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Chapter 26 All-Pairs Shortest Paths

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Title: Chapter 26 All-Pairs Shortest Paths


1
Chapter 26 All-Pairs Shortest Paths
  • Problem definition
  • Shortest paths and matrix multiplication
  • The Floyd-Warshall algorithm

2
Problem definition
  • Real problemWhen to make a table of distances
    between all pairs of cities for a road atlas,how
    can we compute the distances?
  • Model the problem
  • Given a weighted,directed graph G(V,E) with a
    weight function wE R that maps edges to
    real-valued weights.
  • For every pair of vertices u,v V, find a
    shortest(least weight) path from u to v, where
    the weight of a path is the sum of the weights of
    its constituent edges.

3
Solve the problem by Single-Source shortest paths
algorithm
  • If all edge weights are nonnegative, we can use
    Dijkstras algorithm.
  • If negative-weight edges are allowed, then we use
    Bellman-Ford algorithm instead.
  • Can you analysis the complexity of the two
    solutions ? Is it possible for us to find a more
    efficient algorithm?

4
Preliminary knowledge
  • In this chapter, we use an adjacency-matrix
    representation
  • wij
    if ij
  • the weight of directed edge (i,j) if
    i j and (i,j) E

  • if i j and (i,j) E
  • The output of the algorithm presented in this
    chapter is an nn matrix D(dij),where entry dij
    contains the weight of a shortest path from
    vertex i to vertex j. That is , if we let denotes
    the shortest-path weight from vertex i to vertex
    j, then dij at termination.

5
Continue
  • To solve the all-pairs shortest-paths problem on
    an input adjacency matrix, we need to compute not
    only the shortest path weights but also a
    predecessor matrix ( ij), where ij is
    NIL if either ij or there is no path from i to
    j, and otherwise ij is some predecessor of j
    on a shortest path from i.
  • The subgraph induced by the ith row of the
    matrix should be a shortest-paths tree with root
    i. For each vertex i V,we define the predecessor
    subgraph of G for i as G ,i (V ,i,E ,i)
  • V ,ij V i,j NIL i
  • E ,i( ij , j) j V ,i and ij NIL

6
Shortest paths and matrix multiplication
  • This section presents a dynamic-programming
    algorithm for the all-pairs shortest-paths
    problem on a directed graph G(V,E). Each major
    loop of the dynamic program will invoke an
    operation that is very similar to matrix
    multiplication, so that the algorithm will look
    like repeated matrix multiplication. We shall
    start by developing a (V4)-time algorithm for
    the all-pairs shortest-paths problem and the
    improve its running time to (V3lgV).

7
Review the steps to develop a dynamic-programming
algorithm
  • Characterize the structure of an optimal
    solution.
  • Recursively define the value of an optimal
    solution.
  • Computing the value of an optimal solution in a
    bottom-up fashion.
  • Constructing an optimal solution from computed
    information.

8
The structure of a shortest path
  • Consider a shortest path p from vertex i to
    vertex j, and suppose that p contains at most m
    edges.Assuming that there are no negative-weight
    cycles,m is finite.If ij,then p has weight 0 and
    no edges.If vertices i and j are distinct, then
    we decompose path p into i k
    j,where path p now contains at most m-1
    edges.Moreover, p is a shortest path from i to
    k. Thus, we have wkj.

9
A recursive solution to the all-pairs
shortest-paths problem
  • Now,let dij(m) be the minimum weight of any path
    from vertex i to vertex j that contains at most m
    edges.When m0,there is a shortest path from i to
    j with no edges if and only if ij.For mgt1,we
    compute dij(m) as the minimum of dij(m-1) (the
    weight of the shortest path from i to j
    consisting of at most m-1 edges) and the minimum
    weight of any path from i to j consisting of at
    most m edges,obtained by looking at all possible
    predecessors k of j.Thus,we recursively define
  • dij(m)min ( dij(m-1), dik(m-1)wkj)
    dik(m-1)wkj.

10
Continue
  • If the graph contains no negative-weight
    cycles,then all shortest paths are simple and
    thus contain at most n-1 edges.A path from vertex
    i to vertex j with more than n-1 edges cannot
    have less than a shortest path from i to j.The
    actual shortest-path weights are therefore given
    by
  • dij(n-1)dij(n)dij(n1)...

11
Computing the shortest-path weights bottom up
  • We compute the shortest-path weights by extending
    shortest paths edge by edge.Letting AB denote
    the matrix product returned by
    EXTEND-SHORTEST-PATH(A,B) .We compute the
    sequence of n-1 matrices
  • D(1)D(0)WW,
  • D(2)D(1)WW2,
  • D(n-1)D(n-2)WWn-1.
  • As we argued above, the matrix D(n-1)Wn-1
    contains the shortest-path weights.So we get the
    algorithm.

12
Continue
  • SLOW-ALL-PAIRS-SHORTEST-PATH(W)
  • n rowsW
  • D(1) W
  • for m 2 to n-1
  • do D(m) EXTEND-SHORTEST-PATHS(D
    (m-1),W)
  • return D(n-1)

13
Continue
  • EXTEND-SHORTEST-PATH(D,W)
  • n rowsD
  • let D(dij) be a nn matrix
  • for i 1 to n
  • do for j 1 to n
  • do dij
  • for k 1 to n
  • do dij
    min(dij,dikwkj)
  • return D

14
Example
  • Figure 1

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16
Improving the running time
  • Our goal,however, is not to compute all the D(m)
    matrices we are interested in matrix
    D(n-1).Recall that in the absence of
    negative-weight cycles, D(m)D(n-1),for all
    integers mgtn-1.So, we can compute D(n-1) with
    only matrix products by
    computing the sequence
  • D(1)W,
  • D(2)W2WW
  • D(4) W2 W2
  • D(2 )W2 W2
    -1 W2 -1

17
Continue
  • FASTER-ALL-PAIRS-SHORTEST-PATHS(W)
  • n rowsW
  • D(1) W
  • while n-1gtm
  • do D(2m) EXTEND-SHORTEST-PATHS(D(m
    ),D(m))
  • m 2m
  • return D(m)

18
Floyd-Warshall algorithm
  • In this section, we shall use a different
    dynamic-programming formulation to solve the
    all-pairs shortest-paths problem on a directed
    graph G(V,E).The resulting algorithm, known as
    the Floyd-Warshall algorithm, runs in (V3)
    time. As before, negative-weight edges may be
    present, but we shall assume that there are no
    negative-weight cycles.

19
The structure of a shortest path
  • In the Floyd-Warshall algorithm, we use a
    different characterization of the structure of a
    shortest path than we used in the
    matrix-multiplication-based all-pairs
    algorithms.The algorithm considers the
    intermediate vertices of a shortest path, where
    an intermediate vertex of a simple path
    pltv1,v2,,vlgt is any vertex of p other than v1
    or vl, that is, any vertex in the set
    v2,v3,,vl-1

20
Continue
  • Let the vertices of G be V1,2,,n, and
    consider a subset 1,2,,k of vertices for some
    k.For any pair of vertices i,j V,consider
    all paths from i to j whose intermediate vertices
    are all drawn from 1,2,,k,and let p be a
    minimum-weight path from among them.The
    Floyd-Warshall algorithm exploits a relationship
    between path p and shortest paths from i to j
    with all intermediate vertices in the set
    1,2,,k-1.

21
Continue
  • The relationship depends on whether or not k is
    an intermediate vertex of path p.
  • If k is not an intermediate vertex of path p,
    then all intermediate vertices of path p are in
    the set 1,2,,k-1. Thus, a shortest path from
    vertex i to vertex j with all intermediate
    vertices in the set 1,2,,k-1 also a shortest
    path from i to j with all intermediate vertices
    in the set 1,2,,k.
  • If k is an intermediate vertex of path p,then we
    break p down into i k
    j as shown Figure 2.p1 is a shortest path from i
    to k with all intermediate vertices in the set
    1,2,,k-1, so as p2.

22
All intermediate vertices in 1,2,,k-1
p2
k
p1
j
i
Pall intermediate vertices in 1,2,,k
Figure2 Path p is a shortest path from vertex i
to vertex j,and k is the highest-numbered
intermediate vertex of p. Path p1, the portion
of path p from vertex i to vertex k,has all
intermediate vertices in the set 1,2,,k-1.The
same holds for path p2 from vertex k to vertex j.
23
A recursive solution to the all-pairs shortest
paths problem
  • Let dij(k) be the weight of a shortest path from
    vertex i to vertex j with all intermediate
    vertices in the set 1,2,,k. A recursive
    definition is given by
  • dij(k) wij
    if k0,
  • min(dij(k-1),dik(k-1)dkj(k-1))
    if k 1.
  • The matrix D(n)(dij(n)) gives the final
    answer-dij(n) for all i,j
    V-because all intermediate vertices are in the
    set 1,2,,n.

24
Computing the shortest-path weights bottom up
  • FLOYD-WARSHALL(W)
  • n rowsW
  • D(0) W
  • for k 1 to n
  • do for i 1 to n
  • do for j 1 to n
  • dij(k)
    min(dij(k-1),dik(k-1)dkj(k-1))
  • return D(n)

25
Example
  • Figure 3

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D(2)
(2)
(3)
D(3)
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D(4)
(4)
(5)
D(5)
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The End
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