3505. Minimum Operations to Make Elements Within K Subarrays Equal

Difficulty: Hard

Topics: Principal, Array, Hash Table, Math, Dynamic Programming, Sliding Window, Heap (Priority Queue), Weekly Contest 443

You are given an integer array nums and two integers, x and k. You can perform the following operation any number of times (including zero):

Increase or decrease any element of nums by 1.

Return the minimum number of operations needed to have at least k non-overlapping subarrays¹ of size exactly x in nums, where all elements within each subarray are equal.

Example 1:

Input: nums = [5,-2,1,3,7,3,6,4,-1], x = 3, k = 2
Output: 8
Explanation:
- Use 3 operations to add 3 to nums[1] and use 2 operations to subtract 2 from nums[3]. The resulting array is [5, 1, 1, 1, 7, 3, 6, 4, -1].
- Use 1 operation to add 1 to nums[5] and use 2 operations to subtract 2 from nums[6]. The resulting array is [5, 1, 1, 1, 7, 4, 4, 4, -1].
- Now, all elements within each subarray [1, 1, 1] (from indices 1 to 3) and [4, 4, 4] (from indices 5 to 7) are equal. Since 8 total operations were used, 8 is the output.

Example 2:

Input: nums = [9,-2,-2,-2,1,5], x = 2, k = 2
Output: 3
Explanation:
- Use 3 operations to subtract 3 from nums[4]. The resulting array is [9, -2, -2, -2, -2, 5].
- Now, all elements within each subarray [-2, -2] (from indices 1 to 2) and [-2, -2] (from indices 3 to 4) are equal. Since 3 operations were used, 3 is the output. Example 3:
Input: nums = [1,1,1,1,1], x = 2, k = 2
Output: 0

Example 4:

Input: nums = [-5,-5,-5,-5,-5], x = 2, k = 2
Output: 0

Example 5:

Input: nums = [1,2,3,4], x = 2, k = 2
Output: 2

Example 6:

Input: nums = [5,1,3], x = 3, k = 1
Output: 4

Example 7:

Input: nums = [3,1,2,4,5,6], x = 2, k = 3
Output: 5

Example 8:

Input: nums = [1,2,3,4,5,6,7,8], x = 4, k = 2
Output: 8

Example 9:

Input: nums = [-1000000,1000000,-1000000,1000000], x = 2, k = 2
Output: 4000000

Example 10:

Input: nums = [0,0,0,100,100,0,0,0], x = 3, k = 2
Output: 0

Example 11:

Input: nums = [10,-10,10,-10,10,-10], x = 2, k = 3
Output: 60

Example 12:

Input: nums = [1,1,2,2,3,3,4,4], x = 4, k = 2
Output: 8

Constraints:

2 <= nums.length <= 10⁵
-10⁶ <= nums[i] <= 10⁶
2 <= x <= nums.length
1 <= k <= 15
2 <= k * x <= nums.length

Hint:

Making every element of an x-sized window equal to its median is optimal.
Precalculate this for each window.

Similar Questions:

Solution:

We solve this problem by first precomputing the minimum cost to make every x-sized window uniform (using the median as target), then applying a prefix DP to select k non-overlapping windows with minimum total cost. Window costs are computed in O(log n) per position using two Fenwick trees (BITs) — one for counts, one for sums — over coordinate-compressed values, enabling efficient sliding-window median and cost queries.

Approach

Coordinate compress all nums values so BIT indices are bounded by m ≤ n.
Two Fenwick trees (bitCount, bitSum) maintain the multiset of values inside the current window of size x.
Initial window fill: insert the first x elements into both BITs.
Sliding window: for each start index 0 … n-x, do:
- Find the lower median ((x+1)/2-th smallest) via BIT kth query.
- Compute sum |aᵢ − median| using prefix count/sum at the median rank.
- Slide: remove nums[start], insert nums[start+x], updating both BITs and the running window sum.
DP for window selection: prev[i] = min cost using chosen-1 windows among first i elements.
- Transition: cur[i] = min(cur[i-1], prev[i-x] + cost[i-x]).
- Iterate chosen = 1…k, return prev[n].

Let's implement this solution in PHP: 3505. Minimum Operations to Make Elements Within K Subarrays Equal

<?php /** * @param Integer[] $nums * @param Integer $x * @param Integer $k * @return Integer */ function minOperations(array $nums, int $x, int $k): int { ... ... ... /** * go to ./solution.php */ } /** * @param $bitCount * @param $bitSum * @param $idx * @param $deltaCount * @param $deltaSum * @param $m * @return void */ function add(&$bitCount, &$bitSum, $idx, $deltaCount, $deltaSum, $m): void { ... ... ... /** * go to ./solution.php */ } /** * @param $bitCount * @param $k * @param $m * @param $highestPower * @return int|mixed */ function kth($bitCount, $k, $m, $highestPower): mixed { ... ... ... /** * go to ./solution.php */ } // Test cases echo minOperations([5,-2,1,3,7,3,6,4,-1], 3, 2) . "\n"; // Output: 8 echo minOperations([9,-2,-2,-2,1,5], 2, 2) . "\n"; // Output: 3 echo minOperations([1,1,1,1,1], 2, 2) . "\n"; // Output: 0 echo minOperations([-5,-5,-5,-5,-5], 2, 2) . "\n"; // Output: 0 echo minOperations([1,2,3,4], 2, 2) . "\n"; // Output: 2 echo minOperations([5,1,3], 3, 1) . "\n"; // Output: 4 echo minOperations([3,1,2,4,5,6], 2, 3) . "\n"; // Output: 5 echo minOperations([1,2,3,4,5,6,7,8], 4, 2) . "\n"; // Output: 8 echo minOperations([-1000000,1000000,-1000000,1000000], 2, 2) . "\n"; // Output: 4000000 echo minOperations([0,0,0,100,100,0,0,0], 3, 2) . "\n"; // Output: 0 echo minOperations([10,-10,10,-10,10,-10], 2, 3) . "\n"; // Output: 60 echo minOperations([1,1,2,2,3,3,4,4], 4, 2) . "\n"; // Output: 4 ?>

Explanation

Why median? For a fixed window, sum |aᵢ − m| is minimized at the median (hint 1). Any even-sized window can use the lower median without loss.
Why BIT? We need the median and the count/sum split around it in O(log n) while sliding. Two BITs give count and sum for the rank query, and prefix sums up to the median let us split the window into left/right halves.
Cost formula:
- cost = median * cntLE − sumLE + (windowSum − sumLE) − median * (x − cntLE)
- First half: elements ≤ median — sum of (median − aᵢ).
- Second half: elements > median — sum of (aᵢ − median).
Non-overlap enforcement in DP: Placing a window ending at i−1 (starting at i−x) consumes exactly those x slots, so prev[i−x] uses positions before the window starts — guaranteeing non-overlap by construction.
"At least k" vs "exactly k": Since each window cost is non-negative, min over exactly k equals min over ≥ k. The constraint k*x ≤ n guarantees feasibility.
Lower median handling for even x: Using the lower median still yields the minimum L1 cost, so medianRank = (x+1)/2 is correct for both odd and even window sizes.

Complexity Analysis

• Time Complexity:

Coordinate compression: O(n log n)
Initial window fill: O(x log n)
Sliding window (n − x + 1 steps, each O(log n)): O(n log n)
DP: O(k · n) ≤ O(15 · 10⁵)
Total: O(n log n + nk) • Space Complexity:
bitCount, bitSum arrays of size m + 1: O(n)
vals, map, nums, prev, cur, cost: O(n)
Total: O(n)

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