Finding multiple induced disjoint paths in general graphs

Information Processing Letters 111 (2011) 1022–1026

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Information Processing Letters www.elsevier.com/locate/ipl

Finding multiple induced disjoint paths in general graphs Kejia Zhang ∗ , Hong Gao, Jianzhong Li School of Computer Science and Technology, Harbin Institute of Technology, Harbin, Heilongjiang, China

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Article history: Received 29 April 2011 Received in revised form 18 July 2011 Accepted 20 July 2011 Available online 26 July 2011 Communicated by J. Torán Keywords: Induced disjoint paths Graph Approximation algorithm

In an undirected graph, paths P 1 , P 2 , . . . , P k are induced disjoint if each one of them is chordless (i.e., is an induced path) and any two of them have neither common nodes nor adjacent nodes. This paper investigates the Maximum Induced Disjoint Paths (MIDP) problem: in an undirected graph G = ( V , E ), given k node pairs {s1 , t 1 }, . . . , {sk , tk }, connect maximum number of these node pairs via induced disjoint paths. Till now, the only things known about MIDP are: i) it is NP-hard; ii) it is NP-hard even when k = 2; iii) it can be solved in polynomial time when k is a fixed constant and the given graph is a directed planar graph (Kobayashi, 2009 [9]). This paper proves that for general k and any  > 0, it is NP-hard to approximate MIDP within m1/2− , where m = | E |. Two algorithms for MIDP are √ given by this paper: a greedy algorithm whose approximation ratio is m and an on-line algorithm which has a good lower bound. © 2011 Elsevier B.V. All rights reserved.

1. Introduction In an undirected graph G = ( V , E ), paths P 1 , P 2 , . . . , P k are induced disjoint if each one of them is chordless (i.e., is an induced path) and any two of them have neither common vertices nor adjacent vertices [6]. In this paper, we investigate the Maximum Induced Disjoint Paths (MIDP) problem, which is defined as: in an undirected graph G = ( V , E ), given k vertex pairs {s1 , t 1 }, . . . , {sk , tk }, connect maximum number of these vertex pairs via induced disjoint paths. MIDP can be applied to some wireless networks, like wireless ad-hoc networks and wireless sensor networks. It is a natural way to model a network as a graph, in which vertices denote nodes and edges denote wireless links.1 Each given node pair denotes a route request that needs to be connected via a path in the network. Designers always want to connect the given route requests via independent paths, so that no two paths “interfere” with each other.

*

Corresponding author. E-mail addresses: [email protected] (K. Zhang), [email protected] (H. Gao), [email protected] (J. Li). 1 In the following, the terms “vertex” and “node” are completely interchangeable, so are the terms “edge” and “link”. 0020-0190/$ – see front matter doi:10.1016/j.ipl.2011.07.016

© 2011

Elsevier B.V. All rights reserved.

For two paths P 1 and P 2 in an ad-hoc network or sensor network, even if P 1 and P 2 are node-disjoint, serious interference between them can still happen when there are wireless links between the nodes in P 1 and the nodes in P 2 [13,14]. Let u and v be two nodes in node-disjoint paths P 1 and P 2 respectively. There is an edge (wireless link) between u and v. When node u is relaying messages on P 1 , node v cannot relay messages on P 2 , vice versa. If there are many wireless links between the nodes in P 1 and the nodes in P 2 , data transmissions of the two paths will conflict frequently. Therefore, induced disjoint paths are more suitable to depict highly independent paths in such wireless networks. A problem related to MIDP is the Maximum Disjoint Paths (MDP) problem, which is defined as: in a directed or undirected graph G = ( V , E ), given k node pairs {s1 , t 1 }, . . . , {sk , tk }, connect maximum number of these node pairs via node-disjoint or edge-disjoint paths. MDP is a classic NP-hard problem and widely studied [5,8,4,11, 10,12,3,1,2,7]. MIDP can be seen as an extension of the Undirected-graph-Node-disjoint version of MDP (denoted by UN-MDP), because any instance of UN-MDP can be reduced to an instance of MIDP by subdividing every edge into two edges [9]. Thus, MIDP is harder than UN-MDP and it is definitely an NP-hard problem.

K. Zhang et al. / Information Processing Letters 111 (2011) 1022–1026

√

An O ( m )-approximation algorithm is given for the Undirected-graph-Edge-disjoint version of MDP (denoted by UE-MDP) [10], where m = | E |. An on-line O (d )approximation algorithm √ is also proposed for UE-MDP [8], where d = max( D G , m ) and D G is the diameter of G. In [12,3], √ algorithms for UE-MDP with approximation ratio O ( n) are given, where n = | V |. We cannot use these algorithms to solve MIDP, because MIDP is different from UE-MDP and more complicated. In [2,7], another version of MDP is studied, which assumes that G = ( V , E ) is an undirected planar graph and at most c > 1 paths are allowed to share an edge. When c = 2, an O (log n)-approximation algorithm is given in [2]. When c = 4, an algorithm with constant approximation ratio is given in [7]. According to [1], unless NP ⊆ ZPTIME(npoly log n ), for any  > 0, UE-MDP cannot be approximated within

1023

problem on G can be converted to some MIDP problems on the components of G. In the following, the term “graph” means connected undirected graph. Define n = | V | and m = | E |. A path P in G = ( V , E ) can be expressed as a sequence of distinct nodes ( v 1 , v 2 , . . . , v n ), where v i ∈ V and ( v i , v i +1 ) ∈ E. A path with end nodes v 1 and v n is a v 1 ∼ v n path. | P | denotes the number of nodes in path P . An edge (u , v ) ∈ E is a chord of P if u and v are two non-adjacent nodes in P . Path P is chordless if there is no cord of P in G, i.e., P is an induced path in G. Definition 1. In graph G = ( V , E ), paths P 1 , . . . , P k are induced disjoint if each one of them is chordless and for any 1  i < j  k, P i and P j do not have adjacent nodes, i.e., {(u , v ) ∈ E | u ∈ P i , v ∈ P j } = ∅.

1

log 2 − n. According to [5], given any  > 0, it is NP-hard to approximate the Directed-graph-Edge-disjoint version of MDP (denoted by DE-MDP) within m1/2− . These hardness results cannot be applied to MIDP directly because MIDP is an extension of UN-MDP but not an extension of DE-MDP or UE-MDP. However, we borrow the proving idea of [5] when we prove the hardness to approximate MIDP. According to [6,9], MIDP can be solved in polynomial time when k is a fixed constant and the given graph is a directed planar graph. These two constraints are not realistic for routing in large-scale wireless ad-hoc or sensor networks. In a large-scale network, there are usually many route requests that need to be satisfied simultaneously within a period of time, so the number of route requests is more likely an arbitrary number, not a constant. On the other hand, in most cases, a wireless network cannot be modeled as a planar graph. To the best of our knowledge, for general k and general graph, no approximation algorithm is proposed for MIDP so far, and the hardness to approximate MIDP is unknown either.

Paths P i and P j not having adjacent nodes imply that they do not have any common nodes or edges, i.e., {(u , v ) ∈ E | u ∈ P i , v ∈ P j } = ∅ ⇒ P i ∩ P j = ∅. We say that two paths interfere with each other if they have common nodes or adjacent nodes. Definition 2. For a path P in G = ( V , E ), G I [ P ] = ( V I [ P ], E I [ P ]) is P ’s interfering graph, where V I [ P ] = { v | v ∈ P or ∃(u , v ) ∈ E that u ∈ P } and E I [ P ] = {(u , v ) | u ∈ P or v ∈ P }. V I [ P ] is called P ’s interfering node set and E I [ P ] is called P ’s interfering edge set. Lemma 1. For two chordless paths P 1 and P 2 , the following statements are equivalent. 1. 2. 3. 4.

P 1 and P 2 are induced disjoint. E I [ P 1 ] ∩ E I [ P 2 ] = ∅. P 1 does not contain any nodes in V I [ P 2 ]. P 2 does not contain any nodes in V I [ P 1 ].

Main contributions of this paper:

• For general k and any  > 0, proving that it is NP-hard to approximate MIDP within m1/2− , where m = | E |. • Proposing an algorithm for MIDP named Greedily Searching by√Path Weight (GSPW), whose approximation ratio is m. • Proposing an on-line algorithm for MIDP named Online Searching by Bounded Path Weight (OSBPW), which has a good lower bound. The organization of this paper is: Section 2 gives some necessary preliminaries. The hardness to approxi√ mate MIDP is proven in Section 3. Section 4 gives a ( m )approximation algorithm GSPW and an on-line algorithm OSBPW which also has a good lower bound. 2. Preliminaries The given graph G = ( V , E ) is an undirected graph. Moreover, G can be assumed to be connected without loss of generality. This is because if G is disconnected, an MIDP

We can get Lemma 1 by the definition of induced disjoint paths. The Maximum Induced Disjoint Paths problem is formally defined as: Problem MIDP: INPUT: a graph G = ( V , E ); k node pairs {s1 , t 1 }, . . . , {sk , tk }, where si , t i ∈ V for 1  i  k. OUTPUT: η induced disjoint paths P i 1 , P i 2 , . . . , P i η (1  i 1 < i 2 < · · · < i η  k) such that P i x is an si x ∼ t i x path for 1  x  η . OBJECTIVE: Maximizing η . The decision version of MIDP when k = 2 (denoted by 2-IDP) is: in a graph G, given two node pairs {s1 , t 1 } and {s2 , t 2 }, ask for whether there exist induced disjoint s1 ∼ t 1 path and s2 ∼ t 2 path in G. According to [9], we have the following lemma: Lemma 2. 2-IDP is NP-complete.

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K. Zhang et al. / Information Processing Letters 111 (2011) 1022–1026

be the corresponding paths of R 1 and R 2 respectively. In G  , for each node pair {si , t i }, define the path

P i = (t i , v 1,i , v 2,i , . . . , v k−i ,i , u i , v k−i +1,i −1 , . . . , v k−i +1,1 , si ) as shown by bold lines in Fig. 1. In G, si ∼ t i path P i is obtained from P i by replacing each v x,i (1  x  k − i) with path R 2x,i and replacing each v k−i +1, y (i − 1  y  1) with

Fig. 1. The constructed G  (bold lines denote P i ).

3. Hardness to approximate MIDP In this section, we will prove that for general k and any

 > 0, it is NP-hard to approximate MIDP within m1/2− , where m is the number of edges in the given graph. A k × k grid graph is the adjacent graph of a k × k chessboard. It has node set { v i j | i , j ∈ {0, 1, . . . , k − 1}} and edge set {( v i j , v xy ) | |x − i | + | y − j | = 1}. The basic proving idea is: given an instance of 2-IDP and any  > 0, we construct a graph G = ( V , E ) with node pairs {s1 , t 1 }, . . . , {sk , tk }, where k  | E |1/2− and i) the answer of the 2-IDP instance is “YES” iff all these k node pairs can be connected by induced disjoint paths in G; ii) the answer of the 2-IDP instance is “NO” iff at most 1 of these k node pairs can be connected by induced disjoint path in G. Let a graph H = ( V H , E H ) and distinct nodes a, b, c , d ∈ V H be an instance of 2-IDP, which asks whether there exist induced disjoint a ∼ b path and c ∼ d path in H . For any  > 0, we construct a graph G = ( V , E ) in two steps. Let k = | E H |1/ . In the first step, we construct a graph G  = ( V  , E  ), which is a subgraph of (k + 1) × (k + 1) grid. It has node set V  = { v i j | i , j ∈ {0, 1, . . . , k} and 1  i + j  k + 1} and edge set E  = {( v i j , v xy ) | |x − i | + | y − j | = 1 and x + i  1, y + j  1}. We set t i = v 0,i , si = v k+1−i ,0 and u i = v k+1−i ,i for 1  i  k. The constructed graph G  is as shown in Fig. 1. In the second step, graph G is constructed by doing the following modifications to G  . i) Replace each 4-degree node v i j ∈ V  with a copy of H , which is denoted by H i j . The corresponding nodes of a, b, c , d in H i j are denoted by ai j , b i j , c i j , di j respectively. ii) Replace each edge ( v i j , v i +1, j ) ∈ E  with edge (di j , c i +1, j ). Replace each edge ( v i j , v i , j +1 ) ∈ E  with edge (ai j , bi , j+1 ). Here, we assume that v i j = ai j = b i j = c i j = di j for each v i j ∈ V  not replaced by a copy of H , i.e., v i j is sl , tl , or ul for some l = 1, . . . , k. The constructed graph G is as shown in Fig. 2. Lemma 3. All these {s1 , t 1 }, . . . , {sk , tk } can be connected by induced disjoint paths in G iff {a, b} and {c , d} can be connected by induced disjoint paths in H . Proof. “⇐”. There are induced disjoint a ∼ b path R 1 and c ∼ d path R 2 in H . In each copy H i j of H , let R 1i j and R 2i j

path R k1−i +1, y . The obtained P i and P i +1 are as shown by bold lines in Fig. 2. Since R 1 and R 2 are induced disjoint in H , P i and P j are induced disjoint for any 1  i < j  k. “⇒”. There are k induced disjoint paths Q 1 , . . . , Q k in G such that Q i (1  i  k) is an si ∼ t i path. For any 1  i < j  k, since Q i and Q j do not have any common edges, there must be an H xy such that i) one of Q i and Q j contains an axy ∼ b xy path in H xy ; ii) the other of Q i and Q j contains a c xy ∼ dxy path in H xy (see [5]). Q i and Q j being induced disjoint implies that there are induced disjoint a ∼ b path and c ∼ d path in H . 2 Lemma 4. At most 1 of {s1 , t 1 }, . . . , {sk , tk } can be connected by induced disjoint paths in G iff {a, b} and {c , d} cannot be connected by induced disjoint paths in H . We can get Lemma 4 from the proof of Lemma 3. In G  , | E  | = k(k + 1) and the number of 4-degree nodes is k(k − 1)/2. Therefore, for G,

m = | E | = k(k + 1) +

k(k − 1) 2

· | E H |  k2 | E H |

The inequality holds because | E H |  2. Since k = | E H |1/ 1

 | E H |1/ , we have m  k2+ and k  m 2+ . The choice of

 can be arbitrary, so we get Theorem 1. Theorem 1. There exist an infinite family of graphs G = ( V , E ) and k node pairs {s1 , t 1 }, . . . , {sk , tk } in G satisfying: For any  > 0, it is NP-hard to distinguish whether all these node pairs can be connected by induced disjoint paths or at most m1/12− of these node pairs can be connected by induced disjoint paths, where m = | E |.

√

By Theorem 1, for MIDP, we know that m is the best approximation ratio possible in terms of m. 4. Algorithms for MIDP 4.1. Greedily searching by path weight In this part, we will give an algorithm for MIDP named Greedily Searching by Path √ Weight (GSPW). The approximation ratio of GSPW is m. For each node v in graph G, d G ( v ) denotes the degree of v. We define a function w : V → N as w ( v ) = d G ( v ) − 1.  For a path P in G, its w-length is defined as l w ( P ) = v ∈ P w ( v ). The w-shortest path denotes the shortest path w.r.t. w-length. GSPW loops to search induced disjoint paths as shown in Algorithm 1. In each iteration: Find the w-shortest path P i for each unconnected node pair {si , t i } (lines 3–4 in

K. Zhang et al. / Information Processing Letters 111 (2011) 1022–1026

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Fig. 2. The constructed G (bold lines denote P i and P i +1 ).

Algorithm 1). Choose the node pair, say {sλ , t λ }, whose w-shortest path has minimum w-length (line 5 in Algorithm 1). Connect {sλ , t λ } with P λ . Remove the interfering node set of P λ from G (line 10 in Algorithm 1). GSPW is executed until all the node pairs are connected or none of the unconnected node pairs can be connected by a path in G. Clearly, the paths generated by GSPW are all chordless and they are induced disjoint. Algorithm 1: GSPW

1 2 3 4

8 9 10

√

Theorem 2. GSPW is a ( m )-approximation algorithm.

Choose P λ such that l w ( P λ ) = min{l w ( P i ) | {si , t i } ∈ T }; if No such P λ exists then Stop; Output P λ ; T = T \ {{sλ , t λ }}; G = G − V I [ P λ ];

Lemma 5. In G = ( V , E ), let { P 1 , . . . , P η } be any set of induced η disjoint paths. Then i =1 [l w ( P i ) + 1]  m. Proof. By Lemma 1, since P 1 , P 2 , . . . , P η are induced disjoint, their interfering edge sets do not have any common edges. On the other hand, E I [ P 1 ], E I [ P 2 ], . . . , E I [ P η ] are all subsets of E, so η     E I [ P i ]  | E | = m

(1)

i =1

v∈P i

Proof. Let O be an optimal solution of MIDP. Let A be the set of the induced disjoint paths generated by GSPW. Each path in O must interfere with some path in A, otherwise A can be further extended by GSPW. Note that if P ∈ A is also a path in O , P interferes with itself. In A, let P i be the path generated at the ith iteration of GSPW. Oi ⊆ O consists of the paths in O that interfere with P i but not with P 1 , . . . , P i −1 . Set xi = |Oi |, then

 |A |

|O | =

x. i =1 i

By Lemma 6,

xi  l w ( P i ) + 1

(3)

For each path Q ∈ Oi , l w ( Q ) + 1  xi , otherwise we have l w ( Q ) < l w ( P i ) and Q will be selected by GSPW instead of P i . Therefore, for each Oi ,





l w ( Q ) + 1  x2i

For each P i ,

    E I [ P i ] = dG ( v ) − | P i | + 1 = l w ( P i ) + 1

Lemma 6. Let O be an optimal solution of MIDP in a graph G. Then, any chordless path P in G interferes with at most l w ( P ) + 1 paths in O . Proof. By equality (2), we have | E I [ P ]| = l w ( P ) + 1. For any path Q ∈ O which interferes with P , E I [ Q ] and E I [ P ] have at least one common edge. For any two paths Q 1 and Q 2 in O , E I [ Q 1 ] ∩ E I [ Q 2 ] = ∅. Therefore, P interferes with at most | E I [ P ]| = l w ( P ) + 1 paths in O . 2

Input: A graph G; k node pairs {s1 , t 1 }, . . . , {sk , tk } in G Output: Induced disjoint paths connecting some of these node pairs T = {{s1 , t 1 }, . . . , {sk , tk }}; while T = ∅ do for each {si , t i } in T do Find the w-shortest si ∼ t i path P i ;

5 6 7

By adding up d G ( v ) for each v ∈ P i , the edges with both ends in P i are counted twice and the edges with only one end in P i are counted once. Since each P i is chordless, there are | P i | − 1 edges with both ends in P i , and the first equality of (2) holds. 2

(2)

Q ∈Oi

By Lemma 5,

(4)

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K. Zhang et al. / Information Processing Letters 111 (2011) 1022–1026





lw (Q ) + 1 =

Q ∈O

|A|   



lw (Q ) + 1  m

(5)

i =1 Q ∈Oi

|A| 

x2i 

i =1

 



lw (Q ) + 1 >

√

m · |O U |

(7)

Q ∈OU

√

Thus,

m 

m

|A|  |A| 2  2 ( i =1 xi )2 x |O | i =1 i  = |A | |A | |A | (6)

In (6), the first inequality is obtained by combining (4) and (5), the second inequality is an application of Cauchy– Schwarz inequality, the third √inequality holds because |A|  1. Finally, we have |O|  m · |A|. 2

Therefore, |OU | < m. For each path Q ∈ O I , we charge it in a greedy way to some path P ∈ A interfering with it. By Lemma 6, for each path P ∈ A , at most l w ( P ) + 1 paths√in O I are charged to P . OSBPW guarantees that l w ( P )  m − 1 for each P ∈ √ A . Thus, |O I |  m · |A | and

| O | = | O I | + |O U | <

√





m · A  +

√

m=

√





m · A  + 1



(8) The proof is completed.

2

Acknowledgement 4.2. On-line searching by bounded path weight Sometimes, in MIDP, the node pairs are provided one after another and we need to deal with a node pair immediately once it comes. For example, in a wireless network, route requests (each of them is denoted by a node pair) are generated one by one and the users require that each route request is handled without delay. In this case, we need an on-line algorithm for MIDP. In this part, we will give an on-line algorithm for MIDP named On-line Searching by Bounded Path Weight (OSBPW), which has a good lower bound. OSBPW is as shown in Algorithm 2. The given node pairs are executed one by one. For each node pair {si , t i }: find a w-shortest √ si ∼ t i path P i . If the w-length of P i is no more than m − 1, connect {si , t i } with P i and remove the interfering node set of P i from G. Algorithm 2: OSBPW

1 2 3 4 5

Input: A graph G; k node pairs {s1 , t 1 }, . . . , {sk , tk } in G Output: Induced disjoint paths connecting some of these node pairs for i = 1, . . . , k do Find the w-shortest si ∼ t i path P i ; √ if l w ( P i )  m − 1 then Output P i ; G = G − V I [ P i ];

Clearly, the paths generated by OSBPW are all chordless and they are induced disjoint. Theorem 3. Let O be an optimal solution of MIDP. Let A be the set of the induced disjoint paths generated by OSBPW. Then, we |O | have |A | > √ − 1. m

Proof. Define O I ⊆ O as O I = { Q | Q ∈ O and Q interferes with some path in A }. Define OU = O \ O I . Note that for an si ∼ t i path P in A and an si ∼ t i path Q in O , P must interfere with Q because of the common end nodes si and t i . √ For each path Q ∈ OU , we have l w ( Q ) > m − 1, otherwise its corresponding node pair will be connected by OSBPW and it will interfere with a path in A  . By Lemma 5,

This work is supported by the National Natural Science Foundation of China under Grant No. 60831160525; the Key Program of the National Natural Science Foundation of China under Grant No. 61033015; the Key Program of the National Natural Science Foundation of China under Grant No. 60933001. References [1] M. Andrews, J. Chuzhoy, L. Zhang, Hardness of the undirected edgedisjoint paths problem with congestion, in: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005. [2] C. Chekuri, S. Khanna, F.B. Shepherd, Edge-disjoint paths in planar graphs, in: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, 2004. √ [3] C. Chekuri, S. Khanna, F.B. Shepherd, An o( n ) approximation and integrality gap for disjoint paths and unsplittable flow, Theory of Computing 2 (2006) 137–146. [4] D. Ganesan, R. Govindan, S. Shenker, D. Estrin, Highly-resilient, energy-efficient multipath routing in wireless sensor networks, ACM SIGMOBILE Mobile Computing and Communications Review 5 (4) (2001) 11–25. [5] V. Guruswami, S. Khanna, R. Rajaraman, B. Shepherd, M. Yannakakis, Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems, Journal of Computer and System Sciences 67 (3) (2003) 473–496. [6] K. Kawarabayashi, Y. Kobayashi, The induced disjoint paths problem, in: Proceedings of International Conference on Integer Programming and Combinatorial Optimization (IPCO), 2008. [7] C.C.S. Khanna, F.B. Shepherd, Edge-disjoint paths in planar graphs with constant congestion, in: Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, 2006. [8] J.M. Kleinberg, Approximation algorithms for disjoint paths problems, PhD thesis, MIT, 1996. [9] Y. Kobayashi, Induced disjoint paths problem in a planar digraph, Discrete Applied Mathematics 157 (15) (2009) 3231–3238. [10] S. Kolliopoulos, C. Stein, Approximating disjoint-path problems using greedy algorithms and packing integer programs (extended abstract), in: Integer Programming and Combinatorial Optimization, 1998. [11] S. Li, Z. Wu, Node-disjoint parallel multi-path routing in wireless sensor networks, in: Proceedings of the Second International Conference on Embedded Software and Systems, 2005. [12] T. Nguyen, On the disjoint paths problem, Operation Research Letters 35 (1) (2007) 10–16. [13] M.R. Pearlman, Z.J. Haas, P. Sholander, S.S. Tabrizi, On the impact of alternate path routing for load balancing in mobile ad hoc networks, in: Proceedings of International Symposium on Mobile Ad Hoc Networking & Computing, 2000. [14] S. Roy, S. Bandyopadhyay, T. Ueda, K. Hasuike, Multipath routing in ad hoc wireless networks with omni directional and directional antenna: A comparative study, in: Proceedings of International Workshop on Distributed Computing, Mobile and Wireless Computing, 2002.