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1 Maximal Independent Set

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2 Independent Set (IS): In a graph G=(V,E), |V|=n, |E|=m, any set of nodes that are not adjacent

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3 Maximal Independent Set (MIS): An independent set that is no subset of any other independent set

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4 Maximum Independent Set: A MIS of maximum size A graph G……a MIS of G… …a MIS of max size

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5 Applications in Distributed Systems In a network graph consisting of nodes representing processors, a MIS defines a set of processors which can operate in parallel without interference For instance, in wireless ad hoc networks, to avoid interference, a conflict graph is built, and a MIS on that defines a clustering of the nodes enabling efficient routing

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6 Applications in Distributed Systems (2) A MIS is always a Dominating Set (DS) of the graph (the converse in not true), namely every node in G must be at distance at most 1 from at least one node in the MIS In a network graph G consisting of nodes representing processors, a MIS defines a set of processors which can monitor the correct functioning of all the nodes in G

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7 A Sequential Greedy algorithm Suppose that will hold the final MIS Initially

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8 Pick a node and add it to Phase 1:

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9 Remove and neighbors

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10 Remove and neighbors

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11 Pick a node and add it to Phase 2:

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12 Remove and neighbors

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13 Remove and neighbors

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14 Repeat until all nodes are removed Phases 3,4,5,…:

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15 Repeat until all nodes are removed No remaining nodes Phases 3,4,5,…,x:

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16 At the end, set will be an MIS of

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17 Worst case graph (for number of phases): n nodes, n-1 phases Running time of the algorithm: Θ(m) Number of phases of the algorithm: O(n)

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18 Homework Can you see a distributed version of the algorithm just given?

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19 A General Algorithm For Computing MIS Same as the sequential greedy algorithm, but at each phase we may select any independent set (instead of a single node)

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20 Suppose that will hold the final MIS Initially Example:

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21 Find any independent set Phase 1: And insert to :

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22 remove and neighbors

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23 remove and neighbors

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24 remove and neighbors

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25 Phase 2: Find any independent set And insert to : On new graph

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26 remove and neighbors

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27 remove and neighbors

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28 Phase 3: Find any independent set And insert to : On new graph

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29 remove and neighbors

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30 remove and neighbors No nodes are left

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31 Final MIS

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32 The number of phases depends on the choice of independent set in each phase: The larger the subgraph removed at the end of a phase, the smaller the residual graph, and then the faster the algorithm Observation:

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33 Example: If is MIS, 1 phase is needed Example: If each contains one node, phases are needed (sequential greedy algorithm)

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34 A Randomized Sync. Distributed Algorithm Follows the general MIS algorithm paradigm, by choosing randomly at each phase the independent set, in such a way that it is expected to include many nodes of the remaining graph

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35 Let be the maximum node degree in the whole graph 1 2 Suppose that d is known to all the nodes (this may require a pre-processing)

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36 Elected nodes are candidates for independent set Each node elects itself with probability At each phase : 1 2

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37 However, it is possible that neighbor nodes may be elected simultaneously Problematic nodes

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38 All the problematic nodes must be un-elected. The remaining elected nodes form independent set

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39 Success for a node in phase : disappears at end of phase (enters or ) Analysis: 1 2 No neighbor elects itself A good scenario that guarantees success elects itself

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40 Basics of Probability E: finite universe of events; let A and B denote two events in E; then: 1.A B is the event that A or (non-exclusive) B occurs; 2.A B is the event that both A and B occur.

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41 Probability of success in a phase: is at least the probability that a node elects itself and no neighbor elects itself, i.e.: 1 2 No neighbor should elect itself elects itself

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42 Fundamental inequalities

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43 Probability of success in phase: At least For First (left) ineq. with t =-1

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44 Therefore, node disappears at the end of phase with probability at least 1 2

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45 after phases Definition: Bad event for node : node did not disappear This happens with probability (first (right) ineq. with t =-1 and n =2ed) at most:

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46 after phases Bad event for G: at least one node did not disappear This happens with probability: P(OR x G (bad event for x)) ≤

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47 within phases Good event for G: all nodes disappear This happens with probability: (high probability)

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48 Total number of phases: # rounds for each phase: 3 1.In round 1, each node tries to elect itself and notifies neighbors; 2.In round 2, each node receives notifications from neighbors, decide whether is in I k, and notifies neighbors; 3.In round 3, each node receiving notifications from elected neighbors, realizes to be in N(I k ). total # of rounds: (with high probability)

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49 Homework Can you provide a good bound on the number of messages?

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