#3261

Hard

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Count Substrings That Satisfy K-Constraint II

Count Substrings That Satisfy K-Constraint II requires finding valid substrings in a binary string based on a k-constraint.

array string binary search sliding window prefix sum

Problem Statement

You are given a binary string s and an integer k. A binary string satisfies the k-constraint if it satisfies one of the following conditions: the number of '1's or the number of '0's in the substring should not exceed k. The task is to find the number of valid substrings within the given binary string s for each query in the list of queries.

The queries are given as pairs [li, ri] which indicate the left and right bounds of the substring. You need to calculate the number of substrings within s[li..ri] that satisfy the k-constraint. Efficient solutions require binary search over valid answer spaces and possible application of prefix sums or sliding window techniques to manage the constraints and answer queries.

Examples

Example 1

Input: s = "0001111", k = 2, queries = [[0,6]]

Output: [26]

For the query [0, 6] , all substrings of s[0..6] = "0001111" satisfy the k-constraint except for the substrings s[0..5] = "000111" and s[0..6] = "0001111" .

Example 2

Input: s = "010101", k = 1, queries = [[0,5],[1,4],[2,3]]

Output: [15,9,3]

The substrings of s with a length greater than 3 do not satisfy the k-constraint.

Constraints

1 <= s.length <= 105
s[i] is either '0' or '1'.
1 <= k <= s.length
1 <= queries.length <= 105
queries[i] == [li, ri]
0 <= li <= ri < s.length
All queries are distinct.

Solution Approach

Binary Search on Answer Space

Start by using binary search to find the maximum length of substrings that satisfy the k-constraint. This will help optimize the solution by narrowing down the range of valid answers.

Sliding Window Technique

Apply a sliding window approach to quickly count substrings that meet the constraint. This reduces the problem complexity by avoiding repetitive checking of the same substrings multiple times.

Prefix Sum for Efficient Query Handling

Utilize a prefix sum to preprocess the binary string. This helps quickly calculate the number of valid substrings within any query range, reducing time complexity for large inputs.

Complexity Analysis

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

The time complexity depends on the implementation of binary search and the sliding window or prefix sum techniques. Typically, it ranges from O(n log n) to O(n) for preprocessing and answering each query. The space complexity depends on the auxiliary storage used for prefix sums or window state, usually O(n).

What Interviewers Usually Probe

Candidate demonstrates understanding of sliding window and prefix sum techniques.
Candidate approaches binary search over valid answer space for query optimization.
Candidate focuses on reducing time complexity when processing large query sets.

Common Pitfalls or Variants

Common pitfalls

Failing to optimize the solution for large inputs, resulting in slow runtime.
Incorrectly managing query bounds or assuming substrings without proper validation.
Misunderstanding the k-constraint or binary search, leading to incorrect counting of valid substrings.

Follow-up variants

Count substrings for other constraints, such as length or specific substring characters.
Generalize the problem to non-binary strings or extended constraint conditions.
Handle multiple queries simultaneously without preprocessing, testing for a less efficient solution.

FAQ

How can binary search be applied in Count Substrings That Satisfy K-Constraint II?

Binary search is used to find the maximum valid substring length that satisfies the k-constraint, optimizing the solution by narrowing down potential answers.

What is the sliding window approach used for?

The sliding window technique helps to efficiently count valid substrings by dynamically adjusting the window size and checking the constraint without redundant operations.

What role does prefix sum play in solving the problem?

Prefix sum is used to preprocess the binary string, allowing quick retrieval of valid substrings within a specific query range, thus improving efficiency.

What is the primary challenge in solving this problem?

The main challenge is efficiently answering multiple range queries on a large binary string, which requires preprocessing and optimizing with binary search or sliding window.

Can this problem be generalized to other constraints?

Yes, the problem can be adapted to handle different types of constraints, such as substring length or the occurrence of other characters beyond '0' and '1'.

terminal

Solution

Solution 1: Sliding Window + Prefix Sum

We use two variables $\textit{cnt0}$ and $\textit{cnt1}$ to record the number of $0$s and $1$s in the current window, respectively. Pointers $i$ and $j$ mark the left and right boundaries of the window. We use an array $d$ to record the first position to the right of each position $i$ that does not satisfy the $k$ constraint, initially setting $d[i] = n$. Additionally, we use a prefix sum array $\textit{pre}[i]$ of length $n + 1$ to record the number of substrings that satisfy the $k$ constraint with the right boundary at position $i$.

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class Solution:
    def countKConstraintSubstrings(
        self, s: str, k: int, queries: List[List[int]]
    ) -> List[int]:
        cnt = [0, 0]
        i, n = 0, len(s)
        d = [n] * n
        pre = [0] * (n + 1)
        for j, x in enumerate(map(int, s)):
            cnt[x] += 1
            while cnt[0] > k and cnt[1] > k:
                d[i] = j
                cnt[int(s[i])] -= 1
                i += 1
            pre[j + 1] = pre[j] + j - i + 1
        ans = []
        for l, r in queries:
            p = min(r + 1, d[l])
            a = (1 + p - l) * (p - l) // 2
            b = pre[r + 1] - pre[p]
            ans.append(a + b)
        return ans

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