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CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch

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Discrete Algs for Mobile Wireless Sys2 Lecture 25 Topic: Unit Disk Graphs and Relatives Sources: Kuhn, Wattenhofer & Zollinger Barriere, Fraigniaud & Narayanan MIT 6.885 Fall 2008 slides

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Discrete Algs for Mobile Wireless Sys3 Unit Disk Graph Model Popular theoretical model of wireless communication: every node within distance 1 of a transmitting node receives the message no node outside distance 1 receives the message Easy to use Not a very accurate of reality: varying transmission power obstacles atmospheric conditions

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Discrete Algs for Mobile Wireless Sys4 Quasi Unit Disk Graph Quasi Unit Disk Graph (QUDG) model Two nodes can communicate if Euclidean distance between them is < d Two nodes cannot communicate if Euclidean distance is >1 If distance is in the range [d..1], it is unspecified whether nodes can communicate

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Discrete Algs for Mobile Wireless Sys5 Results on QUDGs [KWZ] Lower bound of ((c/d) 2 ) messages to route a message from s to t d is parameter of the QUDG c is length of shortest path from s to t assumes nodes cannot keep routing tables Matching upper bound: restricted flooding algorithm that routes a message from s to t using O((c/d) 2 ) messages in worst case nodes use a topology control graph structure not necessarily efficient in the average case

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Discrete Algs for Mobile Wireless Sys6 Results on QUDGs [KWZ] Greedy routing + restricted flooding algorithm still optimal in worst case more efficient in average case if d > 1/ 2, geometric routing works as well in QUDG as in UDG

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Discrete Algs for Mobile Wireless Sys7 Ruling Out Routing Tables "Volatile Memory Routing Algorithm" each node can hold O(log n) bits on behalf of a message as long as the message is in transit (n = number of nodes) Need the memory to store message ids for flooding Need logarithmic number of bits to distinguish between different messages that are in transit concurrently

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Discrete Algs for Mobile Wireless Sys8 Geometric Routing Algorithm Every node knows the position in the plane of itself and all its neighbors Sender of a message knows position of destination Each message keeps "control info" on at most O(1) nodes

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Discrete Algs for Mobile Wireless Sys9 Message Lower Bound Theorem 4.1: There exist graphs on which any volatile memory routing algorithm uses ((c/d) 2 ) messages. Proof Sketch: Construct a specific graph such that rough structure is a k x k matrix, where k is about n because of lack of global knowledge of the graph, any volatile memory algorithm needs to explore almost all the nodes in the graph to find path from s to t, i.e., uses (n) messages Yet shortest path from s to t has length c = ( n*d) So (c/d) 2 = (n)

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Discrete Algs for Mobile Wireless Sys10 Developing an Optimal Message QUDG Routing Algorithm First step is to find a certain subgraph of a given QUDG, called a backbone graph Backbone graph should not be too dense: Within a region of area A, contains O(A/d 2 ) nodes Backbone graph should not make paths too long (should be a spanner): length of shortest path in b.b. graph is at most O(log (1/d)) times length in original graph

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Discrete Algs for Mobile Wireless Sys11 Constructing Backbone Graph Construct a maximal independent set of G nodes in MIS will be cluster heads Connect cluster heads with connector nodes since a MIS is a dominating set, cluster heads can be connected by "bridges" of at most 2 nodes resulting graph is a constant-stretch spanner, but it can be too dense, (A/d 4 ) nodes in a region of area A Reduce the number of connecting bridges between cluster heads to reduce the density somewhat complex operation that divides the plane into a grid of square cells and calls (3 times!) an algorithm to construct a sparse spanner

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Discrete Algs for Mobile Wireless Sys12 Using Backbone Graph for Routing To route a message from s' to t': 1.route message from s' to its cluster head s' 2.route message on backbone graph from s to t, cluster head for t' 3.route message from t' to t Steps 1 and 3 take constant time and messages So just focus on Step 2

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Discrete Algs for Mobile Wireless Sys13 Message-Optimal Flooding: Echo Algorithm (Assume synchronous system.) TTL := 1 while message is not yet delivered flood message (over backbone graph) to all nodes at distance TTL or less (constructs a breadth-first search tree) leaves of tree initiate sending echo messages back to source TTL := 2*TTL

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Discrete Algs for Mobile Wireless Sys14 Analysis of Echo Algorithm Message complexity is O((c/d) 2 ): Consider phase when TTL = : Since max edge length in BBG is 1, all nodes reached in phase are distance at most from source Relevant circular region has area 2 = O( 2 ) By BBG property, there are O( 2 /d 2 ) nodes in the region So tree constructed in this region has O( 2 /d 2 ) edges So message complexity is O( 2 /d 2 )

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Discrete Algs for Mobile Wireless Sys15 Analysis of Echo Algorithm Since TTL is doubled at each phase, total message complexity is a constant multiple of the message complexity in the last phase in last phase (when destination is reached), t is at most 2c (c is length of shortest path) Message complexity of last phase is O(c 2 /d 2 ) So total message complexity is O(c 2 /d 2 ) which is optimal

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Discrete Algs for Mobile Wireless Sys16 Other Properties of Echo Algorithm In synchronous system, time complexity is O(c log(1/d)) In asynchronous system, message and time complexity are O(log 3 (c/d)) factor larger than in synchronous system use previously known synchronizer construction Are there issues in using synchronizer algorithms in wireless networks?

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Discrete Algs for Mobile Wireless Sys17 Improving Average Case Performance of Echo Algorithm Combine echo scheme with a greedy scheme Use greedy routing If greedy algorithm hits a local minimum with no closer neighbors, then switch to echo algorithm Criterion for switching back to greedy mode: have found a node that is "significantly" closer to destination than local minimum chosen in a way to ensure that worst-case message complexity stays optimal In lucky cases, can stay in greedy mode the whole time, which is much cheaper than flooding

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Discrete Algs for Mobile Wireless Sys18 Large Values of QUDG Parameter d If d > 1/ 2, then all intersecting edges can be detected locally: Lemma 8.2: Let (u 1,v 1 ) and (u 2,v 2 ) be two intersecting edges in QUDG G with d > 1/ 2. Then at least one of the edges (u 1,u 2 ), (u 1,v 2 ), (v 1,u 2 ), and (v 1,v 2 ) is also in G.

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Discrete Algs for Mobile Wireless Sys19 Geometric Routing on QUDGs with Large Values of d Define virtual nodes at all intersections of two edges virtual nodes are managed by the endpoints of the intersection edges By doing this to the backbone graph, obtain a graph that is planar (because of the virtual nodes) only O(A) nodes in any region of area A Since this planar graph is a spanner, known geometric routing algorithms have cost O(c 2 )

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