1. Maximal Independent Set

2. Independent Set (IS):
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. A Sequential Greedy algorithm
Suppose that will hold the final MIS
Initially

5. Phase 1:
Pick a node and add it to

6. Remove and neighbors

7. Remove and neighbors

8. Phase 2:
Pick a node and add it to

9. Remove and neighbors

10. Remove and neighbors

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

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

13. At the end, set will be an MIS of

14. Running time of algorithm:
Worst case graph:
nodes

15. 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)

16. Example:
Suppose that will hold the final MIS
Initially

17. Phase 1:
Find any independent set
And insert to :

18. remove and neighbors

19. remove and neighbors

20. remove and neighbors

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

22. remove and neighbors

23. remove and neighbors

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

25. remove and neighbors

26. remove and neighbors
No nodes are left

27. Final MIS

28. 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

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

30. A Simple 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

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

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

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

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

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

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

37. Fundamental inequalities

38. Probability of success in phase:
At least
For

39. Therefore, node will enter
and disappear in phase
with probability at least 2 1

40. Expected number of phases until node
disappears:
at most
phases

41. Bad event for node :
after phases
node did not disappear
Probability:

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

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

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

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

46. Let be the degree of node2 1

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

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

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

50. Using previous rules, problematic nodes are removed

51. The remaining elected nodes form
independent set

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

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

54. LEMMA:
at least one neighbor of
is elected with probability at least
maximum neighbor degree 2 1

55. PROOF:
No neighbor of is elected with probability
(the elections are independent) 2 1

56. maximum neighbor degree

57. Therefore, at least one neighbor of
is elected with probability at least 2 1

58. Therefore, at least one neighbor of
is elected with probability at least
With a different analysis, this probability
can also be proven to be at least
END OF PROOF

59. if a neighbor is elected, then it enters with probability at least
LEMMA:
if a neighbor is elected,
then it enters with probability
at least 2 1 2 1

60. PROOF:
Node enters if no neighbor
of same or higher degree elects itself 2 1

61. Probability that some neighbor of
with same or higher degree elects itself 2 1
neighbors of same or higher degree

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

63. Thus, if is elected, it enters with probability at least1 2 1 2
END OF PROOF

64. LEMMA:
at least one neighbor of enters 2 1

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

66. The events
are mutually exclusive

67. It holds:
Therefore:

68. is elected
and no node is elected
after is elected, it enters 2 1

69. after is elected, it enters
(we have shown it earlier) 2 2 1 1

71. is elected
and no node is elected
The events are mutually exclusive

72. We showed earlier that:
Therefore:

73. Consequently:

74. Therefore node disappears in phase
with probability at least
An alternative bound is:
END OF PROOF

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

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

77. Consider a node which in initial graph
has degree
Suppose that the degree of remains
at least for the next phases
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 lets than
with probability at least

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

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