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# separators1 - Edge Separators 15-853:Algorithms in the Real...

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1 15-853 Page1 15-853:Algorithms in the Real World Graph Separators – Introduction – Applications – Algorithms 15-853 Page2 Edge Separators 7 8 3 4 0 1 2 5 6 An edge separator : a set of edges E’ E which partitions V into V 1 and V 2 Criteria: |V 1 |, |V 2 | balanced |E’| is small V 1 V 2 E’ 15-853 Page3 Vertex Separators 7 8 3 4 0 1 2 5 6 An vertex separator : a set of vertices V’ V which partitions V into V 1 and V 2 Criteria: |V 1 |, |V 2 | balanced |V’| is small V 1 V 2 15-853 Page4 Compared with Min-cut s t Min-cut : as in the min- cut, max-flow theorem. Min-cut has no balance criteria. Min-cut typically has a source (s) and sink (t). Will tend to find unbalanced cuts. V 1 V 2 E’

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2 15-853 Page5 Other names Sometimes referred to as graph partitioning (probably more common than “graph separators”) – graph bisectors – graph bifurcators – balanced or normalized graph cuts 15-853 Page6 Recursive Separation 7 8 3 4 0 1 2 5 6 8 1 2 5 6 7 3 4 0 2 6 8 1 5 7 4 3 0 3 0 7 4 8 5 1 8 5 2 6 15-853 Page7 What graphs have small separators Planar graphs : O(n 1/2 ) vertex separators 2d meshes, constant genus, excluded minors Almost planar graphs : the internet, power networks, road networks Circuits need to be laid out without too many crossings Social network graphs : phone-call graphs, link structure of the web, citation graphs, “friends graphs” 3d-grids and meshes : O(n 2/3 ) 15-853 Page8 What graphs don’t have small separatos Hypercubes : O(n) edge separators O(n/(log n) 1/2 ) vertex separators Butterfly networks : O(n/log n) separators ? Expander graphs: Graphs such that for any U V, s.t. |U| ≤ α |V|, | neighbors (U)| ≥ β |U|. ( α < 1, β > 0) random graphs are expanders, with high probability It is exactly the fact that they don’t have small separators that make them useful.