DEV Community

Cover image for 3535. Find X Value of Array II
MD ARIFUL HAQUE
MD ARIFUL HAQUE

Posted on

3535. Find X Value of Array II

#php #leetcode #algorithms #programming

3535. Find X Value of Array II

Difficulty: Hard

Topics: Senior Staff, Array, Math, Segment Tree, Weekly Contest 446

You are given an array of positive integers nums and a positive integer k. You are also given a 2D array queries, where queries[i] = [indexᵢ, valueᵢ, startᵢ, xᵢ].

You are allowed to perform an operation once on nums, where you can remove any suffix from nums such that nums remains non-empty.

The x-value of nums for a given x is defined as the number of ways to perform this operation so that the product of the remaining elements leaves a remainder of x modulo k.

For each query in queries you need to determine the x-value of nums for xᵢ after performing the following actions:

  • Update nums[indexᵢ] to valueᵢ. Only this step persists for the rest of the queries.
  • Remove the prefix nums[0..(startᵢ - 1)] (where nums[0..(-1)] will be used to represent the empty prefix).

Return an array result of size queries.length where result[i] is the answer for the iᵗʰ query.

A prefix of an array is a subarray[^1] that starts from the beginning of the array and extends to any point within it.

A suffix of an array is a subarray[^1] that starts at any point within the array and extends to the end of the array.

Note that the prefix and suffix to be chosen for the operation can be empty.

Note that x-value has a different definition in this version.

Example 1:

  • Input: nums = [1,2,3,4,5], k = 3, queries = [[2,2,0,2],[3,3,3,0],[0,1,0,1]]
  • Output: [2,2,2]
  • Explanation:
    • For query 0, nums becomes [1, 2, 2, 4, 5], and the empty prefix must be removed. The possible operations are:
    • Remove the suffix [2, 4, 5]. nums becomes [1, 2].
    • Remove the empty suffix. nums becomes [1, 2, 2, 4, 5] with a product 80, which gives remainder 2 when divided by 3.
    • For query 1, nums becomes [1, 2, 2, 3, 5], and the prefix [1, 2, 2] must be removed. The possible operations are:
    • Remove the empty suffix. nums becomes [3, 5].
    • Remove the suffix [5]. nums becomes [3].
    • For query 2, nums becomes [1, 2, 2, 3, 5], and the empty prefix must be removed. The possible operations are:
    • Remove the suffix [2, 2, 3, 5]. nums becomes [1].
    • Remove the suffix [3, 5]. nums becomes [1, 2, 2].

Example 2:

  • Input: nums = [1,2,4,8,16,32], k = 4, queries = [[0,2,0,2],[0,2,0,1]]
  • Output: [1,0]
  • Explanation:
    • For query 0, nums becomes [2, 2, 4, 8, 16, 32]. The only possible operation is:
    • Remove the suffix [2, 4, 8, 16, 32].
    • For query 1, nums becomes [2, 2, 4, 8, 16, 32]. There is no possible way to perform the operation.

Example 3:

  • Input: nums = [1,1,2,1,1], k = 2, queries = [[2,1,0,1]]
  • Output: [5]

Example 4:

  • Input: nums = [5], k = 2, queries = [[0,1,0,1]]
  • Output: [1]

Example 5:

  • Input: nums = [2,2], k = 3, queries = [[0,2,0,0]]
  • Output: [0]

Example 6:

  • Input: nums = [1,2,3], k = 2, queries = [[1,2,0,0],[1,1,0,1]]
  • Output: [2,3]

Constraints:

  • 1 <= nums[i] <= 10⁹
  • 1 <= nums.length <= 10⁵
  • 1 <= k <= 5
  • 1 <= queries.length <= 2 * 10⁴
  • queries[i] == [indexᵢ, valueᵢ, startᵢ, xᵢ]
  • 0 <= indexᵢ <= nums.length - 1
  • 1 <= valueᵢ <= 10⁹
  • 0 <= startᵢ <= nums.length - 1
  • 0 <= xᵢ <= k - 1

Hint:

  1. Use a segment tree to efficiently maintain and merge product prefix information for the array nums.
  2. In each segment tree node, store a frequency count of prefix product remainders for every x in the range [0, k - 1].
  3. For each query, update nums[index] to value, then merge the segments corresponding to nums[start..n - 1] to compute the x-value for xᵢ.

Solution:

We maintain a segment tree where each node stores the product modulo k of its segment and a frequency array of non-empty prefix product remainders. After each point update, a range query over nums[start..n-1] merges segment nodes in order, and the answer is the frequency of remainder x.

Approach

  • Removing prefix nums[0..start-1] leaves the subarray nums[start..n-1].
  • Removing a suffix while keeping the array non-empty is equivalent to choosing a non-empty prefix of this remaining subarray.
  • So the task becomes: count non-empty prefixes of nums[start..n-1] whose product modulo k equals x.
  • Each segment-tree node stores:
    • prod: product of the whole segment modulo k.
    • cnt[r]: number of non-empty prefixes of this segment whose product modulo k is r.
  • Merge two adjacent segments L then R:
    • New product = prodL * prodR % k.
    • Prefixes are either from L, or the whole L followed by a prefix of R.
    • For every remainder y with cntR[y] > 0, add it to remainder (prodL * y) % k.
  • Build the tree bottom-up.
  • For each query:
    • Update nums[index] = value in the tree and recompute ancestors.
    • Query range [start, n - 1] by merging segment nodes in left-to-right order.
    • Return mergedCnt[x].

Let's implement this solution in PHP: 3535. Find X Value of Array II

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

/**
 * @param $prod
 * @param $cnt
 * @param $i
 * @param $k
 * @return void
 */
function pull(&$prod, &$cnt, $i, $k): void
{
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

/**
 * @param $leftProd
 * @param $leftCnt
 * @param $rightProd
 * @param $rightCnt
 * @param $k
 * @return void
 */
function mergeLeft(&$leftProd, &$leftCnt, $rightProd, $rightCnt, $k): void
{
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

/**
 * @param $nodeProd
 * @param $nodeCnt
 * @param $rightProd
 * @param $rightCnt
 * @param $k
 * @return void
 */
function mergeRight($nodeProd, $nodeCnt, &$rightProd, &$rightCnt, $k): void
{
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
echo resultArray([1,2,3,4,5], 3, [[2,2,0,2],[3,3,3,0],[0,1,0,1]]) .  "\n";      // Output: [2, 2, 2]
echo resultArray([1,2,4,8,16,32], 4, [[0,2,0,2],[0,2,0,1]]) .  "\n";            // Output: [1, 0]
echo resultArray([1,1,2,1,1], 2, [[2,1,0,1]]) .  "\n";                          // Output: [5]
echo resultArray([5], 2, [[0,1,0,1]]) .  "\n";                                  // Output: [1]
echo resultArray([2,2], 3, [[0,2,0,0]]) .  "\n";                                // Output: [0]
echo resultArray([1,2,3], 2, [[1,2,0,0],[1,1,0,1]]) .  "\n";                    // Output: [2,3]
?>
Enter fullscreen mode Exit fullscreen mode

Explanation:

  • The segment tree avoids recomputing prefix products from scratch for every query.
  • Since k <= 5, each merge only loops over at most 5 remainders, making merges very cheap.
  • The update is persistent because the tree is updated before processing the next query.
  • The range query preserves order using left and right accumulators:
    • Left accumulator collects segments from the left side.
    • Right accumulator collects segments from the right side.
    • Finally, they are merged in the correct order.
  • The provided PHP code implements exactly this logic.

Complexity Analysis

  • Let n = nums.length, q = queries.length, and k <= 5.
  • Build: O(n * k)
  • Each update: O(k log n)
  • Each query: O(k log n)
  • Total time: O(k * (n + q log n))
  • Space: O(k * n) for the segment tree frequency arrays.

Contact Links

If you found this series helpful, please consider giving the repository a star on GitHub or sharing the post on your favorite social networks 😍. Your support would mean a lot to me!
Buy Me A Coffee

If you want more helpful content like this, feel free to follow me:

Top comments (0)