3
7 4
2 4 6
8 5 9 3
That is, 3 + 7 + 4 + 9 = 23.
Find the maximum total from top to bottom of the triangle below:
75
95 64
17 47 82
18 35 87 10
20 04 82 47 65
19 01 23 75 03 34
88 02 77 73 07 63 67
99 65 04 28 06 16 70 92
41 41 26 56 83 40 80 70 33
41 48 72 33 47 32 37 16 94 29
53 71 44 65 25 43 91 52 97 51 14
70 11 33 28 77 73 17 78 39 68 17 57
91 71 52 38 17 14 91 43 58 50 27 29 48
63 66 04 68 89 53 67 30 73 16 69 87 40 31
04 62 98 27 23 09 70 98 73 93 38 53 60 04 23
NOTE: As there are only 16384 routes, it is possible to solve this problem by trying every route. However, Problem 67, is the same challenge with a triangle containing one-hundred rows; it cannot be solved by brute force, and requires a clever method! ;o)
最简单的方法就是穷举，从根节点出发，每个节点都有两个分叉，到达底部的路径有估计有2的指数级的数目（有会算的朋友请留言，我的组合数学都还给老师了），不过这道题显然是符合动态规划的特征，往下递增一层的某个节点的最佳结果f[i][j]肯定是上一层两个入口节点对应的最佳结果的最大值，也就是f[i-1][j]或者f[i-1][j+1]，递归的边界就是定点f[0][0]=75。因此我的解答如下，考虑了金字塔边界的情况，数据按照金字塔型存储在numbers.txt中，
import java.io.InputStreamReader;
public class Euler18Problem {
public static void maxSun( int [][] a, int rows, int cols) {
// 结果列表
int [][] f = new int [ 15 ][ 15 ];
// 路径，用于输出计算路径
int [][] path = new int [ 15 ][ 15 ];
// 递归边界
f[ 0 ][ 0 ] = a[ 0 ][ 0 ];
path[ 0 ][ 0 ] = 0 ;
// 递推
for ( int i = 1 ; i < rows; i ++ ) {
int col = i + 1 ;
// 决策
for ( int j = 0 ; j < col; j ++ ) {
// 左边界
if (j - 1 < 0 ) {
f[i][j] = f[i - 1 ][j] + a[i][j];
path[i][j] = j;
} else if (j + 1 > col) { // 右边界
f[i][j] = f[i - 1 ][j - 1 ] + a[i][j];
path[i][j] = j - 1 ;
} else {
// 处于中间位置
if (f[i - 1 ][j] <= f[i - 1 ][j - 1 ]) {
f[i][j] = f[i - 1 ][j - 1 ] + a[i][j];
path[i][j] = j - 1 ;
} else {
f[i][j] = f[i - 1 ][j] + a[i][j];
path[i][j] = j;
}
}
}
}
// 求出结果
int result = 0 , col = 0 ;
for ( int i = 0 ; i < cols; i ++ ) {
if (f[ 14 ][i] > result) {
result = f[ 14 ][i];
col = i;
}
}
// 输出路径
System.out.println( " row=14,col= " + col + " ,value= " + a[ 14 ][col]);
for ( int i = rows - 2 ; i >= 0 ; i -- ) {
col = path[i][col];
System.out.println( " row= " + i + " ,col= " + col + " ,value= "
+ a[i][col]);
}
System.out.println(result);
}
public static void main(String[] args) throws Exception {
int rows = 15 ;
int cols = 15 ;
int [][] a = new int [rows][cols];
BufferedReader reader = new BufferedReader( new InputStreamReader(
Euler18Problem. class .getResourceAsStream( " /numbers.txt " )));
String line = null ;
int row = 0 ;
while ((line = reader.readLine()) != null && ! line.trim().equals( "" )) {
String[] numbers = line.split( " " );
for ( int i = 0 ; i < numbers.length; i ++ ) {
a[row][i] = Integer.parseInt(numbers[i]);
}
row ++ ;
}
reader.close();
maxSun(a, rows, cols);
}
}
row = 13 ,col = 8 ,value = 73
row = 12 ,col = 7 ,value = 43
row = 11 ,col = 6 ,value = 17
row = 10 ,col = 5 ,value = 43
row = 9 ,col = 4 ,value = 47
row = 8 ,col = 3 ,value = 56
row = 7 ,col = 3 ,value = 28
row = 6 ,col = 3 ,value = 73
row = 5 ,col = 2 ,value = 23
row = 4 ,col = 2 ,value = 82
row = 3 ,col = 2 ,value = 87
row = 2 ,col = 1 ,value = 47
row = 1 ,col = 0 ,value = 95
row = 0 ,col = 0 ,value = 75
1074
ps.并非我闲的蛋疼在半夜做题，只是被我儿子折腾的无法睡觉了，崩溃。
文章转自庄周梦蝶 ，原文发布时间2009-09-27
