Selecting Forwarding Neighbors in Wireless Ad Hoc Networks Gruia C&linescu
Ion M~.ndoiu
Peng-Jun Wan
CS Department Illinois Institute of Technology Chicago, IL 60616
[email protected]
CS Department University of California at Los Angeles Los Angeles, CA 90095 mandoiu @cc.gatech.edu
CS Department Illinois Institute of Technology Chicago, IL 60616 wan @cs.iit.edu
ABSTRACT
CS Department Georgia State University Atlanta, GA 30303 alexz @cs.gsu.edu
in many networking tasks such as paging a particular host, sending an alarm signal, and finding a route to a particular host [1][6][13]. A simple broadcasting mechanism, known as flooding, is to let every node retransmit the message to all its 1-hop neighbors when receiving the first copy of the message. Despite its simplicity, flooding has a serious drawback, known as the broadcast storm [12]. First, because the radio propagation is omnidirectional and a physical location may be covered by the transmission ranges of several nodes, many retransmissions are redundant. Second, heavy contention could exist because retransmitting nodes are probably close to each other. Third, collisions are more likely to occur because the R T S / C T S dialogue is inapplicable and the timing of retransmissions is highly correlated.
Broadcasting is a fundamental operation which is frequent in wireless ad hoc networks. A simple broadcasting mechanism, known as flooding, is to let every node retransmit the message to all its 1-hop neighbors when receiving the first copy of the message. Despite its simplicity, flooding is very inefficient and can result in high redundancy, contention, and collision. One approach to reducing the redundancy is to let each node forward the message only to a small subset of 1-hop neighbors that cover all of the node's 2-hop neighbors. In this paper, we propose two practical heuristics for selecting the minimum number of forwarding neighbors: an O(n log n) time algorithm t h a t selects at most 6 times more forwarding neighbors than the optimum, and an O(n 2) time algorithm with an improved approximation ratio of 3, where n is the number of 1- and 2-hop neighbors. The best previously known algorithm, due to Bronnimann and Goodrich [2], guarantees O(1) approximation in O(n 3 log n) time.
1.
Alexander Zelikovsky
The following simple technique was recently exploited [9] [14] to reduce redundant retransmissions: By virtue of beaconing, each node maintains a local topology of its 2-hop neighborhood, and relays the message only to a small subset of 1-hop neighbors which cover (in terms of radio range) all nodes that are two hops away. The subset of 1-hop neighbors selected by each node is referred to as forwarding set [14] or multipoint relaying set [9]. In this paper we consider the problem of finding a forwarding set of minimum size.
INTRODUCTION
Wireless ad hoc networks can be flexibly and quickly deployed for many applications such as automated battlefield, search and rescue, and disaster relief. Unlike wired networks or cellular networks, no wired backbone infrastructure is installed in wireless ad hoc networks. A communication session is achieved either through a single-hop radio transmission if the communication parties are close enough, or through relaying by intermediate nodes otherwise. In this paper, we assume that all nodes in a wireless ad hoc network are distributed in a two-dimensional plane and have an equal maximum transmission range of one unit.
M i n i m u m Forwarding Set P r o b l e m : Given a source A, let 10 and P be the sets of 1- and 2-hop neighbors of A. Find a minimum-size subset 5r of 7) such that every node in P is within the coverage area of at least one node from P .
1.1
Previous work
Jacquet et al. [9] and Sinha et al. [14] considered the Minimum Forwarding Set problem assuming no knowledge of the geographic location of the nodes. In this case, the Minimum Forwarding Set problem is essentially the well-studied Set Cover problem. Not surprisingly, the heuristic proposed in [9] is a translation of ChvAtal's greedy algorithm [3] for Set Cover, and thus guarantees an approximation factor of O(log m), where m is the maximum neighborhood size. The greedy algorithm iteratively selects a 1-hop neighbor covering the maximum number of 2-hop neighbors not yet covered, and terminates when all 2-hop neighbors have been covered. The greedy algorithm does not take into account the geometric properties of the Minimum Forwarding Set problem, and in fact Figure 1 shows a family of instances for which the size of the solution found by the greedy algo-
Broadcasting is a fundamental networking operation in wireless ad hoc networks. It is widely and frequently performed *Partially supported by NSF Grant CCR-9988331 and a C R D F Grant. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page, To copy otherwise, to republish, to post on servers or to redistribute 1o lists, requires prior specific permission and/or a fee. DIAL M 2001 7/01 Rome, Italy © 2001 ACM ISBN 1-58113-421-5101107...$5.00
34
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F i g u r e 1: I n s t a n c e for w h i c h t h e size o f t h e s o l u t i o n c o m p u t e d b y t h e g r e e d y a l g o r i t h m , l a r g e r t h a n t h e o p t i m u m s o l u t i o n , {opt1, opt2}, b y a l o g a r i t h m i c f a c t o r .
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given set of points. The complexity of Minimum Forwarding Set problems is not known. A constant-ratio approximation algorithm for Disk Cover, and therefore also for Minimum Forwarding Set, was given by Bronnimann and Goodrich [2] However, their algorithm - which is a special case of a sophisticated algorithm for spaces with bounded VC-dimension - has impractical running-time and its proven approximation ratio is a very large constant.
rithm is larger than the optimum by a logarithmic factor. Under the assumption that the nodes in the wireless network are distributed in a two-dimensional plane and each node has unit transmission range, the topology of the network is modeled as a unit-disk graph [4]. In this graph, there is an edge between two nodes if and only if their distance is at most one. The Minimum Forwarding Set problem for a given source node s asks for a minimum size set of 1hop neighbors of s dominating 2-hop neighbors of s in the unit-disk graph. The related Dominating Set problem in unit-disk graphs [4] asks for a subset of nodes dominating (i.e., adjacent to) all the other nodes. The Dominating Set problem in unit-disk graphs is NP-hard [4] but admits a PTAS [8]. The Minimum Forwarding Set problem does not reduce to the Dominating Set problem in unit-disk graphs since dominators are restricted to the set of 1-hop neighbors.
1.2
Our contributions
• A 6-approximation algorithm for the Minimum Forwarding Set problem running in O(n log n) time, where n is the total number of 1- and 2-hop neighbors. • A 3-approximation algorithm for the Minimum Forwarding Set problem running in O(n 2) time. • An exact O(n 2) time, and a 2-approximation O(n log n) time algorithm for the special case of the Minimum Forwarding Set problem when all 2-hop neighbors are in the same quadrant with respect to the source node.
The Minimum Forwarding Set problem is also related to the Unit-Disk Cover problem [7], which asks for the minimum number of unit disks covering a given set of points in the plane. The Unit-Disk Cover problem is also NP-hard [4] and admits a PTAS [7]. Since in the Unit-Disk Cover problem disk centers can be chosen arbitrarily in the plane, the algorithms for this problem do not apply to the Minimum Forwarding Set problem where disks must be centered at 1-hop neighbors only.
• A constant-factor approximation for the Minimum Disk Cover problem with disks of the same radius, based on rounding the optimal solution of a linear programming relaxation. • An experimental study of the proposed algorithms for the Minimum Forwarding Set problem.
The Minimum Forwarding Set problem is a special case of the NP-Hard Disk Cover problem [2], which asks for a minimum size subset of a given set of disks covering a
The paper is organized as follows. In next section we reformulate the Minimum Forwarding Set problem in geometric
35
A l g o r i t h m 1: 1 - H o p D i s k C o v e r I n p u t : Unit-disk A, set of unit disks 7) centered inside A, set of points 7~ outside A such that "P C_ U { D E ~ } O u t p u t : Subset .T" C ~ such that 7~ C U { D C .T'}
Q1
1. Partition the exterior of A into four quadrants Q1-Q4 by two orthogonal lines, not containing points in "P, through the center of A (see Figure 2). 2. For q = 1 , . . . ,4, compute a disk cover, bL-q, for the points
in "Pn Qq.
3. Output 9v-- .%-1U.T'2u.T'3 U.~4.
Q:
terms, give a high-level algorithm based on decomposition into quadrants, and establish basic geometric properties of the partitioned sets of 1- and 2-hop neighbors. In Section 3 we describe a 2-approximation O ( n l o g n) time algorithm for covering 2-hop neighbors in a quadrant. An exact O(n 2) time algorithm for the same problem is described in Section 4. In Section 5 we give an extension of our techniques to the Disk Cover problem of [2]. We present preliminary experimental results comparing the proposed algorithms for the Minimum Forwarding Set problem in Section 6 and conclude in Section 7.
2.
PARTITION
BASED
F i g u r e 2: T h e four q u a d r a n t s in A l g o r i t h m 1.
centers in the qth sector of disk A. The key observation is that points in the quadrant Qq cannot be covered by disks in O P T + 2 ( m o d 4)" Therefore, points in 7~ fl Q1 must be covered by disks in OPT4 U OPT1 U OPT2, and thus, by the assumption t h a t ~'q's are within a factor of a of the respective optimum solutions,
ALGORITHM
1
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[11] B.M.E. Moret and H.D. Shapiro. Algorithms from P to NP, Volume I: Design and Efficiency. Benjamin/Cummings, 1991.
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[12] S.-Y. Ni, Y.-C. Tseng, Yuh-Shyan Chen, and J.-P. Sheu, The Broadcast Storm Problem in a Mobile Ad Hoc Network", Proceedings of the fifth annual A CM//IEEE international conference on Mobile computing and networking, 1999, Pages 151 - 162.
ACKNOWLEDGMENTS
We thank Yuchen Wu for his help with the implementation of the proposed algorithms.
9.
REFERENCES
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