动态规划经典题目——滑雪问题(递归+记忆化搜索)

25-03-07 18:21

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一、题目

Michael喜欢滑雪百这并不奇怪，因为滑雪的确很刺激。可是为了获得速度，滑的区域必须向下倾斜，而且当你滑到坡底，你不得不再次走上坡或者等待升降机来载你。Michael想知道载一个区域中最长底滑坡。区域由一个二维数组给出。数组的每个数字代表点的高度。下面是一个例子

1 2 3 4 5
16 17 18 19 6
15 24 25 20 7
14 23 22 21 8
13 12 11 10 9
一个人可以从某个点滑向上下左右相邻四个点之一，当且仅当高度减小。在上面的例子中，一条可滑行的滑坡为24-17-16-1。当然25-24-23-…-3-2-1更长。事实上，这是最长的一条。

二、解题思路

动态规划解题思路可详见另一篇文章。

设二维数组dp[i][j]表示从（i，j）滑到低坡的最大长度。
解题技巧：在矩阵周围加入数值为正无穷大数值(INT_MAX)；
If exist a[i][j] > a[i-1][j] or a[i][j] > a[i][j-1] or a[i][j] > a[i+1][j] or a[i][j] > a[i][j+1] dp[i][j] = max( dp[i-1][j] , dp[i][j-1] , dp[i+1][j] , dp[i][j+1]) + 1; else dp[i][j] = 1 ;

三、代码编写

 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="1">

class="hljs-ln-code"> class="hljs-ln-line">#include

class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="2">

 class="hljs-ln-code"> class="hljs-ln-line">#include  class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="3"> class="hljs-ln-code"> class="hljs-ln-line">#include 
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="4"> class="hljs-ln-code"> class="hljs-ln-line"> 
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="5"> class="hljs-ln-code"> class="hljs-ln-line">#define N 7
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="6"> class="hljs-ln-code"> class="hljs-ln-line">#define max(a,b) ((a>b)?a:b)
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="7"> class="hljs-ln-code"> class="hljs-ln-line"> 
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="8"> class="hljs-ln-code"> class="hljs-ln-line">//定义四个方向，上、左、下、右
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="9"> class="hljs-ln-code"> class="hljs-ln-line">int dir[4][2] ={{-1,0},{0,-1},{1,0},{0,1}};
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="10"> class="hljs-ln-code"> class="hljs-ln-line">//保存最大长度
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="11"> class="hljs-ln-code"> class="hljs-ln-line">int maxRuslt = 0;
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="12"> class="hljs-ln-code"> class="hljs-ln-line">//定义矩阵
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="13"> class="hljs-ln-code"> class="hljs-ln-line">int a[N][N]={{INT_MAX,INT_MAX,INT_MAX,INT_MAX,INT_MAX,INT_MAX,INT_MAX},
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="14"> class="hljs-ln-code"> class="hljs-ln-line">                 {INT_MAX,1,2,3,4,5,INT_MAX},
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="15"> class="hljs-ln-code"> class="hljs-ln-line">                 {INT_MAX,16,17,18,19,6,INT_MAX},
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="16"> class="hljs-ln-code"> class="hljs-ln-line">                 {INT_MAX,15,24,25,20,7,INT_MAX},
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="17"> class="hljs-ln-code"> class="hljs-ln-line">                 {INT_MAX,14,23,22,21,8,INT_MAX},
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="18"> class="hljs-ln-code"> class="hljs-ln-line">                 {INT_MAX,13,12,11,10,9,INT_MAX},
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="19"> class="hljs-ln-code"> class="hljs-ln-line">                 {INT_MAX,INT_MAX,INT_MAX,INT_MAX,INT_MAX,INT_MAX,INT_MAX}};
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="20"> class="hljs-ln-code"> class="hljs-ln-line">//定义dp
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="21"> class="hljs-ln-code"> class="hljs-ln-line">int dp[N][N] = {0};
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="22"> class="hljs-ln-code"> class="hljs-ln-line"> 
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="23"> class="hljs-ln-code"> class="hljs-ln-line">int dfs(int i,int j){
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="24"> class="hljs-ln-code"> class="hljs-ln-line"> 
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="25"> class="hljs-ln-code"> class="hljs-ln-line">    //递归出口
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="26"> class="hljs-ln-code"> class="hljs-ln-line">    if((a[i][j] < a[i-1][j]) && (a[i][j] < a[i][j-1]) && (a[i][j] < a[i+1][j]) && (a[i][j] < a[i][j+1])){
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="27"> class="hljs-ln-code"> class="hljs-ln-line">        return dp[i][j] = 1;
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="28"> class="hljs-ln-code"> class="hljs-ln-line">    }
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="29"> class="hljs-ln-code"> class="hljs-ln-line">    //已经算出来则不需再计算(备忘录)
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="30"> class="hljs-ln-code"> class="hljs-ln-line">    else if(dp[i][j] > 0){
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="31"> class="hljs-ln-code"> class="hljs-ln-line">        return dp[i][j];
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="32"> class="hljs-ln-code"> class="hljs-ln-line">    }
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="33"> class="hljs-ln-code"> class="hljs-ln-line">    else{
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="34"> class="hljs-ln-code"> class="hljs-ln-line">        int k ;
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="35"> class="hljs-ln-code"> class="hljs-ln-line">        int count = 0;
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="36"> class="hljs-ln-code"> class="hljs-ln-line">        int temp = 0;
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="37"> class="hljs-ln-code"> class="hljs-ln-line">        for(k=0 ; k < 4 ;k++){
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="38"> class="hljs-ln-code"> class="hljs-ln-line">            if(a[i][j] > (a[i + dir[k][0]][j + dir[k][1]])){
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="39"> class="hljs-ln-code"> class="hljs-ln-line">                //继续递归
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="40"> class="hljs-ln-code"> class="hljs-ln-line">                temp = dfs(i + dir[k][0],j + dir[k][1]) + 1;
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="41"> class="hljs-ln-code"> class="hljs-ln-line">                count = max(count,temp);
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="42"> class="hljs-ln-code"> class="hljs-ln-line">                dp[i][j] = count;
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="43"> class="hljs-ln-code"> class="hljs-ln-line">                //最大值保存在全局变量
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="44"> class="hljs-ln-code"> class="hljs-ln-line">                maxRuslt = max(maxRuslt,count);
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="45"> class="hljs-ln-code"> class="hljs-ln-line">                //printf("temp = %d;count = %d;maxResult = %d;\n",temp,count,maxRuslt);
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="46"> class="hljs-ln-code"> class="hljs-ln-line">            }
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="47"> class="hljs-ln-code"> class="hljs-ln-line">        }
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="48"> class="hljs-ln-code"> class="hljs-ln-line">        return count;
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="49"> class="hljs-ln-code"> class="hljs-ln-line">    }
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="50"> class="hljs-ln-code"> class="hljs-ln-line">}
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="51"> class="hljs-ln-code"> class="hljs-ln-line"> 
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="52"> class="hljs-ln-code"> class="hljs-ln-line">int main()
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="53"> class="hljs-ln-code"> class="hljs-ln-line">{
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="54"> class="hljs-ln-code"> class="hljs-ln-line">    int i,j;
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="55"> class="hljs-ln-code"> class="hljs-ln-line">    for(i =1 ; i < N-1 ; i++){
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="56"> class="hljs-ln-code"> class="hljs-ln-line">        for(j =  1 ; j < N-1 ; j++){
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="57"> class="hljs-ln-code"> class="hljs-ln-line">            dfs(i,j);
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="58"> class="hljs-ln-code"> class="hljs-ln-line">        }
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="59"> class="hljs-ln-code"> class="hljs-ln-line">    }
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="60"> class="hljs-ln-code"> class="hljs-ln-line">    printf("%d",maxRuslt);
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="61"> class="hljs-ln-code"> class="hljs-ln-line"> 
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="62"> class="hljs-ln-code"> class="hljs-ln-line">    return 0;
 class="hljs-ln-numbers"> class="hljs-ln-line hljs-ln-n" data-line-number="63"> class="hljs-ln-code"> class="hljs-ln-line">}

class="hide-preCode-box">

class="hljs-button signin active def" data-title="登录复制" data-report-click="{"spm":"3001.10243"}" onclick="hljs.signin(event)">' class="code-edithtml " data-title="运行"> data-report-click="{"spm":"3001.10182","extra":{"index":0,"ab":"exp1"}}" class="code-edithtml-box code-edithtml-box0">运行>

四、运行结果

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注：本文转载自blog.csdn.net的仁者乐山智者乐水的文章"https://blog.csdn.net/qq_39559641/article/details/98852768"。版权归原作者所有，此博客不拥有其著作权，亦不承担相应法律责任。如有侵权，请联系我们删除。

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