Mar 3, 1993 - For the case in which the cost of being a reporting center is the same .... largest vicinity is at most Z. We call this case the unweighted C(G; Z), ...
Tracking Mobile Users in Wireless Communications Networks Amotz Bar-Noy
Ilan Kessler
IBM T. J. Watson Research Center Yorktown Heights NY. March 3, 1993
Abstract Tracking strategies for mobile wireless networks are studied. We assume a cellular architecture where base stations that are interconnected by a wired network communicate with mobile units via wireless links. Previous works focused on the cost of utilizing the wired links for management of directories. In this paper, the issue considered is the cost of utilizing the wireless links for the actual tracking of mobile users. We propose a novel tracking strategy in which a subset of all base stations is selected and designated as reporting centers. Mobile users transmit update messages only upon entering cells of reporting centers, while every search for a mobile user is restricted to the vicinity of the reporting center to which the user lastly reported. We rst show that for an arbitrary topology of the cellular network (represented by the mobility graph), nding an optimal set of reporting centers is an NP-complete problem. We then present optimal and near optimal solutions for important special cases of the mobility graph. For the case in which the cost of being a reporting center is the same for all vertices, we present an optimal solution for ring graphs and near optimal solutions for various types of grid graphs (one of which corresponds to the common topology of hexagonal cells). For the case in which di erent vertices may have di erent costs, we present an optimal solution for tree graphs and a simple approximation algorithm for arbitrary graphs.
1 Introduction The goal of future wireless networks is to provide ubiquitous communication services to a large number of mobile users. This requires wireless networks with high capacity that can support hand held terminals. To achieve high capacity under severe limitations on spectrum usage, as well as to be able to support low-power terminals, most designs of such networks are based on a cellular architecture with a very small cell size (microcellular structure) [6], [7], [3]. The cellular architecture comprises two levels { a stationary level and a mobile level. The stationary level consists of xed base stations that are interconnected through a xed network, referred to as the backbone network. The mobile level consists of mobile units that communicate with the base stations via wireless links. The geographic area within which mobile units can communicate with a particular base station is referred to as a cell. Neighboring cells overlap with each other, thus ensuring continuity of communications. The mobile units communicate among themselves, as well as with the xed information networks, through the base stations and the backbone network. An important issue in such networks is the design and analysis of strategies for tracking mobile users. In these networks, whenever there is a need to establish communication with any particular user, the network has rst to nd out which one of the base stations can communicate with that user. This is due to the fact that the users are mobile and could be anywhere within the area covered by the network. This issue was considered in [5], [1]. The problem addressed in these works is to nd ecient data structures for manipulating the information regarding the locations of the mobile users. It is assumed that this information is sent by the mobile users to the backbone network whenever the users move from cell to cell. The focus in these studies is on the trade-o between updating and retrieving information from directories in the backbone network. Thus, the issue considered in these works is the cost of utilizing the wired links of the backbone network for management of directories. In this paper, the issue considered is the cost of utilizing the wireless links for the actual tracking of mobile users. To illustrate the problem, let us consider the following two simple tracking strategies. The rst strategy is the Always-Update strategy, in which each mobile user transmits an update message whenever it moves into a new cell. Clearly, under this strategy the overhead due to transmissions of update messages is very high, especially in networks with a small cell size and a large number of highly mobile users. On the other hand, since the current location of each user is always known, the overhead for nding users is zero. The second strategy is the Never-Update strategy, in which the users never send update messages regarding their location. Clearly, under this strategy there is no overhead for updating. On 1
the other hand, whenever there is a need to nd a particular user, a network-wide search is required, the overhead of which is very high. These two simple strategies demonstrate the basic trade-o that is inherent to the problem. Thus, there are two basic operations that are elemental to tracking mobile users { update and nd. Associated with each one of these operations there is a certain cost. As demonstrated by the two examples above, increasing one cost leads to a decrease in the other one. In fact, the above simple strategies are the two possible extreme strategies, in each of which one cost is minimized (set to zero) and the other cost is maximized. The question that arises is what is the strategy that minimizes the total cost. Existing systems use either one of the two extreme strategies described above, or the following combination of them (used in current cellular telephone networks). Each user is aliated with some geographic region, referred to as its home-system. Within the homesystem, the Never-Update strategy is used. When the user moves into another region, then it must register (update), and then the Never-Update strategy is used within that region. The partition into regions is done according to commercial considerations, and there is no attempt to design this partition in order to minimize costs. In this paper we investigate a novel strategy for tracking mobile users. In Section 2 we propose a strategy in which a subset of all cells is selected and designated as reporting cells. The idea is that mobile users transmit update messages only upon entering a reporting cell, and a search for any user is always restricted to the vicinity of some reporting cell, namely, the one to which the user lastly reported. In Section 3 we present the mathematical model and show that the problem of selecting the reporting cells is in general NP -complete. In Sections 4 and 5 we focus on some common topologies of wireless networks. In Section 4 we present an optimal solution for ring graphs and near optimal solutions for various types of grid graphs, when the cost of being a reporting center is the same for all vertices. One of the grid graphs considered in this section corresponds to the common topology of hexagonal cells. In Section 5 we present an optimal solution for tree graphs and a simple approximation algorithm for arbitrary graphs, when the cost of being a reporting center may be di erent for di erent vertices. Finally, in Section 6 we discuss some open problems.
2 The strategy In this section we describe our strategy for tracking mobile users. The basic idea behind the strategy is to let mobile users transmit update messages only at speci c cells (the number of 2
which is small compared to the total number of cells in the network), while restricting every search for a mobile user to a small subset of cells. To achieve this, a subset of all base stations of the network is selected, and those base stations are designated as reporting centers. The cells associated with these base station are referred to as reporting cells. These base stations periodically transmit on the wireless channel short messages which identify them as reporting centers. Thus, each mobile user can learn whether it is in a reporting cell or not simply by listening to the transmissions of the base stations. We de ne the vicinity of reporting cell x to be the collection of all non-reporting cells that are reachable from cell x without crossing another reporting cell. By de nition, cell x is in the vicinity of itself. The following rules specify the update and the nd operations under the proposed strategy.
Update: Each time a mobile user enters a reporting cell, the user reports (i.e. transmits an
update message) to the reporting center associated with that cell (unless this reporting center is the last one to which the user reported). No update messages are sent to base stations that are not reporting centers.
'$ '$ '$ t '$ '$ '$ &% &% &% t'$ t '$ '$ &% &% &% t &% &% &%
Find: Whenever there is a need to establish communication with a particular mobile user,
the user is searched for in all cells that are in the vicinity of the reporting center to which the user last reported. 1
4
2
5
7
3
? 6 ?? -
8
6
9
Figure 1: An Example
An example that demonstrates this tracking strategy is shown in Figure 1. In this example there are 9 cells denoted by 1,...,9. The cells 2,4,6,8 are designated as reporting centers and all the vicinities are of size four. For example, the vicinity of center 2 consists of the cells 1,2,3,5. Suppose a user m moves along the path 2,5,8,9,6,5,8. Then user m reports when it arrives at cells 2,8,6 and again at cell 8. Now assume that the system is looking for m during each one 3
of its visits to cell 5. In the rst time the system searches for m in cells 1,2,3,5, since this is the vicinity of cell 2 (which is the last reporting center m reported to). In the second time the system searches for m in the vicinity of reporting center 6 which includes the cells 3,5,6,9. The problem that arises is how to select the reporting centers such that the costs of the update and the nd operations are minimized. To address this problem, we rst have to de ne the cost measure associated with each one of these operations. These cost measures are de ned formally in the next section. However, we give here an intuitive motivation for these de nitions. Clearly, the cost of the nd operation increases with the size of the vicinity in which the search is performed. Thus, the cost of the nd operation is taken to be the number of vertices in the largest vicinity in the network. The cost of the update operation increases with the volume of update messages that are transmitted by the mobile users. The volume of update messages in each reporting cell increases with the frequency that mobile users enter that cell. Thus, we associate with each cell x a weight wx, which re ects the frequency that mobile users enter into that cell (as well as other factors on which the cost of update may depend). The cost of the update operation is taken to be the sum of the weights of all the reporting cells. With these de nitions of the costs, the problem is to select reporting centers in such a way that both the size of the largest vicinity and the total weight of the reporting centers are minimized. These are contradicting goals in general, as in order to decrease the sizes of the vicinities, the number of reporting centers should be increased (and vice versa). The approach we take is to bound one of the costs and then to minimize the other cost. This is formalized in the next section.
3 Model In this section we introduce our model for the problem, and present some preliminary results. The mobility graph G of the network is the graph in which each vertex corresponds to a di erent cell, and two vertices are connected by an edge if and only if the corresponding cells overlap. Each vertex i of the mobility graph has a weight wi > 0. Let I be a set of vertices, referred to as centers. The vicinity of center v is the set of all vertices not in I which are reachable from v by a path containing no centers. By de nition, the P vicinity of center v includes v. The weight of I is w(I ) = i2I wi , and the size of the largest vicinity in the graph is denoted by z (I ). 4
The reporting centers problem { C (G; Z ): Given a weighted graph G and an integer Z , select a set of centers S such that z (S ) Z and w(S ) w(S 0 ) for all S 0 such that z (S 0 ) Z . Note that one can de ne a dual problem, in which a weighted graph G and a number W are given, and the goal is to select a set S of centers such that w(S ) W and z (S ) z (S 0 ) for all S 0 such that w(S 0 ) W . Although in this paper we concentrate on C (G; Z ), it will be clear how the results apply to the dual problem. An important special case of C (G; Z ) is the case in which all the weights are equal to one. In this case the weight of a set of vertices is just the cardinality of the set. Thus, in this case the reporting centers problem is to nd the smallest set of centers such that the size of the largest vicinity is at most Z . We call this case the unweighted C (G; Z ), while the more general case is referred to as the weighted C (G; Z ). For an n vertices graph, the following proposition state trivial bounds to the unweighted C (G; Z ). Proposition 1 For any set S of centers, z(nS) jS j n ? z(S ) + 1 We conclude this section by showing that C (G; Z ) is a hard problem.
Theorem 1 C (G; Z ) is an NP -complete problem for any Z 2. Proof sketch: A set J is an independent set if there is no edge between any two vertices in J ,
i.e., the shortest path between any two vertices in J contains at least one vertex that is not in J . A set J is an 2-independent set if the shortest path between any two vertices in J contains at least two vertices that are not in J . De ne MIS to be the problem of nding the maximum size independent set in a graph. De ne MIS 2 to be the problem of nding the maximum size 2-independent set in a graph. It is well known that MIS is an NP -complete problem [2]. The proof proceeds as follows. First it is shown that MIS 2 is an NP-complete problem by a reduction from MIS (see Appendix). Next we show that C (G; 2) is an NP-complete problem by a reduction from MIS 2 . This reduction follows from the observation that a set of vertices J is a 2-independent set if and only if z(J) 2, where J is the set of all vertices in G that are not in J . This implies that S is a solution to C (G; 2) if and only if S is a solution to MIS 2 . The proof is completed by constructing a reduction from C (G; 2) to C (G; Z ) for any Z > 2 (see Appendix).
2
5
4 Unweighted Graphs In this section we consider the case in which wi = 1 for every vertex i of the mobility graph. In Subsection 4.1 we present an optimal solution for ring graphs (the same solution applies also to line graphs). In Subsections 4.2, 4.3 and 4.4, we consider grid graphs, uni-diagonal grid graphs and bi-diagonal grid graphs, respectively. In each case we rst prove a lower bound on the number of centers, and then present a solution in which the number of centers is at most within a constant factor from the lower bound. Note that the mobility graph that corresponds to the common topology of hexagonal cells is a uni-diagonal grid graph.
4.1 Rings We rst derive a lower bound on the number of centers c, given that the largest vicinity contains at most Z vertices. Let 1; 2; : : : ; n be the vertices of the ring. Let R1 ; : : : ; Rc be any set of c centers and let zi Z be the size of the vicinity of center Ri .
Lemma 1 Pci zi = 2n ? c Proof: Each vertex which is not a center belongs to exactly two vicinities, while each center =1
belongs only to its own vicinity. Therefore, in the above summation each non-center vertex is counted twice, whereas each center is counted exactly once. 2 l
m
Proposition 2 c Z n Proof: From Lemma 1 it follows that c = 2n ? Pci zi 2n ? Zc. Therefore, c Z n . The 2 +1
2 +1
=1
result follows since c is an integer. 2 We now show how to achieve the lower bound of Proposition 2. Let p and q be non-negative integers such that n = p(Z + 1) + q and q < Z + 1. To achieve the lower bound select the following vertices to be centers: 1; Z + 1; Z + 2; 2(Z + 1); : : : ; (p ? 1)(Z + 1) + 1; p(Z + 1):
(1)
Clearly, each vicinity contains at most Z vertices, and there are exactly 2p = 2 bn=(Z + 1)c centers. Thus we have the following.
Proposition 3 If the set of centers is as de ned by (1), then c
l
n
2 +1
Z
m
Note that the lower bound can also be achieved by selecting the centers such that the distance between any two centers is about Z=2. 6
4.2 Grids In this subsection we consider a grid graph of size n n. The vertices of the grid are denoted by pairs (x; y) where 1 x; y n. The neighbors of vertex (x; y) are vertices (x + 1; y); (x ? 1; y); (x; y + 1); (x; y ? 1) (if they exist). We rst derive a lower bound on the number of centers c, given that the largest vicinity contains at most Z vertices. Consider an in nite grid, and let A be any connected sub-graph of it. De ne the perimeter of A as the set of all vertices not in A which have an edge to a vertex in A. Let P (a) be the minimum number of vertices in a perimeter of a connected sub-graph with a vertices, where the minimum is taken over all possible sub-graphs with a vertices.
p Proposition 4 P (a) 8a
To prove Proposition 4 we need the following. Given a connected sub-graph of the in nite grid and its perimeter P , de ne Xmin = minfxj(x; y) 2 Pg, and de ne similarly Xmax , Ymin and Ymax . A perimeter P is said to be a generalized diamond if the following holds: (1) Each one of the following sets contains at most two vertices that are contained in P : f(x; y) j x = Xmin g, f(x; y) j x = Xmax g, f(x; y) j y = Ymin g and f(x; y) j y = Ymaxg. (2) For every Xmin < x < Xmax and Ymin < y < Ymax , (x; y) 2 P implies that none of the neighbours of (x; y) is in P . In Figure 2 we give four examples of generalized diamonds that demonstrate the four possible classes of generalized diamonds (the class is de ned by the number and position of corners of the diamond that are \truncated"). α
(a)
β
α
β
α
(b)
β
(c)
α
β
(d)
Figure 2: Example of generalized diamonds. The vertices of the grid graph are represented by the intersections of the grid lines; the thick lines mark the reporting centers.
Lemma 2 For every positive integer a, there exists a connected sub-graph with a0 a vertices, such that its perimeter P 0 has P 0 P (a) vertices and P 0 is a generalized diamond. Proof: The basic idea is that if a perimeter is not a generalized diamond, then it can be
\stretched" to a generalized diamond without increasing the number of vertices in the perime7
ter, such that it surrounds at least all the vertices that were surrounded originally. Moreover, note that the only type of perimeters that cannot be further stretched to surround more vertices is the generalized diamond. The rigorous proof is straightforward but tedious, and therefore is omitted. 2
Lemma 3 Let B be a connected sub-graph p with b vertices, whose perimeter P is a generalized diamond with P vertices. Then P 8b. Proof: For each one of the four classes of generalized diamonds let and be de ned as
shown in Figure 2. Then the perimeter P and the number of vertices b for each one of the cases (a) - (d) is as follows. (a) P = 2 + 2 ? 4 ; b = ( ? 1)( ? 1) + ( ? 2)( ? 2) (b) P = 2 +2 ; b = ( ?1)+( ?1) (c) P = 2 +2 ?2 ; b = ( ?1)( ?1)+( ?1)( ?1) (d) P = 2 + 2 ? 1 ; b = ( ? 1)( ? 1) + ( ? 1). Using these formulas, the result can be veri ed straightforwardly. 2 p Proof of Proposition 4: From Lemmas 2 and 3 it follows that 8 pPa Pp(aa) 2 Proposition 4 enables us to derive a lower bound on the number of centers as follows. Assume an arbitrary selection of centers in an n n grid, and denote the number of centers by c. Let A1 ; A2 ; : : : ; Ak be the connected components that are formed when all c centers are removed from the n n grid. Denote the number of vertices in Ai by zi . 0
0
Lemma 4 c p Pki pzi ? n: Proof: Let Pi be the number of vertices in the perimeter of Ai, i = 1; : : : ; k. Since the P perimeter of the n n grid has 4n vertices, it follows that ki Pi 4c + 4n. The factor of 1 2
=1
=1
4 that precedes the c follows from the fact that each center may be counted up to four times when summing the Pi 's. The result now follows from Proposition 4. 2
Lemma 5 The minimum of Pki pzi under the two constraints zi Z ? 1 for all i 1 and Pk i zi = n ? c is obtained when zi = Z ? 1 for all i 2. Proof: Let a a : : : ak be the values of the zi 's for which the minimum is obtained. Assume that a < Z ? 1. Then zi = a0i , i 1, where a0 q= a ? 1, a0 = a + 1 and a0i = ai P for all i 3, satisfy the above two constraints. But ki a0i < Pki pai, which contradicts the assumption that theqminimum is obtained for zi = ai , i 1. The above inequality is true q 0 0 2 since it is equivalent to a + a < pa + pa , which is equivalent to a < a + 1. =1
=1
2
1
2
2
1
=1
1
1
2
2
1
2
2
=1
1
2
Theorem 2 c n p pZ ? ? n : Proof: Let m and Z 0 be integers such that n ? c = m(Z ? 1) + Z 0 and Z 0 < Z ? 1. Plugging 2
1
1
2
1 +1
2
8
p p the result of Lemma 4 into Lemma 5 yields: c p12 m Z ? 1 + Z 0 ? n: The result p p of the theorem is obtained by noting that m Z ? 1 + Z 0 pnZ2 ??c1 : 2
Figure 3: Solution for unweighted grid We now show a particular selection of centers that achieves the lower bound of Theorem 2 p by a factor of 2 2. The basic idea is to select the centers such that they cover the grid with generalized diamonds of the type shown in Figure 3. In this type of generalized diamonds, all edges of the diamonds contain the same number of vertices, and all four corners of the diamond are \truncated". Such generalized diamonds are referred to as perfect diamonds. A set of centers that covers the grid with perfect diamonds such that the edge of each diamond contains L vertices is referred to as CL . For example, the set of centers shown in Figure 3 is C3 . Recall that for any set of centers I , the size of the largest vicinity is denoted by z(I ).
Proposition 5 Let Z = (2L ? 1) for some integer L 2. Then the set CL satis es the condition z (CL ) Z , and we have c n pZ + 2n: Proof: Each row (and each column) of the grid contains at most 2 dn=(2L)e centers, and therefore the total number of centers in the grid satis es c 2n dn=(2L)e n =L + 2n. The 2
2
2 +1
2
size of the largest vicinity is twice the number of vertices in the interior of each diamond, plus one for the center. This yields z (CL ) = (2L ? 1)2 = Z . Solving the last equality for L and substituting the result in the above upper bound on c yields the result. 2 The ratio of the upper bound of Proposition 5 to the lower bound of Theorem 2 is
pp
1 ?1 ?! 2 2 pZ ? 1 + 2 p 2 + n2 p p 1 ? 2 Z ?1 +1 n Z +1 Z +1
p
as n ! 1. The expression on the right-hand side is bounded from above by 2 2. Thus we have proved the following for any integer L 2. 9
p Theorem 3 For n large enough, the number of centers in the set CL is at most 2 2 times the number of centers in any set of centers I for which z (I ) z (CL ) = (2L ? 1) . 2
Remark: For values of Z other than (2L ? 1) for some integer L, non-perfect generalized 2
diamonds should be used in order to minimize the number of centers. However, if we select the set of centers CL such that L is the largest integer satisfying (2L ? 1)2 Z (which in such cases is less ecient than non-perfect generalized diamonds), then by a slight modi cation of the proof of Proposition 5 it can be shown that c n2 pZ2?1 + 2n. This yields a ratio of the upper p bound to the lower bound that monotonically decreases to 2 2 as Z increases. In particular, this ratio is less than 4:9 for Z 10, and less than 3:9 for Z 30.
4.3 Uni-Diagonal Grids (Hexagonal Cells Topology) In this subsection we consider the uni-diagonal grid graph. The importance of this graph stems from the fact that it is the mobility graph of the well known cellular topology of hexagonal cells. A uni-diagonal grid graph of size n n is a grid graph in which the vertex (x; y) is also connected to the vertices (x ? 1; y ? 1); (x + 1; y + 1) (if they exist). Proceeding similarly to the previous subsection, consider an in nite uni-diagonal grid, and let A be any connected sub-graph of it. De ne the perimeter of A and P (a) as in the previous subsection.
X
Y β α Figure 4: Example of a hexagon
p Proposition 6 P (a) 12a Proof: The proof proceeds similarly to the proof of Proposition 4. A perimeter P is said
to be a hexagon if it has six sides as depicted in Figure 4 (we omit the formal de nition). Note that the hexagon has diagonal sides only along the diagonals of the uni-diagonal grid. 10
Replacing in Lemmas 2 and 3 the generalized diamond with a hexagon, these lemmas hold also p p for uni-diagonal grids (with a factor of 12 instead of 8 in Lemma 3). The proof of Lemma 2 in this case is similar to the original proof. To prove Lemma 3 for the uni-diagonal grid case p (with a hexagon and a factor of 12), let X , Y , and be de ned as shown in Figure 4. Then P = 2(X + Y ) ? ( + + 2) and b = XY ? 21 ( ? 1) ? 21 ( ? 1) ? P . This yields P 2 ?12a = (A?B )+(C ?D) where A = 4x2 +4y2 +7 2 +7 2 +2 , B = 4XY +4( + )(X +Y ), C = 16(X + Y ) and D = 20 + 14( + ). Using the fact that u2 + v2 2uv, it can be shown that A B , and since C D the result follows. 2 Assume now an arbitrary selection of centers in an n n uni-diagonal grid. De ne c, A1; A2; : : : ; Ak , and zi as in the previous subsection.
Lemma 6 c
p2
3
Pk
i=1
pzi ? n:
The proof of Lemma 6 is similar to that of Lemma 4, except that here ki=1 Pi 3c + 4n. Here each center may be counted up to three times, since in the sub-graph that consists of a center and all its neighbors, the maximum size of an independent set is 3. The proof of the next theorem is similar to that of Theorem 2. P
Theorem 4 c n
2
p3pZ2?1 +2 ? n1
(a)
(b)
Figure 5: Solutions for (a) uni-diagonal grid (b) bi-diagonal grid grid graphs A particular selection of centers that achieves the lower bound of Theorem 4 by a factor of p p = 4:5 is shown in Figure 5(a). The important features are (a) all edges of the hexagons have the same lengths, and (b) two vertices from the interior of the hexagon, each of which is the closest to a corner, are also centers. By having (b), the vicinity of each center consists of the interiors of at most two hexagons. A set of centers that covers the uni-diagonal grid as shown in Figure 5(a) such that the edge of each hexagon contains L vertices is referred to as HL . 3 2
11
Proposition 7 Let Z = 6(L ? 1)(L ? 2) ? 1 for some p integer L 4. Then the set HL satis es the condition z (HL ) Z , and we have c n pZ ? : Proof: Assume further that n = 3`(L ? 1) for some integer ` 1. Then, out of every L ? 1 rows we have 1 row with `L centers, 2 rows with 3` centers and L ? 4 rows with 2` centers. This yields nc2 = LL?? 2 . The proposition follows since Z ? (L ? 1)(L ? 2). With the right 6
2
1
3 3(
2 1)
6
1
choice for the positions of the hexagons within the uni-diagonal grid, the claim is valid for all values of n. 2 The ratio of the upper bound of Proposition 7 to the lower bound of Theorem 4 monop p tonically decreases to 4:5 as Z increases. In particular, this ratio is less than 1 + 4:5 for Z 7.
4.4 Bi-Diagonal Grids Using the techniques of the two previous subsections, we can obtain results also for bi-diagonal grid graphs. The proofs of these results are omitted as they are very similar to those in the previous two subsections. A bi-diagonal grid graph of size n n is a grid graph in which the vertex (x; y) is also connected to the vertices (x?1; y ?1); (x+1; y +1); (x?1; y +1); (x+1; y ?1) (if they exist). Using similar de nitions to those in the previous two subsections we obtain that p in a bi-diagonal grid, P (a) 4 a (here a rectangle plays the role of the generalized diamond and the hexagon from the previous subsections). This yields,
Theorem 5 c n
2
pZ ?11 +1 ? n1
:
p
A particular selection of centers that achieves this lower bound by a factor of 8 is shown in Figure 5 (b). Such a set of centers is referred to as SL , where L is the number of vertices on each edge of the squares.
Proposition 8 Let Z = 2(L ? 2) ? 3 for some p integer L 2. Then the set SL satis es the condition z (SL ) Z , and we have c n pZ ? : 2
2
8
1
The ratio of the upper bound of Proposition 8 to the lower bound of Theorem 5 monop p tonically decreases to 8 as Z increases. In particular, this ratio is less than 1 + 8 for Z 9.
12
5 Weighted graphs In this section we consider the case in which the vertices of the mobility graph have arbitrary weights. We rst present an optimal algorithm for nding centers in tree graphs. We begin with the special case of line graphs, since the simplicity of this case enables one to get more insight into the problem. Thereafter, we generalize the results for tree graphs. Finally, we present and analyze an approximation algorithm for general graphs.
5.1 Weighted Lines In this subsection we describe an algorithm that nds an optimal set of centers for a weighted line graph. Let the vertices on the line be numbered consecutively from 1 to n, and let their weights be w1 ; : : : ; wn , respectively. Given an integer 1 < Z < n, the goal is to nd a set of centers S such that the following hold: (a) The largest vicinity contains at most Z vertices, and (b) w(S ) = minI f1;:::;ngfw(I ) j (a) holds for I g. We denote this problem by C (n; Z ) and call a set with the above properties a solution to C (n; Z ). A solution to C (n; Z ) can be found by checking all the possible subsets of f1; : : : ; ng. However, this process requires an exponential time of computation. On the other hand, it seems that there is no simple dynamic programming solution, since knowing the solutions to C (1; Z ); : : : ; C (n ? 1; Z ) does not directly imply the solution to C (n; Z ). The key idea behind our algorithm is the de nition of a related problem which can be solved via a dynamic programming method and which implies the solution to C (n; Z ). The modi ed problem is to nd for a given integer 0 k < Z , a set of centers Sk such that the following hold: (a) The set Sk contains the vertex n ? k and does not contain the vertices n ? k + 1; : : : ; n, (b) the largest vicinity contains at most Z vertices, and (c) w(Sk ) = minI f1;:::;ng fw(I ) j (a) and (b) hold for I g. We denote the modi ed problem by Ck (n; Z ) and call a set with the above properties a solution to Ck (n; Z ). The following proposition shows how to obtain a solution to C (n; Z ) from solutions to Ck (n; Z ); 0 k < Z .
Proposition 9 For every k = 0; : : : ; Z ? 1, let Sk be a solution to Ck (n; Z ), and let k0 be an w(Sk ). Then Sk is a solution to C (n; Z ). index for which w(Sk ) = 0min k 0, the connected component that contains r, denoted by Ur , satis es jUr \ TL j + jUr \ TR j = k ? 1. This implies that at least one of the sets fSi1+l+j (TL ) [ Sj1+l+i (TR ); i + j = k ? 1g must be a solution to Ckl (T; Z ). Here the subscripts i and j correspond to jUr \ TL j and jUr \ TR j, respectively. The superscript in Si1+l+j (TL ) is 1 + l + j since r is not a center, and the connected component of TR which contains the root of TR has j vertices. Therefore the largest vicinity in TL which contains the root of TL , can have at most (Z ? l) ? (1 + j ) vertices. A similarexplanation holdsfor 1+l+j 1+l+i 1+l+i (TR ) is (TL ) + w Sj the superscript in Sj (TR ). The result follows since w Si minimal. 2 The initial values for the recursive calculation are the following. Let r be the root of the tree T , then S0j (T ) = frg for all j such that j + jT j Z . Also, by de nition, Sij (T ) = ; and w Sij (T ) = 1 for i > jT j and i + j Z ? 1. Finally, by de nition, Sij (T ) = ; and w Sij (T ) = 0 for i = jT j and i + j Z ? 1. The complexity of computing the optimal sets Skl (T ) for all non-negative integers k and l such that k + l Z ? 1 is determined by the implementation of the recursive equations in Proposition 12. 0
0
0
0
0
0
Lemma 8 For any binary tree T , the sets Skl (T ) can be computed for all non-negative k and l such that k + l Z ? 1 in O(nZ ) time. Proof: Each min operation in the equation of Proposition 12(a) can be done by O(Z ) op3
2
erations and is computed for at most Z values of k. Each min operation in the equation of Proposition 12(b) can be done by O(Z ) operations and is computed for at most Z 2 values of k and l. The lemma follows since there are n vertices in the tree T . 2 The next theorem follows since computing the solution to C (T; Z ) requires only O(Z ) operations (see Proposition 11).
Theorem 7 For any binary tree T , the algorithm de ned by Proposition 12 computes a solution to C (T; Z ) in O(nZ 3 ) time.
Now assume that T is an arbitrary tree. In this case, the solutions for T are computed after 17
computing the solutions for all the sub-trees of T . Let r be the root of the tree T and denote the sub-trees of T by T1 ; : : : ; Td . Here we just state, without complete proofs, the equivalent to Proposition 12, Lemma 8, and Theorem 7.
Proposition 13 (a) For every l Z ? 1, let i0 ; : : : ; i0d be indices for which w Si1 (T ) + + w Sid (Td ) = i min w S ( T ) + + w S ( T ) : d i i 1 d id lZ ? 1 1
0
0
0
1
0
0
+
+
+
1
0
1
Then S0l (T ) = frg [ Si01 (T1 ) [ [ Si0d (Td ) is a solution to C0l (T; Z ). (b) For every k > 0 and l 0 such that k + l Z ? 1, let let i01 ; : : : ; i0d be indices for which 0
0
w
Sik1+l?i1 (T1 ) + + w
0
0
Sikd+l?id (Td ) 0
0
1
d
Then Skl (T ) = Sik1+l?i1 (T1 ) [ [ Sikd+l?id (Td ) is a solution to Ckl (T; Z ). Lemma 9 For any arbitrary tree T , the sets Skl (T ) can be computed for all non-negative k and l such that k + l Z ? 1 in O(nZ 3 ) time. 0
0
0
= i +min w Sik1+l?i1 (T1 ) + + w Sikd+l?id (Td ) : +i =k
0
Proof: The min operations in the equations of Proposition 13 can be computed in d ? 1 stages.
First they are computed assuming T has only two sub-trees T1 and T2 , next assuming that T has three sub-trees T1 , T2 , and T3 , and so on. When adding the sub-tree Tj we use the results computed assuming T 's sub-trees are T1 ; : : : ; Tj ?1 , using the same method as in the binary tree case. This implies that in each vertex in the tree at most O(dZ 3 ) operations are computed. The lemma follows since the sum of all the degrees in the tree is n ? 1. 2
Theorem 8 For any arbitrary tree T , the algorithm de ned by Proposition 13 computes a solution of C (T; Z ) in O(nZ 3) time.
5.3 Weighted arbitrary graphs In this subsection we describe a greedy algorithm for an arbitrary weighted graph, given that the largest vicinity has at most Z vertices. The weight of the set of centers selected by this algorithm is at most Z 2 times the weight of any optimal set of centers, where is the maximum degree in the graph. Note that this is a worst case performance; there are graphs in which this algorithm produces an optimal solution. The basic idea behind the algorithm is the following. Initially all vertices are designated as centers. Then, the centers are checked in an order of decreasing weights, and if making a 18
center a non-center vertex does not create a vicinity larger than Z , then this center is made a non-center vertex. Let the vertices of the graph be denoted by 1; : : : ; n, and without loss of generality assume that w1 w2 wn . The following is a high-level description of the algorithm. 1. S := f1; : : : ; ng ; 2. x := 1 ; 3. If z (S ? x) Z then S := S ? fxg ; 4. x := x + 1 ; 5. If x n then go to step 3 ; 6. Return the set S ; Two centers are said to be siblings if their vicinities are not disjoint. Let S be the output of the greedy algorithm, and let R be an optimal set of centers.
Lemma 10 . For any greedy center x 2 S there exists an optimal center y 2 R such that wy wx, and x is either in the vicinity of y or in the vicinity of a sibling of y (where the vicinities are with respect to the optimal set R).
Proof: If x 2 S \ R than choose y = x. Otherwise, assume to the contrary that no such
optimal center y exists. Consider step 3 of the greedy algorithm at the iteration in which it was checked whether x should be removed from S . By the contradiction assumption, at this stage wx > wj for all j such that x is either in the vicinity of j or in the vicinity of a sibling of j . Therefore, since R is a solution, removing x from S would not have been created a vicinity larger than Z . But x 2 S which is a contradiction. 2
Lemma 11 . Each center has at most Z siblings. Proof: The lemma follows since each vicinity contains at most Z vertices, and each vertex is 2
connected to at most centers. Note that this result holds with equality for tree graphs.
Theorem 9 Px2S wx Z Py2R wy . Proof: Let x be a greedy center and let y(x) be an optimal center such that wy x yx and 2
( )
x is either in the vicinity of y(x) or x is in the vicinity of a sibling of y(x) (see Lemma 10). It 19
follows that x2S wx x2S wy(x) . Note that the right-hand side of this inequality contains only optimal centers. It follows from Lemma 11 and the fact that the largest vicinity has at most Z vertices, that each optimal center appears at most Z 2 times in the right-hand side P P of the last inequality. Therefore x2S wy(x) Z 2 y2R wy . 2 P
P
6 Discussion In this paper we proposed a novel strategy for tracking mobile users in wireless networks. In this strategy, a subset of all cells is selected and designated as reporting cells. Mobile users send update messages only when entering a reporting cell, and a search for any particular mobile user is conducted only in the vicinity of the reporting center to which the user lastly reported. We rst showed that nding an optimal set of reporting centers is an NP -complete problem for arbitrary mobility graphs. We then presented optimal and near optimal solutions for important special cases of the mobility graph. For the unweighted problem, we presented an optimal solution for ring graphs and near optimal solutions for various types of grid graphs (one of which is the mobility graph of the common topology of hexagonal cells). For the weighted problem, we presented an optimal solution for tree graphs and a simple approximation algorithm for arbitrary graphs. The volume of control trac due to tracking of mobile users depends on the mobility of the users as well as on the frequency with which they need to be located. As discussed in Section 2, the mobility of the users is taken into account by the model through the weights of the cells. The frequency with which users need to be located can be taken into account in our model through the choice of Z . If the mobile users need to be located more frequently, then the vicinities should be chosen smaller, which can be done by choosing a smaller Z . We note that the cost of the nd operation may depend on how a search for a mobile user is implemented. For example, assume that a search for a mobile user always starts from the cell in which the user last reported, then progresses to the neighboring cells of that cell, and so on until the user is found. Then the cost measure can be taken as either the maximum radius of a vicinity (which represents the maximum time to nd a user), or the maximum size of a vicinity (which represents the maximum volume of paging messages), or any combination of the two. Clearly, picking any particular cost measure would implicitly favor certain allocations of reporting centers while ruling out others. However, it seems that the ideas and techniques presented in this paper can be used to solve the problem with other cost measures as well. 20
It is important also to note that the approach of xed reporting centers is most suitable to situations in which most of the users most of the time have no communication with the backbone (bursty trac pro le). The reason is that in practice, whenever a connection is established between a mobile and the backbone, the location of that mobile becomes known to the network. If the trac is not bursty, then the updates due to connections cannot be ignored. In such a case, dynamic allocation of reporting centers is preferable. Such allocations can naturally be optimized for each mobile user rather than for the entire network as in our strategy. For example, we might want that a mobile user who moves more frequently would update the system more times. There are many open problems left, some of which we list here. For each type of grid graph considered in the paper (grid, uni-diagonal grid and bi-diagonal grid), there is still a gap between the performance of our algorithm for selecting centers in the graph and the lower bound we proved. We conjecture that our solutions are optimal and that the lower bounds can be improved. As per weighted graphs, important special cases other than trees could be almost-trees and plannar graphs. Also, we believe that there exists a better approximation algorithm for arbitrary graphs. Finally, in our work we ignore the directory management in the backbone network and assumed that such a mechanism exists already. Previous works ignore the cost on the wireless links. One should device strategies that optimize the cost for the wireless and the wired links together.
Acknowledgment We would like to thank Hamid Ahmadi and Arvind Krishna for helpful discussions. We would also like to thank the referees for their comments, which led to enhancemnt of propositions 5,7 and 8, and to the addition of the results on the hexagonal cells topology.
Appendix Reduction from
MIS
to
MIS
2
Let G be the input graph for the MIS problem. De ne G2 to be the graph obtained from G by rst replacing each edge (vi ; vj ) in G with a new vertex uk that is connected to both vi and vj (by two new edges), and then connecting all the new vertices among themselves. Since the new vertices in G2 are connected in a click, no 2-independent set in G2 that contains a new vertex, has more than one vertex. Moreover, if vi and vj are vertices in G, then the distance 21
between the corresponding vertices in G2 is two if there is an edge (vi ; vj ) in G, and is three otherwise. Ignoring independent sets of size one, it follows that a set is an independent set in G if and only if the set of the corresponding vertices in G2 is a 2-independent set in G2 . Hence, any polynomial solution to MIS 2 implies a polynomial solution to MIS .
Reduction from ( 2) to ( C G;
C G; Z
)
Assume rst that Z = 2k for some integer k > 1. Let G be a graph with n vertices. De ne Gk to be the graph obtained from G by connecting to each vertex v in G (through a new edge) a line graph consisting of k ? 1 vertices (i.e., the graph Gk contains n k vertices). We refer to each such line graph together with the vertex v of G to which it is connected as a chain, and call the vertex v the head of the chain. Let J be a set of vertices in G, and denote the corresponding set of vertices in Gk by Jk . Clearly, if z (J ) 2 in G then z (Jk ) 2k in Gk . On the other hand, for any set C in Gk for which z (C ) 2k, we can construct a set E in G for which z (E ) 2, and such that jE j jC j. This shows that any polynomial solution to C (G; 2k) implies a polynomial solution to C (G; 2). We now show how to construct the above set E from the set C . For every chain whose head is not in C but at least one of the vertices of the chain is in C , add to C the head of the chain and remove from C one of the vertices of the same chain. Call the set thus obtained C 0 . Now, let E 0 be the set of all vertices in C 0 that are heads of chains, and let E be the corresponding set of vertices in G. Clearly, jE j = jE 0 j jC 0 j = jC j. We now show that z (E ) 2. Assume to the contrary that z (E ) > 2. Then, there exists a vertex v and two of its neighbors u and w, such that v is in E and u; w are not in E . Hence, the corresponding vertices u0 and w0 in Gk are not in E 0 , and therefore are also not in C 0. The construction of C 0 implies that none of the vertices of the chain of u0 is in C , and similarly none of the vertices of the chain of w0 is in C . This contradicts the assumption that z(C ) 2k, since the chain of v0 (the corresponding vertex of v in Gk ) contains a center v00 2 C , whose vicinity contains at least 2k + 1 vertices. Using the graph Gk it can be similarly shown that any polynomial solution to C (G; 2k + 1) implies a polynomial solution to C (G; 2).
References [1] B. Awerbuch and D. Peleg, \Concurrent Online Tracking of Mobile Users" in Proc. of SIGCOMM `91 Zurich, September 1991, pp. 221-233. 22
[2] M. R. Garey and D. S. Johnson, Computers and Intractability A guide to the Theory of NP -Completeness, New York: W. H. Freeman and Company, 9999. [3] D. J. Goodman, \Cellular Packet Communications", IEEE Trans. on Communications, vol. 38, pp. 1272 { 1280, August 1990. [4] William Lee, Mobile Cellular Telecommunications Systems, New York: Mcgraw-Hill, 1989. [5] S. J. Mullender and p. B. M. Vitanyi, \Distributed Match-Making," Algorithmica vol. 3, pp. 367 { 391, 1988. [6] Raymond Steele, \The cellular environment of lightweight hand held portables", IEEE Communications Magazine, vol. x, pp. 20 { 29, July 1989. [7] Raymond Steele, \Deploying personal communication networks", IEEE Communications Magazine, vol. x, pp. 12 { 15, September 1990.
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