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Unformatted text preview: Lecture 9 AllPairs Shortest Paths AllPairs Shortest Paths nodes. of } , { pairs all for to from path shortest find ), , ( digraph a Given t s t s E V G = Path Counting Problem nodes. of } , { pair each for to from edges exactly with paths of # count , integer positive a and ) , ( digraph a Given t s t s k k E V G = Adjacent Matrix = = = otherwise , , ) , ( if 1, and } ,..., 2 , 1 { where ) ( ) ( E j i a n V a G A ij n n ij 1 2 3 1 1 1 1 1 2 3 1 2 3 Theorem . to from edges exactly with paths of number the is element each , ) ( In ) ( j i k a G A k ij k Proof . We prove it by induction on k. 1 2 3 1 1 1 1 1 2 3 1 2 3 k =1 True! = + = n h hj k ih k ij a a a 1 ) 1 ( ) ( ) 1 ( Induction Step j h i ch oices ) ( k ih a ch o ices ) 1 ( hj a AllPairs Shortest Paths with at most k edges . to from edges most at path with shortest the of weight the nodes, of } , { pair every for compute, , integer positive a and ) , ( digraph a Given t s k t s k E V G = Weighted Adjacent Matrix = = = otherwise , , ) , ( if , and } ,..., 2 , 1 { where ) ( ) ( E j i c l n V l G L ij ij n n ij 1 2 3 5 6 4 1 2 3 1 2 3 4 6 5 )...
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This note was uploaded on 11/03/2010 for the course COMPUTER S CS 6363 taught by Professor Dingzhudu during the Fall '10 term at University of Texas at Dallas, Richardson.
 Fall '10
 DingzhuDu
 Algorithms

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