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

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

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)

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)

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

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

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.