Maximal Independent Set 1 Independent Set IS In

Maximal Independent Set 1

Independent Set (IS): In a graph, 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

A Sequential Greedy algorithm Suppose that will hold the final MIS Initially 6

Phase 1: Pick a node and add it to 7

Remove and neighbors 8

Remove and neighbors 9

Phase 2: Pick a node and add it to 10

Remove and neighbors 11

Remove and neighbors 12

Phases 3, 4, 5, …: Repeat until all nodes are removed 13

Phases 3, 4, 5, …, x: Repeat until all nodes are removed No remaining nodes 14

At the end, set will be an MIS of 15

Running time of the algorithm: Θ(|E|) Number of phases of the algorithm: O(n) Worst case graph (for number of phases): nodes 16

Homework Can you see a distributed version of the algorithm just given? 17

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) 18

Example: Suppose that will hold the final MIS Initially 19

Phase 1: Find any independent set And insert to : 20

remove and neighbors 21

remove and neighbors 22

remove and neighbors 23

Phase 2: On new graph Find any independent set And insert to : 24

remove and neighbors 25

remove and neighbors 26

Phase 3: On new graph Find any independent set And insert to : 27

remove and neighbors 28

remove and neighbors No nodes are left 29

Final MIS 30

Observation: The number of phases depends on the choice of independent set in each phase: The larger the independent set at each phase, the faster the algorithm 31

Example: If is MIS, 1 phase is needed Example: If each contains one node, phases are needed (sequential greedy algorithm) 32

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 33

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) 34

At each phase : Each node with probability 1 elects itself 2 Elected nodes are candidates for independent set 35

However, it is possible that neighbor nodes may be elected simultaneously Problematic nodes 36

All the problematic nodes must be un-elected. The remaining elected nodes form independent set 37

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 38

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. 39

Probability of success in a phase: At least No neighbor should elect itself 1 2 elects itself 40

Fundamental inequalities 41

Probability of success in phase: At least First (left) ineq. with t =-1 For 42

Therefore, node disappears at the end of phase with probability at least 1 2 43

Expected number of phases until node disappears: at most phases 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): 45

Bad event for G: after phases at least one node did not disappear This happens with probability: P(ORx 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