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All-Pairs Shortest Paths (26.0/25)PowerPoint Presentation

All-Pairs Shortest Paths (26.0/25)

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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}

cmj(m-1)

m

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 ’’
- Runtime O(n3 )
- 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

- w’(u,v) 0 since h(v) h(u) + w(u,v) and

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.

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