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T ODAY Shortest Paths Edge-weighted digraph API Shortest-paths - PowerPoint PPT Presentation

Dec 30, 2022 •270 likes •2.05k views

BBM 202 - ALGORITHMS D EPT . OF C OMPUTER E NGINEERING S HORTEST P ATH Acknowledgement: The course slides are adapted from the slides prepared by R. Sedgewick and K. Wayne of Princeton University. T ODAY Shortest Paths

  1. 
 
 Shortest-paths optimality conditions Proposition. Let G be an edge-weighted digraph. Then distTo[] are the shortest path distances from s iff: • For each vertex v , distTo[v] is the length of some path from s to v . • For each edge e = v → w , distTo[w] ≤ distTo[v] + e.weight() . Pf. ⇒ [ sufficient ] • Suppose that s = v 0 → v 1 → v 2 → … → v k = w is a shortest path from s to w . • Then, 
 distTo[v k ] ≤ distTo[v k-1 ] + e k .weight() e i = i th edge on shortest distTo[v k-1 ≤ distTo[v k-2 ] + e k-1 .weight() path from s to w ] ... ≤ distTo[v 1 ] distTo[v 0 ] + e 1 .weight() • Add inequalities; simplify; and substitute distTo[v 0 ] = distTo[s] = 0 : 
 distTo[w] = distTo[v k ] ≤ e k .weight() + e k-1 .weight() + … + e 1 .weight() 
 weight of shortest path from s to w • Thus, distTo[w] is the weight of shortest path to w . � weight of some path from s to w 23

  2. 
 
 
 
 
 
 
 
 
 Generic shortest-paths algorithm Generic algorithm (to compute SPT from s) Initialize distTo[s] = 0 and distTo[v] = ∞ for all other vertices. Repeat until optimality conditions are satisfied: 
 - Relax any edge. Proposition. Generic algorithm computes SPT (if it exists) from s . Pf sketch. • Throughout algorithm, distTo[v] is the length of a simple path from s 
 to v (and edgeTo[v] is last edge on path). • Each successful relaxation decreases distTo[v] for some v . • The entry distTo[v] can decrease at most a finite number of times. � 24

  3. Generic shortest-paths algorithm Generic algorithm (to compute SPT from s) Initialize distTo[s] = 0 and distTo[v] = ∞ for all other vertices. Repeat until optimality conditions are satisfied: 
 - Relax any edge. Efficient implementations. How to choose which edge to relax? Ex 1. Dijkstra's algorithm (nonnegative weights). Ex 2. Topological sort algorithm (no directed cycles). Ex 3. Bellman-Ford algorithm (no negative cycles). 25

  4. S HORTEST P ATHS ‣ Edge-weighted digraph API ‣ Shortest-paths properties ‣ Dijkstra's algorithm ‣ Edge-weighted DAGs ‣ Negative weights

  5. Edsger W. Dijkstra: select quotes “ Do only what only you can do. ” “ In their capacity as a tool, computers will be but a ripple on the 
 surface of our culture. In their capacity as intellectual challenge, 
 they are without precedent in the cultural history of mankind. ” “ The use of COBOL cripples the mind; its teaching should, 
 therefore, be regarded as a criminal offence. ” Edsger W. Dijkstra Turing award 1972 www.cs.utexas.edu/users/EWD “ It is practically impossible to teach good programming to students that have had a prior exposure to BASIC: as potential programmers they are mentally mutilated beyond hope of regeneration. ” “ APL is a mistake, carried through to perfection. It is the 
 language of the future for the programming techniques 
 of the past: it creates a new generation of coding bums. ” 27

  6. Edsger W. Dijkstra: select quotes 28

  7. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 15 0 → 1 5.0 5 0 → 4 9.0 4 12 0 → 7 8.0 s 0 3 1 → 2 12.0 8 1 → 3 15.0 9 7 2 7 1 → 7 4.0 2 → 3 3.0 9 6 1 2 → 6 11.0 11 3 → 6 9.0 5 5 4 → 5 4.0 4 → 6 20.0 4 13 4 → 7 5.0 6 4 5 → 2 1.0 20 5 → 6 13.0 7 → 5 6.0 7 → 2 7.0 an edge-weighted digraph 29

  8. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 0 2 3 7 2 4 5 6 7 5 6 4 choose source vertex 0 30

  9. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. ∞ 3 1 v distTo[] edgeTo[] 0 0.0 - 0 5 1 0 2 ∞ 8 3 7 2 4 5 9 6 7 5 6 4 ∞ relax all edges incident from 0 31

  10. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. ∞ 5 3 1 v distTo[] edgeTo[] 0 0.0 - 0 5 1 5.0 0 → 1 0 2 ∞ 8 8 3 7 2 4 9.0 0 → 4 5 9 6 7 8.0 0 → 7 5 6 4 ∞ 9 relax all edges incident from 0 32

  11. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 3 7 2 4 9.0 0 → 4 5 6 7 8.0 0 → 7 5 6 4 33

  12. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 3 7 2 4 9.0 0 → 4 5 6 7 8.0 0 → 7 5 6 4 choose vertex 1 34

  13. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. ∞ 5 3 1 15 v distTo[] edgeTo[] 0 0.0 - 4 1 5.0 0 → 1 12 0 2 8 3 ∞ 7 2 4 9.0 0 → 4 5 6 7 8.0 0 → 7 5 6 4 relax all edges incident from 1 35

  14. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. ∞ 5 20 3 1 15 v distTo[] edgeTo[] 0 0.0 - 4 12 1 5.0 0 → 1 0 2 17.0 1 → 2 8 3 20.0 1 → 3 ∞ 7 2 17 4 9.0 0 → 4 5 6 7 8.0 0 → 7 5 ✔ 6 4 relax all edges incident from 1 36

  15. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 17.0 1 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 6 7 8.0 0 → 7 5 6 4 37

  16. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 17.0 1 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 6 7 8.0 0 → 7 5 6 4 choose vertex 7 38

  17. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 17.0 1 → 2 8 3 20.0 1 → 3 17 7 2 7 4 9.0 0 → 4 5 6 6 7 8.0 0 → 7 5 ∞ 6 4 relax all edges incident from 7 39

  18. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 15.0 7 → 2 8 17 15 3 20.0 1 → 3 7 2 7 4 9.0 0 → 4 5 14.0 7 → 5 6 6 7 8.0 0 → 7 5 ∞ 14 6 4 relax all edges incident from 7 40

  19. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 15.0 7 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 14.0 7 → 5 6 7 8.0 0 → 7 5 6 4 41

  20. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 15.0 7 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 14.0 7 → 5 6 7 8.0 0 → 7 5 6 4 select vertex 4 42

  21. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 15.0 7 → 2 8 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 14.0 7 → 5 6 5 14 7 8.0 0 → 7 5 4 ∞ 6 4 9 20 relax all edges incident from 4 43

  22. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 15.0 7 → 2 8 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 29.0 4 → 6 5 13 14 5 7 8.0 0 → 7 ✔ 4 ∞ 6 4 9 29 20 relax all edges incident from 4 44

  23. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 15.0 7 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 29.0 4 → 6 5 7 8.0 0 → 7 6 4 45

  24. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 15.0 7 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 29.0 4 → 6 5 7 8.0 0 → 7 6 4 select vertex 5 46

  25. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 15.0 7 → 2 3 20.0 1 → 3 7 2 15 4 9.0 0 → 4 5 13.0 4 → 5 1 6 29.0 4 → 6 5 7 8.0 0 → 7 13 13 6 4 29 relax all edges incident from 5 47

  26. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 20.0 1 → 3 7 2 15 14 4 9.0 0 → 4 5 13.0 4 → 5 1 6 26.0 5 → 6 5 7 8.0 0 → 7 13 13 6 4 29 26 relax all edges incident from 5 48

  27. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 26.0 5 → 6 5 7 8.0 0 → 7 6 4 49

  28. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 26.0 5 → 6 5 7 8.0 0 → 7 6 4 select vertex 2 50

  29. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 20 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 3 2 14.0 5 → 2 3 20.0 1 → 3 7 2 14 4 9.0 0 → 4 5 13.0 4 → 5 11 6 26.0 5 → 6 5 7 8.0 0 → 7 6 4 26 relax all edges incident from 2 51

  30. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 20 17 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 3 2 14.0 5 → 2 3 17.0 2 → 3 7 2 14 4 9.0 0 → 4 5 13.0 4 → 5 11 6 25.0 2 → 6 5 7 8.0 0 → 7 6 4 26 25 relax all edges incident from 2 52

  31. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 17.0 2 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 25.0 2 → 6 5 7 8.0 0 → 7 6 4 53

  32. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 17.0 2 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 25.0 2 → 6 5 7 8.0 0 → 7 6 4 select vertex 3 54

  33. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 20 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 17.0 2 → 3 9 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 25.0 2 → 6 5 7 8.0 0 → 7 6 4 25 relax all edges incident from 3 55

  34. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 20 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 17.0 2 → 3 9 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 25.0 2 → 6 ✔ 5 7 8.0 0 → 7 6 4 25 relax all edges incident from 3 56

  35. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 17.0 2 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 25.0 2 → 6 5 7 8.0 0 → 7 6 4 57

  36. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 17.0 2 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 25.0 2 → 6 5 7 8.0 0 → 7 6 4 select vertex 6 58

  37. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 17.0 2 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 25.0 2 → 6 5 7 8.0 0 → 7 6 4 relax all edges incident from 6 59

  38. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 17.0 2 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 25.0 2 → 6 5 7 8.0 0 → 7 6 4 60

  39. Dijkstra's algorithm • Consider vertices in increasing order of distance from s 
 (non-tree vertex with the lowest distTo[] value). • Add vertex to tree and relax all edges incident from that vertex. 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 s 0 2 14.0 5 → 2 3 17.0 2 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 25.0 2 → 6 5 7 8.0 0 → 7 6 4 shortest-paths tree from vertex s 61

  40. Dijkstra’s algorithm visualization 62

  41. Dijkstra’s algorithm visualization 63

  42. 
 Dijkstra's algorithm: correctness proof Proposition. Dijkstra's algorithm computes a SPT in any edge-weighted digraph with nonnegative weights. Pf. • Each edge e = v → w is relaxed exactly once (when v is relaxed), 
 leaving distTo[w] ≤ distTo[v] + e.weight() . • Inequality holds until algorithm terminates because: - distTo[w] cannot increase distTo[] values are monotone decreasing - distTo[v] will not change 
 edge weights are nonnegative and we choose 
 lowest distTo[] value at each step • Thus, upon termination, shortest-paths optimality conditions hold. � 64

  43. Dijkstra's algorithm: Java implementation public class DijkstraSP { private DirectedEdge[] edgeTo; private double[] distTo; private IndexMinPQ<Double> pq; public DijkstraSP(EdgeWeightedDigraph G, int s) { edgeTo = new DirectedEdge[G.V()]; distTo = new double[G.V()]; pq = new IndexMinPQ<Double>(G.V()); for (int v = 0; v < G.V(); v++) distTo[v] = Double.POSITIVE_INFINITY; distTo[s] = 0.0; pq.insert(s, 0.0); relax vertices in order 
 while (!pq.isEmpty()) of distance from s { int v = pq.delMin(); for (DirectedEdge e : G.adj(v)) relax(e); } } } 65

  44. Dijkstra's algorithm: Java implementation private void relax(DirectedEdge e) { int v = e.from(), w = e.to(); if (distTo[w] > distTo[v] + e.weight()) { distTo[w] = distTo[v] + e.weight(); edgeTo[w] = e; if (pq.contains(w)) pq.decreaseKey(w, distTo[w]); update PQ else pq.insert (w, distTo[w]); } } 66

  45. 
 
 
 
 
 
 
 
 
 
 Dijkstra's algorithm: which priority queue? Depends on PQ implementation: V insert, V delete-min, E decrease-key. PQ implementation insert delete-min decrease-key total array 1 V 1 V 2 binary heap log V log V log V E log V d-way heap 
 d log d V d log d V log d V E log E/V V (Johnson 1975) Fibonacci heap 
 1 † log V † 1 † E + V log V (Fredman-Tarjan 1984) † amortized Bottom line. • Array implementation optimal for dense graphs. • Binary heap much faster for sparse graphs. • d-way heap worth the trouble in performance-critical situations. • Fibonacci heap best in theory, but not worth implementing. 67

  46. Priority-first search Insight. Four of our graph-search methods are the same algorithm! • Maintain a set of explored vertices S . • Grow S by exploring edges with exactly one endpoint leaving S . DFS. Take edge from vertex which was discovered most recently. BFS. Take edge from vertex which was discovered least recently. Prim. Take edge of minimum weight. Dijkstra. Take edge to vertex that is closest to S . e w v S s Challenge. Express this insight in reusable Java code. 68

  47. S HORTEST P ATHS ‣ Edge-weighted digraph API ‣ Shortest-paths properties ‣ Dijkstra's algorithm ‣ Edge-weighted DAGs ‣ Negative weights

  48. 
 Acyclic edge-weighted digraphs Q. Suppose that an edge-weighted digraph has no directed cycles. 
 Is it easier to find shortest paths than in a general digraph? 
 A. Yes! 70

  49. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 → 1 5.0 3 1 15 0 → 4 9.0 5 0 → 7 8.0 4 12 3 s 1 → 2 12.0 0 1 → 3 15.0 8 9 7 1 → 7 4.0 2 7 2 → 3 3.0 9 6 1 2 → 6 11.0 11 3 → 6 9.0 5 5 4 → 5 4.0 4 13 4 → 6 20.0 4 → 7 5.0 6 4 20 5 → 2 1.0 5 → 6 13.0 7 → 5 6.0 an edge-weighted DAG 7 → 2 7.0 71

  50. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 3 1 0 7 2 5 6 4 topological order: 0 1 4 7 5 2 3 6 72

  51. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 0 2 3 7 2 4 5 6 7 5 6 4 choose vertex 0 73

  52. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 ∞ 3 1 v distTo[] edgeTo[] 0 0.0 - 5 0 1 0 2 ∞ 8 3 7 2 4 5 9 6 7 5 6 4 ∞ relax all edges incident from 0 74

  53. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 ∞ 5 3 1 v distTo[] edgeTo[] 0 0.0 - 5 0 1 5.0 0 → 1 0 2 ∞ 8 8 3 7 2 4 9.0 0 → 4 9 5 6 7 8.0 0 → 7 5 6 4 ∞ 9 relax all edges incident from 0 75

  54. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 3 7 2 4 9.0 0 → 4 5 6 7 8.0 0 → 7 5 6 4 76

  55. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 3 7 2 4 9.0 0 → 4 5 6 7 8.0 0 → 7 5 6 4 choose vertex 1 77

  56. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 ∞ 5 3 1 15 v distTo[] edgeTo[] 0 0.0 - 4 1 5.0 0 → 1 12 0 2 3 ∞ 7 2 4 9.0 0 → 4 8 5 6 7 8.0 0 → 7 5 6 4 relax all edges incident from 1 78

  57. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 ∞ 5 20 3 1 15 v distTo[] edgeTo[] 0 0.0 - 4 12 1 5.0 0 → 1 0 2 17.0 1 → 2 3 20.0 1 → 3 ∞ 7 2 17 4 9.0 0 → 4 8 5 6 7 8.0 0 → 7 5 ✔ 6 4 relax all edges incident from 1 79

  58. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 17.0 1 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 6 7 8.0 0 → 7 5 6 4 80

  59. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 17.0 1 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 6 7 8.0 0 → 7 5 6 4 select vertex 4 (Dijkstra would have selected vertex 7) 81

  60. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 17.0 1 → 2 8 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 6 5 ∞ 7 8.0 0 → 7 5 4 6 ∞ 4 9 20 relax all edges incident from 4 82

  61. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 17.0 1 → 2 8 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 29.0 4 → 6 5 ∞ 13 5 7 8.0 0 → 7 ✔ 4 6 ∞ 4 9 20 29 relax all edges incident from 4 83

  62. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 17.0 1 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 29.0 4 → 6 5 7 8.0 0 → 7 6 4 84

  63. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 17.0 1 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 29.0 4 → 6 5 7 8.0 0 → 7 6 4 choose vertex 7 85

  64. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 17.0 1 → 2 8 17 3 20.0 1 → 3 7 2 7 4 9.0 0 → 4 6 5 13.0 4 → 5 6 29.0 4 → 6 5 7 8.0 0 → 7 13 6 4 relax all edges incident from 7 86

  65. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 15.0 7 → 2 8 17 15 3 20.0 1 → 3 7 2 7 4 9.0 0 → 4 6 5 13.0 4 → 5 6 29.0 4 → 6 5 7 8.0 0 → 7 13 6 4 relax all edges incident from 7 87

  66. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 15.0 7 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 29.0 4 → 6 5 7 8.0 0 → 7 6 4 88

  67. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 15.0 7 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 29.0 4 → 6 5 7 8.0 0 → 7 6 4 select vertex 5 89

  68. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 15.0 7 → 2 3 20.0 1 → 3 7 2 15 4 9.0 0 → 4 5 13.0 4 → 5 1 6 29.0 4 → 6 5 7 8.0 0 → 7 13 13 6 4 29 relax all edges incident from 5 90

  69. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 20.0 1 → 3 7 2 15 14 4 9.0 0 → 4 5 13.0 4 → 5 1 6 26.0 5 → 6 5 7 8.0 0 → 7 13 13 6 4 29 26 relax all edges incident from 5 91

  70. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 26.0 5 → 6 5 7 8.0 0 → 7 6 4 92

  71. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 20.0 1 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 26.0 5 → 6 5 7 8.0 0 → 7 6 4 select vertex 2 93

  72. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 20 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 3 0 2 14.0 5 → 2 3 20.0 1 → 3 7 2 14 4 9.0 0 → 4 5 13.0 4 → 5 11 6 26.0 5 → 6 5 7 8.0 0 → 7 6 4 26 relax all edges incident from 2 94

  73. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 20 17 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 3 0 2 14.0 5 → 2 3 17.0 2 → 3 7 2 14 4 9.0 0 → 4 5 13.0 4 → 5 11 6 25.0 2 → 6 5 7 8.0 0 → 7 6 4 26 25 relax all edges incident from 2 95

  74. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 17.0 2 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 25.0 2 → 6 5 7 8.0 0 → 7 6 4 96

  75. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 17.0 2 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 25.0 2 → 6 5 7 8.0 0 → 7 6 4 select vertex 3 97

  76. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 20 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 9 3 17.0 2 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 25.0 2 → 6 5 7 8.0 0 → 7 6 4 25 relax all edges incident from 3 98

  77. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 20 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 9 3 17.0 2 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 25.0 2 → 6 ✔ 5 7 8.0 0 → 7 6 4 25 relax all edges incident from 3 99

  78. Topological sort algorithm • Consider vertices in topological order. • Relax all edges incident from that vertex. 0 1 4 7 5 2 3 6 3 1 v distTo[] edgeTo[] 0 0.0 - 1 5.0 0 → 1 0 2 14.0 5 → 2 3 17.0 2 → 3 7 2 4 9.0 0 → 4 5 13.0 4 → 5 6 25.0 2 → 6 5 7 8.0 0 → 7 6 4 100

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