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