图的遍历

图的遍历指的是从图中的任一顶点出发，对图中的所有顶点访问一次且只访问一次。
图的遍历分两种，深度优先和广度优先。

深度优先

深度优先搜索法是树的先根遍历的推广，它的基本思想是：从图G的某个顶点v0出发，访问v0，然后选择一个与v0相邻且没被访问过的顶点vi访问，再从vi出发选择一个与vi相邻且未被访问的顶点vj进行访问，依次继续。如果当前被访问过的顶点的所有邻接顶点都已被访问，则退回到已被访问的顶点序列中最后一个拥有未被访问的相邻顶点的顶点w，从w出发按同样的方法向前遍历，直到图中所有顶点都被访问。

Depth-first search (DFS) is an algorithm for traversing or searching tree or graph data structures. The algorithm starts at the root node (selecting some arbitrary node as the root node in the case of a graph) and explores as far as possible along each branch before backtracking.

The time and space analysis of DFS differs according to its application area. In theoretical computer science, DFS is typically used to traverse an entire graph, and takes time {\displaystyle O(|V|+|E|)}O(|V| + |E|),[4] where {\displaystyle |V|}|V| is the number of vertices and {\displaystyle |E|}|E| the number of edges. This is linear in the size of the graph. In these applications it also uses space {\displaystyle O(|V|)}O(|V|) in the worst case to store the stack of vertices on the current search path as well as the set of already-visited vertices. Thus, in this setting, the time and space bounds are the same as for breadth-first search and the choice of which of these two algorithms to use depends less on their complexity and more on the different properties of the vertex orderings the two algorithms produce.

Stack structure is preferred.

a non-recursive implementation with stack:

procedure DFS_iterative(G, v) is
    let S be a stack
    S.push(v)
    while S is not empty do
        v = S.pop()
        if v is not labeled as discovered then
            label v as discovered
            for all edges from v to w in G.adjacentEdges(v) do 
                S.push(w)

广度优先

图的广度优先搜索是树的按层次遍历的推广，它的基本思想是：首先访问初始点vi，并将其标记为已访问过，接着访问vi的所有未被访问过的邻接点vi1,vi2,…, vi t，并均标记已访问过，然后再按照vi1,vi2,…, vi t的次序，访问每一个顶点的所有未被访问过的邻接点，并均标记为已访问过，依次类推，直到图中所有和初始点vi有路径相通的顶点都被访问过为止。

Breadth-first search (BFS) is an algorithm for searching a tree data structure for a node that satisfies a given property. It starts at the tree root and explores all nodes at the present depth prior to moving on to the nodes at the next depth level. Extra memory, usually a queue, is needed to keep track of the child nodes that were encountered but not yet explored.

Queue is preferred.

 1  procedure BFS(G, root) is
 2      let Q be a queue
 3      label root as explored
 4      Q.enqueue(root)
 5      while Q is not empty do
 6          v := Q.dequeue()
 7          if v is the goal then
 8              return v
 9          for all edges from v to w in G.adjacentEdges(v) do
10              if w is not labeled as explored then
11                  label w as explored
12                  Q.enqueue(w)

(完)