You are given an integer array nums and two integers a and b.
For a subarray, let:
x be the number of even elements.y be the number of odd elements.The ratio of even to odd elements in a subarray is defined as x / y, where ratios are compared by their exact rational values.
A subarray is considered valid if:
y > 0, andx / y <= a / b.Return the number of valid subarrays in nums.
Example 1:
Input: nums = [1,2,1,2], a = 3, b = 2
Output: 7
Explanation:
The following are the valid subarrays:
| Subarray | Values | Even Count | Odd Count | Ratio |
|---|---|---|---|---|
nums[0..0] |
[1] |
0 | 1 | 0 / 1 |
nums[0..1] |
[1, 2] |
1 | 1 | 1 / 1 |
nums[0..2] |
[1, 2, 1] |
1 | 2 | 1 / 2 |
nums[0..3] |
[1, 2, 1, 2] |
2 | 2 | 2 / 2 |
nums[1..2] |
[2, 1] |
1 | 1 | 1 / 1 |
nums[2..2] |
[1] |
0 | 1 | 0 / 1 |
nums[2..3] |
[1, 2] |
1 | 1 | 1 / 1 |
Thus, the number of valid subarrays is 7.
Example 2:
Input: nums = [2,2,1], a = 2, b = 1
Output: 3
Explanation:
The following are the valid subarrays:
| Subarray | Values | Even Count | Odd Count | Ratio |
|---|---|---|---|---|
nums[0..2] |
[2, 2, 1] |
2 | 1 | 2 / 1 |
nums[1..2] |
[2, 1] |
1 | 1 | 1 / 1 |
nums[2..2] |
[1] |
0 | 1 | 0 / 1 |
Thus, the number of valid subarrays is 3.
Example 3:
Input: nums = [2,2,2], a = 1, b = 1
Output: 0
Explanation:
Every subarray contains 0 odd numbers, so no subarray is valid.
Constraints:
1 <= nums.length <= 1051 <= nums[i] <= 1091 <= a, b <= 109​​​​​​​Replace every even element with b and every odd element with -a. A subarray is valid exactly when its transformed sum is at most 0.
The condition y > 0 is then automatic, because a non-empty subarray containing only even elements has a positive transformed sum.
Let pref[i] be the prefix sum of the transformed array. A subarray [l, r] is valid when pref[r + 1] <= pref[l].
Scan the prefix sums from left to right and count how many previous prefix sums are greater than or equal to the current one using coordinate compression and a Fenwick tree.
[1,2,1,2] 3 2
[2,2,1] 2 1
[2,2,2] 1 1