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) = mink{dik(m-1) + wkj}
Runtime = O( n4)
because n-1 passes
each computing n2 d’s 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 Strassen’s 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} m
cmj(m-1)
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 m=1..n do for i=1..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 ’’
Useful in many problems

6. Johnson’s Algorithm (26.3/25.3)
Johnson’s = Bellman-Ford + Dijkstra’s (or Thorup’s)
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. Johnson’s Algorithm (26.3/25.3)
Since all weights are nonnegative,
apply Dijkstra’s 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.Thorup’s algorithm
O(V(E+V log V)) = V times Dijkstra’s with Fibonacci heaps
O(VE log V) = V times Dijkstra’s with binary heaps (faster for sparse graphs)
To find s-p for a single pair = s-p from single source = all pairs shortest paths.