1. Fundamental Structures of Computer Science II
Graph Algorithms - 2
CS 202 – Fundamental Structures of Computer Science II
Bilkent University
Computer Engineering Department
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

2. Shortest Path Algorithms
Input is weighted graph:
Associated with edge (vi, vj), there is a cost ci,j to traverse the edge.
The cost of a path v1v2…vN is:
Unweighted path length is simply the number of edges on the path, namely N-1.
Weighted Path Length
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

3. Single-Source Shortest Path Problem
Given as input a weighted graph, G = (V,E), and a distinguished vertex, s, find the shortest weighted path from s to every other vertex in G. 2 v1 v2 10 4 1 3 2 v3 v4 2 v5 8 4 6 5 1 v6 v7
Shortest (Weighted) Path from V1 to V6 is: v1, v4, v7, v6. Cost=6
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

4. Single-Source Shortest Path Problem
We will examine different algorithms
1) Unweighted shortest path algorithm
2) Weighted shortest path algorithm
Assuming no negative edges.
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

5. Unweighted Shortest Pathsv1 v2 v3 v4 v5 v6 v7
Assume all edges are unweighted.
We are interested in all shortest paths to all nodes from a given node, s.
Example: assume v3 is the node from where we will find all the shortest paths.
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

6. Unweighted Shortest Pathsv1 v2 v3 v4 v5 v6 v7
Assume all edges are unweighted.
We are interested in all shortest path to all nodes from a given node, s.
Example: assume v3 is the node from where we will find all the shortest paths.
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

7. Fundamental Structures of Computer Science II
Algorithm Sketch
1. Start with initial node s.
Mark the distance of s to s as 0.
2. Find all nodes adjacent to s. For all these nodes:
Mark their distances to s as distances of previous nodes + 1.
3. Repeat step 2 for all adjacent nodes.
Stop when you exhausted all the nodes (vertices).
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

8. Example We want to find out all the shortest paths to all nodes from
node v3 v1 v2 S v3 v4 v5 v6 v7
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

9. Fundamental Structures of Computer Science II
Vertex
Adjacent Vertices v3
v1, v6 1 v1 v2 S v3 v3 v4 v5 v6 v7 1
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

10. Fundamental Structures of Computer Science II
Vertex
Adjacent Vertices v1
v2, v4 2 v1 v1 v2 1 S v3 v3 v4 v5 2 1 v6 v7
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

11. Fundamental Structures of Computer Science II
Vertex
Adjacent Vertices v2
v4, v5 2 v1 v1 v2 v2 1 S v3 v3 v4 v5 3 2 1 v6 v7
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

12. Fundamental Structures of Computer Science II
Vertex
Adjacent Vertices v4
v6, v7 2 v1 v1 v2 v2 1 S v3 v3 v4 v4 v5 3 2 1 v6 v7 3
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

13. Fundamental Structures of Computer Science II
Vertex
Adjacent Vertices v5
v4, v7 2 v1 v1 v2 v2 1 S v3 v3 v4 v4 v5 v5 3 2 1 v6 v7 3
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

14. Fundamental Structures of Computer Science II
Vertex
Adjacent Vertices v6
none v7 2 v1 v1 v2 v2 1 S v3 v3 v4 v4 v5 v5 3 2 1 v6 v6 v7 v7 3
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

15. Algorithm – Initial Configuration
vertex
known
Distance to S
Previous node v1 F ∞ v2 v3 v4 V5 V6 V7
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

16. Fundamental Structures of Computer Science II
void Graph::unweighted_shortest_paths(vertex s) {
Queue q(NUM_VERTICES);
Vertex v,w;
q.enqueue(s);
s.dist = 0;
while (!q.isEmpty())
v= q.dequeue();
v.known = true; // not needed anymore
for each w adjascent to v
if (w.dist == INFINITY)
w.dist = v.dist + 1;
w.path = v;
q.enqueue(w); }
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

17. Fundamental Structures of Computer Science II
Final Configuration
vertex
known
Distance to S
Previous node v1 T 1 v3 v2 2 v4 V5 3 V6 V7
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

18. Weighted Shortest Paths
Dijkstra’s Algorithm
Example of a greedy algorithm
Do the best thing in each step.
At each state, the algorithm chooses a vertex v, which has the smallest distance to s (dv) among all the unknown vertices.
Then the adjacent nodes of v (which are denoted as w) are updated with new distance information.
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

19. Example: Weighted Directed Graph G is shown!2 v1 v2 10 4 1 3 2 v3 v4 2 v5 8 4 6 5 1 v6 v7
We are interested in all shortest paths to all nodes from
node v1
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

20. Initial Configuration
vertex
known
Distance to S
Previous node v1 F v2 ∞ v3 v4 V5 V6 V7
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

21. Fundamental Structures of Computer Science II∞ 2 v1 v2 10 4 1 3 ∞ 2 2 v3 v4 v5 ∞ ∞ 8 4 6 5 1 v6 v7 ∞ ∞
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

22. Fundamental Structures of Computer Science II2 2 v1 v1 v2 10 4 1 3 ∞ 2 2 v3 v4 v5 ∞ 1 8 4 6 5 1 v6 v7 ∞ ∞
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

23. Fundamental Structures of Computer Science II2 2 v1 v1 v2 10 4 1 3 3 2 2 v3 v4 v4 v5 3 1 8 4 6 5 1 v6 v7 9 5
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

24. Fundamental Structures of Computer Science II2 2 v1 v1 v2 v2 10 4 1 3 3 2 2 v3 v4 v4 v5 3 1 8 4 6 5 1 v6 v7 9 5
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

25. Fundamental Structures of Computer Science II2 2 v1 v1 v2 v2 10 4 1 3 3 2 2 v3 v4 v4 v5 v5 3 1 8 4 6 5 1 v6 v7 9 5
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

26. Fundamental Structures of Computer Science II2 2 v1 v1 v2 v2 10 4 1 3 3 2 2 v3 v3 v4 v4 v5 v5 3 1 8 4 6 5 1 v6 v7 8 9 5
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

27. Fundamental Structures of Computer Science II2 2 v1 v1 v2 v2 10 4 1 3 3 2 2 v3 v3 v4 v4 v5 v5 3 1 8 4 6 5 1 v6 v7 v7 8 6 5
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

28. Fundamental Structures of Computer Science II2 2 v1 v1 v2 v2 10 4 1 3 3 2 2 v3 v3 v4 v4 v5 v5 3 1 8 4 6 5 1 v6 v6 v7 v7 6 5
Finished!
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

29. Fundamental Structures of Computer Science II
Final Configuration
vertex
known
Distance to S
Previous node v1 T v2 2 v3 3 v4 1 V5 V6 6 v7 V7 5
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

30. Implementation of Algorithm
struct Vertex {
List adj; // adjacency list
boolean known; // if we are finished processing the node
int dist; // distance to source.
Vertex path; // the previous node towards the source }
void Graph::createTable( vector<Vertex> &t) {
readGraph(t);
for (int i=0; i<t.size(); i++)
t[i].known = FALSE;
t[i].dist = INFINITY;
t[i].path = NOT_A_VERTEX; //NULL }
NUM_VERTICES = t.size();
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

31. Fundamental Structures of Computer Science II
void Graph::dijkstra( vector<Vertex> &s) {
Vertex v, w;
s.dist = 0;
for ( ; ; )
v = an unknown vertex whose distance to
s is minimum;
if (v == NOT_A_VERTEX)
break; // we are finished
v.known = TRUE;
for each w adjacent to v
if (w.known == FALSE)
if (v.dist + cost_v_w < w.dist)
// update w
w.dist = v.dist + cost_v_w;
w.path = v; }
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University