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

2. 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

3. 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.

4. 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)

5. 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.

6. 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.

7. 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.

8. 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.

9. 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…

10. 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

11. 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.

12. 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…

13. 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

14. 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}.

15. Discrete Algs for Mobile Wireless Sys15 o(n) Algorithm for D = o(log log n)  Complicated algorithm, uses many techniques

16. 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

17. Discrete Algs for Mobile Wireless Sys17 Discussion  What is the impact of these results?