1. CSE 421 Algorithms Richard Anderson Lecture 8 Optimal Caching Dijkstra’s algorithm

2. Today’s Lecture Optimal Caching (Section 4.3) Dijkstra’s Algorithm (Section 4.4)

3. Optimal Caching Caching problem: –Maintain collection of items in local memory –Minimize number of items fetched

4. Caching example A, B, C, D, A, E, B, A, D, A, C, B, D, A

5. Optimal Caching If you know the sequence of requests, what is the optimal replacement pattern? Note – it is rare to know what the requests are in advance – but we still might want to do this: –Some specific applications, the sequence is known –Competitive analysis, compare performance on an online algorithm with an optimal offline algorithm

6. Farthest in the future algorithm Discard element used farthest in the future A, B, C, A, C, D, C, B, C, A, D

7. Correctness Proof Sketch Start with Optimal Solution O Convert to Farthest in the Future Solution F-F Look at the first place where they differ Convert O to evict F-F element –There are some technicalities here to ensure the caches have the same configuration...

8. Single Source Shortest Path Problem Given a graph and a start vertex s –Determine distance of every vertex from s –Identify shortest paths to each vertex Express concisely as a “shortest paths tree” Each vertex has a pointer to a predecessor on shortest path s v x u 1 2 5 3 4 s v x u 1 3 3

9. Construct Shortest Path Tree from s a b c s e g f d 4 2 -3 2 1 5 4 -2 3 3 6 3 7 4 a b c s e g f d

10. Warmup If P is a shortest path from s to v, and if t is on the path P, the segment from s to t is a shortest path between s and t WHY? s t v

11. Dijkstra’s Algorithm S = {}; d[s] = 0; d[v] = infinity for v != s While S != V Choose v in V-S with minimum d[v] Add v to S For each w in the neighborhood of v d[w] = min(d[w], d[v] + c(v, w)) s u v z y x 1 4 3 2 3 2 1 2 1 0 1 2 2 5 4 Assume all edges have non-negative cost

12. Simulate Dijkstra’s algorithm (strarting from s) on the graph 1 2 3 4 5 Round Vertex Added sabcd bd c a 1 1 1 23 4 6 1 3 4 s

13. Dijkstra’s Algorithm as a greedy algorithm Elements committed to the solution by order of minimum distance

14. Correctness Proof Elements in S have the correct label Key to proof: when v is added to S, it has the correct distance label. s y v x u

15. Proof Let v be a vertex in V-S with minimum d[v] Let P v be a path of length d[v], with an edge (u,v) Let P be some other path to v. Suppose P first leaves S on the edge (x, y) –P = P sx + c(x,y) + P yv –Len(P sx ) + c(x,y) >= d[y] –Len(P yv ) >= 0 –Len(P) >= d[y] + 0 >= d[v] s y v x u

16. Negative Cost Edges Draw a small example a negative cost edge and show that Dijkstra’s algorithm fails on this example

17. Bottleneck Shortest Path Define the bottleneck distance for a path to be the maximum cost edge along the path s v x u 6 5 5 3 4 2

18. Compute the bottleneck shortest paths a b c s e g f d 4 2 -3 6 6 5 4 -2 3 4 6 3 7 4 a b c s e g f d

19. How do you adapt Dijkstra’s algorithm to handle bottleneck distances Does the correctness proof still apply?