1 Maximal Independent Set

2 Independent Set (IS): In a graph, any set of nodes that are not adjacent

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

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 Pick a node and add it to Phase 1:

7 Remove and neighbors

8

9 Pick a node and add it to Phase 2:

10 Remove and neighbors

11 Remove and neighbors

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

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

14 At the end, set will be an MIS of

15 Worst case graph (for number of phases): nodes Running time of algorithm: Number of phases of the algorithm:

16 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)

17 Suppose that will hold the final MIS Initially Example:

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

19 remove and neighbors

20 remove and neighbors

21 remove and neighbors

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

23 remove and neighbors

24 remove and neighbors

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

26 remove and neighbors

27 remove and neighbors No nodes are left

28 Final MIS

29 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 Observation:

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

31 A Randomized Sync. Distributed Algorithm Same with the general MIS algorithm At each phase the independent set is chosen randomly so that it includes many nodes of the remaining graph

32 Let be the maximum node degree in the whole graph 1 2 Suppose that is known to all the nodes

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

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

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

36 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

37 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 either A or B occurs; 2.A  B is the event that both A and B occur.

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

39 Fundamental inequalities

40 Probability of success in phase: At least For First ineq. with t =-1

41 Therefore, node will enter and disappear at the end of phase with probability at least 1 2

42 Expected number of phases until node disappears: at most phases

43 after phases Bad event for node : node did not disappear Probability (First ineq. with t =-1 and n=2ed):

44 after phases Bad event for any node in : at least one node did not disappear Probability:

45 within phases Good event for all nodes in : all nodes disappear Probability: (high probability)

46 Total number of phases: Time duration of each phase: Total time: with high probability

47 Luby’s MIS Distributed Algorithm Runs in time in expected case with high probability this algorithm is asymptotically better than the previous

48 Let be the degree of node 1 2

49 Each node elects itself with probability At each phase : 1 2 degree of in Elected nodes are candidates for the independent set

50 If two neighbors are elected simultaneously, then the higher degree node wins Example: if

51 If both have the same degree, ties are broken arbitrarily Example: if

52 Problematic nodes Using previous rules, problematic nodes are removed

53 The remaining elected nodes form independent set

54 at least one neighbor enters Analysis A good event for node 1 2 Consider phase

55 At end of phase If is true, then and will disappear at end of current phase

56 At least one neighbor of elects itself with probability at least 1 2 LEMMA 1: maximum neighbor degree

No neighbor of elects itself with probability PROOF: (the elections are independent)

58 maximum neighbor degree

Therefore, at least one neighbor of Elects itself with probability at least END OF PROOF

60 If a node elects itself, then it enters with probability at least LEMMA 2:

Node enters if no neighbor of same or higher degree elect itself PROOF:

Probability that some neighbor of with same or higher degree elects itself neighbors of same or higher degree

63 Probability that that no neighbor of with same or higher degree elects itself neighbors of same or higher degree 1 2

Thus, if elects itself, it enters with probability at least 1 2 END OF PROOF

at least one neighbor of enters LEMMA 3:

66 and no node is elected neighbor is in 1 2 New event PROOF:

67 The events are mutually exclusive

68 It holds: Therefore:

69 and no node elects itself elects itself 1 2 after elects itself, it enters

70 (from Lemma 2) after elects itself, it enters

71

72 and no node elects itself elects itself The events are mutually exclusive

73 We showed earlier (Lemma 1) that: Therefore:

74 Therefore node disappears in phase with probability at least END OF PROOF

75 Let be the maximum node degree in the graph Suppose that in : Then, constant

76 (thus, nodes with high degree will disappear fast) a node with degree with probability at least Thus, in phase disappears

77 Suppose that the degree of remains at least for the next phases Consider a node which in initial graph has degree Node does not disappear within phases with probability at most

78 Take Node does not disappear within phases with probability at most

79 Thus, within phases either disappears or its degree gets less than with probability at least

80 by the end of phases there is no node of degree higher than with probability at least (ineq. 2) Therefore,

81 In every phases, the maximum degree of the graph reduces by at least half, with probability at least

82 Maximum number of phases until degree drops to 0 (MIS has formed) with probability at least (ineq. 2)