Maximal Independent Set 1 Independent Set IS In

Maximal Independent Set 1

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

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

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

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 5

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 6

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

Phase 1: Pick a node and add it to 8

Remove and neighbors 9

Remove and neighbors 10

Phase 2: Pick a node and add it to 11

Remove and neighbors 12

Remove and neighbors 13

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

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

At the end, set will be an MIS of 16

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

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

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) 19

Example: Suppose that will hold the final MIS Initially 20

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

remove and neighbors 22

remove and neighbors 23

remove and neighbors 24

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

remove and neighbors 26

remove and neighbors 27

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

remove and neighbors 29

remove and neighbors No nodes are left 30

Final MIS 31

Observation: 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 32

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

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 34

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) 35

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

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

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

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

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. 40

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

Fundamental inequalities 42

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

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

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

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

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

Total number of phases: (with high probability) # 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 Ik, and notifies neighbors; 3. In round 3, each node receiving notifications from elected neighbors, realizes to be in N(Ik). total # of rounds: 48

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