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Monday, April 25, 2005
Last time we talked about Dijkstra's Algorithm: The shortest path from a source vertex to all other vertices.
(We saw an example in the last class.)
Now, we will give pseudocode for Dijkstra's Shortest Path Algorithm
Algorithm shortestPath(G, S, D)
Input: weighted graph, G; source vertex, S.
Output: Array D[ ] such that D[U] will be the length of the shortest path from S to U in G.
Initialize D[S] = 0;
Initialize D[U] = ∞; // For each U ≠ S
Initialize Q = {all vertices in G};
// Q = list or queue; keeps track of unvisited vertices
while(Q not empty)
{
Get a vertex U from Q such that D[U] is the minimum among all vertices in Q.
For each vertex Z adjacent to U such that Z is in Q
{
if(D[U] + W(U,Z) < D[Z])
{
D[Z] = D[U] + W(U,Z)
}
}
}
W(U,Z) is the weight of the edge from U to Z
m = # edges
n = # vertices
Complexity:
The for loop is O(m) independent of the while loop.
The while loop is O(n
2
) or O(n log n).
getMin( ) takes O(n) if we are using a list or queue, O(log n) if we are using a heap.
O(n
2
+ O(m) = O(n
2
+ m) = O(n
2
) for a simple list or queue.
O(n log n) + O(m) = O(n log n + m) for a heap
Minimum Spanning Tree
 Spanning tree with minimum total weight
 Example: connect all computers in the building with minimum amount of cable.
What is the minimum weight necessary to connect all the vertices?
Minimum Spanning Tree Property
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This note was uploaded on 02/02/2012 for the course CS 251 taught by Professor Staff during the Fall '08 term at Purdue UniversityWest Lafayette.
 Fall '08
 Staff
 Data Structures

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