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

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
1 15-853 Page1 15-853:Algorithms in the Real World Graph Separators –Introduct ion – Applications –A lgor ithms 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’
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 15-853 Page5 Other names Sometimes referred to as graph partitioning (probably more common than “graph separators”) –graph b isectors b ifurcators – 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.
Background image of page 2
3 15-853 Page9 Applications of Separators Circuit Layout (dates back to the 60s) VLSI layout Solving linear systems (nested dissection) n 3/2 time for planar graphs Partitioning for parallel algorithms Approximations to certain NP hard problems TSP, maximum-independent-set Clustering and machine learning Machine vision 15-853 Page10 More Applications of Separators Out of core algorithms Register allocation Shortest Paths Graph compression Graph embeddings 15-853 Page11 Available Software METIS : U. Minnessota PARTY : University of Paderborn CHACO : Sandia national labs JOSTLE : U. Greenwich SCOTCH : U. Bordeaux GNU : Popinet Benchmarks : Graph Partitioning Archive 15-853 Page12 Different Balance Criteria Bisectors: 50/50 Constant fraction cuts: e.g. 1/3, 2/3 Trading off cut size for balance: min cut criteria: min quotient separator: All versions are NP-hard 2 1 ' ' min V V V V V ) , min( ' min 2 1 ' V V V V V
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
4 15-853 Page13 Other Variants of Separators k-Partitioning: Might be done with recursive partitioning, but direct solution can give better answers.
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/09/2008 for the course COMPUTER S 15853 taught by Professor Guyblelloch during the Fall '07 term at Carnegie Mellon.

Page1 / 15

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online