【算法导论笔记】所有结点对的最短路径问题

基于矩阵乘法的动态规划算法求解所有最短路径

EXTEND_SHORTEST_PATHS(L, W)
  n = L.rows
  let L' = l'(i,j) be a new n*n matrix
  for i=1 to n
      for j=1 to n
          l'(i,j) = ∞
  return L'

SLOW-ALL-PATHS-SHORTEST-PATHS(W)
  n = W.rows
  L[1] = W
  for m=2 to n-1
      let L[m] be a new n*n matrix
      L[m] = EXTEND_SHORTEST_PATHS(L(m-1), W)
  return L[n-1]

使用重复平方技术来计算上述矩阵序列

FASTER-ALL-PAIRS-SHORTEST-PATHS(W)
  n = W.rows
  L[1] = W
  m = 1
  while m

Floyd-Warshall算法（佛洛依德算法）

FLOYD-WARSHALL(W)
  n = W.rows
  D[0] = W
  for k=1 to n
      let D[k] = d[k](i,j) be a new n*n matrix
   for i=1 to n
      for j=1 to n
          d[k](i,j) = min(d[k-1](i,j), d[k-1](i,k)+d[k-1](k,j))
  return D[n]

有向图的传递闭包算法

TRANSITIVE-CLOSURE(G)
  n = G.V
  let T[0] = {t[0](i,j)} be a new n*n matrix
  for i=1 to n
      for j=1 to n
          if i==j or (i,j)∈G.E
              t[0](i,j) = 1
          else  t[0](i,j) = 0
  for k=1 to n
      let T[k] = {t[k](i,j)} be a new n*n matrix
      for i=1 to n
          for j=1 to n
              t[k](i,j) = t[k-1](i,j) ∨ (t[k-1](i,k)∧t[k-1](k,j))
  return T[n]

用于稀疏图的Johnson算法

JOHNSON(G, w)
  compute G', where G'.V = G.V∪{s},
      G'.E = G.V∪{(s,v):v∈G.V}, and
      w(s,v)=0 for all v∈G.V
  if BELLMAN-FORD(G', w, s) == FALSE
      print"the input graph contain a negative-weigh cycle"
  else  for each vertex v∈G'.V
          set h(v) to the value if δ(s,v)
              computed by the Bellman-Ford algorithm
        for each edge(u,v)∈G'.E
            w'(u,v) = w(u,v) + h(u) + h(v)
        let D={d(u,v)} be a new n*n matrix
        for each vertex u∈G'.V
            run DIJKSTRA(G,w',u) to compute δ'(u,v) for all δ∈G.V
            for each vertex v∈G.V
                d(u,v) = δ'(u,v) + h(v) - h(u)
  return D