Find X Value of Array I

This problem requires finding the x-value of an array by counting the number of ways to trim prefixes and suffixes while maintaining non-empty arrays. Using state transition dynamic programming efficiently tracks remainders modulo k across all subarrays. The solution balances time and space by iterating through nums and updating DP states for each operation, capturing every valid configuration for the remainder.

Problem Statement

Given an array of positive integers nums and a positive integer k, you can perform a single operation removing any non-overlapping prefix and suffix. The remaining subarray must be non-empty.

Your task is to compute the x-value of nums, defined as the number of ways this operation can be applied such that the product of the remaining elements modulo k equals each possible remainder.

Examples

Example 1

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

Output: [9,2,4]

Example 2

Input: nums = [1,2,4,8,16,32], k = 4

Output: [18,1,2,0]

Example 3

Input: nums = [1,1,2,1,1], k = 2

Output: [9,6]

Example details omitted.

Constraints

• 1 <= nums[i] <= 109
• 1 <= nums.length <= 105
• 1 <= k <= 5

Solution Approach

Use dynamic programming for remainder tracking

Initialize a DP table where dp[i][r] stores the number of ways the prefix up to i produces remainder r modulo k. Iterate through the array updating DP states using state transition rules for each element.

Combine prefix and suffix counts

After computing prefix remainders, similarly compute suffix remainders. Merge counts from prefixes and suffixes to determine total valid operations for each possible remainder.

Optimize with small k

Since k is small (<=5), store only k states per prefix and suffix to save space. Use modulo properties to avoid large number multiplications and keep the DP efficient.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time and space complexity depend on array length n and modulus k. Each element updates k DP states, resulting in O(n*k) time and O(k) additional space when optimized.

What Interviewers Usually Probe

• Check if candidate recognizes prefix-suffix state transitions.
• Observe whether modulo arithmetic is correctly applied in DP updates.
• Evaluate understanding of space optimization when k is small.

Common Pitfalls or Variants

Common pitfalls

• Forgetting to keep the remaining subarray non-empty.
• Incorrectly merging prefix and suffix DP counts.
• Not applying modulo operations at every DP update, causing overflow or wrong counts.

Follow-up variants

• Change k to a larger number, requiring more careful DP and potential pruning.
• Allow multiple operations, increasing state tracking complexity.
• Compute x-value for arrays with negative numbers, requiring adjusted modulo handling.

FAQ

What is the main pattern used in Find X Value of Array I?

The problem relies on state transition dynamic programming to track remainders of products modulo k across subarrays.

How do I ensure the remaining subarray is non-empty?

Always validate that prefix and suffix removals leave at least one element when updating DP counts.

Can this method handle large arrays?

Yes, with O(n*k) time and O(k) space using optimized DP for small k values.

Why is modulo operation applied at every step?

To prevent integer overflow and correctly count remainder occurrences for each DP state.

Is it necessary to track both prefix and suffix separately?

Yes, because valid operations depend on combinations of prefix and suffix removals to compute x-value correctly.