lect19-20

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

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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
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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
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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
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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
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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)
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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)
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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
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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
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Analysis of the algorithm C the min-cut of size k G has at least kn/2 edges Why?
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cut)
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lect19-20 - Graph Clustering Why graph clustering is...

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