1. 1 Dijkstra’s Algorithm Dr. Ying Lu ylu@cse.unl.edu RAIK 283 Data Structures & Algorithms

2. 2  Giving credit where credit is due: »Most of slides for this lecture are based on slides created by Dr. John Vergara at Ateneo De Manila UniversityDr. John VergaraAteneo De Manila University »I have modified them and added new slides

3. 3 Single-Source Shortest-Paths Problem  Problem: given a connected graph G with non-negative weights on the edges and a source vertex s in G, determine the shortest paths from s to all other vertices in G.  Useful in many applications (e.g., road map applications) 14 10 3 64 5 2 9 15 8

4. 4 Single-Source Shortest-Paths  For instance, the following edges form the shortest paths from node A HBC GED F A 14 10 3 64 5 2 9 15 8

5. 5 Single-Source Shortest-Paths HBC GED F A 14 10 3 64 5 2 9 15 8 Hint: does this problem looks similar? Can we modify Prim’s algorithm to solve this problem?

6. 6 Dijkstra’s Algorithm  Solves the single-source shortest paths problem  Involves keeping a table of current shortest path lengths from source vertex (initialize to infinity for all vertices except s, which has length 0)  Repeatedly select the vertex u with shortest path length, and update other lengths by considering the path that passes through that vertex  Stop when all vertices have been selected  Could use a priority queue to facilitate selection of shortest lengths »Here, priority is defined differently from that of the Prim’s algorithm »Like Prim’s algorithm, it needs to refine data structure so that the update of key-values in priority queue is allowed »Like Prim’s algorithm, time complexity is O( (n + m) log n )

7. 7 Dijkstra’s algorithm ORD BOS PVD JFK BWI MIA DFW LAX SFO 0 849 867 187 144 184 1258 1090 946 740 621 1391 802 1121 2342 1235 1464 337 1846 2704

8. 8 Dijkstra’s algorithm ORD BOS PVD JFK BWI MIA DFW LAX SFO 184 0 946 621 849 867 187 144 184 1258 1090 946 740 621 1391 802 1121 2342 1235 1464 337 1846 2704

9. 9 Dijkstra’s algorithm ORD BOS PVD JFK BWI MIA DFW LAX SFO 184 0 946 621 849 867 187 144 184 1258 1090 946 740 621 1391 802 1121 2342 1235 1464 337 1846 2704 JFK

10. 10 Dijkstra’s algorithm ORD BOS PVD JFK BWI MIA DFW LAX SFO 371 328 184 0 946 621 1575 849 867 187 144 184 1258 1090 946 740 621 1391 802 1121 2342 1235 1464 337 1846 2704 PVD

11. 11 Dijkstra’s algorithm ORD BOS PVD JFK BWI MIA DFW LAX SFO 371 328 184 0 946 621 1575 3075 849 867 187 144 184 1258 1090 946 740 621 1391 802 1121 2342 1235 1464 337 1846 2704 BOS

12. 12 3075 1575 Dijkstra’s algorithm ORD BOS PVD JFK BWI MIA DFW LAX SFO 371 328 184 0 946 621 1423 2467 849 867 187 144 184 1258 1090 946 740 621 1391 802 1121 2342 1235 1464 337 1846 2704 ORD

13. 13 Dijkstra’s algorithm (cont) ORD BOS PVD JFK BWI MIA DFW LAX SFO 371 328 184 0 946 621 1423 3288 2467 849 867 187 144 184 1258 1090 946 740 621 1391 802 1121 2342 1235 1464 337 1846 2704 MIA

14. 14 3288 Dijkstra’s algorithm (cont) ORD BOS PVD JFK MIA DFW LAX SFO 371 328 184 0 946 621 1423 2658 2467 849 867 187 144 184 1258 1090 946 740 621 1391 802 1121 2342 1235 1464 337 1846 2704 BWI DFW

15. 15 Dijkstra’s algorithm (cont) LAX SFO 371 328 184 0 946 621 1423 2658 2467 849 867 187 144 184 1258 1090 946 740 621 1391 802 1121 2342 1235 1464 337 1846 2704 ORD BOS PVD JFK BWI MIA DFW SFO

16. 16 Dijkstra’s algorithm (cont) LAX SFO 371 328 184 0 946 621 1423 2658 2467 849 867 187 144 184 1258 1090 946 740 621 1391 802 1121 2342 1235 1464 337 1846 2704 ORD BOS PVD JFK BWI MIA DFW LAX

17. 17 In-Class Exercises

18. 18 In-Class Exercises  Design a linear-time, (i.e.,  ( n + m)) algorithm for solving the single-source shortest-paths problem for dags (directed acyclic graphs) represented by their adjacency lists.  Hint: topologically sort the dag’s vertices first.

19. 19 In-Class Exercise  Suppose a person is making a travel plan driving from city 1 to city n, n > 1, following a route that will go through cities 2 through n –1 in between. The person knows the mileages between adjacent cities, and knows how many miles a full tank of gasoline can travel. Based on this information, the problem is to minimize the number of stops for filling up the gas tank, assuming there is exactly one gas station in each of the cities. Design a greedy algorithm to solve this problem and analyze its time complexity.

20. fill up the gas tank in City 1 for c = 2 to n –1 do // decide whether to stop in City c for gas while approaching (2.1) if there is still enough gas to go to City (c+1) then don’t stop in City c (2.2) else fill up the gas tank in City c It can be proved that this greedy algorithm minimizes the number of gas stops. The time complexity is O(n) because the loop in Step (2) runs O(n) iterations in which each iteration uses O(1) time (in Steps (2.1) and (2.2)). Proofs Techniques for Greedy Algorithms