POJ1163 The Triangle

題目：

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

解答：

//POJ1163 
//A
//B C
//D E F
//G H I J
//K L M N O
//P Q R S T U

/*
#include<iostream>
#include<algorithm>
#define MAX 101
using namespace std;
int D[MAX][MAX];
int n;
int MaxSum(int i,int j){
    if(i==n)return D[i][j];
    int x=MaxSum(i+1,j),y=MaxSum(i+1,j+1);
    return max(x,y)+D[i][j];
}
int main(){
    int i,j;
    cin>>n;
    for(i=1;i<=n;i++)
        for(j=1;j<=i;j++){
            cin>>D[i][j];
        }

    cout<<MaxSum(1,1)<<endl;
    return 0;
}//超時 
*/
/*
#include<iostream>
#include<algorithm>
#define MAX 101
using namespace std;
int D[MAX][MAX];
int maxSum[MAX][MAX];/////////////////////////////////
int n;
int MaxSum(int i,int j){
    if(maxSum[i][j]!=-1)return maxSum[i][j];//////////
    if(i==n)return D[i][j];
    int x=MaxSum(i+1,j),y=MaxSum(i+1,j+1);
    return max(x,y)+D[i][j];
}
int main(){
    int i,j;
    cin>>n;
    for(i=1;i<=n;i++)
        for(j=1;j<=i;j++){
            cin>>D[i][j];
            maxSum[i][j]=-1;//////////////////////////
        }

    cout<<MaxSum(1,1)<<endl;
    return 0;
}////////數字三角形的記憶遞歸型動態規劃程序。 
*/
/* 
//遞歸變遞推：
#include<iostream>
#include<algorithm>
#define MAX 101
using namespace std;
int D[MAX][MAX];
int maxSum[MAX][MAX];
int n;
int main(){
    int i,j;
    cin>>n;
    for(int i=1;i<=n;i++)for(int j=1;j<=i;j++)cin>>D[i][j];
    for(int i=1;i<=n;i++)maxSum[n][i]=D[n][i];
    for(int i=n-1;i>=1;i--)for(int j=1;j<=i;j++)maxSum[i][j]=max(maxSum[i+1][j],maxSum[i+1][j+1])+D[i][j];
    cout<<maxSum[1][1]<<endl;
    return 0;
}*/ 
//繼續空間優化 //將maxSum二維數組換成一維 
//利用滾動數組，把原本上一行的運算數值保存在這一行。 
#include<iostream>
#include<algorithm>
#define MAX 101
using namespace std;
int D[MAX][MAX];
int maxSum[MAX];
int n;
int main(){
    int i,j;
    cin>>n;
    for(int i=1;i<=n;i++)for(int j=1;j<=i;j++)cin>>D[i][j];
    for(int i=1;i<=n;i++)maxSum[i]=D[n][i];
    for(int i=n-1;i>=1;i--)for(int j=1;j<=i;j++)maxSum[j]=max(maxSum[j],maxSum[j+1])+D[i][j];
    cout<<maxSum[1]<<endl;
    return 0;
}