Maximal Independent Set

Independent Set (IS): Any set of nodes that are not adjacent

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

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

Phase 1: Pick a node and add it to

Remove and neighbors

Remove and neighbors

Phase 2: Pick a node and add it to

Remove and neighbors

Remove and neighbors

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

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

At the end, set will be an MIS of

Running time of algorithm: Worst case graph: nodes

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)

Example: Suppose that will hold the final MIS Initially

Phase 1: Find any independent set And insert to :

remove and neighbors

remove and neighbors

remove and neighbors

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

remove and neighbors

remove and neighbors

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

remove and neighbors

remove and neighbors No nodes are left

Final MIS

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

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

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

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

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

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

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

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

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

Fundamental inequalities

Probability of success in phase: At least For

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

Expected number of phases until node disappears: at most phases

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

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

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

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

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

Let be the degree of node 2 1

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

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

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

Using previous rules, problematic nodes are removed

The remaining elected nodes form independent set

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

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

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

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

maximum neighbor degree

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

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

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

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

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

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

Thus, if is elected, it enters with probability at least 1 2 1 2 END OF PROOF

LEMMA: at least one neighbor of enters 2 1

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

The events are mutually exclusive

It holds: Therefore:

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

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

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

We showed earlier that: Therefore:

Consequently:

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

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

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

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

Take Node does not disappear within phases with probability at most

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

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

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

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