计算前 N 个字母中可能按字典顺序递增的 K 长度字符串

给定两个正整数N和K，该任务是找到能够从第一N字母，使得该字符串中的字符字典顺序排序来生成的K个长度的字符串的数目。

例子：

Input: N = 5, K = 2
Output: 15
Explanation: All possible strings are {“AA”, “AB”, “AC”, “AD”, “AE”, “BB”, “BC”, “BD”, “BE”, “CC”, “CD”, “CE”, “DD”, “DE”, “EE”}.

Input: N = 7, K = 10
Output: 8008

朴素的方法：解决问题的最简单方法是使用递归和回溯来生成所有可能的字符排列，并检查每个字符按照字典顺序递增。打印所有此类字符串的计数。

时间复杂度： O(K ^N )
辅助空间： O(1)

高效方法：为了优化上述方法，想法是使用动态规划，因为存在重叠的子问题，可以通过使用辅助二维数组dp[][]在递归调用中记忆或列表，并计算每个状态的值自下而上的方法。

dp[i][j] represents the number of ways to arrange “i” length strings with the “j” distinct letters.
dp[i][j] =  dp[i][j – 1] (Choose not to start with first letter) 
                 +  dp[i – 1][j – 1] (Choose first 1 letter in string as first letter) 
                 +  dp[i – 2][j – 1] (Choose first 2 letters in string as first letter) 
                 + …. 
                 + ….               
                 + dp[0][j – 1] (Choose first i letters in string as first letter)
dp[i][j] = Sum of all values of (j-1)th column for “i” rows

请按照以下步骤解决此问题：

• 初始化大小为(N+1)的数组columnSum[] ，其中columnSum[i]是数组dp[][] 的列“j”中所有值的总和。
• 初始化一个大小为(K + 1)*(N + 1)的dp[][]表。
• 将dp[0][i]初始化为1并随后更新数组columnSum[] 。
• 在K 上迭代两个嵌套循环并分别使用变量i和j ：
  • 将dp[i][j] 存储为columnSum[j – 1] 。
  • 将columnSum[j]更新为columnSum[j] + dp[i][j] 。
• 完成上述步骤后，将dp[K][N] 的值作为结果的方式数打印出来。

下面是上述方法的实现：

C++

// C++ program for the above approach
 
#include 
using namespace std;
 
// Function to count K-length
// strings from first N alphabets
void waysToArrangeKLengthStrings(
    int N, int K)
{
    // To keep track of column sum in dp
    int column_sum[N + 1] = { 0 }, i, j;
 
    // Auxiliary 2d dp array
    int dp[K + 1][N + 1] = { 0 };
 
    // Initialize dp[0][i] = 1 and
    // update the column_sum
    for (i = 0; i <= N; i++) {
        dp[0][i] = 1;
        column_sum[i] = 1;
    }
 
    // Iterate for K times
    for (i = 1; i <= K; i++) {
 
        // Iterate for N times
        for (j = 1; j <= N; j++) {
 
            // dp[i][j]: Stores the number
            // of ways to form i-length
            // strings consisting of j letters
            dp[i][j] += column_sum[j - 1];
 
            // Update the column_sum
            column_sum[j] += dp[i][j];
        }
    }
 
    // Print number of ways to arrange
    // K-length strings with N alphabets
    cout << dp[K][N];
}
 
// Driver Code
int main()
{
    // Given N and K
    int N = 5, K = 2;
 
    // Function Call
    waysToArrangeKLengthStrings(N, K);
 
    return 0;
}

Java

// Java program for the above approach
import java.io.*;
import java.util.*;
   
class GFG{
   
// Function to count K-length
// strings from first N alphabets
static void waysToArrangeKLengthStrings(
    int N, int K)
{
     
    // To keep track of column sum in dp
    int[] column_sum = new int[N + 1];
    int i, j;
     
    for(i = 1; i < N + 1; i++)
    {
        column_sum[i] = 0;
    }
  
    // Auxiliary 2d dp array
    int dp[][] = new int[K + 1][N + 1];
     
    for(i = 1; i < K + 1; i++)
    {
        for(j = 1; j < N + 1; j++)
        {
            dp[i][j] = 0;
        }
    }
  
    // Initialize dp[0][i] = 1 and
    // update the column_sum
    for(i = 0; i <= N; i++)
    {
        dp[0][i] = 1;
        column_sum[i] = 1;
    }
  
    // Iterate for K times
    for(i = 1; i <= K; i++)
    {
         
        // Iterate for N times
        for(j = 1; j <= N; j++)
        {
             
            // dp[i][j]: Stores the number
            // of ways to form i-length
            // strings consisting of j letters
            dp[i][j] += column_sum[j - 1];
  
            // Update the column_sum
            column_sum[j] += dp[i][j];
        }
    }
     
    // Print number of ways to arrange
    // K-length strings with N alphabets
    System.out.print(dp[K][N]);
}
   
// Driver Code
public static void main(String[] args)
{
     
    // Given N and K
    int N = 5, K = 2;
  
    // Function Call
    waysToArrangeKLengthStrings(N, K);
}
}
   
// This code is contributed by susmitakundugoaldanga

Python3

# Python3 program for the above approach
 
# Function to count K-length
# strings from first N alphabets
def waysToArrangeKLengthStrings(N, K):
     
    # To keep track of column sum in dp
    column_sum = [0 for i in range(N + 1)]
    i = 0
    j = 0
 
    # Auxiliary 2d dp array
    dp = [[0 for i in range(N + 1)]
             for j in range(K + 1)]
              
    # Initialize dp[0][i] = 1 and
    # update the column_sum
    for i in range(N + 1):
        dp[0][i] = 1
        column_sum[i] = 1
         
    # Iterate for K times
    for i in range(1, K + 1):
         
        # Iterate for N times
        for j in range(1, N + 1):
             
            # dp[i][j]: Stores the number
            # of ways to form i-length
            # strings consisting of j letters
            dp[i][j] += column_sum[j - 1]
 
            # Update the column_sum
            column_sum[j] += dp[i][j]
 
    # Print number of ways to arrange
    # K-length strings with N alphabets
    print(dp[K][N])
 
# Driver Code
if __name__ == '__main__':
     
    # Given N and K
    N = 5
    K = 2
 
    # Function Call
    waysToArrangeKLengthStrings(N, K)
 
# This code is contributed by SURENDRA_GANGWAR

C#

// C# program for the above approach
using System;
    
class GFG{
     
// Function to count K-length
// strings from first N alphabets
static void waysToArrangeKLengthStrings(int N,
                                        int K)
{
   
    // To keep track of column sum in dp
    int[] column_sum = new int[N + 1];
    int i, j;
      
    for(i = 1; i < N + 1; i++)
    {
        column_sum[i] = 0;
    }
   
    // Auxiliary 2d dp array
    int[,] dp = new int[K + 1, N + 1];
      
    for(i = 1; i < K + 1; i++)
    {
        for(j = 1; j < N + 1; j++)
        {
            dp[i, j] = 0;
        }
    }
   
    // Initialize dp[0][i] = 1 and
    // update the column_sum
    for(i = 0; i <= N; i++)
    {
        dp[0, i] = 1;
        column_sum[i] = 1;
    }
   
    // Iterate for K times
    for(i = 1; i <= K; i++)
    {
          
        // Iterate for N times
        for(j = 1; j <= N; j++)
        {
              
            // dp[i][j]: Stores the number
            // of ways to form i-length
            // strings consisting of j letters
            dp[i, j] += column_sum[j - 1];
   
            // Update the column_sum
            column_sum[j] += dp[i, j];
        }
    }
      
    // Print number of ways to arrange
    // K-length strings with N alphabets
    Console.Write(dp[K, N]);
}
  
// Driver Code
public static void Main()
{
   
    // Given N and K
    int N = 5, K = 2;
   
    // Function Call
    waysToArrangeKLengthStrings(N, K);
}
}
 
// This code is contributed by code_hunt

Javascript

 
// Javascript program to implement
// the above approach
 
// Function to count K-length
// strings from first N alphabets
function waysToArrangeKLengthStrings(N, K)
{
      
    // To keep track of column sum in dp
    let column_sum = [];
    let i, j;
      
    for(i = 1; i < N + 1; i++)
    {
        column_sum[i] = 0;
    }
   
    // Auxiliary 2d dp array
    let dp = new Array(K + 1);
    // Loop to create 2D array using 1D array
    for (i = 0; i < dp.length; i++) {
        dp[i] = new Array(2);
    }
      
    for(i = 1; i < K + 1; i++)
    {
        for(j = 1; j < N + 1; j++)
        {
            dp[i][j] = 0;
        }
    }
   
    // Initialize dp[0][i] = 1 and
    // update the column_sum
    for(i = 0; i <= N; i++)
    {
        dp[0][i] = 1;
        column_sum[i] = 1;
    }
   
    // Iterate for K times
    for(i = 1; i <= K; i++)
    {
          
        // Iterate for N times
        for(j = 1; j <= N; j++)
        {
              
            // dp[i][j]: Stores the number
            // of ways to form i-length
            // strings consisting of j letters
            dp[i][j] += column_sum[j - 1];
   
            // Update the column_sum
            column_sum[j] += dp[i][j];
        }
    }
      
    // Print number of ways to arrange
    // K-length strings with N alphabets
    document.write(dp[K][N]);
}
 
    // Driver Code
     
    // Given N and K
    let N = 5, K = 2;
   
    // Function Call
    waysToArrangeKLengthStrings(N, K);
 
// This code is contributed by splevel62.

输出：

15

时间复杂度： O(N*K)
辅助空间： O(N*K)

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