[英]Need help to optimize dynamic programming problem solution
The problem statement:问题陈述:
Given two arrays
a
andb
with sizesn
andm
respectively.给定两个 arraysa
和b
,大小分别为n
和m
。 All numbers in these arrays are in the range of 0 to 9 inclusive.这些 arrays 中的所有数字都在 0 到 9 的范围内。 Lets create a matrix with size ofnxm
where values in rowi
and columnj
is equal toai * 10^9 + bj
.让我们创建一个大小为nxm
的矩阵,其中第i
行和第j
列的值等于ai * 10^9 + bj
。 Find the path from square1,1
ton,m
with the maxium sum.找到从方块1,1
到n,m
的路径,总和最大。 You're allowed to move forward or down.你可以向前或向下移动。Input parameters: The first line contains
n
andm
(1 <= n, m <= 100 000)输入参数:第一行包含n
和m
(1 <= n, m <= 100 000)The second line contains values of array
a
第二行包含数组a
值The third line contains values of array
b
第三行包含数组b
的值Output Print the maximum sum Output 打印最大总和
Time limit: 1 second时间限制:1秒
Memory limit: 512MB Memory 限制:512MB
Example:例子:
input:输入:
7 4
0 7 1 7 6 7 6
4 1 9 7
output: 55000000068
output: 55000000068
I tried to solve this problem with dynamic programming, but my solution works in O(n * m)
and can't pass time limit:我试图用动态规划来解决这个问题,但我的解决方案在O(n * m)
中有效并且无法通过时间限制:
#include <iostream>
#include <vector>
#include <cmath>
using namespace std;
int main() {
int n, m; cin >> n >> m;
vector<uint64_t> a(n), b(m);
for(int i = 0; i < n; i++) {
int tmp;
cin >> tmp;
a[i] = tmp * 10e8;
}
for(int i = 0; i < m; i++) cin >> b[i];
vector<uint64_t> dp(m);
dp[0] = a[0] + b[0];
for (int i = 1; i < m; i++)
dp[i] = dp[i-1] + a[0] + b[i];
for (int i = 1; i < n; ++i) {
for(int j = 0; j < m; ++j) {
if (j == 0)
dp[j] = dp[j] + a[i] + b[j];
else
dp[j] = max(dp[j], dp[j-1]) + a[i] + b[j];
}
}
cout << dp.back() << endl;
return 0;
}
It's possible to solve this problem without dynamic programming, and with O(n+m) memory and time requirements.无需动态规划即可解决此问题,并且需要 O(n+m) memory 和时间要求。
As pointed out by @Botje in the comments, the reward for higher values in a is overwhelmingly large.正如@Botje在评论中指出的那样, a中更高值的回报是非常大的。 An optimal path will therefore remain in the leftmost column until it reaches the largest value in a (which is 7 in the above example).因此,最佳路径将保留在最左边的列中,直到它达到a中的最大值(在上面的示例中为 7)。 If this maximum value appears only once in a , then the best option would be to consume the whole of this row, followed by the last elements of all the following rows until we reach the bottom right corner.如果这个最大值在a中只出现一次,那么最好的选择是消耗整行,然后是所有后续行的最后一个元素,直到我们到达右下角。
However, if this maximum value appears more than once, we can get a better score by moving right along the first row with the maximum value of a until we reach a column with the maximum value of b , then moving down this column to the last row containing the maximum value of a .但是,如果这个最大值出现不止一次,我们可以通过沿着具有最大值a的第一行向右移动直到我们到达具有最大值b的列,然后将该列向下移动到最后一个来获得更好的分数包含a最大值的行。 We can then consume the rest of this row followed by the last elements of all following rows as before.然后,我们可以像以前一样使用此行的 rest,然后是所有后续行的最后一个元素。
Perhaps an illustration will help:也许一个例子会有所帮助:
a = [ 0, 6, 9, 9, 0, 9, 3, 1 ]
b = [ 1, 3, 2, 8, 4, 8, 1, 6 ]
Col: 0 1 2 3 4 5 6 7
Row:
0 0,1 0,3 0,2 0,8 0,4 0,8 0,1 0,6
v
1 6,1 6,3 6,2 6,8 6,4 6,8 6,1 6,6
v
2 9,1 > 9,3 > 9,2 > 9,8 9,4 9,8 9,1 9,6
v
3 9,1 9,3 9,2 9,8 9,4 9,8 9,1 9,6
v
4 0,1 0,3 0,2 0,8 0,4 0,8 0,1 0,6
v
5 9,1 9,3 9,2 9,8 > 9,4 > 9,8 > 9,1 > 9,6
v
6 3,1 3,3 3,2 3,8 3,4 3,8 3,1 3,6
v
7 1,1 1,3 1,2 1,8 1,4 1,8 1,1 1,6
In this example, there are three rows where a = 9, which are rows 2, 3 and 5. To get the maximum score, we need to follow the first of these rows (ie row 2) until we reach the column with the maximum value of b (either column 3 or column 5, it makes no difference).在这个例子中,有三行a = 9,分别是第 2、3 和 5 行。为了获得最大分数,我们需要跟随这些行中的第一行(即第 2 行),直到到达最大值的列b的值(第 3 列或第 5 列,没有区别)。 Then move down to the last row where a =9 (row 5), step right to the end of this row, and finally down to the bottom right corner.然后向下移动到a = 9的最后一行(第5行),向右移动到这一行的末尾,最后向下移动到右下角。
I've converted the Python code from an earlier version of this answer into C++. In tests with 10 5 random values in arrays a and b , it produces a result in about 0.3 seconds on my system.我已将此答案的早期版本的 Python 代码转换为 C++。在 arrays a和b中使用 10 5 个随机值进行的测试中,它在我的系统上大约 0.3 秒内产生结果。 The dynamic programming solution above gives identical results, but takes about 4 minutes:上面的动态规划解决方案给出了相同的结果,但需要大约 4 分钟:
#include <iostream>
#include <vector>
using namespace std;
int main() {
int n, m;
cin >> n >> m;
vector<int64_t> a(n), b(m);
int tmp, astart, aend, bmax, i, j;
// Read in arrays a[] and b[]. At the same time,
// find the first and last indices of the maximum
// value in a[] (astart and aend) and any index
// corresponding to the maximum value of b[] (bmax)
for (tmp = -1, i = 0; i < n; i++) {
cin >> a[i];
if (a[i] >= tmp) {
aend = i;
if (a[i] > tmp) {
astart = i;
tmp = a[i];
}
}
a[i] *= 1000000000LL;
}
for (tmp = -1, j = 0; j < m; j++) {
cin >> b[j];
if (b[j] > tmp) {
tmp = b[j];
bmax = j;
}
}
// Trace through the matrix. First work down as far as
// astart, then right until column bmax. Then work down
// as far as row aend, add the remaining elements in this
// row, and finally add the last element of each remaining
// rows until we reach the bottom right corner.
i = j = 0;
int64_t tot = a[i] + b[j];
while (i < astart) tot += a[++i] + b[j];
while (j < bmax) tot += a[i] + b[++j];
while (i < aend) tot += a[++i] + b[j];
while (j < m-1) tot += a[i] + b[++j];
while (i < n-1) tot += a[++i] + b[j];
cout << tot << endl;
return 0;
}
声明:本站的技术帖子网页,遵循CC BY-SA 4.0协议,如果您需要转载,请注明本站网址或者原文地址。任何问题请咨询:yoyou2525@163.com.