1. CS223 Advanced Data Structures and Algorithms 1 The Bellman-Ford Shortest Path Algorithm Neil Tang 03/11/2010

2. CS223 Advanced Data Structures and Algorithms 2 Class Overview  The shortest path problem  Differences  The Bellman-Ford algorithm  Time complexity

3. CS223 Advanced Data Structures and Algorithms 3 Shortest Path Problem  Weighted path length (cost): The sum of the weights of all links on the path.  The single-source shortest path problem: Given a weighted graph G and a source vertex s, find the shortest (minimum cost) path from s to every other vertex in G.

4. CS223 Advanced Data Structures and Algorithms 4 Differences  Negative link weight: The Bellman-Ford algorithm works; Dijkstra’s algorithm doesn’t.  Distributed implementation: The Bellman-Ford algorithm can be easily implemented in a distributed way. Dijkstra’s algorithm cannot.  Time complexity: The Bellman-Ford algorithm is higher than Dijkstra’s algorithm.

5. CS223 Advanced Data Structures and Algorithms 5 The Bellman-Ford Algorithm ,nil 0 6 7 9 5 -3 8 7-4 2 ,nil s y z x t -2

6. CS223 Advanced Data Structures and Algorithms 6 The Bellman-Ford Algorithm ,nil 0 6 7 9 5 -3 8 7-4 2 ,nil s y z x t -2 6,s 7,s ,nil 0 6 7 9 5 -3 8 7-4 2 ,nil s y z x t -2 InitializationAfter pass 1 6,s 7,s 4,y 0 6 7 9 5 -3 8 7-4 2 2,t s y z x t -2 After pass 2 2,x 7,s 4,y 0 6 7 9 5 -3 8 7-4 2 2,t s y z x t -2 After pass 3 The order of edges examined in each pass: (t, x), (t, z), (x, t), (y, x), (y, t), (y, z), (z, x), (z, s), (s, t), (s, y)

7. CS223 Advanced Data Structures and Algorithms 7 The Bellman-Ford Algorithm After pass 4 2,x 7,s 4,y 0 6 7 9 5 -3 8 7-4 2 -2,t s y z x t -2 The order of edges examined in each pass: (t, x), (t, z), (x, t), (y, x), (y, t), (y, z), (z, x), (z, s), (s, t), (s, y)

8. CS223 Advanced Data Structures and Algorithms 8 The Bellman-Ford Algorithm Bellman-Ford(G, w, s) 1. Initialize-Single-Source(G, s) 2. for i := 1 to |V| - 1 do 3. for each edge (u, v)  E do 4. Relax(u, v, w) 5. for each vertex v  u.adj do 6. if d[v] > d[u] + w(u, v) 7. then return False// there is a negative cycle 8. return True Relax(u, v, w) if d[v] > d[u] + w(u, v) then d[v] := d[u] + w(u, v) parent[v] := u

9. CS223 Advanced Data Structures and Algorithms 9 Time Complexity Bellman-Ford(G, w, s) 1. Initialize-Single-Source(G, s) 2. for i := 1 to |V| - 1 do 3. for each edge (u, v)  E do 4. Relax(u, v, w) 5. for each vertex v  u.adj do 6. if d[v] > d[u] + w(u, v) 7. then return False// there is a negative cycle 8. return True O(|V|) O(|V||E|) O(|E|) Time complexity: O(|V||E|)