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Single-Source Shortest Path

Single-Source Shortest Path. How does Google Maps work?. Single-Source Shortest Path. How many ways can I go from SPSU to KSU? How can I represent the map? What are vertices? What are edges? What are weights?. Single-Source Shortest Path. For this problem Disallow cycles

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Single-Source Shortest Path

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  1. Single-Source Shortest Path Jeff Chastine

  2. How does Google Maps work? Jeff Chastine

  3. Single-Source Shortest Path • How many ways can I go from SPSU to KSU? • How can I represent the map? • What are vertices? • What are edges? • What are weights? Jeff Chastine

  4. Single-Source Shortest Path • For this problem • Disallow cycles • Still have weight function • Calculate path p and shortest path where • Note: Breadth works for non-weighted edges Jeff Chastine

  5. Single-Source Shortest Path • Has optimal sub-structure. Why? • Path is is fastest • Assume faster way between x and w • Cut and paste faster way and you get optimal! • Graph can’t have a negative cycle. Why? • Can the graph have a positive cycle? • Can the shortest path contain a positive cycle? Jeff Chastine

  6. How it’s Done in General • A node v has a predecessor node • Tells us how we got there • NIL for first node • End with a directed graph rooted at start node s • Relaxation (pay attention!) • Maintain an upper-bound cost for each node d[v] • Initially, all nodes are marked as ∞ • If we find a cheaper path to node v, relax (update) the cost and predecessor Jeff Chastine

  7. Relaxation Example(using a breadth-first traversal) b ∞ 2 5 11 a d 0 ∞ 3 99 c ∞ Original Graph Predecessor and Cost INITIALIZE-SINGLE-SOURCE(G, s) Jeff Chastine

  8. Relaxation Example b ∞ 2 5 11 a d 0 ∞ 3 99 c ∞ Original Graph Predecessor and Cost Start with source node Jeff Chastine

  9. Relaxation Example b 2 2 5 11 a d 0 ∞ 3 99 c 99 Original Graph Predecessor and Cost Update connected nodes Jeff Chastine

  10. Relaxation Example b 2 2 5 11 a d 0 ∞ 3 99 c 99 Original Graph Predecessor and Cost Continue with Jeff Chastine

  11. Relaxation Example b 2 2 5 11 a d 0 7 3 99 c 99 Original Graph Predecessor and Cost Jeff Chastine

  12. Relaxation Example b 2 2 5 11 a d 0 7 3 99 c 99 Original Graph Predecessor and Cost This guy can chill out… RELAX! Jeff Chastine

  13. Relaxation Example b 2 2 5 11 a d 0 7 3 99 c 13 Original Graph Predecessor and Cost Jeff Chastine

  14. Relaxation Example b 2 2 5 11 a d 0 7 3 99 c 13 Original Graph Predecessor and Cost But wait! We’re not done! Jeff Chastine

  15. Relaxation Example b 2 2 5 11 a d 0 7 3 99 c 13 Original Graph Predecessor and Cost No relaxation Jeff Chastine

  16. Relaxation Example b 2 2 5 11 a d 0 7 3 99 c 13 Original Graph Predecessor and Cost Jeff Chastine

  17. Relaxation Example b 2 2 5 11 a d 0 7 3 99 c 13 Original Graph Predecessor and Cost Needs to relax again! Jeff Chastine

  18. Relaxation Example b 2 2 5 11 a d 0 7 3 99 c 10 Original Graph Predecessor and Cost Needs to relax again! Jeff Chastine

  19. Relaxation Example b 2 2 5 11 a d 0 7 3 99 c 10 Original Graph Predecessor and Cost Final Predecessor and Cost Jeff Chastine

  20. How does Google Maps work? PWND! Jeff Chastine

  21. Bellman-Ford Algorithm • Works with negatively weighted edges • Detects if negative cycle exists • Consistently uses relaxation • Runs in Jeff Chastine

  22. Bellman-Ford Algorithm BELLMAN-FORD(G, w, s) • INITIALIZE-SINGLE-SOURCE(G, s) • fori← 1 to |V[G]| -1 • foreach edge (u, v) E[G] • RELAX(u, v, w) • foreach edge (u, v) E[G] • if d[v] > d[u] + w(u, v) • then return FALSE • return TRUE Jeff Chastine

  23. Dijkstra’s Algorithm • Is more efficient than Bellman-Ford • Doesn’t work with negative edges • Has a set S of vertices that it has already traversed. Heapifies V. • In general • Picks vertex u from V - S with minimum estimate • Relaxes everything connected to u • Adds u to S • Repeats untilV - Sis the empty set Jeff Chastine

  24. Dijkstra’s Algorithm DIJKSTRA (G, w, s) • INITIALIZE-SINGLE-SOURCE(G, s) • S ← • Q ← V[G] • while Q • do u ← EXTRACT-MIN(Q) • S←S {u} • foreach vertex v Adj[u] • do RELAX(u, v, w) Jeff Chastine

  25. Summary • Both algorithms use • A predecessor graph • Costs to each node • Relaxation • Bellman-Ford () • Works with negative weights • Detects negative cycles • Dijkstra () • More efficient • Doesn’t work with negative weights Jeff Chastine

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