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Unformatted text preview: r < 2 always short paths, but no efcient algorithm to nd them probability of encountering edge that gets you closer too low r = 2 alogrithm (ln n ) 2 take edge that gets you closest to destination each time r > 2 there may or may not be a short path, but no algorithm exists to nd them probability of encountering long edge too low r = no long edges Proof of algorithm: s start point, t end point FIGURE 3 A j (2 j , 2 j +1 ] at most log 2 n phases, E ( time ) = ln n Time of algorithm is (ln n ) 2 Lemma: For r = 2 constant c such that the probability that a long distance edge from u goes to v is at least c d2 ( u,v ) ln n . Proof: Pr [( u,v )] dr ( u,v ) P w 6 = v dr ( u,w ) To Be Continued . .....
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 Spring '09

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