CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

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

2. Lecture 30 • Topic: • Compulsory Protocols • Sources: • Hatzis,Pentaris, Spirakis, Tampakas & Tan • Chatzigiannakis, Nikoletseas & Spirakis • MIT 6.885 Fall 2008 slides Discrete Algs for Mobile Wireless Sys

3. Algorithms for Mobile Ad Hoc Networks • Focus on the mobility of the nodes • Classify protocols into 3 types: • Non-compulsory: nodes move however they like (controlled by other protocols/ applications/ …) • Compulsory: movement of all nodes is directed by the protocol • Semi-compulsory: movement of some nodes is directed by the protocol • Consider protocols for leader election, message routing, and counting the number of nodes in the system Discrete Algs for Mobile Wireless Sys

4. Network Model • m mobile nodes in 3-D space • Space is partitioned into n cells • approximate the broadcast sphere of a node with a regular polyhedron (e.g., a cube) • Graph G = (V,E) where |V| = n cells • degree of G bounded by a constant • ex: in a 3-D grid, number of neighbors is 6 • Transmission between 2 nodes in same cell is guaranteed • lower layer handles frequency allocation issues • Negligible transmission delays Discrete Algs for Mobile Wireless Sys

5. Leader Election & Node Counting [HPSTT] • Problem: • Elect a unique leader among the m nodes • All other nodes should know they are not the leader • Leader should also learn the total number of nodes • Assumptions: • Nodes know their geographic location and thus what cell they are in • Each node has a unique id (relaxed later) • Number of nodes does not change Discrete Algs for Mobile Wireless Sys

6. Non-Compulsory Algorithm • Each node keeps a local counter, initialized to 1 • When two nodes meet (are in the same cell / within communication range of each other): • exchange ids • node with larger id wins • node with smaller id becomes "inactive" • winner adds the loser's count to its own count • Optimization: nodes bcast their entire list of ids encountered so far • ultimate leader knows all the nodes • larger messages Discrete Algs for Mobile Wireless Sys

7. Analysis • No guaranteed termination: • only terminates if node with max id meets all the other nodes • No termination detection: • Even if node with max id meets all the other nodes, it can't know that it is finished • How to analyze? • Assume movement of each node is a random walk on the graph G of the 3-D space Discrete Algs for Mobile Wireless Sys

8. Random Walk • Random walk is a path in the graph: • Node begins at some graph vertex • Randomly decides which neighbor to visit next • equal probability of choosing each neighbor • Continue • Can analyze various properties: • how long until all the vertices are visited • how long until you return to your start • how long to get from vertex i to vertex j • etc. Discrete Algs for Mobile Wireless Sys

9. Analysis of LE Algorithm • Assume one node in each cell (m = n) • Model a node's random walk as a "continuous time Markov chain" • Let Mij be first meeting time of 2 nodes that start at cells (vertices) i and j • Let Cf be time when only one node is still active Discrete Algs for Mobile Wireless Sys

10. Analysis • Lemma 3.2: Pr[Mij > t] decreases exponentially in t • As time goes by, it is more and more likely that the nodes that started at cells i and j will meet (be in the same cell) • Theorem 1: E[Cf] = O(log n maxij(Ei(Tj))) • Ei(Tj) is expected time for a node that starts in cell i to reach cell j • uses L. 3.2 and random walk theory to relate E[Mij] to Ei(Tj) • Corollary 3.1: E[Cf] = O(n log n) • appeals to full paper to show maxij(Ei(Tj)) = 2*number of edges in G Discrete Algs for Mobile Wireless Sys

11. Improved Analysis • Notice that when two hosts meet, the counter of the winner, on average, will be doubled. • Then use some probability to show that the log n term in the previous expression becomes log of some constant and we get: • Theorem 2: E[Cf] = O(n) Discrete Algs for Mobile Wireless Sys

12. Relaxing the m = n Assumption • Suppose m > n. • Then some cells start with multiple nodes. • A single winner per cell will emerge after the first step • Suppose m < n. • Worst case is when they are spread around. • Augment the algorithm to get n nodes with dummy nodes whose ids are smaller than those of the real nodes. • In both cases, expected time to finish is still O(n). • Since n is ratio of the volume of the space to the transmission radius volume, increasing the transmission radius will reduce n and lead to faster termination Discrete Algs for Mobile Wireless Sys

13. Termination Detection • Inspired by the previous analysis, we can simply have all the nodes stop after a certain amount of time • There is some probability the algorithm will be wrong (have more than one leader) • but it is bounded by the length of time the protocol ran • Lemma 3.5: if protocol terminates after time t, then the probability of failure (i.e., probability Cf > t) is at most O(n/t). • depends on n (really on |E|) • so error probability depends on how many cells the space is divided into Discrete Algs for Mobile Wireless Sys

14. Anonymous Networks • What if nodes do not have unique ids? • When two nodes meet and compete: • choose ids at random and exchange them • keep doing this until they choose different ids • How to count the total number of nodes in this case? Discrete Algs for Mobile Wireless Sys

15. Simulation Results • Simulation results were actually even better than the analysis • Probably because the analysis did not assume anything about the underlying graph • but in the quantization scheme, the graph is regular and thus random walks "mix" faster Discrete Algs for Mobile Wireless Sys

16. Semi-Compulsory Protocols [CNS] • Same set-up as in previous paper • 3-D space is quantized into cubic cells • Problem of interest is routing: • deliver a message from node S to node R • Possible solutions: • sender gives message to every node it meets in the hopes of eventually finding R • form paths of intermediate nodes lying w/in each other's transmission range • impractical in large sparse ad hoc networks; formation of long paths is infeasible; link failures are very likely Discrete Algs for Mobile Wireless Sys

17. "Support" Approach • They consider networks where the highly dynamic movement of nodes makes maintaining valid paths impractical • Instead, take advantage of nodes' natural movement • exchange info when they meet • Also, force a small subset of nodes (the support) to move according to the needs of the protocol • The support acts as a moving intermediate pool fo rmessages • sweeps through the network area Discrete Algs for Mobile Wireless Sys

18. Definitions •  : subset of mobile nodes (the support) whose motion is determined by the protocol of interest P • Me : the support management sub-protocol of P controlling how the support nodes move and communication • Me is k-fault tolerant if it executes correctly in the presence of at most k faults (crash failures) of nodes in  Discrete Algs for Mobile Wireless Sys

19. Snake Protocol • Support nodes sweep the entire space quickly (think of a moving backbone in the network) • When a support node is within communication range of a sender, the sender is notified that it can send a message • Messages are stored in the support (e.g., copied to all nodes of the support) • When a receiver comes within range of a support node, the receiver is notified that a message is waiting for it and the message is forwarded to the receiver Discrete Algs for Mobile Wireless Sys

20. Snake Protocol Details • Set-up phase in which k nodes become the support: • elect a leader MS0 • MS0 then identifies k-1 additional nodes MS1, …, MSk-1 and coordinates the support topology and movement • Snake: • all nodes start in the same cell • MS0 randomly chooses direction of next move, sends info to MS1, and the moves • MS1 sends direction info to MS2 and follows MS0 • etc. Discrete Algs for Mobile Wireless Sys

21. Snake Protocol Discussion • Advantages: • They claim it achieves fast communication between any two mobile nodes • low communication overhead, no elaborate state • nodes don't use any location information • Disadvantage: • requires compulsory motion of support nodes Discrete Algs for Mobile Wireless Sys

22. Analysis • Theorem 3.1: The Snake Protocol guarantees reliable communication between any sender-receiver pair in finite time, whose expected value is bounded only by a function of the size of the motion space (graph representing the quantization). • independent of the number of nodes • tolerates arbitrary motion of non-support nodes as long as they are not deliberately trying to avoid the support nodes Discrete Algs for Mobile Wireless Sys

23. Fault-Tolerance • Snake Protocol can be made 1-fault tolerant. • Suppose last node fails. • no changes needed • Suppose head fails. • 2nd node can detect and take over as head. • Suppose an internal node fails. • Snake is broken into two. • The two new snakes meet eventually. • When they do (say x in S1 and y in S2 meet), they merge into one: • x follows the node ahead of y in S2 • rest of S1 waits to follow tail of S2. Discrete Algs for Mobile Wireless Sys

24. Multiple Faults • Difficulties in applying the previous idea to more than 1 failure • Can get a deadlock situation with 4 snakes in a square Discrete Algs for Mobile Wireless Sys

25. Choosing Size of Support • How big should k (size of support) be? • Analysis shows best choice is k = sqrt(2|E|c) • c is approximately the "separation threshold time" of the random walk on G Discrete Algs for Mobile Wireless Sys

26. Extension to Snake Protocol • Have the head of the snake do a regular random walk on a spanning graph of G. • Improves performance in analysis and simulation. Discrete Algs for Mobile Wireless Sys

27. Runners Protocol • Each support node performs an independent random walk on G. When two runners meet, they exchange any information given to them by senders • Each support node keeps track of undelivered messages and receipt lists • When two runners meet, they use 2-phase commit to synchronize the two lists • Performs better than Snake Protocol in experiments • Is k-1 fault tolerant Discrete Algs for Mobile Wireless Sys