最短路径算法

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最新推荐文章于 2024-11-16 13:30:41 发布

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原文链接：http://www.cnblogs.com/ITfresh/archive/2012/02/17/2355726.html

松弛技术：

RELAX(u,v,w)
 if(d[v]>d[u]+w(u,v))
{
   d[v]=d[u]+w(u,v);
   p[v]=u;
}

松弛是改变最短路径和前趋的唯一方式。算法之间的区别在于对每条边进行松弛的次数，以及对边执行松弛操作的次序有所不同。

Bellman–Ford is in its basic structure very similar to Dijkstra's algorithm, but instead of greedily selecting the minimum-weight node not yet processed to relax, it simply relaxes all the edges, and does this |V | − 1 times, where |V | is the number of vertices in the graph. The repetitions allow minimum distances to accurately propagate throughout the graph, since, in the absence of negative cycles, the shortest path can only visit each node at most once. Unlike the greedy approach, which depends on certain structural assumptions derived from positive weights, this straightforward approach extends to the general case.

Bellman–Ford runs in O(|V|·|E|) time, where |V| and |E| are the number of vertices and edges respectively.

BELLMAN-FORD(G, w, s)
1  INITIALIZE-SINGLE-SOURCE(G, s)
2  for i ← 1 to |V[G]| - 1
3       do for each edge (u, v) ∈ E[G]
4              do RELAX(u, v, w)
5  for each edge (u, v) ∈ E[G]
6       do if d[v] > d[u] + w(u, v)
7             then return FALSE
8  return TRUE

有向无回路图中的单源最短路径DAG-SHORTEST-PATHS

DAG-SHORTEST-PATHS(G, w, s)
1  topologically sort the vertices of G
2  INITIALIZE-SINGLE-SOURCE(G, s)
3  for each vertex u, taken in topologically sorted order
4       do for each vertex v ∈ Adj[u]
5              do RELAX(u, v, w)

DIJKSTRA(G, w, s)
1  INITIALIZE-SINGLE-SOURCE(G, s)
2  S ← Ø
3  Q ← V[G]
4  while Q ≠ Ø
5      do u ← EXTRACT-MIN(Q)
6         S ← S ∪{u}
7         for each vertex v ∈ Adj[u]
8             do RELAX(u, v, w)