[poj 1163]The Triangle

Posted 2020-10-23 studyingfather

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

30

Source

IOI 1994

大意

数字三角形问题，给出一个数字三角形，求从顶点开始，到达最下面一行的最大路径（每次只能向下方或向右下方走）。

题解

DP中最简单的一道问题，每次因为只有两种决策（往下方走或者往右下方走），所以状态转移方程很容易可以得出：f[i][j]=max(f[i-1][j]+a[i][j]（下方）,f[i-1][j-1]+a[i][j]（右下方）)，时间复杂度为O(n^2)。

 1 #include <cstdio>
 2 #include <algorithm>
 3 #include <cstring>
 4 #include <iostream>
 5 using namespace std;
 6 int n,a[205][205],f[205][205],maxn;
 7 int main()
 8 {
 9  scanf("%d",&n);
10  for(int i=1;i<=n;i++)
11   for(int j=1;j<=i;j++)
12    scanf("%d",&a[i][j]);
13  for(int i=1;i<=n;i++)
14   for(int j=1;j<=i;j++)
15    f[i][j]=max(f[i-1][j]+a[i][j],f[i-1][j-1]+a[i][j]);
16  for(int i=1;i<=n;i++)
17   maxn=max(f[n][i],maxn);
18  printf("%d",maxn);
19 }

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