115-853Page115-853:Algorithms in the Real WorldGraph Separators– Introduction– Applications– Algorithms15-853Page2Edge Separators783401256An edge separator: a set of edges E’⊆E which partitions V into V1and V2Criteria:|V1|, |V2| balanced|E’| is smallV1V2E’15-853Page3Vertex Separators783401256An vertex separator: a set of vertices V’⊆V which partitions V into V1and V2Criteria:|V1|, |V2| balanced|V’| is smallV1V215-853Page4Compared with Min-cutstMin-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. V1V2E’
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215-853Page5Other namesSometimes referred to as –graph partitioning(probably more common than “graph separators”)– graph bisectors– graph bifurcators– balanced or normalized graph cuts15-853Page6Recursive Separation7834012568125673402681574303074851852615-853Page7What graphs have small separatorsPlanar graphs: O(n1/2) vertex separators2d meshes, constant genus, excluded minorsAlmost planar graphs:the internet, power networks, road networksCircuitsneed to be laid out without too many crossingsSocial network graphs:phone-call graphs, link structure of the web, citation graphs, “friends graphs”3d-grids and meshes: O(n2/3)15-853Page8What graphs don’t have small separatosHypercubes: O(n) edge separators O(n/(log n)1/2) vertex separatorsButterfly 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 probabilityIt is exactly the fact that they don’t have small separators that make them useful.