2011-02-25 CLRS Chapter22 Elementary Graph Algorithms 图的几个基础算法_算法导论 elementary graph algorithms-优快云博客

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本文介绍了图的两种主要表示方法：邻接表和邻接矩阵，并探讨了它们的应用场景。此外，还详细阐述了广度优先搜索（BFS）、深度优先搜索（DFS）、拓扑排序及强连通分量等关键图算法的工作原理和实现步骤。

Representations of graphs:

There are two standard ways to represent a graph G = (V, E): as a collection of adjacency lists(O(V+E)) or as an adjacency matrix(O(V^2)).

The adjacency-list representation is usually preferred, because it provides a compact way to represent sparse graphs-those for which |E| is much less than |V|2.

An adjacency-matrix representation may be preferred, however, when the graph is dense-|E| is close to |V|2-or when we need to be able to tell quickly if there is an edge connecting two given vertices.

the simplicity of an adjacency matrix may make it preferable when graphs are reasonably small. Moreover, if the graph is unweighted, there is an additional advantage in storage for the adjacency-matrix representation. Rather than using one word of computer memory for each matrix entry, the adjacency matrix uses only one bit per entry.

Breadth-first search:

BFS(G, s)

1 for each vertex u ∈ V [G] - {s}

2 do color[u] ← WHITE

3 d[u] ← ∞

4 π[u] ← NIL

5 color[s] ← GRAY

6 d[s] ← 0

7 π[s] ← NIL

8 Q ← Ø

9 ENQUEUE(Q, s)

10 while Q ≠ Ø

11 do u ← DEQUEUE(Q)

12 for each v ∈ Adj[u]

13 do if color[v] = WHITE

14 then color[v] ← GRAY

15 d[v] ← d[u] + 1

16 π[v] ← u

17 ENQUEUE(Q, v)

18 color[u] ← BLACK

Required running time: O(V+E).

Printing shortest path:

PRINT-PATH(G, s, v)

1 if v = s

2 then print s

3 else if π[v] = NIL

4 then print "no path from" s "to" v "exists"

5 else PRINT-PATH(G, s, π[v])

6 print v

This procedure runs in time linear

Depth-first search：

DFS(G)
1  for each vertex u ∈ V [G]
2       do color[u] ← WHITE
3          π[u] ← NIL
4  time ← 0
5  for each vertex u ∈ V [G]
6       do if color[u] = WHITE
7             then DFS-VISIT(u)

DFS-VISIT(u)
1  color[u] ← GRAY     ▹White vertex u has just been discovered.
2  time ← time +1
3  d[u] time
4  for each v ∈ Adj[u]  ▹Explore edge(u, v).
5       do if color[v] = WHITE
6             then π[v] ← u
7                         DFS-VISIT(v)
8  color[u] BLACK      ▹ Blacken u; it is finished.
9  f [u] ▹ time ← time +1

Topological sort：

TOPOLOGICAL-SORT(G)
1  call DFS(G) to compute finishing times f[v] for each vertex v
2  as each vertex is finished, insert it onto the front of a linked list
3  return the linked list of vertices
Strongly connected components：
STRONGLY-CONNECTED-COMPONENTS (G)
1  call DFS (G) to compute finishing times f[u] for each vertex u
2  compute G^T
3  call DFS (G^T), but in the main loop of DFS, consider the vertices
          in order of decreasing f[u] (as computed in line 1)
4  output the vertices of each tree in the depth-first forest formed in line 3 as a
          separate strongly connected component