使用O(n)时间的栈的滑动窗口最大值(大小为k的所有子数组的最大值)

2021年4月2日10:32:16 发表评论 618 次浏览

本文概述

给出一个包含N个整数和另一个k≤N的整数的数组arr[]，任务是找到每个大小为k的子数组的最大元素。

例子：

Input: arr[] = {9, 7, 2, 4, 6, 8, 2, 1, 5}
 k = 3
Output: 9 7 6 8 8 8 5
Explanation:
Window 1: {9, 7, 2}, max = 9
Window 2: {7, 2, 4}, max = 7
Window 3: {2, 4, 6}, max = 6
Window 4: {4, 6, 8}, max = 8
Window 5: {6, 8, 2}, max = 8
Window 6: {8, 2, 1}, max = 8
Window 7: {2, 1, 5}, max = 5

Input: arr[] = {6, 7, 5, 2, 1, 7, 2, 1, 10}
 k = 2
Output: 7 7 5 2 7 7 2 10
Explanation:
Window 1: {6, 7}, max = 7
Window 2: {7, 5}, max = 7
Window 3: {5, 2}, max = 5
Window 4: {2, 1}, max = 2
Window 5: {1, 7}, max = 7
Window 6: {7, 2}, max = 7
Window 7: {2, 1}, max = 2
Window 8: {1, 10}, max = 10

先决条件： 下一个更大的元素

方法：

对于每个索引，计算子数组从这个索引开始时当前元素最大的索引，即对于每个索引i，一个索引j≥i，使max(a[i]， a[i + 1]，…a[j]) = a[i]。我们把它命名为max_upto[i]。

然后，通过检查从i到i + k - 1, max_upto[i]≥i + k - 1(该窗口的最后一个索引)的每个索引，可以找到长度为k的子数组中从第i个索引开始的最大元素。

栈数据结构可以用来在一个窗口中存储值，即最后访问的或之前插入的元素将在顶部(与当前元素的索引最近的元素)。

算法：

  1. 创建一个数组max_upto和一个用于存储索引的堆栈。在堆栈中压入0。
  2. 运行从索引1到索引n-1的循环。
  3. 从堆栈中弹出所有索引, 其中哪些元素(array 展开)小于当前元素并更新max_upto 展开 = i – 1然后将i插入堆栈。
  4. 从堆栈中弹出所有索引并分配max_upto 展开 = n – 1.
  5. 创建一个变量j = 0
  6. 运行从0到n – k的循环, 循环计数器为i
  7. 运行一个嵌套循环, 直到j <i或max_upto [j] <i + k – 1, 在每次迭代中递增j。
  8. 打印第j个数组元素。

实现

C ++

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to print the maximum for
// every k size sub-array
void print_max( int a[], int n, int k)
{
     // max_upto array stores the index
     // upto which the maximum element is a[i]
     // i.e. max(a[i], a[i + 1], ... a[max_upto[i]]) = a[i]
  
     int max_upto[n];
  
     // Update max_upto array similar to
     // finding next greater element
     stack< int > s;
     s.push(0);
     for ( int i = 1; i < n; i++) {
         while (!s.empty() && a展开 < a[i]) {
             max_upto展开 = i - 1;
             s.pop();
         }
         s.push(i);
     }
     while (!s.empty()) {
         max_upto展开 = n - 1;
         s.pop();
     }
     int j = 0;
     for ( int i = 0; i <= n - k; i++) {
  
         // j < i is to check whether the
         // jth element is outside the window
         while (j < i || max_upto[j] < i + k - 1)
             j++;
         cout << a[j] << " " ;
     }
     cout << endl;
}
  
// Driver code
int main()
{
     int a[] = { 9, 7, 2, 4, 6, 8, 2, 1, 5 };
     int n = sizeof (a) / sizeof ( int );
     int k = 3;
     print_max(a, n, k);
  
     return 0;
}

Java

// Java implementation of the approach
import java.util.*;
  
class GFG 
{
  
     // Function to print the maximum for
     // every k size sub-array
     static void print_max( int a[], int n, int k)
     {
         // max_upto array stores the index
         // upto which the maximum element is a[i]
         // i.e. max(a[i], a[i + 1], ... a[max_upto[i]]) = a[i]
  
         int [] max_upto = new int [n];
  
         // Update max_upto array similar to
         // finding next greater element
         Stack<Integer> s = new Stack<>();
         s.push( 0 );
         for ( int i = 1 ; i < n; i++)
         {
             while (!s.empty() && a展开 < a[i])
             {
                 max_upto展开 = i - 1 ;
                 s.pop();
             }
             s.push(i);
         }
         while (!s.empty())
         {
             max_upto展开 = n - 1 ;
             s.pop();
         }
         int j = 0 ;
         for ( int i = 0 ; i <= n - k; i++)
         {
  
             // j < i is to check whether the
             // jth element is outside the window
             while (j < i || max_upto[j] < i + k - 1 )
             {
                 j++;
             }
             System.out.print(a[j] + " " );
         }
         System.out.println();
     }
  
     // Driver code
     public static void main(String[] args) 
     {
         int a[] = { 9 , 7 , 2 , 4 , 6 , 8 , 2 , 1 , 5 };
         int n = a.length;
         int k = 3 ;
         print_max(a, n, k);
  
     }
}
  
// This code has been contributed by 29AjayKumar

Python3

# Python3 implementation of the approach
  
# Function to print the maximum for
# every k size sub-array
def print_max(a, n, k):
      
     # max_upto array stores the index
     # upto which the maximum element is a[i]
     # i.e. max(a[i], a[i + 1], ... a[max_upto[i]]) = a[i]
  
     max_upto = [ 0 for i in range (n)]
  
     # Update max_upto array similar to
     # finding next greater element
     s = []
     s.append( 0 )
  
     for i in range ( 1 , n):
         while ( len (s) > 0 and a展开] < a[i]):
             max_upto展开] = i - 1
             del s[ - 1 ]
          
         s.append(i)
  
     while ( len (s) > 0 ):
         max_upto展开] = n - 1
         del s[ - 1 ]
  
     j = 0
     for i in range (n - k + 1 ):
  
         # j < i is to check whether the
         # jth element is outside the window
         while (j < i or max_upto[j] < i + k - 1 ):
             j + = 1
         print (a[j], end = " " )
     print () 
  
# Driver code
  
a = [ 9 , 7 , 2 , 4 , 6 , 8 , 2 , 1 , 5 ]
n = len (a)
k = 3
print_max(a, n, k)
  
# This code is contributed by mohit kumar

C#

// C# implementation of the approach
using System;
using System.Collections.Generic;
  
class GFG 
{
  
     // Function to print the maximum for
     // every k size sub-array
     static void print_max( int []a, int n, int k)
     {
         // max_upto array stores the index
         // upto which the maximum element is a[i]
         // i.e. max(a[i], a[i + 1], ... a[max_upto[i]]) = a[i]
  
         int [] max_upto = new int [n];
  
         // Update max_upto array similar to
         // finding next greater element
         Stack< int > s = new Stack< int >();
         s.Push(0);
         for ( int i = 1; i < n; i++)
         {
             while (s.Count!=0 && a展开 < a[i])
             {
                 max_upto展开 = i - 1;
                 s.Pop();
             }
             s.Push(i);
         }
         while (s.Count!=0)
         {
             max_upto展开 = n - 1;
             s.Pop();
         }
         int j = 0;
         for ( int i = 0; i <= n - k; i++)
         {
  
             // j < i is to check whether the
             // jth element is outside the window
             while (j < i || max_upto[j] < i + k - 1)
             {
                 j++;
             }
             Console.Write(a[j] + " " );
         }
         Console.WriteLine();
     }
  
     // Driver code
     public static void Main(String[] args) 
     {
         int []a = {9, 7, 2, 4, 6, 8, 2, 1, 5};
         int n = a.Length;
         int k = 3;
         print_max(a, n, k);
  
     }
}
  
// This code has been contributed by 29AjayKumar

输出如下：

9 7 6 8 8 8 5

复杂度分析：

  • 时间复杂度：O(n)。
    仅需要两次遍历数组。因此, 时间复杂度为O(n)。
  • 空间复杂度：O(n)。
    需要两个大小为n的额外空间。
木子山

发表评论

:?: :razz: :sad: :evil: :!: :smile: :oops: :grin: :eek: :shock: :???: :cool: :lol: :mad: :twisted: :roll: :wink: :idea: :arrow: :neutral: :cry: :mrgreen: