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Analysis and Design of Algorithm

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- 1. Analysis and Design of Algorithm B. Tech. (CSE), VII Semester Roshan Tailor Amity University Rajasthan
- 2. 2 This Lecture • Single-source shortest paths in weighted graphs – Shortest-Path Problems – Properties of Shortest Paths, Relaxation – Dijkstra’s Algorithm – Bellman-Ford Algorithm
- 3. 3 Shortest Path • Generalize distance to weighted setting • Digraph G = (V,E) with weight function W: E → R (assigning real values to edges) • Weight of path p = v1 → v2 → … → vk is • Shortest path = a path of the minimum weight • Applications – static/dynamic network routing – robot motion planning – map/route generation in traffic 1 1 1 ( ) ( , ) k i i i w p w v v − + = = ∑
- 4. 4 Shortest-Path Problems • Shortest-Path problems – Single-source (single-destination). Find a shortest path from a given source (vertex s) to each of the vertices. – Single-pair. Given two vertices, find a shortest path between them. Solution to single-source problem solves this problem efficiently, too. – All-pairs. Find shortest-paths for every pair of vertices. Dynamic programming algorithm.
- 5. 5 Negative Weights and Cycles? • Negative edges are OK, as long as there are no negative weight cycles (otherwise paths with arbitrary small “lengths” would be possible) • Shortest-paths can have no cycles (otherwise we could improve them by removing cycles) – Any shortest-path in graph G can be no longer than n – 1 edges, where n is the number of vertices
- 6. 6 Relaxation • For each vertex v in the graph, we maintain v.d(), the estimate of the shortest path from s, initialized to ∞ at the start • Relaxing an edge (u,v) means testing whether we can improve the shortest path to v found so far by going through u 5 u v vu 2 2 9 5 7 Relax(u,v) 5 u v vu 2 2 6 5 6 Relax(u,v) Relax (u,v,G) if v.d > u.d()+ w(u,v) then v.d = (u.d()+ w(u,v)) v.setparent = u
- 7. 7 Dijkstra's Algorithm • Non-negative edge weights • Greedy, similar to Prim's algorithm for MST • Like breadth-first search (if all weights = 1, one can simply use BFS) • Use Q, a priority queue ADT keyed by v.d() (BFS used FIFO queue, here we use a PQ, which is re-organized whenever some d decreases) • Basic idea – maintain a set S of solved vertices – at each step select "closest" vertex u, add it to S, and relax all edges from u
- 8. 8 Dijkstra’s Pseudo Code
- 9. 9 Dijkstra’s Example ∞ ∞ ∞ ∞ 0s u v yx 10 5 1 2 3 9 4 6 7 2 10 ∞ 5 ∞ 0s u v yx 10 5 1 2 3 9 4 6 7 2
- 10. 10 Dijkstra’s Example (2) u v 8 14 5 7 0s yx 10 5 1 2 3 9 4 6 7 2 8 13 5 7 0s u v yx 10 5 1 2 3 9 4 6 7 2
- 11. 11 Dijkstra’s Example (3) 8 9 5 7 0 u v yx 10 5 1 2 3 9 4 6 7 2 8 9 5 7 0 u v yx 10 5 1 2 3 9 4 6 7 2
- 12. 12 Dijkstra’s Running Time • Extract-Min executed |V| time • Decrease-Key executed |E| time • Time = |V| TExtract-Min + |E| TDecrease-Key • T depends on different Q implementations Q T(Extract- Min) T(Decrease-Key) Total array Ο(V) Ο(1) Ο(V 2 ) binary heap Ο(lg V) Ο(lg V) Ο(E lg V) Fibonacci heap Ο(lg V) Ο(1) (amort.) Ο(V lgV + E)
- 13. 13 Bellman-Ford Algorithm • Dijkstra’s doesn’t work when there are negative edges: – Intuition – we can not be greedy any more on the assumption that the lengths of paths will only increase in the future • Bellman-Ford algorithm detects negative cycles (returns false) or returns the shortest path-tree
- 14. 14 Bellman-Ford Algorithm
- 15. 15 Bellman-Ford Example 5 ∞ ∞ ∞ ∞ 0s zy 6 7 8 -3 7 2 9 -2 xt -4 6 ∞ 7 ∞ 0s zy 6 7 8 -3 7 2 9 -2 xt -4 5 6 4 7 2 0s zy 6 7 8 -3 7 2 9 -2 xt -4 5 2 4 7 2 0s zy 6 7 8 -3 7 2 9 -2 xt -4 5
- 16. 16 Bellman-Ford Example 2 4 7 −2 0s zy 6 7 8 -3 7 2 9 -2 xt -4 • Bellman-Ford running time: – (|V|-1)|E| + |E| = Θ(VE) 5
- 17. Bellman-Ford Example 17

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