AllPairs Shortest Paths

1
All-Pairs Shortest Paths

• ITSD4312

2
Outline

• Review of graph theory
• Problem definition
• Properties of interest
• Recurrence
• Example
• Recent Work
• References

3
Graph Terminology

• G (V, E)
• W weight matrix
• wij weight/length of edge (vi, vj)
• wij 8 if vi and vj are not connected by an edge
• wii 0
• Assume W has positive, 0, and negative values
• For this problem, we cannot have a negative-sum
  cycle in G

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Weighted Graph and Weight Matrix
v1
v0
5
-4
v2
3
1
2
7
9
6
v3
v4
5
Directed Weighted Graph and Weight Matrix
v3
v0
-2
1
7
v2
v1
-1
2
5
9
6
3
4
0 1 2 3 4 5
v4
v5
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All-Pairs Shortest Paths Problem Defined

• For every pair of vertices vi and vj in V, it is
  required to find the length of the shortest path
  from vi to vj along edges in E.
• Specifically, a matrix D is to be constructed
  such that dij is the length of the shortest path
  from vi to vj in G, for all i and j.
• Length of a path (or cycle) is the sum of the
  lengths (weights) of the edges forming it.

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Sample Shortest Path
v3
v0
-2
1
7
v2
v1
2
-1
5
9
6
3
4
v4
v5
Shortest path from v0 to v4 is along edges (v0,
v1), (v1, v2), (v2, v4) and has length 6
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Disallowing Negative-length Cycles

• APSP does not allow for input to contain
  negative-length cycles
• This is necessary because
• If such a cycle were to exist within a path from
  vi to vj, then one could traverse this cycle
  indefinitely, producing paths of ever shorter
  lengths from vi to vj.
• If a negative-length cycle exists, then all paths
  which contain this cycle would have a length of
  -8.

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Properties of Interest

• Let denote the length of the shortest path
  from vi to vj that goes through at most k - 1
  intermediate vertices (k hops)
• wij (edge length from vi to vj)
• If i ? j and there is no edge from vi to vj, then
• Also,
• Given that there are no negative weighted cycles
  in G, there is no advantage in visiting any
  vertex more than once in the shortest path from
  vi to vj.
• Since there are only n vertices in G,

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Guaranteeing Shortest Paths

• If the shortest path from vi to vj contains vr
  and vs (where vr precedes vs)
• The path from vr to vs must be minimal (or it
  wouldnt exist in the shortest path)
• Thus, to obtain the shortest path from vi to vj,
  we can compute all combinations of optimal
  sub-paths (whose concatenation is a path from vi
  to vj), and then select the shortest one

vi
vs
vj
vr
MIN
MIN
MIN
? MINs
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Growing Path Length

• Iteratively build longer paths (similar to
  Dijkstras algorithm for single-source shortest
  path)
• From 0 to v-1 lengths
• Runs in O(v4)

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Recurrence Definition

• For k gt 1,
• Guarantees O(log k) steps to calculate
• Improves to O(v3 lg v)

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Floyd Worshall

• Let Dk matrix with entries dij for paths among
  vertices v1, v2, , vk.
• D0 W
• Since we arent allowing any intermediate
  vertices (k0), dij ? i and j wij

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Example
0 1 2 3 4
D0 Wjk
v1
3
4
8
v2
7
v0
1
-5
-4
2
6
v3
v4
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Recent Work on Sequential Algorithms

• Floyd-Warshall algorithm is T(V3)
• Appropriate for dense graphs E O(V2)
• Johnsons algorithm
• Appropriate for sparse graphs E O(V)
• O(V2 log V V E) if using a Fibonacci heap
• O(V E log V) if using binary min-heap
• Shoshan and Zwick (1999)
• Integer edge weights in 1, 2, , W
• O(W V? p(V W)) where ? 2.376 and p is a polylog
  function
• Pettie (2002)
• Allows real-weighted edges
• O(V2 log log V V E)

Strassens Algorithm (matrix multiplication)
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References

• Akl S. G. Parallel Computation Models and
  Methods. Prentice Hall, Upper Saddle River NJ,
  pp. 381-384,1997.
• Cormen T. H., Leiserson C. E., Rivest R. L., and
  Stein C. Introduction to Algorithms (2nd
  Edition). The MIT Press, Cambridge MA, pp.
  620-642, 2001.