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Chapter 24: Single-Source Shortest Paths

Comp 122, Fall 2003. Single-source SPs - 2. Outline. General Lemmas and Theorems.CLRS now does this last. We'll still do it first.Bellman-Ford algorithm.DAG algorithm.Dijkstra's algorithm.We will skip Section 24.4.. Comp 122, Fall 2003. Single-source SPs - 3. Corollary: Let p = SP from s t

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Chapter 24: Single-Source Shortest Paths

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    1. Comp 122, Fall 2003 Single-source SPs - 1 Chapter 24: Single-Source Shortest Paths Given: A single source vertex in a weighted, directed graph. Want to compute a shortest path for each possible destination. Similar to BFS. We will assume either no negative-weight edges, or no reachable negative-weight cycles. Algorithm will compute a shortest-path tree. Similar to BFS tree.

    2. Comp 122, Fall 2003 Single-source SPs - 2 Outline General Lemmas and Theorems. CLRS now does this last. Well still do it first. Bellman-Ford algorithm. DAG algorithm. Dijkstras algorithm. We will skip Section 24.4.

    3. Comp 122, Fall 2003 Single-source SPs - 3 General Results (Relaxation)

    4. Lemma 24.1 holds because one edge gives the shortest path, so the other edges must give sums that are at least as large. Comp 122, Fall 2003 Single-source SPs - 4

    5. Comp 122, Fall 2003 Single-source SPs - 5 Relaxation

    6. Comp 122, Fall 2003 Single-source SPs - 6 Properties of Relaxation d[v], if not ?, is the length of some path from s to v. d[v] either stays the same or decreases with time Therefore, if d[v] = ?(s, v) at any time, this holds thereafter Note that d[v] ? ?(s, v) always After i iterations of relaxing on all (u,v), if the shortest path to v has i edges, then d[v] = ?(s, v).

    7. Comp 122, Fall 2003 Single-source SPs - 7 Properties of Relaxation

    8. For lemma 24.11, note that initialization makes the invariant true at the beginning. Comp 122, Fall 2003 Single-source SPs - 8

    9. Comp 122, Fall 2003 Single-source SPs - 9 More Properties

    10. Lemma 24.13 follows simply from the structure of Relax. Lemma 24.14 shows that the shortest path will be found one vertex at a time, if not faster. Thus after a number of iterations of Relax equal to V(G) - 1, all shortest paths will be found. Comp 122, Fall 2003 Single-source SPs - 10

    11. Comp 122, Fall 2003 Single-source SPs - 11 Predecessor Subgraph

    12. Comp 122, Fall 2003 Single-source SPs - 12 Proof of (1) (Continued)

    13. Comment on Proof d[vi] ? d[vi-1] + w(vi-1, vi) for i = 1, , k1 because when Relax(vi-1 , vi , w) was called, there was an equality, and d[vi-1] may have gotten smaller by further calls to Relax. d[vk] > d[vk-1] + w(vk-1, vk) before the last call to Relax because that last call changed d[vk]. Comp 122, Fall 2003 Single-source SPs - 13

    14. Comp 122, Fall 2003 Single-source SPs - 14 Proof of (2)

    15. Comp 122, Fall 2003 Single-source SPs - 15 Lemma 24.17

    16. Comp 122, Fall 2003 Single-source SPs - 16 Proof (Continued)

    17. And note that this shortest path tree will be found after V(G) - 1 iterations of Relax. Comp 122, Fall 2003 Single-source SPs - 17

    18. Comp 122, Fall 2003 Single-source SPs - 18 Bellman-Ford Algorithm

    19. So if Bellman-Ford has not converged after V(G) - 1 iterations, then there cannot be a shortest path tree, so there must be a negative weight cycle. Comp 122, Fall 2003 Single-source SPs - 19

    20. Comp 122, Fall 2003 Single-source SPs - 20 Example

    21. Comp 122, Fall 2003 Single-source SPs - 21 Example

    22. Comp 122, Fall 2003 Single-source SPs - 22 Example

    23. Comp 122, Fall 2003 Single-source SPs - 23 Example

    24. Comp 122, Fall 2003 Single-source SPs - 24 Example

    25. Comp 122, Fall 2003 Single-source SPs - 25 Another Look

    26. Comp 122, Fall 2003 Single-source SPs - 26 Lemma 24.2

    27. Comp 122, Fall 2003 Single-source SPs - 27 Correctness

    28. Comp 122, Fall 2003 Single-source SPs - 28 Case 2

    29. Comp 122, Fall 2003 Single-source SPs - 29 Shortest Paths in DAGs

    30. Comp 122, Fall 2003 Single-source SPs - 30 Example

    31. Comp 122, Fall 2003 Single-source SPs - 31 Example

    32. Comp 122, Fall 2003 Single-source SPs - 32 Example

    33. Comp 122, Fall 2003 Single-source SPs - 33 Example

    34. Comp 122, Fall 2003 Single-source SPs - 34 Example

    35. Comp 122, Fall 2003 Single-source SPs - 35 Example

    36. Comp 122, Fall 2003 Single-source SPs - 36 Example

    37. Comp 122, Fall 2003 Single-source SPs - 37 Dijkstras Algorithm

    38. Comp 122, Fall 2003 Single-source SPs - 38 Example

    39. Comp 122, Fall 2003 Single-source SPs - 39 Example

    40. Comp 122, Fall 2003 Single-source SPs - 40 Example

    41. Comp 122, Fall 2003 Single-source SPs - 41 Example

    42. Comp 122, Fall 2003 Single-source SPs - 42 Example

    43. Comp 122, Fall 2003 Single-source SPs - 43 Example

    44. Comp 122, Fall 2003 Single-source SPs - 44 Correctness

    45. Comp 122, Fall 2003 Single-source SPs - 45 Proof (Continued)

    46. Comp 122, Fall 2003 Single-source SPs - 46 Proof (Continued)

    47. Comp 122, Fall 2003 Single-source SPs - 47 Complexity

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