3201. Find the Maximum Length of Valid Subsequence I

Difficulty: Medium

Topics: Array, Dynamic Programming

You are given an integer array nums.

A subsequence¹ sub of nums with length x is called valid if it satisfies:

• (sub[0] + sub[1]) % 2 == (sub[1] + sub[2]) % 2 == ... == (sub[x - 2] + sub[x - 1]) % 2.

Return the length of the longest valid subsequence of nums.

A subsequence is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.

Example 1:

• Input: nums = [1,2,3,4]
• Output: 4
• Explanation: The longest valid subsequence is [1, 2, 3, 4].

Example 2:

• Input: nums = [1,2,1,1,2,1,2]
• Output: 6
• Explanation: The longest valid subsequence is [1, 2, 1, 2, 1, 2].

Example 3:

• Input: nums = [1,3]
• Output: 2
• Explanation: The longest valid subsequence is [1, 3].

Constraints:

• 2 <= nums.length <= 2 * 105
• 1 <= nums[i] <= 107

Hint:

1. The possible sequence either contains all even elements, all odd elements, alternate even odd, or alternate odd even elements.
2. Considering only the parity of elements, there are only 4 possibilities and we can try all of them.
3. When selecting an element with any parity, try to select the earliest one.

Solution:

We need to find the length of the longest valid subsequence in an array where the sum of every pair of consecutive elements in the subsequence has the same parity (either all even or all odd). The solution involves analyzing the array based on the parity of its elements and leveraging dynamic programming to efficiently compute the result.

Approach

1. Problem Analysis: The problem requires finding a subsequence where the sum of every consecutive pair of elements has the same parity. This condition can be satisfied in two scenarios:

   • Same Parity Subsequence: All elements in the subsequence are either even or odd. The sum of any two consecutive even or odd numbers is even, satisfying the condition.
   • Alternating Parity Subsequence: The subsequence alternates between even and odd elements. The sum of an even and an odd number is odd, which also satisfies the condition if all consecutive pairs have odd sums.

2. Key Insights:

   • Same Parity: The longest valid subsequence of all even or all odd elements is simply the count of all even or all odd elements in the array, provided there are at least two such elements.
   • Alternating Parity: The longest alternating subsequence can be found using dynamic programming by maintaining two variables:
     • even_len: Length of the longest alternating subsequence ending with an even number.
     • odd_len: Length of the longest alternating subsequence ending with an odd number.
   • For each element in the array, update even_len or odd_len based on the element's parity. If the element is even, even_len is updated to the maximum of its current value or odd_len + 1 (indicating extending an alternating sequence ending with an odd number). Similarly, if the element is odd, odd_len is updated to the maximum of its current value or even_len + 1.

3. Algorithm Selection:

   • Traverse the array once to count the number of even and odd elements.
   • During the traversal, dynamically update even_len and odd_len to track the longest alternating subsequence.
   • Compute the candidate lengths for same parity (if feasible) and alternating parity subsequences.
   • The result is the maximum among the longest same parity subsequence and the longest alternating subsequence (if its length is at least 2).

4. Complexity Analysis:

   • Time Complexity: O(n), where n is the number of elements in the array. The solution processes each element exactly once.
   • Space Complexity: O(1), as only a few variables are used to store intermediate results.

Let's implement this solution in PHP: 3201. Find the Maximum Length of Valid Subsequence I

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

// Test cases
$nums1 = [1, 2, 3, 4];            // Output: 4
$nums2 = [1, 2, 1, 1, 2, 1, 2];   // Output: 6
$nums3 = [1, 3];                  // Output: 2

echo maximumLength($nums1) . "\n";
echo maximumLength($nums2) . "\n";
echo maximumLength($nums3) . "\n";
?>

Explanation:

1. Initialization: Variables count_even and count_odd are initialized to zero to count the number of even and odd elements. even_len and odd_len are initialized to zero to track the lengths of the longest alternating subsequences ending with an even or odd number, respectively.
2. Traversal: The array is traversed once. For each element:
   • If the element is even, count_even is incremented, and even_len is updated to the maximum of its current value or odd_len + 1 (indicating the extension of an alternating sequence ending with an odd number).
   • If the element is odd, count_odd is incremented, and odd_len is updated to the maximum of its current value or even_len + 1 (indicating the extension of an alternating sequence ending with an even number).
3. Candidate Calculation:
   • candidate_same is computed as the maximum count of even or odd elements, provided there are at least two such elements.
   • candidate_alt is the maximum of even_len and odd_len, set to zero if less than 2 (since a valid subsequence must have at least two elements).
4. Result: The result is the maximum value between candidate_same and candidate_alt, representing the longest valid subsequence satisfying the given conditions.

This approach efficiently checks both same parity and alternating parity scenarios in linear time, ensuring optimal performance even for large input sizes.

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1. Subsequence: A subsequence is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements. ↩