广度优先搜索

宽度优先遍历，又称广度优先搜索算法。

首先，让我们回忆一下什么是“深度”：

更学术点的说法，可以看做“单位距离下，离起始状态的长度”

那么广度是什么呢？ 
个人觉得，可以这么归纳：

何为广度？ 可以看做“距离初始状态距离相等的结点”的集合

那么BFS的核心思想就是：从初始结点开始，搜索生成第一层结点，检查目标结点是否在这些结点中，若没有，再将所有第一层的结点逐一进行搜索，得到第二层结点，并逐一检查第二层结点中是否包含目标结点。若没有，再继续搜索第二层所有结点……，如此依次扩展，直到发现目标结点为止。

这样就保证了：如果我现在找到了目标结点（也可以称作当前问题的解），那么我之前肯定没有发现过目标结点，而且由于是广度优先搜索，所以当前的解一定是距离起始结点最近的，也就是最优解。由于第一个解就是最优解，那么我们就可以尝试打印出道路。

伪代码如下：

<code class="hljs cpp has-numbering" style="display: block; padding: 0px; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">queue</span> 结点队列<span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">queue</span>.push(起始节点)<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">while</span>(结点队列不为空){    <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">queue</span>.front();    <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">queue</span>.pop();<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//取出头结点</span>    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(头结点是目标结点)    {        跳出循环；    }    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(对下一层结点进行判断)     {        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(结点满足筛选条件)        {            <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">queue</span>.push(满足帅选条件的结点)        }    }}</code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li><li style="box-sizing: border-box; padding: 0px 5px;">19</li></ul>

那么对于上一篇文章中果园的地图，我们的部分状态是这样的： 
S 0 1 1 0 
0 1 0 0 0 
0 0 1 1 0 
0 1 0 0 0 
0 0 0 0 G

从S点开始，设S点坐标为（1,1） 
(1,1) 
->(1,2)//第一层 
->(2,1)//第一层 
->(3,1)//第二层 
->(3,2)//第三层 
->(4,1)//第三层 
…… 
->(5,5)//找到目标结点，结束

BFS因为维护了一个队列，所以节省了递归需要消耗的时间，但是空间上却比递归消耗的多的多，如果需要找多个解或者每个节点的状态比较多，可能会面临空间超限的情况。

所以：

1. 求多个解，首先考虑DFS;
2. 求单个解，首先考虑BFS

---

代表题目：HDU 1072 Nightmare

题目大意：在迷宫中有一个炸弹，过六个单位时间就会爆炸，要你求一个起点到迷宫的终点的最短距离，迷宫中有时间重置器，当你走到这个格子，炸弹的爆炸时间重新置为0，迷宫中标识为墙壁的格子不能走，到达任意一个格子时，炸弹计数器为0时，则失败。

解题思路:最短距离，就想到用宽度优先遍历，但是要一个辅助的数组来保存每个格子的时间信息。因为每个格子前进一步的花费时间单位是一样的，所以宽度遍历的结果自然是最短距离，由于时间单位有限制，只能走六个单元，只有在这六个时间单位里到达了终点，或者走到了时间设置开关把炸弹计时器置为0才能继续走下去，所以要一个辅助数组来保存每个格子剩余的最大时间单位，只有走到当前的格子的剩余时间单位比之前的大时，这个状态才能加入队列中。

下面是代码：

<code class="hljs cpp has-numbering" style="display: block; padding: 0px; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include <iostream></span><span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include <cstdio></span><span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include <cstring></span><span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include <cstdlib></span><span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include <queue></span><span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include <algorithm></span><span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">using</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">namespace</span> <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">std</span>;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">struct</span> node{    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> x, y;    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> step;    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> t;};<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">const</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> maxn = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">9</span>;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> dir[<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">4</span>][<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>] = {{<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>, <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>}, {-<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>, <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>}, {<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>, <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>}, {<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>, -<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>}};<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> maze[maxn][maxn], graph[maxn][maxn];<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> n, m, ex, ey, ans;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">bool</span> bfs(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> x, <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> y);<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> main(){    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> test;    <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">scanf</span>(<span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"%d"</span>, &test);    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">while</span>(test-- != <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)    {        <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">scanf</span>(<span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"%d %d"</span>, &n, &m);        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> sx, sy;        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>; i < n; i++)        {            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> j = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>; j < m; j++)            {                <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">scanf</span>(<span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"%d"</span>, &maze[i][j]);                <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(maze[i][j] == <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>)                    sx = i, sy = j;                <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(maze[i][j] == <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">3</span>)                    ex = i, ey = j;                graph[i][j] = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;            }        }        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(bfs(sx, sy))            <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">printf</span>(<span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"%d\n"</span>, ans);        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">else</span>            <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">printf</span>(<span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"-1\n"</span>);    }    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">return</span> <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;}<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">bool</span> bfs(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> x, <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> y){    <span class="hljs-stl_container" style="box-sizing: border-box;"><span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">queue</span><node></span> que;    node s;    s.x = x;    s.y = y;    s.step = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;    s.t = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">6</span>;    graph[x][y] = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">6</span>;    que.push(s);    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">while</span>(!que.empty())    {        node st = que.front();        que.pop();        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(st.x == ex && st.y == ey)        {            ans = st.step;            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">return</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">true</span>;        }        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(st.t == <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>)            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">continue</span>;        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>; i < <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">4</span>; i++)        {            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> dx = st.x + dir[i][<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>];            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> dy = st.y + dir[i][<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>];            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(dx >= <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span> && dx < n && dy >= <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span> && dy < m && maze[dx][dy] != <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)            {                node tmp;                tmp.x = dx; tmp.y = dy;                tmp.step = st.step + <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>;                tmp.t  = st.t - <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>;                <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(maze[dx][dy] == <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">4</span>)                    tmp.t = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">6</span>;                <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(tmp.t > graph[dx][dy])                {                    graph[dx][dy] = tmp.t;                    que.push(tmp);                }            }        }    }    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">return</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">false</span>;}</code>