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Lecture 7 All-Pairs Shortest Paths
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All-Pairs Shortest Paths
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Path Counting Problem
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Adjacency Matrix
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1 3 2 1 2 3 1 2 3
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Theorem Proof. We prove it by induction on k.
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k=1 True! 1 3 2 1 2 3 1 2 3
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Induction Step
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All-Pairs Shortest Paths with at most k edges
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Recursive formula Proof.
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Case 1. The path with length contains
at most edges. Case 2. the path with length contains exactly edges.
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Key Observation
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Dynamic Program
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Speed Up dynamic Program
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Idea
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Weighted Adjacency Matrix
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1 4 5 6 3 2 1 2 3 1 2 3
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A New Multiplication
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Associative Law
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Theorem Proof. We prove it by induction on k.
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1 k=1 True! 4 5 6 3 2 1 2 3 1 2 3
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Induction Step
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All-Pairs Shortest Paths
Theorem Proof
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How to Compute
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Lemma
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Theorem
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Theorem
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Floyd-Warshall Algorithm
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Observation
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Dynamic Program
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Theorem
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Theorem
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What we learnt in this lecture?
The relationship between shortest path and matrix multiplication. Faster-All-Pairs-Shortest-Paths algorithm Floyd-Warshall algorithm.
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Puzzle 1
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Puzzle 2
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Puzzle 3
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