vb 导出整数 科学计数法
Problem statement:
问题陈述：
Given two positive integer n and m, find how many arrays of size n that can be formed such that:
给定两个正整数n和m ，找出可以形成多少个大小为n的数组：
Each element of the array is in the range [1, m]
数组的每个元素都在[1，m]范围内
Any adjacent element pair is divisible, i.e., that one of them divides another. Either element A[i] divides A[i + 1] or A[i + 1] divides A[i].
任何相邻的元素对都是可分割的 ，即，其中一个元素对另一个元素 。 元素A [i]除以A [i + 1]或A [i + 1]除以A [i] 。
Input:
输入：
Only one line with two integer, n & m respectively.
只有一行包含两个整数，分别为n和m 。
Output:
输出：
Print number of different possible ways to create the array. Since the output could be long take modulo 10^9+7.
打印创建数组的各种可能方式的数量。 由于输出可能很长，取模10 ^ 9 + 7 。
Constraints:
限制条件：
1<=n, m<=100
Example:
例：
Input:
n = 3, m = 2.
Output:
8
Explanation:
{1,1,1},{1, 1, 2}, {1, 2, 1},
{1, 2, 2}, {2, 1, 1},
{2,1,2}, {2,2,1}, {2,2,2} are possible arrays.
Input:
n = 1, m = 5.
Output:
5
Explanation:
{1}, {2}, {3}, {4}, {5}
Solution Approach:
解决方法：
The above problem is a great example of recursion. What can be the recursive function and how we can formulate.
上面的问题是递归的一个很好的例子。 什么是递归函数，以及我们如何公式化。
Say,
说，
Let
让
F(n, m) = number of ways for array size n and range 1 to m
F(n，m) =数组大小为n且范围为1到m的路径数
Now,
现在，
We actually can try picking every element from raging 1 to m and try recurring for other elements
实际上，我们可以尝试从1到m范围内选取每个元素，然后尝试对其他元素进行重复
So, the function can be written like:
因此，该函数可以这样写：
Function: NumberofWays(cur_index,lastelement,n,m)
So, to describe the arguments,
因此，为了描述这些论点，
cur_index is your current index and the last element is the previous index value assigned. So basically need to check which value with range 1 to m fits in the cur_index such that the divisibility constraint satisfies.
cur_index是当前索引，最后一个元素是分配的前一个索引值。 因此，基本上需要检查范围在1到m之间的哪个值适合cur_index ，以便除数约束满足。
So, to describe the body of the function
因此，要描述功能的主体
Function NumberofWays(cur_index,lastelement,n,m)
// formed the array completely
if(cur_index==n)
return 1;
sum=0
// any element in the range
for j=1 to m
// if divisible then lastelement,
// j can be adjacent pair
if(j%lastelement==0 || lastelement%j==0)
// recur for rest of the elments
sum=(sum%MOD+ NumberofWays(cur_index+1,j,n,m)%MOD)%MOD;
end if
end for
End function
Now the above recursive function generates many overlapping sub-problem and that's why we use the top-down DP approach to store already computed sub-problem results.
Below is the implementation with adding memoization.
现在，上面的递归函数会生成许多重叠的子问题，这就是为什么我们使用自上而下的DP方法来存储已经计算出的子问题结果的原因。
下面是添加备忘录的实现。
Initiate a 2D DP array with -1
使用-1启动2D DP阵列
Function NumberofWays(cur_index,lastelement,n,m)
// formed the array completely
if(cur_index==n)
return 1;
// if solution to sub problem already exits
if(dp[cur_index][lastelement]!=-1)
return dpdp[cur_index][lastelement];
sum=0
for j=1 to m // any element in the range
// if divisible then lastelement,j can be adjacent pair
if(j%lastelement==0 || lastelement%j==0)
// recur for rest of the elments
sum=(sum%MOD+ NumberofWays(cur_index+1,j,n,m)%MOD)%MOD;
end if
end for
Dp[curindex][lastelement]=sum
Return Dp[curindex][lastelement]
End function
C++ Implementation:
C ++实现：
#include <bits/stdc++.h>
using namespace std;
#define MOD 1000000007
int dp[101][101];
int countarray(int index, int i, int n, int m)
{
// if solution to sub problem already exits
if (dp[index][i] != -1)
return dp[index][i];
if (index == n)
return 1;
int sum = 0;
//any element in the range
for (int j = 1; j <= m; j++) {
// if divisible then i,j can be adjacent pair
if (j % i == 0 || i % j == 0) {
// recur for rest of the elments
sum = (sum % MOD + countarray(index + 1, j, n, m) % MOD) % MOD;
}
}
dp[index][i] = sum;
return dp[index][i];
}
int main()
{
int n, m;
cout << "Enter value of n:\n";
cin >> n;
cout << "Enter value of m:\n";
cin >> m;
// initialize DP matrix
for (int i = 0; i <= n; i++) {
for (int j = 0; j <= m; j++) {
dp[i][j] = -1;
}
}
cout << "number of ways are: " << countarray(0, 1, n, m) << endl;
return 0;
}
Output:
输出：
Enter value of n:
3
Enter value of m:
2
number of ways are: 8
翻译自: https://www.includehelp.com/icp/count-of-divisible-array.aspx
vb 导出整数 科学计数法
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