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Single-Source Shortest Paths (25/24)

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Single-Source Shortest Paths (25/24) HW: 25-2 and 25-3 p. 546/24-2 and 24-3 p.615 Given a graph G=(V,E) and w: E weight of is w(p) = w(v[i],v[i+1]) – PowerPoint PPT presentation

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Title: Single-Source Shortest Paths (25/24)


1
Single-Source Shortest Paths (25/24)
  • HW 25-2 and 25-3 p. 546/24-2 and 24-3 p.615
  • Given a graph G(V,E) and w E ? ?
  • weight of ltv1,...,vkgt is w(p) ?
    w(vi,vi1)
  • Single-source shortest-paths
  • find a shortest path from a source s to every
    vertex v ?V
  • Single pair shortest path problem asks for a u-v
    shortest path for some pair of vertices
  • All pairs shortest-paths problem will be next time

2
Shortest Paths (25/24)
  • Predecessor subgraph for restoring shortest paths
  • Shortest-paths tree rooted at source s
  • Th Subpath of a shortest path is a shortest path
  • Triangle Inequality
  • ?(u,v) ? ?(u,x) ?(x,v)
  • Well defined some paths may not exist if there
    is a negative-weight cycle in graph

u
v
x
lt0
3
Bellman-Ford (25.3/24.1)
  • Most basic (BFS based) algorithm, shortest paths
    (tree) easy to reconstruct.
  • for each v ?V do dv ? ? ds ? 0
  • Relaxation
  • for i 1,...,V-1 do
  • for each edge (u,v) ? E do
  • dv ? mindv, duw(u,v)
  • Negative cycle checking
  • for each v ?V do if dvgt du w(u,v) then no
    solution
  • Finally dv ?(s,v)

4
Bellman-Ford Analysis (25.3/24.1)
  • Runtime O(VE)
  • Correctness
  • Lemma dv ? ?(s,v)
  • Initially true
  • Let dv du w(u,v)
  • by triangle inequality for first violation
    dv lt ?(s,v) ? ?(s,u)w(u,v) ?
    d(u)w(u,v)
  • After V-1 passes all d values are ?s if there
    are no negative cycles
  • s ? v1 ? v2 ? ... ? v (some shortest path)
  • After i-th iteration ds,vi is correct and
    final

5
Dag Shortest Paths (25.4/24.2)
  • DAG shortest paths
  • Bellman-Ford O(VE)
  • Topological sort O(VE) (DFS)
  • Will never relaxed edges out of vertex v until
    have done all edges in v
  • Runtime O(VE)
  • Application PERT (program evaluation and review
    technique)
  • Critical path is the longest path through the dag
  • Negating the edge weights and running dag
    shortest paths algorithm

6
Dijkstras Shortest Paths (25.2/24.3)
  • Better than BF since non-negative weights.
  • Like BFS but uses priority queue.
  • for each v ?V do dv ? ? ds ? 0
  • S ? ? Q ? V
  • While Q ? ? do
  • u ? Extract-Min(Q)
  • S ? S u
  • for v adjacent to u do
  • dv ? mindv, duw(u,v)
  • (relaxation Decrease-Key)

7
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For each v ? V do dv lt- ? ds lt- 0S lt-
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While Q ? ? do U Extract-Min(Q) S
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While Q ? ? do U Extract-Min(Q) S
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While Q ? ? do U Extract-Min(Q) S
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While Q ? ? do U Extract-Min(Q) S
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While Q ? ? do U Extract-Min(Q) S
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mindv, duw(u,v)
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While Q ? ? do U Extract-Min(Q) S
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mindv, duw(u,v)
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While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
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While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
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While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
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While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
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While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
29
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19
Dijkstras Runtime (25.2/24.3)
  • Extract-Min executed V times
  • Decrease-Key executed E times
  • Time V?T(Extract-Min) (finddelete O(log
    V))
  • E ?T(Decrease-Key) (deleteadd
    O(log V))
  • Binary Heap E ? log V (30
    years ago)
  • Fibonacci Heap E V ? log V (10 years
    ago)
  • Optimal time algorithm found 1/2 year ago. It
    runs in time O(E) (Mikel Thorup)

20
Dijkstras Correctness (25.2/24.3)
  • The same Lemma as for BF dv ? ?(s,v)
  • Th Whenever u is added to S, du ?(s,u)
  • Let u be first s.t. du gt ?(s,u)
  • Let y be first ?V-S on actual
  • shortest s-u path ? dy ?(s,y)
  • For ys predecessor x, dx ?(s,x)
  • when put x in S dy gets ?(s,y)
  • du gt ?(s,u) ?(s,y) ?(y,u) dy ?(y,u) ?
    dy

S
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