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**Unformatted text preview: **Robert Sedgewick and Kevin Wayne Copyright 2006 http://www.Princeton.EDU/~cos226 Minimum Spanning Tree Reference: Chapter 20, Algorithms in Java, 3 rd Edition, Robert Sedgewick 2 Minimum Spanning Tree MST. Given connected graph G with positive edge weights, find a min weight set of edges that connects all of the vertices. 23 10 21 14 24 16 4 18 9 7 11 8 G 5 6 3 Minimum Spanning Tree MST. Given connected graph G with positive edge weights, find a min weight set of edges that connects all of the vertices. Theorem. [Cayley, 1889] There are V V-2 spanning trees on the complete graph on V vertices. can't solve by brute force 23 10 21 14 24 16 4 18 9 7 11 8 cost(T) = 50 5 6 4 MST Origin Otakar Boruvka (1926). ! Electrical Power Company of Western Moravia in Brno. ! Most economical construction of electrical power network. ! Concrete engineering problem is now a cornerstone problem in combinatorial optimization. Otakar Boruvka 5 Applications MST is fundamental problem with diverse applications. ! Network design. telephone, electrical, hydraulic, TV cable, computer, road ! Approximation algorithms for NP-hard problems. traveling salesperson problem, Steiner tree ! Indirect applications. max bottleneck paths LDPC codes for error correction image registration with Renyi entropy learning salient features for real-time face verification reducing data storage in sequencing amino acids in a protein model locality of particle interactions in turbulent fluid flows autoconfig protocol for Ethernet bridging to avoid cycles in a network ! Cluster analysis. 6 Medical Image Processing MST describes arrangement of nuclei in the epithelium for cancer research http://www.bccrc.ca/ci/ta01_archlevel.html 7 http://ginger.indstate.edu/ge/gfx 8 Two Greedy Algorithms Kruskal's algorithm. Consider edges in ascending order of cost. Add the next edge to T unless doing so would create a cycle. Prim's algorithm. Start with any vertex s and greedily grow a tree T from s. At each step, add the cheapest edge to T that has exactly one endpoint in T. Theorem. Both greedy algorithms compute an MST. Greed is good. Greed is right. Greed works. Greed clarifies, cuts through, and captures the essence of the evolutionary spirit." - Gordon Gecko 5 Applications MST is fundamental problem with diverse applications. ! Network design. telephone, electrical, hydraulic, TV cable, computer, road ! Approximation algorithms for NP-hard problems. traveling salesperson problem, Steiner tree ! Indirect applications. max bottleneck paths LDPC codes for error correction image registration with Renyi entropy learning salient features for real-time face verification reducing data storage in sequencing amino acids in a protein model locality of particle interactions in turbulent fluid flows autoconfig protocol for Ethernet bridging to avoid cycles in a network ! Cluster analysis....

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