lect19-20

# lect19-20 - Graph Clustering Why graph clustering is useful...

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Graph Clustering

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Why graph clustering is useful? Distance matrices are graphs as useful as any other clustering Identification of communities in social networks Webpage clustering for better data management of web data
Outline Min s-t cut problem Min cut problem Multiway cut Minimum k-cut Other normalized cuts and spectral graph partitionings

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Min s-t cut Weighted graph G(V,E) An s-t cut C = (S,T) of a graph G = (V, E) is a cut partition of V into S and T such that s S and t T
Max flow problem Flow network Abstraction for material flowing through the edges G = (V,E) directed graph with no parallel edges Two distinguished nodes: s = source , t= sink c(e) = capacity of edge e

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Cuts An s-t cut is a partition (S,T) of V with s Є S and t Є T
Flows An s-t flow is a function that satisfies For each e Є E 0 f(e) c(e) [capacity]

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Max flow problem Find s-t flow of maximum value
Flows and cuts Flow value lemma: Let f be any flow and let (S,T) be any s-t cut. Then, the net flow sent across the cut is equal to the amount leaving s

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Flows and cuts Weak duality: Let f be any flow and let (S,T) be any s-t cut. Then the value of the flow is at most the capacity of the cut defined by (S,T): v(f) cap(S,T)
Certificate of optimality Let f be any flow and let (S,T) be any cut. If v(f) = cap(S,T) then f is a max flow and (S,T) is a min cut. The min-cut max-flow problems can be solved optimally in polynomial time!

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Setting Connected, undirected graph G=(V,E)
Min cut problem Can we solve the min-cut problem using an algorithm for s-t cut?

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Randomized min-cut algorithm Repeat : pick an edge uniformly at random and merge the two vertices at its end-points If as a result there are several edges between some pairs of (newly-formed) vertices retain them all Edges between vertices that are merged are removed ( no self-loops ) Until only two vertices remain The set of edges between these two vertices is a cut in G and is output as a candidate min-cut
Example of contraction e

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Observations on the algorithm Every cut in the graph at any intermediate stage is a cut in the original graph
Analysis of the algorithm C the min-cut of size k G has at least kn/2 edges Why?

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lect19-20 - Graph Clustering Why graph clustering is useful...

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