1. 1 Sections 2.5 and 4.4 Priority Queues and Dijkstra’s Algorithm For Shortest Paths Some Slides by Kevin Wayne. Copyright © 2005 Pearson-Addison Wesley. All rights reserved.

2. A Problem with Greed Primitive Greedy Algorithm Strategy: 1.Sort the set of items by some criteria 2.Select the items in sorted order Problem: Sometimes the criteria is non-static! Consider: 1.Distance from current position to point X changes if I move. 2.In cases of uncertainty, the world will change—stock values, political polls, etc. All issues of optimality aside, these situations make the above greedy approach unnecessarily slow. In these situations, the strategy becomes: 1.For each choice, 1.Search for the best choice and pick it. 2.Update dynamic criteria values 2

3. NEVER FORGET DATA STRUCTURES AFFECT RUNTIME!!! 3

4. Priority Queues: A Greedy Data Structure If sorting is impractical, and searching is inevitable, then we need to speed up the search: ◦ Binary search trees are good, but use extra memory, and are fully ordered –We only want to find the smallest value New abstract data type: Priority Queue ◦ Stores items based on some pre-determined key ◦ Required operations:  Initialize Queue  Insert new item  Extract ‘smallest’ item  Update a an item’s key Find Smallest Item Delete Item Find Item Change Key

5. Binary Heaps A binary heap is similar to a binary search tree, but is not fully sorted Heap Property: For any item in a heap, its key value v must be greater than or equal to its parent’s key value Usefulness of heap property: ◦ Smallest item is at the root (O(1) time to locate) Cost of heap property: ◦ Adding/deleting/changing keys has a cost for restoring heap property 5

6. Array-Based Binary Heap Requires maximum number of elements to be fixed Never any gaps—all empty space is at the end Root at H[0] For a node at position H[i]: ◦ Left child = H[2*i] ◦ Right child = H[2*i + 1] Parent = H[  i /2  ] For convenience, length(H) = number of items, NOT length of array 6

7. Fixing the Heap Property: Heapify-Up 7

8. 8 Fixing the Heap Property: Heapify-Down

9. Insert Item Simple strategy: ◦ Place new item at the end of the heap ◦ Fix heap property using Heapify-up algorithm 9

10. Delete Item And Extract Min Function delete_item(H,i): ◦ Clear the item from the heap at position i ◦ Move the right-most item into its place ◦ If this item is too small, use Heapify-Up to fix it ◦ Else if it is too big, use new algorithm Heapify-Down (next slide) to fix heap property Function extract_min(H) follows: ◦ Return item at root, then delete_item(H,0) 10

11. Initialize Queue Filling the Queue (2 scenarios) 1.Mostly distinct key values:  Use repeated calls to insert_item(H,v), 2.All keys are initially equal:  Just copy values into the queue 11 init_queue(N) return Array of size N

12. Change Key Algorithm First, find the location of the item in the heap: ◦ Throw memory at problem to get constant time  Dictionary object location[v] keeps track of the location of v in the heap  Need to gently modify heapify algorithms to keep track of locations Then, use heapify-up or heapify-down depending on the new value 12

13. Runtime of Heap Operations Everything depends on Heapify-up or Heapify-down: ◦ Both functions are recursive from a specific start node, and spend constant time at each call ◦ Worst case: new key travels full distance between root and leaf ◦ Key observation: heaps are always balanced ◦ Therefore, worst-case performance of Heapify-up and Heapify-down is O(log 2 n) 13

14. New Greedy Algorithm Form 1. Add all items to priority queue PQ where the key value is the greedy selection criteria 2. While PQ not empty: 1.Take extract_min(PQ) as the next selection 2.For any items whose key values change, call change_key(PQ,item,value) 14

15. 4.4 Shortest Paths in a Graph shortest path from Princeton CS department to Einstein's house

16. Shortest Path Problem Shortest path network. ◦ 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. 16 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 cost of path = sum of edge costs in path

17. 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). 17 s v u d(u) S e shortest path to some u in explored part, followed by a single edge (u, v)

18. 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). 18 s v u d(u) shortest path to some u in explored part, followed by a single edge (u, v) S e

19. 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. 19 (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'

20. 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). 20 † Individual ops are amortized bounds PQ Operation Insert ExtractMin ChangeKey Binary heap log n Fib heap † 1 log n 1 Array n n 1 IsEmpty111 Priority Queue Totalm log nm + n log nn2n2 Dijkstra n n m n d-way Heap d log d n log d n 1 m log m/n n

21. Edsger W. Dijkstra 21 The question of whether computers can think is like the question of whether submarines can swim. Do only what only you can do. In their capacity as a tool, computers will be but a ripple on the surface of our culture. In their capacity as intellectual challenge, they are without precedent in the cultural history of mankind. The use of COBOL cripples the mind; its teaching should, therefore, be regarded as a criminal offence. APL is a mistake, carried through to perfection. It is the language of the future for the programming techniques of the past: it creates a new generation of coding bums.