1. Shortest Path Algorithms

2. Single-pair shortest paths. Path from a single vertex to another one.
Dijkstra  Single-source shortest paths.
Bellman-Ford  Single-source shortest paths (having negative edges).
Floyd-Warshall  All-pairs shortest paths.

3. The shortest path algorithms are effected by:
Directed/Undirected graphs.
Weights of the edges (positive or negative).
Representation of the graph in the memory, matrix or a linked list (effecting the performance).
Single or all pairs paths.
Dense or sparse graphs...

5. Shortest Path
Unweighted path length is simply the number of edges on the path, namely N-1.

6. 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

7. Single-Source Shortest Path Algorithms
We will examine different algorithms:
Unweighted shortest path algorithm
Weighted shortest path algorithm
Assuming no negative edges.

8. 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(d[v]) among all the unknown vertices.
Then, the adjacent nodes of v (which are denoted as u) are updated with new distance information.

12. Using Array Data Structure

24. Shortest Path Example: Using Heap Structure
Weighted Directed Graph G is shown! v1 v2 v5 v3 v6 v7 4 2 1 5 8 3 10 6 v4
We are interested in all shortest paths to all nodes from node v1

25. Initial Configuration
vertex
known
Distance to S
Previous node
(parent) v1 F v2 ∞ v3 v4 V5 V6 V7

26. v1 v2 v5 v3 v6 v7 4 2 1 5 8 3 10 6 v4 ∞
V1 0

27. V4 1
V2 2 2 2 v1 v1 v2 10 4 1 3 ∞ 2 2 v3 v4 v5 ∞ 1 8 4 6 5 1 v6 v7 ∞ ∞

28. v1 v2 v5 v3 v6 v7 4 2 1 5 8 3 10 6 v4 9
V2 2
V5 3
V3 3
V7 5

29. V5 3
V3 3
V7 5
V6 9 v1 v2 v5 v3 v6 v7 4 2 1 5 8 3 10 6 v4 9

30. v1 v2 v5 v3 v6 v7 4 2 1 5 8 3 10 6 v4 9
V3 3
V7 5
V6 9

31. v1 v2 v5 v3 v6 v7 4 2 1 5 8 3 10 6 v4 9
V7 5
V6 8

32. v1 v2 v5 v3 v6 v7 4 2 1 5 8 3 10 6 v4
V6 6

33. v1 v2 v5 v3 v6 v7 4 2 1 5 8 3 10 6 v4
Finished! Empty Heap

34. Final Configuration vertex known Distance to S Previous node v1 T v2 2v2 2 v3 3 v4 1 V5 V6 6 v7 V7 5

35. Unweighted Shortest Paths

36. 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.

43. Example
Assume that we want to find out all the shortest paths to all nodes from node v3. v1 v2 S v3 v4 v5 v6 v7

44. Vertex
Adjacent Vertices v3
v1, v6 1 v1 v2 S v3 v3 v4 v5 v6 v7 1

45. Vertex Adjacent Vertices v1 v2, v4 2 1 2 1 v1 v1 v2 S v3 v3 v4 v5 v62 1 v6 v7

46. Vertex Adjacent Vertices v2 v4, v5 2 1 3 2 1 v1 v1 v2 v2 S v3 v3 v4 v52 1 v6 v7

47. Vertex Adjacent Vertices v4 v6, v7 2 1 3 2 1 3 v1 v1 v2 v2 S v3 v3 v42 1 v6 v7 3

48. Vertex Adjacent Vertices v5 v4, v7 2 1 3 2 1 3 v1 v1 v2 v2 S v3 v3 v42 1 v6 v7 3

49. Vertex Adjacent Vertices v6 none v7 2 1 3 2 1 3 v1 v1 v2 v2 S v3 v3 v42 1 v6 v6 v7 v7 3

50. Algorithm – Initial Configuration
vertex
known
Distance to S
Previous node v1 F ∞ v2 v3 v4 V5 V6 V7

51. Last Configuration vertex known Distance to S Previous node v1 T ? v2