The Bellman-Ford Shortest Path Algorithm Neil Tang 03/27/2008

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Presentation transcript:

The Bellman-Ford Shortest Path Algorithm Neil Tang 03/27/2008 CS223 Advanced Data Structures and Algorithms

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

CS223 Advanced Data Structures and Algorithms 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. CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Differences Negative link cost: 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. CS223 Advanced Data Structures and Algorithms

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

The Bellman-Ford Algorithm ,nil 6 7 9 5 -3 8 -4 2 s y z x t -2 6,s 7,s ,nil 6 7 9 5 -3 8 -4 2 s y z x t -2 Initialization After pass 1 6,s 7,s 4,y 6 7 9 5 -3 8 -4 2 2,t s y z x t -2 2,x 7,s 4,y 6 7 9 5 -3 8 -4 2 2,t s y z x t -2 After pass 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) CS223 Advanced Data Structures and Algorithms

The Bellman-Ford Algorithm 2,x 7,s 4,y 6 7 9 5 -3 8 -4 2 -2,t s y z x t -2 After pass 4 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) CS223 Advanced Data Structures and Algorithms

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

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