poj 1163 The Triangle

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本文介绍了一种经典的算法问题——寻找数字三角形中从顶点到底部的最大路径和。提供了两种解决方案，一种是使用记忆性递归来避免重复计算，另一种是采用动态规划的方法进行迭代求解，同时讨论了空间优化技巧。

The Triangle

Time Limit: 1000MS Memory Limit: 10000K
Total Submissions: 44961 Accepted: 27147
Description

7
3 8
8 1 0
2 7 4 4
4 5 2 6 5

(Figure 1)
Figure 1 shows a number triangle. Write a program that calculates the highest sum of numbers passed on a route that starts at the top and ends somewhere on the base. Each step can go either diagonally down to the left or diagonally down to the right.
Input

Your program is to read from standard input. The first line contains one integer N: the number of rows in the triangle. The following N lines describe the data of the triangle. The number of rows in the triangle is > 1 but <= 100. The numbers in the triangle, all integers, are between 0 and 99.
Output

Your program is to write to standard output. The highest sum is written as an integer.
Sample Input

5
7
3 8
8 1 0
2 7 4 4

4 5 2 6 5

Sample Output

方法1：记忆性递归程序

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>

using namespace std;


const int maxn = 110;

int a[maxn][maxn];
int maxsum[maxn][maxn];
int n;

int solve(int i,int j)
{
    if(maxsum[i][j]!=-1)
        return maxsum[i][j];
    if(i==n){
        return a[i][j];
    }else{
        maxsum[i][j] = max(solve(i+1,j),solve(i+1,j+1))+a[i][j];
        return maxsum[i][j];
    }
}

int main()
{
    scanf("%d",&n);
    for(int i=1;i<=n;i++){
        for(int j=1;j<=i;j++){
            scanf("%d",&a[i][j]);
            maxsum[i][j] = -1;
        }
    }
    printf("%d\n",solve(1,1));
    return 0;
}

方法二：“人人为我”递推型动归程序

#include <iostream>
#include <cstring>
#include <cstdio>
#include <algorithm>

using namespace std;

//“人人为我”空间优化版递推型动归

const int maxn = 110;

int a[maxn][maxn];
int n;

int main()
{
scanf("%d",&n);
for(int i=1;i<=n;i++){
for(int j=1;j<=i;j++)
scanf("%d",&a[i][j]);
}
for(int i=n-1;i>=1;--i){
for(int j=1;j<=i;++j){
a[i][j] = a[i][j]+max(a[i+1][j],a[i+1][j+1]);
}
}
printf("%d\n",a[1][1]);
return 0;
}

此外还可以加上滚动数组来进行空间优化。