3737. Count Subarrays With Majority Element I

3737. Count Subarrays With Majority Element I

Difficulty: Medium

Topics: Senior, Array, Hash Table, Divide and Conquer, Segment Tree, Merge Sort, Counting, Prefix Sum, Biweekly Contest 169

You are given an integer array nums and an integer target.

Return the number of subarrays¹ of nums in which target is the majority element.

The majority element of a subarray is the element that appears strictly more than half of the times in that subarray.

Example 1:

• Input: nums = [1,2,2,3], target = 2
• Output: 5
• Explanation:
  • Valid subarrays with target = 2 as the majority element:
    • nums[1..1] = [2]
    • nums[2..2] = [2]
    • nums[1..2] = [2,2]
    • nums[0..2] = [1,2,2]
    • nums[1..3] = [2,2,3]
  • So there are 5 such subarrays.

Example 2:

• Input: nums = [1,1,1,1], target = 1
• Output: 10
• Explanation: All 10 subarrays have 1 as the majority element.

Example 3:

• Input: nums = [1,2,3], target = 4
• Output: 0
• Explanation: target = 4 does not appear in nums at all. Therefore, there cannot be any subarray where 4 is the majority element. Hence the answer is 0.

Constraints:

• 1 <= nums.length <= 1000
• 1 <= nums[i] <= 10⁹
• 1 <= target <= 10⁹

Hint:

1. Use brute force
2. Count all subarrays where 2 * count(target) > length

Solution:

We have developed an efficient brute-force solution that counts all subarrays where a given target appears as the majority element. Our approach iterates through every possible subarray, maintains a running count of target occurrences, and checks the majority condition using integer arithmetic to avoid floating-point precision issues. This solution handles the constraints effectively and produces correct results for all edge cases.

Approach

• Iterate all starting positions: For each index i in the array, we consider all subarrays that begin at i.
• Expand subarrays: For each starting index i, we extend the subarray to every possible ending index j (from i to n-1).
• Maintain running statistics: Keep track of:
  • targetCount: number of times target appears in the current subarray
  • length: total number of elements in the current subarray
• Check majority condition: After adding each new element, verify if targetCount * 2 > length. This condition is equivalent to targetCount > length / 2 and avoids floating-point division.
• Count valid subarrays: Increment the answer counter whenever the majority condition is satisfied.

Let's implement this solution in PHP: 3737. Count Subarrays With Majority Element I

<?php
/**
 * @param Integer[] $nums
 * @param Integer $target
 * @return Integer
 */
function countMajoritySubarrays(array $nums, int $target): int
{
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
echo countMajoritySubarrays([1,2,2,3], 2) . "\n"; // Output: 5
echo countMajoritySubarrays([1,1,1,1], 1) . "\n"; // Output: 10
echo countMajoritySubarrays([1,2,3], 4) . "\n";   // Output: 0
?>

Explanation:

• Brute-force strategy: Since the maximum array length is 1000, the total number of subarrays is at most 500,500, which is well within computational limits for PHP. This allows us to use a straightforward nested loop approach without needing complex optimizations.
• Running count technique: Instead of recomputing counts for each subarray from scratch, we maintain a cumulative count as we extend the subarray to the right. This reduces the inner loop work to just incrementing counters and performing a single comparison.
• Integer comparison trick: The condition targetCount * 2 > length is mathematically equivalent to targetCount > length / 2. We use multiplication by 2 instead of division to avoid floating-point arithmetic and maintain precision with integer operations.
• All subarrays are considered: Our double loop systematically examines every contiguous non-empty sequence, ensuring no valid subarray is missed.
• Zero-count handling: When target doesn't appear in the array, targetCount remains 0 throughout all iterations, so no subarray satisfies the majority condition, correctly yielding 0.

Complexity Analysis

• Time Complexity: O(n²) where n is the length of nums. The outer loop runs n times, and the inner loop runs n-i times for each i, resulting in n(n+1)/2 iterations total. For n = 1000, this is approximately 500,500 operations, which is efficient.
• Space Complexity: O(1) — We only use a constant amount of extra space for variables (count, targetCount, length, loop indices). No additional data structures are allocated that scale with input size.

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1. Subarray: A subarray is a contiguous non-empty sequence of elements within an array. ↩