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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 13 Topic: More Broadcast in General Networks Sources: Bar-Yehuda, Goldreich, Itai. Time complexity of broadcast. Bar-Yehuda, Goldreich, Itai. Efficient emulation of single-hop radio network on multi-hop radio network. Kowalski, Pelc. Time of deterministic broadcasting. MIT 6.885 Fall 2008 slides
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Discrete Algs for Mobile Wireless Sys3 [BGI 2] Use their randomized Decay and Broadcast algorithms to solve other problems: Emulate an algorithm designed for single-hop networks with collision detection, in an arbitrary network without collision detection. Each round expands to O( log( ) (D + sqrt(D log(N/ )) + log(N/ ) ) ) rounds, succeeds with probability 1- . So the entire emulation works with high probability. Apply this to run a known single-hop leader election algorithm [Willard] in a multi-hop network.
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Discrete Algs for Mobile Wireless Sys4 The Two Models Multi-hop: Same as the [BGI1] weak collision model: Arbitrary connected undirected graph, synchronous rounds. In each round, processor chooses whether to send or receive. Weak collision behavior (may receive a message). Single-hop: Complete graph, synchronous rounds. Reliable collision detection, (0,m,c)
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Discrete Algs for Mobile Wireless Sys5 Emulating a Single Round Three synchronized phases: Propagation: All nodes that want to send choose random tags and Broadcast their messages. Every node remembers first received message (initiator remembers its own). Different nodes may remember different messages. Collision detection: If > 1 initiators, two neighbors end up with different tags. Nodes compare tags with neighbors, bit by bit. For each bit, use Decay: all nodes having 1 transmit, all with 0 receive. If any node actually receives a message, detects a collision. Notification: Each node that detected a collision Broadcasts a “collision” message.
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Discrete Algs for Mobile Wireless Sys6 Behavior of Single-Round Emulation Properties of Decay and Broadcast imply: If there is a single initiator, WHP everyone receives the message. Otherwise, WHP everyone detects a conflict.
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Discrete Algs for Mobile Wireless Sys7 Emulating an Entire Algorithm Emulate round-by-round, using the single- round emulation. Must choose appropriately. Apply this to Willard’s Ethernet algorithm for leader election, to run it in a multi-hop network.
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Discrete Algs for Mobile Wireless Sys8 [Kowalski, Pelc] Assumes the strong collision model: collision is indistinguishable from idle Show that [BGI1] deterministic lower bound is incorrect, with a special case O(log n) algorithm. Sublinear (in n) time deterministic algorithms for all graphs of small diameter D = o(log log n) Deterministic lower bound of Ω(n 1/4 ); again claim exponential gap between randomized and deterministic complexity.
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Discrete Algs for Mobile Wireless Sys9 0 sink, n+1 1 2 n Layer 1Layer 2Layer 0 S Graphs from [BGI1] Lower Bound For sink to receive message, must ensure eventually exactly one node in S transmits For weaker collision model, [BGI1] showed (n) rounds necessary What about for stronger collision model? Show O(log n) algorithm…
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Discrete Algs for Mobile Wireless Sys10 High Level Idea Emulate collision detection capability at the source Use the "collision detection" to choose one node in S to communicate with the sink Algorithm uses subroutine Echo(i,A) i is the id of one node in Layer 1 A is a set of ids of nodes in Layer 1
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Discrete Algs for Mobile Wireless Sys11 Echo Subroutine Code: Step 1: Every node in A transmits its id. Step 2: Every node in A U { i } transmits its id. Behavior: Case 1: Message received in Step 1 but not in Step 2 A has 1 node, and its id is known to the source. Case 2: Message received in Step 2 but not in Step 1 A is empty. Case 3: No message received in either step. A has at least 2 nodes.
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Discrete Algs for Mobile Wireless Sys12 Binary Selection Broadcast Phase 0: Source transmits message and the lowest id i of its neighbors (i = 1). Phase 1: Node with id 1 transmits the source message and its degree. If degree is 2, then 1 in S. Therefore, sink receives the message. DONE! If degree is 1, then continue…
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Discrete Algs for Mobile Wireless Sys13 Binary Selection Broadcast Do binary search on the range of ids in Layer 1, looking for a segment containing exactly 1 node in S. 1...8Suppose n = 16. 1...49…12 >1 Done 1 0 1...2 1 Done 5…6 >1 0 9…10 1 Done 13…14 >1 0 5...5 1 Done 7…7 >1 0 1...1 1 Done 3…3 >1 0 9...9 1 Done 11…11 >1 0 13..13 1 Done 15…15 >1 0 Done 1 1 1 1 1 1 1 1 0 16…16 Done 1
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Discrete Algs for Mobile Wireless Sys14 Binary Selection Broadcast Phase 3, 4, …: Step 1: source transmits a range R = {x, …, y} of ids; initially R = {1,2,... n/2} Steps 2 and 3: nodes in Layer 1 execute ECHO(1, R∩S). Case 1: R∩S has one node. DONE! Case 2: R∩S is empty. In Step 1 of next phase source transmits R = {y+1,... y+(y-x+1)/2}. Case 3: R∩S has > 1 node. In Step 1 of next phase source transmits R = {x,..., (y+x-1)/2}.
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Discrete Algs for Mobile Wireless Sys15 o(n) Algorithm for D = o(log log n) Complicated algorithm, uses many techniques
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Discrete Algs for Mobile Wireless Sys16 (n 1/4 ) Lower Bound Similar to (n) lower bound for the weaker collision model Uses same class of graphs (with three layers) Given any algorithm, construct the set S (nodes in Layer 1 that are connected to the sink) so that lots of time must elapse until the sink gets the message
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Discrete Algs for Mobile Wireless Sys17 Discussion What is the impact of these results?
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