DP algorithms for all-pairs Shortest Path

1. DP algorithms for all-pairs Shortest Path Tandy Warnow

2. “Shortest Path” • Given graph G=(V,E) with positive weights on the edges (w:E->R+), and given two vertices a and b. • Find the “shortest path” from a to b (where the length of the path is the sum of the edge weights on the path). Perhaps we should call this the minimum weight path!

3. Greedy algorithm • Start at a, and greedily construct a path that goes to w by adding vertices that are closest to the current endpoint, until you reach b. • Problem: it doesn’t work correctly! Sometimes you can’t reach b at all, and would have to backtrack… and sometimes you get a path that doesn’t have the minimum weight.

4. Designing a DP solution • How are the subproblems defined? • Where is the answer stored? • How are the boundary values computed? • How do we compute each entry from other entries? • What is the order in which we fill in the matrix?

5. Two DP algorithms • Both are correct. Both produce correct values for all-pairs shortest paths. • The difference is the subproblem formulation, and hence in the running time. • The reason both algorithms are given is to teach you how to do DP algorithms! • But, be prepared to provide one or both of these algorithms, and to be able to apply it to an input (on some exam, for example).

6. Dynamic programming First attempt: let {1,2,…,n} denote the set of vertices. Subproblem formulation: • M[i,j,k] = min length of any path from i to j that uses at most k edges. All paths have at most n-1 edges, so 1 ≤ k ≤ n-1.)

7. First DP approach • M[i,j,k] = min length of any path from i to j that uses at most k edges. • How to set the boundary case (k=1)? • How to set M[i,j,k] from other entries?

8. How to set the boundary case (k=1)? • Easy: M[x,y,1] = • 0 if x=y • w(x,y) if x and y are different, and (x,y) is an edge, and • infinity otherwise

9. How to set M[i,j,k] from other entries? • Consider a minimum weight path from i to j that has at most k edges. How many edges does it have?

10. How to set M[i,j,k] from other entries, for k>1? • Consider a minimum weight path from i to j that has at most k edges. • Case 1: The minimum weight path has at most k-1 edges. • Case 2: The minimum weight path has exactly k edges.

11. How to set M[i,j,k] from other entries, for k>1? • Case 1: The minimum weight path has at most most k-1 edges. Then M[i,j,k] = M[i,j,k-1]

12. How to set M[i,j,k] from other entries, for k>1? • Case 2: The minimum weight path has exactly k edges. Then we break the path into two paths: one with k-1 edges, and one with 1 edge. And so: • M[i,j,k] = min{M[i,x,k-1] + w(x,j): x in V-{i,j}}

13. Since the minimum weight path from i to j with at most k>0 edges will satisfy one of these two properties, we can set M[i,j,k] from other entries, as follows: • M[i,j,k] = min{ min{M[i,x,k-1] + w(x,j): x in V}, M[i,j,k-1]} Note that the third index goes down!

14. Finishing the design • Where is the answer stored? • How are the boundary values computed? • How do we compute each entry from other entries? • What is the order in which we fill in the matrix? • Running time?

15. Running time analysis • How many entries do we need to compute? O(n3) 0 ≤ i ≤ n 0 ≤ j ≤ n 1 ≤ k ≤ n-1 • How much time does it take to compute each entry, once the “previous” ones are computed? O(n)

16. Running time analysis (cont) • O(n3) elements, • each takes O(n) time to compute, if computed in the correct order! • so O(n4) time

17. Next DP approach • Try a new subproblem formulation! • Q[i,j,k] = minimum weight of any path from i to j that uses internal vertices drawn from {1,2,…,k}.

18. Designing a DP solution • How are the subproblems defined? • Where is the answer stored? • How are the boundary values computed? • How do we compute each entry from other entries? • What is the order in which we fill in the matrix?

19. Q[i,j,k] = minimum weight of any path from i to j that uses internal vertices drawn from {1,2,…,k}. • Boundary cases: Q[i,j,0] for all i,j • Solution to minimum found in Q[i,j,n] for minimum weight path from i to j • Once again, O(n3) entries in the matrix

20. Solving subproblems • Q[i,j,k] = minimum weight of any path from i to j that uses internal vertices drawn from {1,2,…,k}. • The minimum cost such path either includes vertex k or does not include vertex k.

21. Solving subproblems • Q[i,j,k] = minimum weight of any path from i to j that uses internal vertices drawn from {1,2,…,k}. • If the minimum cost path P includes vertex k, then you can divide P into the path P1 from i to p, and P2 from p to j. • What is the weight of P1? • What is the weight of P2?

22. Solving subproblems • Q[i,j,k] = minimum weight of any path from i to j that uses internal vertices drawn from {1,2,…,k}. • P is a minimum cost path from I to j that uses vertex k, and has all internal vertices from {1,2,…k}. • Path P1 from i to p, and P2 from p to j. • The weight of P1 is Q[i,p,p-1] (why??). • The weight of P2 is Q[p,j,p-1] (why??). • Thus the weight of P is Q[i,p,p-1] + Q[p,j,p-1].

23. New DP algorithm For k=0 up to n DO Compute Q[i,j,k] for each i,j End for

24. Running time analysis • Each entry only takes O(1) time to compute • There are O(n3) entries • Hence, O(n3) time.