1590. Make Sum Divisible by P

Difficulty: Medium

Topics: Array, Hash Table, Prefix Sum

Given an array of positive integers nums, remove the smallest subarray (possibly empty) such that the sum of the remaining elements is divisible by p. It is not allowed to remove the whole array.

Return the length of the smallest subarray that you need to remove, or -1 if it's impossible.

A subarray is defined as a contiguous block of elements in the array.

Example 1:

• Input: nums = [3,1,4,2], p = 6
• Output: 1
• Explanation: The sum of the elements in nums is 10, which is not divisible by 6. We can remove the subarray [4], and the sum of the remaining elements is 6, which is divisible by 6.

Example 2:

• Input: nums = [6,3,5,2], p = 9
• Output: 2
• Explanation: We cannot remove a single element to get a sum divisible by 9. The best way is to remove the subarray [5,2], leaving us with [6,3] with sum 9.

Example 3:

• Input: nums = [1,2,3], p = 3
• Output: 0
• Explanation: Here the sum is 6. which is already divisible by 3. Thus we do not need to remove anything.

Constraints:

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

Hint:

1. Use prefix sums to calculate the subarray sums.
2. Suppose you know the remainder for the sum of the entire array. How does removing a subarray affect that remainder? What remainder does the subarray need to have in order to make the rest of the array sum up to be divisible by k?
3. Use a map to keep track of the rightmost index for every prefix sum % p.

Solution:

We can use a combination of prefix sums and a hash table to efficiently compute the smallest subarray that needs to be removed such that the sum of the remaining elements is divisible by p.

Key Insights:

1. Prefix Sum Modulo: The sum of the entire array modulo p gives us the remainder when the array sum is divided by p. If this remainder is zero, the sum is already divisible by p, and no subarray needs to be removed. Otherwise, the goal is to remove a subarray that brings the sum modulo p to zero.

2. Target Remainder: If the total sum modulo p is r, we need to find a subarray whose sum modulo p is also r. Removing this subarray will result in the remaining sum being divisible by p.

3. Efficient Search Using a Hash Map: We can use a hash map to store the prefix sum modulo p and the index at which that sum occurs. This allows us to quickly find if we can remove a subarray that satisfies the required condition.

Approach:

1. Compute the total sum of the array, and find its remainder when divided by p. Call this remainder r.
2. Traverse through the array while maintaining the current prefix sum and looking for a previously seen prefix sum that satisfies the condition (i.e., removing the elements between the current index and the previous index will result in a sum divisible by p).
3. Use a hash map to store the last occurrence of each prefix sum modulo p.

Let's implement this solution in PHP: 1590. Make Sum Divisible by P

<?php
/**
 * @param Integer[] $nums
 * @param Integer $p
 * @return Integer
 */
function minSubarray($nums, $p) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
$nums1 = [3, 1, 4, 2];
$p1 = 6;
echo minSubarray($nums1, $p1) . "\n";  // Output: 1

$nums2 = [6, 3, 5, 2];
$p2 = 9;
echo minSubarray($nums2, $p2) . "\n";  // Output: 2

$nums3 = [1, 2, 3];
$p3 = 3;
echo minSubarray($nums3, $p3) . "\n";  // Output: 0
?>

Explanation:

1. Initialization:

   • We calculate the total sum of the array and its remainder when divided by p. If the remainder r is zero, the array is already divisible by p, so we return 0.
   • We initialize a hash map (prefixMap) to store the last occurrence of each prefix sum modulo p. The map starts with 0 => -1 to handle cases where the subarray starts from the beginning.

2. Prefix Sum Calculation:

   • As we iterate through the array, we maintain a running prefixSum. At each step, we compute prefixSum % p.

3. Finding the Target:

   • We need to check if there exists a previous prefix sum that, when subtracted from the current prefix sum, results in a sum divisible by p. This is achieved by calculating the target = (prefixSum - r + p) % p.

4. Subarray Removal:

   • If the target is found in the prefixMap, we compute the length of the subarray that would need to be removed to achieve the desired sum and update the minLength accordingly.

5. Final Result:

   • If no valid subarray is found (minLength remains PHP_INT_MAX), we return -1. Otherwise, we return the smallest subarray length found.

Time Complexity:

• The solution runs in O(n) time, where n is the length of the array, since we traverse the array once and each operation (like hash map lookup and insertion) takes constant time.

Space Complexity:

• The space complexity is O(n) due to the hash map storing at most n prefix sums.

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