1. CSCI-256 Data Structures & Algorithm Analysis Lecture Note: Some slides by Kevin Wayne. Copyright © 2005 Pearson-Addison Wesley. All rights reserved. 10

2. Extensions What if the objective is to minimize the sum of the lateness? –EDF does not seem to work If the tasks have release times and deadlines, and are non-preemptable, the problem is NP- complete

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. Greedy Analysis Strategies Greedy algorithm stays ahead: Show that after each step of the greedy algorithm, its solution is at least as good as any other algorithm's Exchange argument: Gradually transform any solution to the one found by the greedy algorithm without hurting its quality

9. Shortest Path Problem Directed graph G = (V, E) –Source s, destination t –Length e = length of edge e Shortest path problem: find shortest directed path from s to t cost of path = sum of edge costs in path Cost of path s-2-3-5-t = 9 + 23 + 2 + 16 = 48. s 3 t 2 6 7 4 5 23 18 2 9 14 15 5 30 20 44 16 11 6 19 6

10. Dijkstra's Algorithm Dijkstra's algorithm –Maintain a set of explored nodes S for which we have determined the shortest path distance d(u) from s to u –Initialize S = { s }, d(s) = 0 –Repeatedly choose unexplored node v which minimizes add v to S, and set d(v) =  (v) shortest path to some u in explored part, followed by a single edge (u, v) s v u d(u) S e

11. Simulate Dijkstra’s algorithm (starting 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

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

13. 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

14. 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

15. Dijkstra's Algorithm: Proof of Correctness Invariant: For each node u  S, d(u) is the length of the shortest s-u path Pf. (by induction on |S|) –Base case: |S| = 1 is trivial –Inductive hypothesis: Assume true for |S| = k  1 Let v be next node added to S, and let u-v be the chosen edge The shortest s-u path plus (u, v) is an s-v path of length  (v) Consider any s-v path P. We'll see that it's no shorter than  (v) Let x-y be the first edge in P that leaves S, and let P' be the subpath to x P is already too long as soon as it leaves S (P)  (P') + (x,y)  d(x) + (x, y)   (y)   (v) nonnegative weights inductive hypothesis defn of  (y) Dijkstra chose v instead of y S s y v x P u P'

16. Dijkstra's Algorithm: Implementation For each unexplored node, explicitly maintain Next node to explore = node with minimum  (v) When exploring v, for each incident edge e = (v, w), update Efficient implementation: Maintain a priority queue of unexplored nodes, prioritized by  (v)