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CSCE 411 Design and Analysis of Algorithms

CSCE 411 Design and Analysis of Algorithms. Set 5a: Bellman-Ford SSSP Algorithm Prof. Jennifer Welch Spring 2013. Bellman-Ford Idea. Consider each edge (u,v) and see if u offers v a cheaper path from s compare d[v] to d[u] + w(u,v)

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CSCE 411 Design and Analysis of Algorithms

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  1. CSCE 411Design and Analysis of Algorithms Set 5a: Bellman-Ford SSSP Algorithm Prof. Jennifer Welch Spring 2013 CSCE 411, Spring 2013: Set 5a

  2. Bellman-Ford Idea • Consider each edge (u,v) and see if u offers v a cheaper path from s • compare d[v] to d[u] + w(u,v) • Repeat this process |V| - 1 times to ensure that accurate information propgates from s, no matter what order the edges are considered in CSCE 411, Spring 2013: Set 5a

  3. Bellman-Ford SSSP Algorithm • input: directed or undirected graph G = (V,E,w) //initialization • initialize d[v] to infinity and parent[v] to nil for all v in V other than the source • initialize d[s] to 0 and parent[s] to s // main body • for i := 1 to |V| - 1 do • for each (u,v) in E do // consider in arbitrary order • if d[u] + w(u,v) < d[v] then • d[v] := d[u] + w(u,v) • parent[v] := u CSCE 411, Spring 2013: Set 5a

  4. Bellman-Ford SSSP Algorithm // check for negative weight cycles • for each (u,v) in E do • if d[u] + w(u,v) < d[v] then • output "negative weight cycle exists" CSCE 411, Spring 2013: Set 5a

  5. Running Time of Bellman-Ford • O(V) iterations of outer for loop • O(E) iterations of inner for loop • O(VE) time total CSCE 411, Spring 2013: Set 5a

  6. s c a b Bellman-Ford Example process edges in order (c,b) (a,b) (c,a) (s,a) (s,c) 3 2 —4 4 1 <board work> CSCE 411, Spring 2013: Set 5a

  7. Correctness of Bellman-Ford Assume no negative-weight cycles. Lemma:d[v] is never an underestimate of the actual shortest path distance from s to v. Lemma:If there is a shortest s-to-v path containing at most i edges, then after iteration i of the outer for loop, d[v] is at most the actual shortest path distance from s to v. Theorem:Bellman-Ford is correct. Proof:Follows from these 2 lemmas and fact that every shortest path has at most |V| - 1 edges. CSCE 411, Spring 2013: Set 5a

  8. Correctness of Bellman-Ford • Suppose there is a negative weight cycle. • Then the distance will decrease even after iteration |V| - 1 • shortest path distance is negative infinity • This is what the last part of the code checks for. CSCE 411, Spring 2013: Set 5a

  9. Speeding Up Bellman-Ford • The previous example would have converged faster if we had considered the edges in a different order in the for loop • move outward from s • If the graph is a DAG (no cycles), we can fully exploit this idea to speed up the running time CSCE 411, Spring 2013: Set 5a

  10. DAG Shortest Path Algorithm • input: directed graph G = (V,E,w) and source vertex s in V • topologically sort G • d[v] := infinity for all v in V • d[s] := 0 • for each u in V in topological sort order do • for each neighbor v of u do • d[v] := min{d[v],d[u] + w(u,v)} CSCE 411, Spring 2013: Set 5a

  11. DAG Shortest Path Algorithm • Running Time is O(V + E). • Example: <board work> CSCE 411, Spring 2013: Set 5a

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