1. Vladimir Kulyukin Computer Science Department Utah State University
CS2420: Lecture 39
Vladimir Kulyukin
Computer Science Department
Utah State University

2. Outline
Graph Algorithms (Chapter 9)

3. Priority Queue with Dynamic Updates
Recall that a priority queue is a min (or max) heap.
To implement Dijkstra’s algorithm, we need to modify a standard priority queue to allow for dynamic priority updates, because the dist member variables of vertices (V.Dist) change dynamically as we run the algorithm.
To allow for dynamic updates, we can have two member variables in a priority queue class: an array of nodes (a hash table would do just as well if we have a lot of nodes) and a heap of nodes.

4. Min Priority Queue: Example10 10 B B C 21 21 14 C G 14 D E F 15 20 22 25 D 22 E 25 F 15
Priority Queue = Hash Table + Heap.
Thus, we implement a Min Priority Queue
with 2 arrays: 1 for the table; 1 for the heap. G 20

5. Min Priority Queue: Example10 10 B B C 21 21 14 C G 14 D E F 15 20 22 25 D 22 E 25
Suppose we want to update the priority of G by
changing it from 20 to 8. F 15 G 20

6. Min Priority Queue: Example10 10 B B C 21 21 14 C G 14 D E F 15 8 22 25 D 22 E 25 F
Only 1 update is needed. Once we update the priority in the hash table, the priority in the heap is also updated, because it is THE SAME POINTER stored in 2 arrays. 15 G 8

7. Min Priority Queue: Example10 10 B B C 21 21 14 C G 14 D E F 15 8 22 25 D 22 E 25 F 15
Now we have to restore the min heap
property by pushing G as far up as it
can go. G 8

8. Min Priority Queue: Example10 10 B B G 21 21 8 C C 14 D E F 14 15 22 25 D 22
Swap G with C. E 25 F 15 G 8

9. Min Priority Queue: ExampleG 10 8 B B A 21 21 10 C C 14 D E F 14 15 22 25 D 22
Swap G with A, and we are done. E 25 F 15 G 8

10. Dijkstra’s Algorithm Dijkstra(G, s) { // G is a graph, s is a source
Initialize Q; // Q is the min priority queue;
For every vertex v in V { v.Dist = INF; v.Prev = NULL; Q.Insert(v); }
s.Dist = 0; Q.UpdatePriority(s);
For i from 0 upto |V| - 1 {
u* = Q.DeleteMin(); // 1st iteration removes s from Q
Add u* to visitation tree VT; // 1st iteration puts s into visitation tree
For every vertex u adjacent to u* and not in VT {
if ( u*.Dist + w(u*, u) < u.Dist ) {
u.Dist = u*.Dist + w(u*, u); u.Prev = u*;
Q.UpdatePriority(u); }

11. Dijkstra’s Algorithm: Asymptotic Analysis