N位数的计数，相邻位数的绝对差不超过K |套装2

给定两个整数N和K ，任务是找到N位数字的计数，以使数字中相邻数字的绝对差不大于K。
例子：

Input: N = 2, K = 1 
Output: 26 
Explanation: The numbers are 10, 11, 12, 21, 22, 23, 32, 33, 34, 43, 44, 45, 54, 55, 56, 65, 66, 67, 76, 77, 78, 87, 88, 89, 98, 99
Input: N = 3, K = 2 
Output: 188

天真的方法和动态规划方法：最简单的解决方案和需要O(N)辅助空间的动态规划方法，是指相邻位数的绝对差不超过K的N位数字的计数。
节省空间的方法：
请按照以下步骤优化上述方法：

• 在以上方法中，初始化了2D数组dp [] [] ，其中dp [i] [j]存储具有i个数字并以j结尾的数字的计数。
• 可以看出，对于任何长度i的答案仅取决于为i – 1生成的计数。因此，代替2D数组，初始化大小为所有可能数字的数组dp []，对于每一个i，直到N ， dp [j]存储以数字j结尾的长度为i的此类数字的计数。
• 初始化另一个数组，该数组的所有可能位数的大小为next [] 。
• 由于以值j结尾的一位数字的计数始终为1 ，因此最初在所有索引处用1填充dp [] 。
• 迭代[2，N]范围，对于范围i中的每个值，检查最后一位是否为j ，则该位置允许的位数在(max(0，jk)，min(9， j + k)) 。执行范围更新：

next[l] = next[l] + dp[j] 
next[r + 1] = next[r + 1] – dp[j] 
where l and r are max(0, j – k) and min(9, j + k) respectively.

• 对于特定的i值完成上述步骤后，计算next []的前缀总和，并用next []的值更新dp [] 。

下面是上述方法的实现：

C

// C program to implement the above approach
#include 
 
// Function to find maximum between two numbers
int max(int num1, int num2)
{
    return (num1 > num2 ) ? num1 : num2;
}
 
// Function to find minimum between two numbers
int min(int num1, int num2)
{
    return (num1 > num2 ) ? num2 : num1;
}
 
// Function to return the count
// of such numbers
int getCount(int n, int k)
{
     
    // For 1-digit numbers, the count
    // is 10 irrespective of K
    if (n == 1)
        return 10;
 
    // dp[j] stores the number
    // of such i-digit numbers
    // ending with j
    int dp[11] = {0};
 
    // Stores the results of length i
    int next[11] = {0};
         
    // Initialize count for
    // 1-digit numbers
    for(int i = 1; i <= 9; i++)
        dp[i] = 1;
 
    // Compute values for count of
    // digits greater than 1
    for(int i = 2; i <= n; i++)
    {
        for(int j = 0; j <= 9; j++)
        {
 
            // Find the range of allowed
            // numbers if last digit is j
            int l = max(0, (j - k));
            int r = min(9, (j + k));
 
            // Perform Range update
            next[l] += dp[j];
            next[r + 1] -= dp[j];
        }
 
        // Prefix sum to find actual count
        // of i-digit numbers ending with j
        for(int j = 1; j <= 9; j++)
            next[j] += next[j - 1];
 
        // Update dp[]
        for(int j = 0; j < 10; j++)
        {
            dp[j] = next[j];
            next[j] = 0;
        }
    }
 
    // Stores the final answer
    int count = 0;
    for(int i = 0; i <= 9; i++)
        count += dp[i];
 
    // Return the final answer
    return count;
}
 
// Driver Code
int main()
{
    int n = 2, k = 1;
    printf("%d", getCount(n, k));
}
 
// This code is contributed by piyush3010.

C++

// C++ Program to implement
// the above approach
#include 
using namespace std;
 
// Function to find maximum between two numbers
int max(int num1, int num2)
{
    return (num1 > num2 ) ? num1 : num2;
}
  
// Function to find minimum between two numbers
int min(int num1, int num2)
{
    return (num1 > num2 ) ? num2 : num1;
}
  
// Function to return the count
// of such numbers
int getCount(int n, int k)
{
      
    // For 1-digit numbers, the count
    // is 10 irrespective of K
    if (n == 1)
        return 10;
  
    // dp[j] stores the number
    // of such i-digit numbers
    // ending with j
    int dp[11] = {0};
  
    // Stores the results of length i
    int next[11] = {0};
          
    // Initialize count for
    // 1-digit numbers
    for(int i = 1; i <= 9; i++)
        dp[i] = 1;
  
    // Compute values for count of
    // digits greater than 1
    for(int i = 2; i <= n; i++)
    {
        for(int j = 0; j <= 9; j++)
        {
  
            // Find the range of allowed
            // numbers if last digit is j
            int l = max(0, (j - k));
            int r = min(9, (j + k));
  
            // Perform Range update
            next[l] += dp[j];
            next[r + 1] -= dp[j];
        }
  
        // Prefix sum to find actual count
        // of i-digit numbers ending with j
        for(int j = 1; j <= 9; j++)
            next[j] += next[j - 1];
  
        // Update dp[]
        for(int j = 0; j < 10; j++)
        {
            dp[j] = next[j];
            next[j] = 0;
        }
    }
  
    // Stores the final answer
    int count = 0;
    for(int i = 0; i <= 9; i++)
        count += dp[i];
  
    // Return the final answer
    return count;
}
 
// Driver Code
int main()
{
    int n = 2, k = 1;
    cout <<  getCount(n, k);
 
    return 0;
}

Java

// Java Program to implement
// the above approach
import java.util.*;
class GFG {
 
    // Function to return the count
    // of such numbers
    public static long getCount(
        int n, int k)
    {
        // For 1-digit numbers, the count
        // is 10 irrespective of K
        if (n == 1)
            return 10;
 
        // dp[j] stores the number
        // of such i-digit numbers
        // ending with j
        long dp[] = new long[11];
 
        // Stores the results of length i
        long next[] = new long[11];
 
        // Initialize count for
        // 1-digit numbers
        for (int i = 1; i <= 9; i++)
            dp[i] = 1;
 
        // Compute values for count of
        // digits greater than 1
        for (int i = 2; i <= n; i++) {
            for (int j = 0; j <= 9; j++) {
 
                // Find the range of allowed
                // numbers if last digit is j
                int l = Math.max(0, j - k);
                int r = Math.min(9, j + k);
 
                // Perform Range update
                next[l] += dp[j];
                next[r + 1] -= dp[j];
            }
 
            // Prefix sum to find actual count
            // of i-digit numbers ending with j
            for (int j = 1; j <= 9; j++)
                next[j] += next[j - 1];
 
            // Update dp[]
            for (int j = 0; j < 10; j++) {
                dp[j] = next[j];
                next[j] = 0;
            }
        }
 
        // Stores the final answer
        long count = 0;
        for (int i = 0; i <= 9; i++)
            count += dp[i];
 
        // Return the final answer
        return count;
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        int n = 2, k = 1;
        System.out.println(getCount(n, k));
    }
}

Python3

# Python3 program to implement
# the above approach
 
# Function to return the count
# of such numbers
def getCount(n, K):
 
    # For 1-digit numbers, the count
    # is 10 irrespective of K
    if(n == 1):
        return 10
 
    # dp[j] stores the number
    # of such i-digit numbers
    # ending with j
    dp = [0] * 11
 
    # Stores the results of length i
    next = [0] * 11
 
    # Initialize count for
    # 1-digit numbers
    for i in range(1, 9 + 1):
        dp[i] = 1
 
    # Compute values for count of
    # digits greater than 1
    for i in range(2, n + 1):
        for j in range(9 + 1):
 
            # Find the range of allowed
            # numbers if last digit is j
            l = max(0, j - k)
            r = min(9, j + k)
 
            # Perform Range update
            next[l] += dp[j]
            next[r + 1] -= dp[j]
 
        # Prefix sum to find actual count
        # of i-digit numbers ending with j
        for j in range(1, 9 + 1):
            next[j] += next[j - 1]
 
        # Update dp[]
        for j in range(10):
            dp[j] = next[j]
            next[j] = 0
 
    # Stores the final answer
    count = 0
    for i in range(9 + 1):
        count += dp[i]
 
    # Return the final answer
    return count
 
# Driver code
if __name__ == '__main__':
 
    n = 2
    k = 1
     
    print(getCount(n, k))
 
# This code is contributed by Shivam Singh

C#

// C# program to implement
// the above approach
using System;
 
class GFG{
 
// Function to return the count
// of such numbers
public static long getCount(int n, int k)
{
     
    // For 1-digit numbers, the count
    // is 10 irrespective of K
    if (n == 1)
        return 10;
 
    // dp[j] stores the number
    // of such i-digit numbers
    // ending with j
    long []dp = new long[11];
 
    // Stores the results of length i
    long []next = new long[11];
 
    // Initialize count for
    // 1-digit numbers
    for(int i = 1; i <= 9; i++)
        dp[i] = 1;
 
    // Compute values for count of
    // digits greater than 1
    for(int i = 2; i <= n; i++)
    {
        for(int j = 0; j <= 9; j++)
        {
             
            // Find the range of allowed
            // numbers if last digit is j
            int l = Math.Max(0, j - k);
            int r = Math.Min(9, j + k);
 
            // Perform Range update
            next[l] += dp[j];
            next[r + 1] -= dp[j];
        }
 
        // Prefix sum to find actual count
        // of i-digit numbers ending with j
        for(int j = 1; j <= 9; j++)
            next[j] += next[j - 1];
 
        // Update []dp
        for(int j = 0; j < 10; j++)
        {
            dp[j] = next[j];
            next[j] = 0;
        }
    }
 
    // Stores the readonly answer
    long count = 0;
    for(int i = 0; i <= 9; i++)
        count += dp[i];
 
    // Return the readonly answer
    return count;
}
 
// Driver Code
public static void Main(String[] args)
{
    int n = 2, k = 1;
    Console.WriteLine(getCount(n, k));
}
}
 
// This code is contributed by amal kumar choubey

Javascript

 
// Javascript program to implement the above approach
 
// Function to find maximum between two numbers
function max(num1, num2)
{
    return (num1 > num2 ) ? num1 : num2;
}
 
// Function to find minimum between two numbers
function min(num1, num2)
{
    return (num1 > num2 ) ? num2 : num1;
}
 
// Function to return the count
// of such numbers
function getCount(n, k)
{
     
    // For 1-digit numbers, the count
    // is 10 irrespective of K
    if (n == 1)
        return 10;
 
    // dp[j] stores the number
    // of such i-digit numbers
    // ending with j
    var dp = Array(11).fill(0);
 
    // Stores the results of length i
    var next = Array(11).fill(0);
         
    // Initialize count for
    // 1-digit numbers
    for(var i = 1; i <= 9; i++)
        dp[i] = 1;
 
    // Compute values for count of
    // digits greater than 1
    for(var i = 2; i <= n; i++)
    {
        for(var j = 0; j <= 9; j++)
        {
 
            // Find the range of allowed
            // numbers if last digit is j
            var l = Math.max(0, (j - k));
            var r = Math.min(9, (j + k));
 
            // Perform Range update
            next[l] += dp[j];
            next[r + 1] -= dp[j];
        }
 
        // Prefix sum to find actual count
        // of i-digit numbers ending with j
        for(var j = 1; j <= 9; j++)
            next[j] += next[j - 1];
 
        // Update dp[]
        for(var j = 0; j < 10; j++)
        {
            dp[j] = next[j];
            next[j] = 0;
        }
    }
 
    // Stores the final answer
    var count = 0;
    for(var i = 0; i <= 9; i++)
        count += dp[i];
 
    // Return the final answer
    return count;
}
 
// Driver Code
var n = 2, k = 1;
document.write( getCount(n, k));
 
// This code is contriuted by chinmoy1997pal.

输出：

26

时间复杂度： O(N)
辅助空间： O(1)