3354. Make Array Elements Equal to Zero

3354. Make Array Elements Equal to Zero

Difficulty: Easy

Topics: Array, Simulation, Prefix Sum, Weekly Contest 424

You are given an integer array nums.

Start by selecting a starting position curr such that nums[curr] == 0, and choose a movement direction of either left or right.

After that, you repeat the following process:

• If curr is out of the range [0, n - 1], this process ends.
• If nums[curr] == 0, move in the current direction by incrementing curr if you are moving right, or decrementing curr if you are moving left.
• Else if nums[curr] > 0:
  • Decrement nums[curr] by 1.
  • Reverse your movement direction (left becomes right and vice versa).
  • Take a step in your new direction.

A selection of the initial position curr and movement direction is considered valid if every element in nums becomes 0 by the end of the process.

Return the number of possible valid selections.

Example 1:

• Input: nums = [1,0,2,0,3]
• Output: 2
• Explanation: The only possible valid selections are the following:
  • Choose curr = 3, and a movement direction to the left.
    • [1,0,2,0,3] -> [1,0,2,0,3] -> [1,0,1,0,3] -> [1,0,1,0,3] -> [1,0,1,0,2] -> [1,0,1,0,2] -> [1,0,0,0,2] -> [1,0,0,0,2] -> [1,0,0,0,1] -> [1,0,0,0,1] -> [1,0,0,0,1] -> [1,0,0,0,1] -> [0,0,0,0,1] -> [0,0,0,0,1] -> [0,0,0,0,1] -> [0,0,0,0,1] -> [0,0,0,0,0].
  • Choose curr = 3, and a movement direction to the right.
    • [1,0,2,0,3] -> [1,0,2,0,3] -> [1,0,2,0,2] -> [1,0,2,0,2] -> [1,0,1,0,2] -> [1,0,1,0,2] -> [1,0,1,0,1] -> [1,0,1,0,1] -> [1,0,0,0,1] -> [1,0,0,0,1] -> [1,0,0,0,0] -> [1,0,0,0,0] -> [1,0,0,0,0] -> [1,0,0,0,0] -> [0,0,0,0,0].

Example 2:

• Input: nums = [2,3,4,0,4,1,0]
• Output: 0
• Explanation: There are no possible valid selections.

Example 3:

• Input: [81,100,95,100,98,90,91,97,100,93,95,93,97,82,100,90,100,92,91,100,0,0,0,0,0,0,38,47,33,44,41,53,47,43,41,49,44,52,44,41,45,42,43,42,45,39,52,49,42,38,43,53,48,47,38,41,42,47,45,41,43,45,39,40,50,48,34,44,42]
• Output: 6

Constraints:

• 1 <= nums.length <= 100
• 0 <= nums[i] <= 100
• There is at least one element i where nums[i] == 0.

Hint:

1. Since the constraints are very small, you can simulate the process described.

Solution:

We need to find all starting positions and directions where beginning at a zero and following the described process will eventually make all array elements become zero.

Let me analyze the examples to understand the pattern:

Key Observations:

1. The process can only start at positions where nums[i] == 0
2. When we encounter a non-zero element, we decrement it and reverse direction
3. The process ends when we move out of bounds

Let me think about what makes a valid selection:

• The total number of times we visit each position must equal its initial value
• The path must be "balanced" - we need to visit each position the exact right number of times

Approach:
I'll simulate the process for each possible starting position (where nums[i] == 0) and both directions, counting how many lead to all zeros.

Let's implement this solution in PHP: 3354. Make Array Elements Equal to Zero

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

/**
 * Helper function to simulate the process
 *
 * @param $nums
 * @param $start
 * @param $dir
 * @return bool
 */
function simulateProcess($nums, $start, $dir): bool
{
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
$nums1 = [1, 0, 2, 0, 3];
$nums2 = [2, 3, 4, 0, 4, 1, 0];
$nums3 = [81,100,95,100,98,90,91,97,100,93,95,93,97,82,100,90,100,92,91,100,0,0,0,0,0,0,38,47,33,44,41,53,47,43,41,49,44,52,44,41,45,42,43,42,45,39,52,49,42,38,43,53,48,47,38,41,42,47,45,41,43,45,39,40,50,48,34,44,42];

echo countValidSelections($nums1) . PHP_EOL; // Output: 2
echo countValidSelections($nums2) . PHP_EOL; // Output: 0
echo countValidSelections($nums3) . PHP_EOL; // Output: 6
?>

Explanation:

1. countValidSelections function:

   • Iterates through all positions in the array
   • For each position that contains zero, tries both left and right directions
   • Calls simulateProcess to check if the selection is valid

2. simulateProcess function:

   • Simulates the entire process according to the given rules
   • Starts at the specified position with the specified direction
   • Follows the movement rules until going out of bounds
   • Checks if all array elements become zero by the end

3. Movement Rules:

   • If current element is zero: move in current direction
   • If current element is non-zero: decrement it, reverse direction, and move

Time Complexity: O(n² × max_value) in worst case, but acceptable for the given constraints (n ≤ 100)
Space Complexity: O(n) for the copy of the array

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