将 N 拆分为给定范围内 K 个数字的总和的方法数

给定四个正整数N、K、L和R ，任务是将 N 拆分为[L, R]范围内 K 个数字的总和。
注意：由于方式的数量可能非常大。输出答案模1000000007 。
例子：

Input : N = 12, K = 3, L = 1, R = 5 
Output : 10 
{2, 5, 5} 
{3, 4, 5} 
{3, 5, 4} 
{4, 3, 5} 
{4, 4, 4} 
{4, 5, 3} 
{5, 2, 5} 
{5, 3, 4} 
{5, 4, 3} 
{5, 5, 2}

Input : N = 23, K = 4, L = 2, R = 10 
Output : 480 

天真的方法
我们可以使用递归来解决这个问题。在递归的每一步尝试使一组大小为 L 到 R 所有
递归中有两个参数会发生变化：

• pos – 组号
• left – 剩下多少 N 需要分配

下面是上述方法的实现：

C++

// C++ implementation to count the
// number of ways to divide N in K
// groups such that each group
// has elements in range [L, R]
 
#include 
 
using namespace std;
 
const int mod = 1000000007;
 
// Function to count the number
// of ways to divide the number N
// in K groups such that each group
// has number of elements in range [L, R]
int calculate(int pos, int left,
              int k, int L, int R)
{
    // Base Case
    if (pos == k) {
        if (left == 0)
            return 1;
        else
            return 0;
    }
 
    // if N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    int answer = 0;
 
    // put all possible values
    // greater equal to prev
    for (int i = L; i <= R; i++) {
        if (i > left)
            break;
        answer = (answer + calculate(pos + 1,
                                     left - i, k, L, R))
                 % mod;
    }
    return answer;
}
 
// Function to count the number of
// ways to divide the number N
int countWaystoDivide(int n, int k,
                      int L, int R)
{
    return calculate(0, n, k, L, R);
}
 
// Driver Code
int main()
{
    int N = 12;
    int K = 3;
    int L = 1;
    int R = 5;
 
    cout << countWaystoDivide(N, K, L, R);
    return 0;
}

Java

// Java implementation to count the
// number of ways to divide N in K
// groups such that each group
// has elements in range [L, R]
class GFG{
 
static int mod = 1000000007;
 
// Function to count the number
// of ways to divide the number N
// in K groups such that each group
// has number of elements in range [L, R]
static int calculate(int pos, int left,
                     int k, int L, int R)
{
    // Base Case
    if (pos == k)
    {
        if (left == 0)
            return 1;
        else
            return 0;
    }
 
    // If N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    int answer = 0;
 
    // Put all possible values
    // greater equal to prev
    for(int i = L; i <= R; i++)
    {
       if (i > left)
           break;
       answer = ((answer +
                  calculate(pos + 1,left - i,
                            k, L, R)) % mod);
    }
    return answer;
}
 
// Function to count the number of
// ways to divide the number N
static int countWaystoDivide(int n, int k,
                             int L, int R)
{
    return calculate(0, n, k, L, R);
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 12;
    int K = 3;
    int L = 1;
    int R = 5;
 
    System.out.print(countWaystoDivide(N, K, L, R));
}
}
 
// This code is contributed by Amit Katiyar

Python3

# Python3 implementation to count the
# number of ways to divide N in K
# groups such that each group
# has elements in range [L, R]
 
mod = 1000000007
 
# Function to count the number
# of ways to divide the number N
# in K groups such that each group
# has number of elements in range [L, R]
def calculate(pos, left, k, L, R):
     
    # Base Case
    if (pos == k):
        if (left == 0):
            return 1
        else:
            return 0
 
    # If N is divides completely
    # into less than k groups
    if (left == 0):
        return 0
         
    answer = 0
 
    # Put all possible values
    # greater equal to prev
    for i in range(L, R + 1):
        if (i > left):
            break
        answer = (answer +
                  calculate(pos + 1,
                            left - i, k, L, R)) % mod
    return answer
 
# Function to count the number of
# ways to divide the number N
def countWaystoDivide(n, k, L, R):
    return calculate(0, n, k, L, R)
 
# Driver Code
N = 12
K = 3
L = 1
R = 5
 
print(countWaystoDivide(N, K, L, R))
     
# This code is contributed by divyamohan123

C#

// C# implementation to count the
// number of ways to divide N in K
// groups such that each group
// has elements in range [L, R]
using System;
 
class GFG{
 
static int mod = 1000000007;
 
// Function to count the number
// of ways to divide the number N
// in K groups such that each group
// has number of elements in range [L, R]
static int calculate(int pos, int left,
                     int k, int L, int R)
{
     
    // Base Case
    if (pos == k)
    {
        if (left == 0)
            return 1;
        else
            return 0;
    }
 
    // If N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    int answer = 0;
 
    // Put all possible values
    // greater equal to prev
    for(int i = L; i <= R; i++)
    {
       if (i > left)
           break;
       answer = ((answer +
                  calculate(pos + 1,left - i,
                            k, L, R)) % mod);
    }
    return answer;
}
 
// Function to count the number of
// ways to divide the number N
static int countWaystoDivide(int n, int k,
                             int L, int R)
{
    return calculate(0, n, k, L, R);
}
 
// Driver Code
public static void Main(String[] args)
{
    int N = 12;
    int K = 3;
    int L = 1;
    int R = 5;
 
    Console.Write(countWaystoDivide(N, K, L, R));
}
}
 
// This code is contributed by Amit Katiyar

Javascript

 
// Javascript implementation to count the
// number of ways to divide N in K
// groups such that each group
// has elements in range [L, R]
 
var mod = 1000000007;
 
// Function to count the number
// of ways to divide the number N
// in K groups such that each group
// has number of elements in range [L, R]
function calculate(pos, left, k, L, R)
{
    // Base Case
    if (pos == k) {
        if (left == 0)
            return 1;
        else
            return 0;
    }
 
    // if N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    var answer = 0;
 
    // put all possible values
    // greater equal to prev
    for (var i = L; i <= R; i++) {
        if (i > left)
            break;
        answer = (answer + calculate(pos + 1,
                                     left - i, k, L, R))
                 % mod;
    }
    return answer;
}
 
// Function to count the number of
// ways to divide the number N
function countWaystoDivide(n, k, L, R)
{
    return calculate(0, n, k, L, R);
}
 
// Driver Code
var N = 12;
var K = 3;
var L = 1;
var R = 5;
document.write( countWaystoDivide(N, K, L, R));
 
// This code is contributed by noob2000.

输出：

10

时间复杂度： O(N ^K ) 。

高效方法：在前面的方法中，我们可以看到我们在重复解决子问题，即它遵循重叠子问题的性质。所以我们可以使用 DP 表记住相同的内容。

下面是上述方法的实现：

C++

// C++ implementation to count the
// number of ways to divide N in K
// groups such that each group
// has elements in range [L, R]
 
#include 
 
using namespace std;
 
const int mod = 1000000007;
 
// DP Table
int dp[1000][1000];
 
// Function to count the number
// of ways to divide the number N
// in K groups such that each group
// has number of elements in range [L, R]
int calculate(int pos, int left,
              int k, int L, int R)
{
    // Base Case
    if (pos == k) {
        if (left == 0)
            return 1;
        else
            return 0;
    }
 
    // if N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    // If the subproblem has been
    // solved, use the value
    if (dp[pos][left] != -1)
        return dp[pos][left];
 
    int answer = 0;
 
    // put all possible values
    // greater equal to prev
    for (int i = L; i <= R; i++) {
        if (i > left)
            break;
        answer = (answer + calculate(pos + 1,
                                     left - i, k, L, R))
                 % mod;
    }
    return dp[pos][left] = answer;
}
 
// Function to count the number of
// ways to divide the number N
int countWaystoDivide(int n, int k,
                      int L, int R)
{
    // Initialize DP Table as -1
    memset(dp, -1, sizeof(dp));
    return calculate(0, n, k, L, R);
}
 
// Driver Code
int main()
{
    int N = 12;
    int K = 3;
    int L = 1;
    int R = 5;
 
    cout << countWaystoDivide(N, K, L, R);
    return 0;
}

Java

// Java implementation to count the
// number of ways to divide N in K
// groups such that each group
// has elements in range [L, R]
class GFG{
 
static int mod = 1000000007;
 
// DP Table
static int [][]dp = new int[1000][1000];
 
// Function to count the number
// of ways to divide the number N
// in K groups such that each group
// has number of elements in range [L, R]
static int calculate(int pos, int left,
                     int k, int L, int R)
{
     
    // Base Case
    if (pos == k)
    {
        if (left == 0)
            return 1;
        else
            return 0;
    }
 
    // If N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    // If the subproblem has been
    // solved, use the value
    if (dp[pos][left] != -1)
        return dp[pos][left];
 
    int answer = 0;
 
    // Put all possible values
    // greater equal to prev
    for(int i = L; i <= R; i++)
    {
       if (i > left)
           break;
       answer = (answer + calculate(pos + 1, left - i,
                                      k, L, R)) % mod;
    }
    return dp[pos][left] = answer;
}
 
// Function to count the number of
// ways to divide the number N
static int countWaystoDivide(int n, int k,
                             int L, int R)
{
     
    // Initialize DP Table as -1
    for(int i = 0; i < 1000; i++)
    {
       for(int j = 0; j < 1000; j++)
       {
          dp[i][j] = -1;
       }
    }
    return calculate(0, n, k, L, R);
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 12;
    int K = 3;
    int L = 1;
    int R = 5;
 
    System.out.print(countWaystoDivide(N, K, L, R));
}
}
 
// This code is contributed by 29AjayKumar

蟒蛇3

# Python3 implementation to count the
# number of ways to divide N in K
# groups such that each group
# has elements in range [L, R]
mod = 1000000007
  
# DP Table
dp = [[-1 for j in range(1000)]
          for i in range(1000)]
  
# Function to count the number
# of ways to divide the number N
# in K groups such that each group
# has number of elements in range [L, R]
def calculate(pos, left, k, L, R):
 
    # Base Case
    if (pos == k):
        if (left == 0):
            return 1
        else:
            return 0
 
    # if N is divides completely
    # into less than k groups
    if (left == 0):
        return 0
  
    # If the subproblem has been
    # solved, use the value
    if (dp[pos][left] != -1):
        return dp[pos][left]
  
    answer = 0
  
    # put all possible values
    # greater equal to prev
    for i in range(L, R + 1):
        if (i > left):
            break
         
        answer = (answer +
                  calculate(pos + 1,
                           left - i,
                           k, L, R)) % mod
     
    dp[pos][left] = answer
     
    return answer
 
# Function to count the number of
# ways to divide the number N
def countWaystoDivide(n, k, L, R):
     
    return calculate(0, n, k, L, R)
 
# Driver code
if __name__=="__main__":
     
    N = 12
    K = 3
    L = 1
    R = 5
  
    print(countWaystoDivide(N, K, L, R))
     
# This code is contributed by rutvik_56

C＃

// C# implementation to count the
// number of ways to divide N in K
// groups such that each group
// has elements in range [L, R]
using System;
class GFG{
 
static int mod = 1000000007;
 
// DP Table
static int [,]dp = new int[1000, 1000];
 
// Function to count the number
// of ways to divide the number N
// in K groups such that each group
// has number of elements in range [L, R]
static int calculate(int pos, int left,
                     int k, int L, int R)
{
     
    // Base Case
    if (pos == k)
    {
        if (left == 0)
            return 1;
        else
            return 0;
    }
 
    // If N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    // If the subproblem has been
    // solved, use the value
    if (dp[pos, left] != -1)
        return dp[pos, left];
 
    int answer = 0;
 
    // Put all possible values
    // greater equal to prev
    for(int i = L; i <= R; i++)
    {
        if (i > left)
            break;
        answer = (answer + calculate(pos + 1, left - i,
                                       k, L, R)) % mod;
    }
    return dp[pos, left] = answer;
}
 
// Function to count the number of
// ways to divide the number N
static int countWaystoDivide(int n, int k,
                             int L, int R)
{
     
    // Initialize DP Table as -1
    for(int i = 0; i < 1000; i++)
    {
        for(int j = 0; j < 1000; j++)
        {
            dp[i, j] = -1;
        }
    }
    return calculate(0, n, k, L, R);
}
 
// Driver Code
public static void Main()
{
    int N = 12;
    int K = 3;
    int L = 1;
    int R = 5;
 
    Console.Write(countWaystoDivide(N, K, L, R));
}
}
 
// This code is contributed by Code_Mech

Javascript

 
// JavaScript implementation to count the
// number of ways to divide N in K
// groups such that each group
// has elements in range [L, R]
 
var mod = 1000000007;
 
// DP Table
var dp = Array.from(Array(1000), ()=>Array(1000));
 
// Function to count the number
// of ways to divide the number N
// in K groups such that each group
// has number of elements in range [L, R]
function calculate(pos, left, k, L, R)
{
    // Base Case
    if (pos == k) {
        if (left == 0)
            return 1;
        else
            return 0;
    }
 
    // if N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    // If the subproblem has been
    // solved, use the value
    if (dp[pos][left] != -1)
        return dp[pos][left];
 
    var answer = 0;
 
    // put all possible values
    // greater equal to prev
    for (var i = L; i <= R; i++) {
        if (i > left)
            break;
        answer = (answer + calculate(pos + 1,
                                     left - i, k, L, R))
                 % mod;
    }
    return dp[pos][left] = answer;
}
 
// Function to count the number of
// ways to divide the number N
function countWaystoDivide(n, k, L, R)
{
    // Initialize DP Table as -1
    dp = Array.from(Array(1000), ()=>Array(1000).fill(-1));
    return calculate(0, n, k, L, R);
}
 
// Driver Code
var N = 12;
var K = 3;
var L = 1;
var R = 5;
document.write( countWaystoDivide(N, K, L, R));
 
 

输出：

10

时间复杂度： O(N ² ) 。