Unformatted text preview: s ; v path. • pred( v ) is the predecessor of v in the shortest s ; v path. We call an edge tense if dist( u ) + w ( u → v ) < dist( v ). Our generic algorithm repeatedly ﬁnds a tense edge in the graph and relaxes it: Relax( u → v ): dist( v ) ← dist( u ) + w ( u → v ) pred( v ) ← u If there are no tense edges, our algorithm is ﬁnished, and we have our desired shortest path tree. The correctness of the relaxation algorithm follows directly from three simple claims. The ﬁrst of these is below. Prove it. • When the algorithm halts, if dist( v ) 6 = ∞ , then dist( v ) is the total weight of the predecessor chain ending at v : s → ··· → ( pred ( pred ( v )) → pred ( v ) → v. 3. Can’t ﬁnd a Cut-edge A cut-edge is an edge which when deleted disconnects the graph. Prove or disprove the following. Every 3-regular graph has no cut-edge. (A common approach is induction.) 1...
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- Spring '11
- Graph Theory, Shortest path problem, shortest path tree, Abduction Mulder, tentative shortest path