Super Jumping! Jumping! Jumping!（DP 最长不下降子序列）超详细题解

Dear 丶 2021-07-24 12:07 342阅读 0赞

### 目录 ###

*   *   *  1.题目
         *  2.题意
         *  3.解题思路
         *  4.代码

### 1.题目 ###

Nowadays, a kind of chess game called “Super Jumping! Jumping! Jumping!” is very popular in HDU. Maybe you are a good boy, and know little about this game, so I introduce it to you now.  
The game can be played by two or more than two players. It consists of a chessboard（棋盘）and some chessmen（棋子）, and all chessmen are marked by a positive integer or “start” or “end”. The player starts from start-point and must jumps into end-point finally. In the course of jumping, the player will visit the chessmen in the path, but everyone must jumps from one chessman to another absolutely bigger (you can assume start-point is a minimum and end-point is a maximum.). And all players cannot go backwards. One jumping can go from a chessman to next, also can go across many chessmen, and even you can straightly get to end-point from start-point. Of course you get zero point in this situation. A player is a winner if and only if he can get a bigger score according to his jumping solution. Note that your score comes from the sum of value on the chessmen in you jumping path.  
Your task is to output the maximum value according to the given chessmen list.  
Input  
Input contains multiple test cases. Each test case is described in a line as follow:  
N value\_1 value\_2 …value\_N  
It is guarantied that N is not more than 1000 and all value\_i are in the range of 32-int.  
A test case starting with 0 terminates the input and this test case is not to be processed.  
Output  
For each case, print the maximum according to rules, and one line one case.  
Sample Input

3 1 3 2
    4 1 2 3 4
    4 3 3 2 1
    0

Sample Output

4
    10
    3

### 2.题意 ###

非连续递增子序列累加和的最大值，类似于经典最长不下降子序列题。

### 3.解题思路 ###

动态规划  
1.状态转移方程 dp\[i\]=max(a\[i\],dp\[j\]+a\[i\])(j<i)。dp\[i\]表示以a\[i\]结尾的最大递增子序列值(非连续)。  
2.边界dp\[i\]=a\[i\] (1<=i<=n).  
3.遍历dp\[i…n\]求出最大dp\[i\]即为答案。

### 4.代码 ###

#include<iostream>
    #include<cstdio>
    #include<algorithm>
    using namespace std;
    const int maxn = 1e4 + 10;
    int dp[maxn],a[maxn]; //dp[i]表示以a[i]结尾的最大递增子序列值(非连续)
    int main()
    { 
    	int n;
    	while (cin >> n)
    	{ 
    		memset(dp, 0, sizeof(dp));  每次初始化dp[i]数组
    		if (n == 0) break;
    		for (int i = 1; i <= n; i++)
    		{ 
    			cin >> a[i];
    			dp[i] = a[i];   //边界
    		}
    		int ans = -1;   ///记录最大dp[i];
    		for (int i = 1; i<= n; i++)
    		{ 
    			for (int j = 1; j < i; j++)
    			{ 
    				if (a[j] < a[i] && (dp[j] + a[i] > dp[i]))
    				{ 
    					dp[i]=dp[j]+a[i]; 状态转移方程，用于更新dp[i];
    				}
    			}
    			ans = max(ans, dp[i]);
    		}
    		cout << ans << endl;
    	}
    	return 0;
    }