1 Dijkstra’s Algorithm Dr. Ying Lu RAIK 283 Data Structures & Algorithms

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 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)

4 Single-Source Shortest-Paths  For instance, the following edges form the shortest paths from node A HBC GED F A

5 Single-Source Shortest-Paths HBC GED F A Hint: does this problem looks similar? Can we modify Prim’s algorithm to solve this problem?

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 Dijkstra’s algorithm ORD BOS PVD JFK BWI MIA DFW LAX SFO

8 Dijkstra’s algorithm ORD BOS PVD JFK BWI MIA DFW LAX SFO

9 Dijkstra’s algorithm ORD BOS PVD JFK BWI MIA DFW LAX SFO JFK

10 Dijkstra’s algorithm ORD BOS PVD JFK BWI MIA DFW LAX SFO PVD

11 Dijkstra’s algorithm ORD BOS PVD JFK BWI MIA DFW LAX SFO BOS

Dijkstra’s algorithm ORD BOS PVD JFK BWI MIA DFW LAX SFO ORD

13 Dijkstra’s algorithm (cont) ORD BOS PVD JFK BWI MIA DFW LAX SFO MIA

Dijkstra’s algorithm (cont) ORD BOS PVD JFK MIA DFW LAX SFO BWI DFW

15 Dijkstra’s algorithm (cont) LAX SFO ORD BOS PVD JFK BWI MIA DFW SFO

16 Dijkstra’s algorithm (cont) LAX SFO ORD BOS PVD JFK BWI MIA DFW LAX

17 In-Class Exercises

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 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.

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