将包含相同元素的最长子数组的长度最大化，最多递减 K 个

给定的阵列ARR大小N和整数的K []中，任务是找到由相同的元件，可以由至多K倍递减1的数组元素中获得的最长子阵列的长度。

例子：

Input: arr[] = { 1, 2, 3 }, K = 1
Output: 2
Explanation:
Decrementing arr[0] by 1 modifies arr[] to { 1, 1, 3 }
The longest subarray with equal elements is { 1, 1 }.
Therefore, the required output is 2.

Input: arr[] = { 1, 7, 3, 4, 5, 6 }, K = 6
Output: 4

方法：该问题可以使用段树和二进制搜索技术来解决。这个想法是使用以下观察结果：

Total number of decrements operations required to make all array elements of the subarray { arr[start], …, arr[end] } equal
= (Σ(start, end)) – (end – start + 1) * (min_value)

where, start = index of the starting point of the subarray
end = index of end point of subarray
min_value = smallest value from index i to j
Σ(start, end) = sum of all elements from index i to j

请按照以下步骤解决上述问题：

1. 初始化一个段树来计算数组子数组中的最小元素和一个前缀和数组来计算子数组的和元素。
2. 遍历数组arr[] 。对于每个^第i^个元素执行以下操作：
   • 初始化两个变量，例如start = i , end = N – 1并在范围[start, end]上应用二分查找来检查子数组{ arr[start], …, arr[end] }的所有元素是否可以通过从上述观察中最多减少K 个操作来使它们相等或不相等。
   • 如果子数组{ arr[start], …, arr[end] } 的所有元素可以通过最多递减K 个操作而变得相等，则更新start = (start + end) / 2 + 1 。
   • 否则，更新结束 = (开始 + 结束) / 2 – 1
3. 最后，打印上述操作得到的最长子数组的长度。

下面是上述方法的实现：

C++

// C++ program to implement
// the above appraoch
 
#include 
using namespace std;
 
// Function to construct Segment Tree
// to return the minimum element in a range
int build(int tree[], int* A, int start,
          int end, int node)
{
    // If leaf nodes of
    // the tree are found
    if (start == end) {
 
        // Update the value in segment
        // tree from given array
        tree[node] = A[start];
 
        return tree[node];
    }
 
    // Divide left and right subtree
    int mid = (start + end) / 2;
 
    // Stores smallest element in
    // subarray { arr[start], arr[mid] }
    int X = build(tree, A, start, mid,
                  2 * node + 1);
 
    // Stores smallest element in
    // subarray { arr[mid + 1], arr[end] }
    int Y = build(tree, A, mid + 1,
                  end, 2 * node + 2);
 
    // Stores smallest element in
    // subarray { arr[start], arr[end] }
    return tree[node] = min(X, Y);
}
 
// Function to find the smallest
// element present in a subarray
int query(int tree[], int start, int end,
          int l, int r, int node)
{
    // If elements of the subarray
    // are not in the range [l, r]
    if (start > r || end < l)
        return INT_MAX;
 
    // If all the elements of the
    // subarray are in the range [l, r]
    if (start >= l && end <= r)
        return tree[node];
 
    // Divide tree into left
    // and right subtree
    int mid = (start + end) / 2;
 
    // Stores smallest element
    // in left subtree
    int X = query(tree, start, mid, l,
                  r, 2 * node + 1);
 
    // Stores smallest element in
    // right subtree
    int Y = query(tree, mid + 1, end, l,
                  r, 2 * node + 2);
 
    return min(X, Y);
}
 
// Function that find length of longest
// subarray with all equal elements in
// atmost K decrements
int longestSubArray(int* A, int N, int K)
{
    // Stores length of longest subarray
    // with all equal elements in atmost
    // K decrements.
    int res = 1;
 
    // Store the prefix sum array
    int preSum[N + 1];
 
    // Calculate the prefix sum array
    preSum[0] = A[0];
    for (int i = 0; i < N; i++)
        preSum[i + 1] = preSum[i] + A[i];
 
    int tree[4 * N + 5];
 
    // Build the segment tree
    // for range min query
    build(tree, A, 0, N - 1, 0);
 
    // Traverse the array
    for (int i = 0; i < N; i++) {
 
        // Stores start index
        // of the subarray
        int start = i;
 
        // Stores end index
        // of the subarray
        int end = N - 1;
 
        int mid;
 
        // Stores end index of
        // the longest subarray
        int max_index = i;
 
        // Performing the binary search
        // to find the endpoint
        // for the selected range
        while (start <= end) {
 
            // Find the mid for binary search
            mid = (start + end) / 2;
 
            // Find the smallest element in
            // range [i, mid] using Segment Tree
            int min_element
                = query(tree, 0, N - 1, i, mid, 0);
 
            // Stores total sum of subarray
            // after K decrements
            int expected_sum
                = (mid - i + 1) * min_element;
 
            // Stores sum of elements of
            // subarray before K decrements
            int actual_sum
                = preSum[mid + 1] - preSum[i];
 
            // If subarray found with
            // all equal elements
            if (actual_sum - expected_sum <= K) {
 
                // Update start
                start = mid + 1;
 
                // Update max_index
                max_index = max(max_index, mid);
            }
 
            // If false, it means that
            // the selected range is invalid
            else {
 
                // Update end
                end = mid - 1;
            }
        }
 
        // Store the length of longest subarray
        res = max(res, max_index - i + 1);
    }
 
    // Return result
    return res;
}
 
// Driver Code
int main()
{
    int arr[] = { 1, 7, 3, 4, 5, 6 };
    int k = 6;
    int n = 6;
    cout << longestSubArray(arr, n, k);
 
    return 0;
}

Java

// Java program to implement
// the above appraoch
import java.util.*;
  
class GFG{
 
// Function to construct Segment Tree
// to return the minimum element in a range
static int build(int tree[], int[] A, int start,
                 int end, int node)
{
     
    // If leaf nodes of
    // the tree are found
    if (start == end)
    {
         
        // Update the value in segment
        // tree from given array
        tree[node] = A[start];
  
        return tree[node];
    }
  
    // Divide left and right subtree
    int mid = (start + end) / 2;
  
    // Stores smallest element in
    // subarray { arr[start], arr[mid] }
    int X = build(tree, A, start, mid,
                  2 * node + 1);
  
    // Stores smallest element in
    // subarray { arr[mid + 1], arr[end] }
    int Y = build(tree, A, mid + 1,
                 end, 2 * node + 2);
  
    // Stores smallest element in
    // subarray { arr[start], arr[end] }
    return (tree[node] = Math.min(X, Y));
}
  
// Function to find the smallest
// element present in a subarray
static int query(int tree[], int start, int end,
                 int l, int r, int node)
{
     
    // If elements of the subarray
    // are not in the range [l, r]
    if (start > r || end < l)
        return Integer.MAX_VALUE;
  
    // If all the elements of the
    // subarray are in the range [l, r]
    if (start >= l && end <= r)
        return tree[node];
  
    // Divide tree into left
    // and right subtree
    int mid = (start + end) / 2;
  
    // Stores smallest element
    // in left subtree
    int X = query(tree, start, mid, l,
                  r, 2 * node + 1);
  
    // Stores smallest element in
    // right subtree
    int Y = query(tree, mid + 1, end, l,
                  r, 2 * node + 2);
  
    return Math.min(X, Y);
}
  
// Function that find length of longest
// subarray with all equal elements in
// atmost K decrements
static int longestSubArray(int[] A, int N, int K)
{
     
    // Stores length of longest subarray
    // with all equal elements in atmost
    // K decrements.
    int res = 1;
  
    // Store the prefix sum array
    int preSum[] = new int[N + 1];
  
    // Calculate the prefix sum array
    preSum[0] = A[0];
    for(int i = 0; i < N; i++)
        preSum[i + 1] = preSum[i] + A[i];
  
    int tree[] = new int[4 * N + 5];
  
    // Build the segment tree
    // for range min query
    build(tree, A, 0, N - 1, 0);
  
    // Traverse the array
    for(int i = 0; i < N; i++)
    {
         
        // Stores start index
        // of the subarray
        int start = i;
  
        // Stores end index
        // of the subarray
        int end = N - 1;
  
        int mid;
  
        // Stores end index of
        // the longest subarray
        int max_index = i;
  
        // Performing the binary search
        // to find the endpoint
        // for the selected range
        while (start <= end)
        {
             
            // Find the mid for binary search
            mid = (start + end) / 2;
  
            // Find the smallest element in
            // range [i, mid] using Segment Tree
            int min_element = query(tree, 0, N - 1,
                                    i, mid, 0);
  
            // Stores total sum of subarray
            // after K decrements
            int expected_sum = (mid - i + 1) *
                                min_element;
  
            // Stores sum of elements of
            // subarray before K decrements
            int actual_sum = preSum[mid + 1] -
                             preSum[i];
  
            // If subarray found with
            // all equal elements
            if (actual_sum - expected_sum <= K)
            {
                 
                // Update start
                start = mid + 1;
  
                // Update max_index
                max_index = Math.max(max_index, mid);
            }
  
            // If false, it means that
            // the selected range is invalid
            else
            {
                 
                // Update end
                end = mid - 1;
            }
        }
  
        // Store the length of longest subarray
        res = Math.max(res, max_index - i + 1);
    }
  
    // Return result
    return res;
}
  
// Driver Code
static public void main(String args[])
{
    int arr[] = { 1, 7, 3, 4, 5, 6 };
    int k = 6;
    int n = 6;
     
    System.out.print(longestSubArray(arr, n, k));
}
}
 
// This code is contributed by sanjoy_62

Python3

# Python3 program to implement
# the above appraoch
import sys
 
# Function to construct Segment Tree
# to return the minimum element in a range
def build(tree, A, start, end, node):
     
    # If leaf nodes of
    # the tree are found
    if (start == end):
         
        # Update the value in segment
        # tree from given array
        tree[node] = A[start]
  
        return tree[node]
     
    # Divide left and right subtree
    mid = (int)((start + end) / 2)
  
    # Stores smallest element in
    # subarray : arr[start], arr[mid]
    X = build(tree, A, start, mid,
              2 * node + 1)
 
    # Stores smallest element in
    # subarray : arr[mid + 1], arr[end]
    Y = build(tree, A, mid + 1,
             end, 2 * node + 2)
  
    # Stores smallest element in
    # subarray : arr[start], arr[end]
    return (tree[node] == min(X, Y))
 
# Function to find the smallest
# element present in a subarray
def query(tree, start, end, l, r, node):
               
    # If elements of the subarray
    # are not in the range [l, r]
    if (start > r or end < l) :
        return sys.maxsize
  
    # If all the elements of the
    # subarray are in the range [l, r]
    if (start >= l and end <= r):
        return tree[node]
  
    # Divide tree into left
    # and right subtree
    mid = (int)((start + end) / 2)
  
    # Stores smallest element
    # in left subtree
    X = query(tree, start, mid, l,
              r, 2 * node + 1)
  
    # Stores smallest element in
    # right subtree
    Y = query(tree, mid + 1, end, l,
            r, 2 * node + 2)
  
    return min(X, Y)
 
# Function that find length of longest
# subarray with all equal elements in
# atmost K decrements
def longestSubArray(A, N, K):
     
    # Stores length of longest subarray
    # with all equal elements in atmost
    # K decrements.
    res = 1
  
    # Store the prefix sum array
    preSum = [0] * (N + 1)
  
    # Calculate the prefix sum array
    preSum[0] = A[0]
    for i in range(N):
        preSum[i + 1] = preSum[i] + A[i]
  
    tree = [0] * (4 * N + 5)
  
    # Build the segment tree
    # for range min query
    build(tree, A, 0, N - 1, 0)
  
    # Traverse the array
    for i in range(N):
  
        # Stores start index
        # of the subarray
        start = i
  
        # Stores end index
        # of the subarray
        end = N - 1
  
        # Stores end index of
        # the longest subarray
        max_index = i
  
        # Performing the binary search
        # to find the endpoint
        # for the selected range
        while (start <= end):
  
            # Find the mid for binary search
            mid = (int)((start + end) / 2)
  
            # Find the smallest element in
            # range [i, mid] using Segment Tree
            min_element = query(tree, 0, N - 1, i, mid, 0)
  
            # Stores total sum of subarray
            # after K decrements
            expected_sum = (mid - i + 1) * min_element
  
            # Stores sum of elements of
            # subarray before K decrements
            actual_sum = preSum[mid + 1] - preSum[i]
  
            # If subarray found with
            # all equal elements
            if (actual_sum - expected_sum <= K):
                 
                # Update start
                start = mid + 1
  
                # Update max_index
                max_index = max(max_index, mid)
             
            # If false, it means that
            # the selected range is invalid
            else:
  
                # Update end
                end = mid - 1
             
        # Store the length of longest subarray
        res = max(res, max_index - i + 2)
 
    # Return result
    return res
 
# Driver Code
arr = [ 1, 7, 3, 4, 5, 6 ]
k = 6
n = 6
 
print(longestSubArray(arr, n, k))
 
# This code is contributed by splevel62

C#

// C# program to implement
// the above appraoch
using System;
  
class GFG{
      
// Function to construct Segment Tree
// to return the minimum element in a range
static int build(int[] tree, int[] A, int start,
                 int end, int node)
{
     
    // If leaf nodes of
    // the tree are found
    if (start == end)
    {
         
        // Update the value in segment
        // tree from given array
        tree[node] = A[start];
   
        return tree[node];
    }
   
    // Divide left and right subtree
    int mid = (start + end) / 2;
   
    // Stores smallest element in
    // subarray { arr[start], arr[mid] }
    int X = build(tree, A, start, mid,
                  2 * node + 1);
   
    // Stores smallest element in
    // subarray { arr[mid + 1], arr[end] }
    int Y = build(tree, A, mid + 1,
                  end, 2 * node + 2);
   
    // Stores smallest element in
    // subarray { arr[start], arr[end] }
    return (tree[node] = Math.Min(X, Y));
}
   
// Function to find the smallest
// element present in a subarray
static int query(int[] tree, int start, int end,
                 int l, int r, int node)
{
      
    // If elements of the subarray
    // are not in the range [l, r]
    if (start > r || end < l)
        return Int32.MaxValue;
   
    // If all the elements of the
    // subarray are in the range [l, r]
    if (start >= l && end <= r)
        return tree[node];
   
    // Divide tree into left
    // and right subtree
    int mid = (start + end) / 2;
   
    // Stores smallest element
    // in left subtree
    int X = query(tree, start, mid, l,
                  r, 2 * node + 1);
   
    // Stores smallest element in
    // right subtree
    int Y = query(tree, mid + 1, end, l,
                  r, 2 * node + 2);
   
    return Math.Min(X, Y);
}
   
// Function that find length of longest
// subarray with all equal elements in
// atmost K decrements
static int longestSubArray(int[] A, int N, int K)
{
      
    // Stores length of longest subarray
    // with all equal elements in atmost
    // K decrements.
    int res = 1;
   
    // Store the prefix sum array
    int[] preSum = new int[N + 1];
   
    // Calculate the prefix sum array
    preSum[0] = A[0];
    for(int i = 0; i < N; i++)
        preSum[i + 1] = preSum[i] + A[i];
   
    int[] tree = new int[4 * N + 5];
   
    // Build the segment tree
    // for range min query
    build(tree, A, 0, N - 1, 0);
   
    // Traverse the array
    for(int i = 0; i < N; i++)
    {
          
        // Stores start index
        // of the subarray
        int start = i;
   
        // Stores end index
        // of the subarray
        int end = N - 1;
   
        int mid;
   
        // Stores end index of
        // the longest subarray
        int max_index = i;
   
        // Performing the binary search
        // to find the endpoint
        // for the selected range
        while (start <= end)
        {
              
            // Find the mid for binary search
            mid = (start + end) / 2;
   
            // Find the smallest element in
            // range [i, mid] using Segment Tree
            int min_element = query(tree, 0, N - 1,
                                    i, mid, 0);
   
            // Stores total sum of subarray
            // after K decrements
            int expected_sum = (mid - i + 1) *
                                min_element;
   
            // Stores sum of elements of
            // subarray before K decrements
            int actual_sum = preSum[mid + 1] -
                             preSum[i];
   
            // If subarray found with
            // all equal elements
            if (actual_sum - expected_sum <= K)
            {
                  
                // Update start
                start = mid + 1;
   
                // Update max_index
                max_index = Math.Max(max_index, mid);
            }
   
            // If false, it means that
            // the selected range is invalid
            else
            {
                  
                // Update end
                end = mid - 1;
            }
        }
   
        // Store the length of longest subarray
        res = Math.Max(res, max_index - i + 1);
    }
   
    // Return result
    return res;
}
  
// Driver Code
static void Main()
{
    int[] arr = { 1, 7, 3, 4, 5, 6 };
    int k = 6;
    int n = 6;
      
    Console.WriteLine(longestSubArray(arr, n, k));
}
}
 
// This code is contributed by susmitakundugoaldanga

输出：

4

时间复杂度： O(N * (log(N)) ² )
辅助空间：O(N)

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