长度为 N 的二进制字符串的计数，最多有 M 个连续的 1 或 0，或者正好是 K 次

给定三个整数， N、K和M。任务是找出长度为N的二进制字符串的数量，这些字符串总是以1开头，其中最多可以有M个连续的 1 或 0，并且它们正好交替K次。
例子：

Input: N = 5, K = 3, M = 2 
Output: 3 
The 3 configurations are: 
11001 
10011 
11011 
Explanation: 
Notice that the groups of 1’s and 0’s alternate exactly K times
Input: N = 7, K = 4, M = 3 
Output: 16 
 

方法：由于这个问题涉及重叠子问题和最优子结构。所以，这个问题可以用动态规划来解决。

• 子问题： DP[i][j]表示长度为i的二进制字符串的数量，直到现在有j 个交替组。所以，计算dp[N][K]，如果我们知道dp[nj][k-1]的值，那么我们可以很容易地通过将j=1到m(DP [N][K] 代表最终答案)。
  如下递归树图中所示，观察到许多子问题重叠。因此，需要缓存结果以避免冗余计算。

• 最佳子结构：
  $$dp[i][j]=$$\sum_{j=1}^{M} f(N-j, K-1)$
• 通过遵循自上而下的 DP 方法：
  因为我们可以有一个最多长度为M 的组，所以我们迭代每个可能的长度并使用新的N递归，并将K减少 1，作为一个新的组。子问题的解决方案被缓存并总结为最终结果dp[N][K]。
• 基本情况：
  1. 当 N 为 0 且 K 为 0 时，则返回 1
  2. 当 N 为 0 但 K 不为 0 时，则返回 0
  3. 当 N 不为 0 但 K 为 0 时，则返回 0
  4. 当两者都为负时，返回 0

下面是上述方法的实现：

C++

// C++ program to find the count
// of Binary strings of length N
// having atmost M consecutive 1s or 0s
// alternatively exactly K times
 
#include 
using namespace std;
 
// Array to contain the final result
int dp[1000][1000];
 
// Function to get the number
// of desirable binary strings
int solve(int n, int k, int m)
{
 
    // if we reach end of string
    // and groups are exhausted,
    // return 1
    if (n == 0 && k == 0)
        return 1;
 
    // if length is exhausted but
    // groups are still to be made,
    // return 0
    if (n == 0 && k != 0)
        return 0;
 
    // if length is not exhausted
    // but groups are exhausted,
    // return 0
    if (n != 0 && k == 0)
        return 0;
 
    // if both are negative
    // just return 0
    if (n < 0 || k < 0)
        return 0;
 
    // if already calculated,
    // return it
    if (dp[n][k])
        return dp[n][k];
 
    // initialise answer
    // for each state
    int ans = 0;
 
    // loop through every
    // possible m
    for (int j = 1; j <= m; j++) {
        ans += solve(n - j, k - 1, m);
    }
    return dp[n][k] = ans;
}
 
// Driver code
int main()
{
 
    int N = 7, K = 4, M = 3;
    cout << solve(N, K, M);
}

Java

// Java program to find the count of
// Binary Strings of length N having
// atmost M consecutive 1s or 0s
// alternatively exactly K times
import java.util.*;
 
class GFG{
 
// Array to contain the final result
static int [][]dp = new int[1000][1000];
 
// Function to get the number
// of desirable binary strings
static int solve(int n, int k, int m)
{
 
    // If we reach end of string
    // and groups are exhausted,
    // return 1
    if (n == 0 && k == 0)
        return 1;
 
    // If length is exhausted but
    // groups are still to be made,
    // return 0
    if (n == 0 && k != 0)
        return 0;
 
    // If length is not exhausted
    // but groups are exhausted,
    // return 0
    if (n != 0 && k == 0)
        return 0;
 
    // If both are negative
    // just return 0
    if (n < 0 || k < 0)
        return 0;
 
    // If already calculated,
    // return it
    if (dp[n][k] > 0)
        return dp[n][k];
 
    // Initialise answer
    // for each state
    int ans = 0;
 
    // Loop through every
    // possible m
    for(int j = 1; j <= m; j++)
    {
       ans += solve(n - j, k - 1, m);
    }
    return dp[n][k] = ans;
}
 
// Driver code
public static void main(String[] args)
{
    int N = 7, K = 4, M = 3;
    System.out.print(solve(N, K, M));
}
}
 
// This code is contributed by Rajput-Ji

Python 3

# Python3 program to find the count
# of Binary strings of length N
# having atmost M consecutive 1s or
# 0s alternatively exactly K times
 
# List to contain the final result
rows, cols = (1000, 1000)
dp = [[0 for i in range(cols)]
         for j in range(rows)]
 
# Function to get the number
# of desirable binary strings
def solve(n, k, m):
     
    # If we reach end of string
    # and groups are exhausted,
    # return 1
    if n == 0 and k == 0:
        return 1
 
    # If length is exhausted but
    # groups are still to be made,
    # return 0
    if n == 0 and k != 0:
        return 0
 
    # If length is not exhausted
    # but groups are exhausted,
    # return 0
    if n != 0 and k == 0:
        return 0
 
    # If both are negative
    # just return 0
    if n < 0 or k < 0:
        return 0
 
    # If already calculated,
    # return it
    if dp[n][k]:
        return dp[n][k]
 
    # Initialise answer
    # for each state
    ans = 0
 
    # Loop through every
    # possible m
    for j in range(1, m + 1):
        ans = ans + solve(n - j,
                          k - 1, m)
    dp[n][k] = ans
     
    return dp[n][k]
 
# Driver code
N = 7
K = 4
M = 3
 
print(solve(N, K, M))
 
# This code is contributed by ishayadav181

C#

// C# program to find the count of
// binary strings of length N having
// atmost M consecutive 1s or 0s
// alternatively exactly K times
using System;
 
class GFG{
 
// Array to contain the readonly result
static int [,]dp = new int[1000, 1000];
 
// Function to get the number
// of desirable binary strings
static int solve(int n, int k, int m)
{
 
    // If we reach end of string
    // and groups are exhausted,
    // return 1
    if (n == 0 && k == 0)
        return 1;
 
    // If length is exhausted but
    // groups are still to be made,
    // return 0
    if (n == 0 && k != 0)
        return 0;
 
    // If length is not exhausted
    // but groups are exhausted,
    // return 0
    if (n != 0 && k == 0)
        return 0;
 
    // If both are negative
    // just return 0
    if (n < 0 || k < 0)
        return 0;
 
    // If already calculated,
    // return it
    if (dp[n, k] > 0)
        return dp[n, k];
 
    // Initialise answer
    // for each state
    int ans = 0;
 
    // Loop through every
    // possible m
    for(int j = 1; j <= m; j++)
    {
       ans += solve(n - j, k - 1, m);
    }
    return dp[n, k] = ans;
}
 
// Driver code
public static void Main(String[] args)
{
    int N = 7, K = 4, M = 3;
     
    Console.Write(solve(N, K, M));
}
}
 
// This code is contributed by gauravrajput1

Javascript

 
// Javascript program to find the count of
// Binary Strings of length N having
// atmost M consecutive 1s or 0s
// alternatively exactly K times
 
    // Array to contain the final result
     dp = Array(1000);
     for(i =0;i<1000;i++)
         dp[i] = Array(1000).fill(0);
 
    // Function to get the number
    // of desirable binary strings
    function solve(n , k , m)
    {
 
        // If we reach end of string
        // and groups are exhausted,
        // return 1
        if (n == 0 && k == 0)
            return 1;
 
        // If length is exhausted but
        // groups are still to be made,
        // return 0
        if (n == 0 && k != 0)
            return 0;
 
        // If length is not exhausted
        // but groups are exhausted,
        // return 0
        if (n != 0 && k == 0){
            return 0;
            }
 
        // If both are negative
        // just return 0
        if (n < 0 || k < 0)
            return 0;
 
        // If already calculated,
        // return it
        if (dp[n][k] > 0)
            return dp[n][k];
 
        // Initialise answer
        // for each state
        var ans = 0;
 
        // Loop through every
        // possible m
        for (var j = 1; j <= m; j++) {
            ans += solve(n - j, k - 1, m);
           // document.write(ans);
        }
        return dp[n][k] = ans;
    }
 
    // Driver code
     
        var N = 7, K = 4, M = 3;
        document.write(solve(N, K, M));
 
// This code contributed by umadevi9616
 

输出：

16

时间复杂度： O(N*K*M)

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