All-Pairs Shortest Paths (26.0/25)

1
All-Pairs Shortest Paths (26.0/25)

• HW problem 26-1, p. 576/25-1, p. 641
• Directed graph G (V,E), weight E ? ?
• Goal Create n ? n matrix of s-p distances ?(u,v)
• Running Bellman-Ford once from each vertex
• O( ) O( ) on dense
  graphs
• Adjacency-matrix representation of graph
• n ? n matrix W (wij) of edge weights
• assume wii 0 ?i,
• s-p to self has no edges in absence of negative
  cycles

2
Dynamic Programming (26.1/25.1)

• dij(m) weight of s-p from i to j with ? m edges
• dij(0) 0 if i j and dij(0) ? if i ?
  j
• dij(m) minkdik(m-1) wkj
• Runtime O( n4)
• because n-1 passes
• each computing n2 ds in O(n) time

? m-1
j
i
? m-1
3
Matrix Multiplication (26.1/25.1)

• Similar C A ? B, two n ? n matrices
• cij ?k aik ? bkj O(n3) operations
• replacing ? min
• ? ?
  
• gives cij mink aik bkj
• D(m) D(m-1) ? W
• identity matrix is D(0)
• Cannot use Strassens because no subtraction
• Time is still O(n ? n3 ) O(n4 )
• Repeated squaring W2n Wn ? Wn
• Compute W, W2 , W4 ,..., W2k , k log n,
  O(n3 log n)

4
Floyd-Warshall Algorithm (26.2/25.2)

• Also dynamic programming but faster (by log n)
• cij(m) weight of s-p from i to j with
  intermediate vertices in the set 1, 2, ..., m
  ? ?(i, j) cij(n)
• DP compute cij(n) in terms of smaller cij(n-1)
• cij(0) wij
• cij(m) min cij(m) , cim(m-1) cmj(m-1)
• intermediate nodes in 1,
  2, ..., m

cmj(m-1)
m
cim(m-1)
j
i
cij(m-1)
5
Floyd-Warshall Algorithm (26.2/25.2)

• Difference from previous we do not check all
  possible intermediate vertices.
• Code for m1..n do for i1..n do for j 1..n do
• cij(m) min cij(m-1) , cim(m-1)
  cmj(m-1)
• Runtime O(n3 )
• Transitive Closure G of graph G
• (i,j) ? G iff ? path from i to j in G
• Adjacency matrix, elements on 0,1
• Floyd-Warshall with min ? OR ,
  ? AND
• Runtime O(n3 )
• Useful in many problems

6
Johnsons Algorithm (26.3/25.3)

• Johnsons Bellman-Ford Dijkstras (or
  Thorups)
• Re-weighting of the graph G(V,E,w)
• assign weights to all vertices h V ? ?
• change weight of edges w(u,v) w(u,v) h(u) -
  h(v)
• then for any u,v ?(u, v) ?(u,v) h(u) -
  h(v)
• Addition of auxiliary vertex s
• V ? V s
• E ? V (s,v), v ? V , w(s,v) 0
• Run Bellman-Ford find h(v) ?(s,v) (or negat.
  cycle)
• w(u,v) ? 0 since h(v) ? h(u) w(u,v) and
• w(u,v) w(u,v) h(u) - h(v) ? 0

7
Johnsons Algorithm (26.3/25.3)

• Since all weights are nonnegative,
• apply Dijkstras for each source and
• find all modified distances ?(u, v)
• Restore actual distances from modified using
• ?(u, v) ?(u,v) - h(u) h(v)
• Runtime
• O(VE) V times O(E) - M.Thorups algorithm
• O(V(EV log V)) V times Dijkstras with
  Fibonacci heaps
• O(VE log V) V times Dijkstras with binary
  heaps (faster for sparse graphs)
• To find s-p for a single pair s-p from single
  source all pairs shortest paths.