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Unformatted text preview: r < 2 always short paths, but no efﬁcient algorithm to ﬁnd 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 ﬁnd 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
 Graph Theory, Vertex, 2J, DI

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