] for the homogeneous case works in the non-homogeneous case as well. More precisely, if costi (j ) is the cost of station i to directly reach station j , we say that costi is a monotone cost function if costi (j ) < costi (k ) if and only if dist(i; j ) < dist(i; k ). In can be proven that the algorithm in [17] is optimal if costi is monotone for each i 2 S . This is sufficient to state that the WMinRange(SC) problem on the line can be solved within the same time-complexity of the homogeneous case, i.e., in O(n4 ) time.
Our contribution. In this paper we investigate the WMinRange() problem in both one-dimensional and multi-dimensional Euclidean space. Section 2 is devoted to the one-dimensional Euclidean space. The network can be modeled as a set of stations located on a line (linear radio network). This is not a simplification of the model; indeed it represents a general framework for the analysis of real environments. For example, it can be necessary to implement an ad-hoc network along an highway or a railway. As for the strong connectivity property, we observe that the algorithm in [17] works also in the non-homogeneous case. As for the broadcast property, we first explain why the algorithm in [13] for the homogeneous case can not be extended to the nonhomogeneous one. We then provide new exact polynomial time algorithms for the unbounded and bounded diameter version. Finally, we show how the algorithm for the broadcast can be also used to solve the WMinRange(MSB) problem. We want to emphasize that our algorithms can be used also in a selfish environment in which a central authority asks to the stations for their energy unit costs while their position on the line is public. This can be done by directly using the VCG mechanism since the algorithms are optimal (see [18, 19]). The assumption that the energy unit cost is a private information and the locations of the stations are known is realistic. Indeed, since radio stations are static it can be reasonable that their locations is established by the central authority. Section 3 is devoted to the multi-dimensional case.
Broadcast. In [13] it is proposed an algorithm for the homogeneous case that runs in O(n2 ) time. When the source eccentricity is required to be at most h, the algorithm runs in O(h n2 ). It relies on the fact that in the optimal solution of the homogeneous case there exists at most one station v whose range is necessary in order to cover some other stations lying in the opposite side of v with respect to the source station s (see Figure 1). On the contrary, in the non-homogeneous case, there are instances of the WMinRange(B RO) problem whose optimal solutions require an arbitrary number of stations that perform this kind of “backwardjumps”: Indeed, consider the instance in Figure 2. If the costs of the stations are (s) = 10; 000, (x) = 100, (y) = 1 and (z ) = 0:01, the optimal solution contains three backward-jumps as reported in the figure. Moreover, the cost of this solution is 10; 000 + 100 9 + 1 49 + 0:01 225 < 10960. This is optimal because (i) if s reaches y instead of x we obtain a solution of cost at least 40; 000, (ii) if x reaches z instead of y we obtain a solution of cost at 3
Lemma 1 Let i and j be integers such that 1 j n; it holds that
/
w
y
s
8 for i = 1, j = n; 0 > > > > > > < min Cost [i; j ℄; p + M [i0; j 0 ℄ M [i; j ℄ = p62[i;j℄ > 0 0 > : [i ; j ℄ = [i; j ℄ [ Cov([i; j ℄; p) > > > > : otherwise
w
U
x
z
Figure 1. The optimal solution for this instance assigns to the station a – on the right of the source station s – a range that is necessary in order to cover also the station d on the left of s.
(3) P ROOF S KETCH . The case i = 1 and j Let us now assume that i 6= 1 or j 6= n. in
)
w
8
9 w y - s - x -
2
1
4
w -z
Observe that there exists at least one station [1; i 1℄ [ [j + 1; n℄ covered in one hop by
then MSB [i; j ℄; [1; i 1℄ [ [j + 1; n℄ must contain a multi-source broadcast from the stations in [i0 ; j 0 ℄ to the stations in [1; i0 1℄ [ [j 0 + 1; n℄ where i0 = p or j 0 = p. 2
least 10; 000+1600 and (iii) if y reaches w instead of z we obtain a solution of cost at least 10; 000+900+64. We now provide dynamic programming based polynomial time algorithms for the non-homogeneous unbounded and bounded case. The complexity of the algorithm is O(n3 ) for the unbounded case and O(h n4 ) for the h-bounded case. Let us denote the n stations on the line as 1; : : : ; n from left to right. Let s 2 f1; : : : ; ng be the source station and let [i; j ℄, 1 i j n, be the set of stations k such that i k j . Consider two sets R and T in [1; n℄ with R \ T 6= ;. Let r( R; T ) be a minimum cost range assignment to stations in R such that, for any station t 2 T , there exists a station s 2 R with rR;T (s) dist(s; t). Denote by Cost(R; T ) the cost of rR;T and by Cov(R; T ) the set of stations (not necessarily in T ) p such that dist(s; p) rR;T (s) for some s 2 R. Finally, we denote ad MSB(R; T ) any optimal multi-source broadcast where the source set is R and the target set T . Let M [i; j ℄, with 1 i j n, be the cost of
= n is trivial.
MSB [i; j ℄; [1; i 1℄ [ [j + 1; n℄ . Without loss of generality, assume at least one of such station belongs to [1; i 1℄ and let p be the leftmost one. Since stations in [i;j ℄ [ Cov([i; j ℄; p) are covered by one link,
Figure 2. A instance for the WMinRange(B RO) problem whose optimal solution contains 3 backwards jumps.
MSB
i
Theorem 1 The WMinRange(B RO) problem on the line can be solved in O(n3 ) time. P ROOF S KETCH . The table M of Lemma 1 can be computed in O(n3 ) time by setting M [1; n℄ = 0 and applying Equation (3) for all the other entries. This is possible since Cost and Cov can be pre-computed in O(n3 ) time by using dynamic programming. The cost of the optimal solution M [s; s℄. 2 We can generalize the previous technique to the hbounded version of the WMinRange(B RO) problem. Let MSBh (R; T ) be an optimal multi-source broadcast from R to T in which every node in T is connected to some node in R by means of a path of length (number of hops) at most h. We also define Mh [i; j ℄, 1 i j n and h 2 [1; n 1℄, as the cost of MSBh
Notice that the case h = n considered in Theorem 1.
[i; j ℄; [1; i 1℄ [ [j + 1; n℄ : 4
[i; j ℄; [1; i 1℄ [ [j + 1; n℄ : 1 is the unbounded case
Lemma 2 Let i be an integer such that holds that
Similarly to Lemma 1, it can be proven that
8 for i = 1, j = n 0 > > ; > > > > min Cost [i; j ℄; fq g + Mh 1 [1; q ℄ > > q>j > > > > for i = 1, j < n; > > > >
1, j = n; min Cost [i; j ℄; fp; q g pj
Mh [i; j ℄= min p > > > > > > > > > > > > > > > :
B [i℄ =
Cost
+Mh 1[p; q℄
8 > > > > >
> > > > :
j sx
22min [ +1 \[ i R
℄
;n i; j
℄
i
2 [1; n℄; it
[i; n℄ \ R = ;; i = n; B RO [i; sx ; j ℄ + B [j ℄ otherwise. (5)
P ROOF S KETCH . The cases [i; n℄ \ R = ; and i = n are trivial. The other case follows by observing that in an optimal multi-source broadcast of [i; n℄, station i is covered by a spanning tree rooted at some station sx in R \ [i; n℄. This spanning tree covers also stations on the right of i till some station j . 2
otherwise.
(4) for all i, j , such that 1 i j n and 1 h n 1. Roughly speaking, in the above formula we need p and q outside [i; j ℄ (on the left and on the right respectively) because we have to be sure that the number of hops on the left and on the right of [i; j ℄ increases simultaneously.
Theorem 3 The WMinRange(MSB) problem on the line can be solved in O(n6 ) time. P ROOF S KETCH . The cost of the optimal solution is
B [1℄. The time complexity for the construction of the table is given by the complexity for computing the n3 broadcasts B RO [i; sx ; j ℄. 2
Theorem 2 The h-bounded version of the WMinRange(B RO) problem on the line can be solved in O(h n4 ) time.
Notice that Theorem 1 Theorem 2 and Theorem 3 hold for any monoton cost function.
P ROOF S KETCH . We need to compute all the tables M1 ; : : : ; Mh. Table M1 is directly computed at first. Then, by using dynamic programming we can precompute all the Cost([i; j ℄; fp; q g) in time O(n4 ). Finally, each entry of M` , 1 ` h, is computed in time O(n2 ) by using pre-computed Cost and M` 1 . The algorithm returns Mh [s; s℄. 2
3 The Multi-Dimensional Case The WMinRange() problem inherits all the hardness results of the homogeneous version. More precisely, for the strong connectivity property, it is N Phard for any 1, while for the broadcast property, it is N P-hard for all > 1. In the remaining of this section, we shall present some positive results.
Multi-Source Broadcast. By using the algorithm for the WMinRange(B RO) problem, it is possible to provide an algorithm also for the WMinRange(MSB) problem on the line. What follows let R = fs1 ; : : : sk g, s1 : : : sk . Without loss of generality we assume T = S R is the target set. Let B RO [i; sx ; j ℄ with 1 i sx j n be an optimal broadcast of the interval [i; j ℄ starting from the source station sx . The algorithm computes B RO [i; sx ; j ℄ for all sources sx and all 1 i sx j n. After this, it combines all the computed broadcasts in an optimal way by using a dynamic programming algorithm. Let B [i℄, with 1 i n, be the cost of the optimal multi-source broadcast for [i; n℄ by using stations R \ [i; n℄ as source set.
3.1 Case = 1 It is easy to define a 2-approximation algorithm for the strong connectivity property. In fact, it is sufficient that the cheapest station u cover all the stations and the other stations reach u with a minimum directed spanning tree towards u. The 2-approximation comes from the fact that, since all the stations have to reach u, the cost of any optimal solution is at least the cost of an minimum directed spanning tree towards u. The claim follows because u has to reach all the other stations and it has minimum cost. For the broadcast we have the following algorithm (greedyBro): The source station s reaches the cheapest station vm by using a shortest path (SP(s; vm )). Station vm has a range that is sufficient to cover all the other stations. 5
f
Theorem 4 The greedyBro algorithm guarantees a 3-approximation for the WMinRange(B RO) problem when = 1.
s
opt maxfcost(SP(s; vm )); cost(SP(s; x))g (6) where SP(s; x) is any shortest path from s to x. The cost of the returned solution is cost(SP(s; vm )) + (vm ) dist(vm ; x):
r = k3=2
k4=3
(a)
(7)
(b)
Figure 3. An instance for which the mst based algorithm does not guarantee a constant approximation factor.
(vm ) dist(s; vm )+
(vm ) dist(s; x) cost(SP(s; vm ))+ cost(SP(s; x)):
convex hull Hv all stations are connected with v and vice-versa. The proof that this algorithm provides a q approximation follows by observing that the cost of the optimal solution is at least the cost of mst (s) for any sink stations s (see [17]). Then the performance ratio of this algorithm is
The assertion directly follows by combining this last inequality with Equations (7) and (6). 2
3.2
k4=3
f
Observe that,
(vm ) dist(vm ; x)
k17=12
ai
P ROOF. Let x be the farthest station from vm . The cost of the optimal solution (opt) is
s 1=2 a1 k1=2 k .. . k 1=2 k1=2 1=2 ak k
Case > 1
We now consider the WMinRange(SC) problem the for the case in which the layout of the stations satisfy a specific property of well-spreadness. Intuitively, this property establishes that the density of stations can not dramatically vary from place to place. For any station v , let H (v ) be a maximum convex hull that contains stations in S fv g as vertices and v as the only internal node. We can now define the two quantities:
P2
(v)rv cost(mst (s)) v
S
P
P 22 v
S
v
S
P
(v)rv
(v)v
P 22
vvq = q: v
v
S
S
( ) ( )
v
v
2
The analysis of the previous algorithm is strongly based on the fact that the solution is comparable to the cost of the minimum directed spanning tree towards some sink station. Let us consider the following algorithm for the general case of the WMinRange(SC) problem (for non q -spread instance): Given a station s, we assign to a generic station v the maximum range between the one coming from the mst (s) and the one coming from the minimum directed spanning tree rooted at s (mst! (s)). Note that this mst-based algorithm is a generalization of the algorithm in [17] that provides a 2-approximation for the homogeneous version of the problem. It is possible to prove that the analysis in [17] of mst-based algorithm for the homogeneous case can not be exploited in the nonhomogeneous case. The reason is that we can not compare mst! (s) with mst (s). More precisely, there exists a family of instances for which
r(H (v)) = max fdist(v; u) g u2H (v) (H (v)) = min fdist(v; u) g: u2H (v) Finally, let Hv any convex hull for v such that r(H (v )) is maximum. Let rv = r(Hv )) and v = (Hv ). We say that the instance S is q -spread if rv q v for any v 2 S . In order to prove the following theorem we need some additional notations: Given a station s, we denote with mst (s) as the minimum directed spanning tree oriented toward the sink station s.
Theorem 5 If S is q-spread then the WMinRange(SC) problem on S can be approximated within q .
cost(mst! (x)) cost(mst (x))
P ROOF. The algorithm assigns to every station v the range rv . This returns a feasible solution since in every 6
kp
(8)
d=1 >0
where x is any station, k is an arbitrary parameter and p is a constant. An instance of the family is shown in Figure 3. It is composed by k 1=12 gadgets reported in Figure 3(b) arranged like in Figure 3(a). The distances between nodes are illustrated in the picture and the unit energy costs are: (s) = (f ) = k 1=6 and (ai ) = k 1=3 for all i = 1; : : : ; k . In order to verify that Equation (8) holds for this instance we have to consider all the possibility for the source/sink station. Note that we are assuming = 2.
4
The source/sink station is f : cost(mst (f )) = O(k3 ) since one ak must reach f (then k3 ) and all the other communications cost k 35=12 (see the previous item). As for the cost(mst! (s)) we have the cost of reaching all the f s. It is easy to check that the more convenient way to do that is using all the k 1=12 ak s. This costs at least k 1=12 k3 .
The source/sink station is ai for some i: Similar to the case in the first item.
d>1
SC
P
N P-hard [17] 2-Apx
B RO
P
N P -hard ? 3-Apx
>1
N P-hard [17] log-apx [3, 5] O(1)-apx?? N P-hard [10] log-apx [3, 5] O(1)-apx??
Table 1. What is known and what is unknown depending on the dimension d and on the gradient .
The source/sink station is s: In this case cost(mst (s)) = O(k 35=12 ) since, in all the gadgets, the stations f must communicate with the station ak . The cost of all such communications is k 1=12 k 1=6 k 8=3 = k 35=12 . All the other communications are asymptotically less expensive. The cost of the mst! (s) is (k 3 ) since to reach a station f it is necessary to pay at least k1=3 k8=3 = k3 .
=1
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Conclusions and Open Questions
In this paper we have introduced a new family of range assignment problems for static ad-hoc wireless networks. We have provided optimal polynomial time algorithms for linear networks. For multi-dimensional space, since the WMinRange() problem is a generalization of the more classical MinRange() problem, all the hardness results for the last one (see Section 1) follow also for the WMinRange(). For = 1 we have constant approximation algorithms, but for > 1 the existence of an approximation algorithm is open. For this last case we have only a q -approximation for q -spread instances of the WMinRange(SC) problem. We have summarized the status of the art in Table 1.
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