What is the time complexity of Number of Subarrays That Match a Pattern II?

The optimal solution has a time complexity of O(n+m).

What data structures are used to solve Number of Subarrays That Match a Pattern II?

This problem uses Array, Rolling Hash, String Matching, Hash Function. Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving Number of Subarrays That Match a Pattern II?

Insight 1: The problem can be transformed into a string matching problem where we convert the differences between consecutive numbers in `nums` into a sequence of -1, 0, and 1. Insight 2: The KMP (Knuth-Morris-Pratt) algorithm is an efficient way to search for occurrences of the `pattern` within the transformed `nums` sequence.

Which companies ask LeetCode 3036. Number of Subarrays That Match a Pattern II in interviews?

This problem is frequently asked at Autodesk, ThoughtWorks.

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Number of Subarrays That Match a Pattern II - LeetCode 3036 Solution

Q: How to solve Number of Subarrays That Match a Pattern II?

The provided solution uses the KMP algorithm to efficiently find matching subarrays. First, the differences between consecutive elements in `nums` are converted into a sequence of -1, 0, and 1. Then, the KMP algorithm is applied to search for the `pattern` within this transformed sequence. The LPS array is precomputed for the `pattern` to optimize the search process by allowing the algorithm to avoid re-comparing portions of the text that have already been matched.

Number of Subarrays That Match a Pattern II - Complete Solution Guide

Number of Subarrays That Match a Pattern II is LeetCode problem 3036, a Hard level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

You are given a 0-indexed integer array nums of size n , and a 0-indexed integer array pattern of size m consisting of integers -1 , 0 , and 1 . A subarray nums[i..j] of size m + 1 is said to match the pattern if the following conditions hold for each element pattern[k] : nums[i + k + 1] > nums[i + k] if pattern[k] == 1 . nums[i + k + 1] == nums[i + k] if pattern[k] == 0 . nums[i + k + 1] < nums[i + k] if pattern[k] == -1 . Return the count of subarrays in nums that match the pattern . Example 1

Detailed Explanation

The problem requires us to find the number of subarrays within a given integer array `nums` that match a specified `pattern` array. The `pattern` array consists of -1, 0, and 1, representing the relationship between consecutive elements in the `nums` array. A subarray `nums[i..j]` matches the `pattern` if, for each element `pattern[k]`, the relationship between `nums[i + k + 1]` and `nums[i + k]` satisfies the condition defined by `pattern[k]`. Specifically, if `pattern[k]` is 1, then `nums[i + k + 1]` must be greater than `nums[i + k]`. If `pattern[k]` is 0, they must be equal, and if `pattern[k]` is -1, `nums[i + k + 1]` must be less than `nums[i + k]`. The goal is to count how many such matching subarrays exist within `nums`.

Solution Approach

The provided solution uses the KMP algorithm to efficiently find matching subarrays. First, the differences between consecutive elements in `nums` are converted into a sequence of -1, 0, and 1. Then, the KMP algorithm is applied to search for the `pattern` within this transformed sequence. The LPS array is precomputed for the `pattern` to optimize the search process by allowing the algorithm to avoid re-comparing portions of the text that have already been matched.

Step-by-Step Algorithm

Step 1: Construct the LPS (Longest Proper Prefix Suffix) array for the `pattern` array. The LPS array stores, for each index `i` in `pattern`, the length of the longest proper prefix of `pattern[0...i]` that is also a suffix of `pattern[0...i]`.
Step 2: Iterate through the `nums` array, calculating the difference between consecutive elements and converting them into -1, 0, or 1 based on whether `nums[i+1]` is less than, equal to, or greater than `nums[i]`, respectively.
Step 3: Use the KMP algorithm to search for the `pattern` in the sequence of differences generated in Step 2. Keep track of the current index `j` in the `pattern`.
Step 4: If a mismatch is found during the KMP matching, update `j` using the LPS array to shift the pattern to the right, avoiding unnecessary comparisons.
Step 5: If a complete match of the `pattern` is found (i.e., `j` reaches the length of `pattern`), increment the count of matching subarrays and reset `j` using the LPS array to continue searching for further matches.
Step 6: Return the final count of matching subarrays.

Key Insights

Insight 1: The problem can be transformed into a string matching problem where we convert the differences between consecutive numbers in `nums` into a sequence of -1, 0, and 1.
Insight 2: The KMP (Knuth-Morris-Pratt) algorithm is an efficient way to search for occurrences of the `pattern` within the transformed `nums` sequence.
Insight 3: The Longest Proper Prefix Suffix (LPS) array helps to avoid unnecessary comparisons during the pattern matching phase of the KMP algorithm.

Complexity Analysis

Time Complexity: O(n+m)

Space Complexity: O(m)

Topics

This problem involves: Array, Rolling Hash, String Matching, Hash Function.

Companies

Asked at: Autodesk, ThoughtWorks.

Number of Subarrays That Match a Pattern II | LeetCode 3036

Hard

Array Rolling Hash String Matching Hash Function

Time: O(n+m)Space: O(m)

Solution Code

Available in:

class Solution {
    public int countMatchingSubarrays(int[] nums, int[] pattern) {
        int n = nums.length;
        int m = pattern.length;

        int[] lps = new int[m];
        int length = 0;
        int i = 1;
        while (i < m) {
            if (pattern[i] == pattern[length]) {
                length++;
                lps[i] = length;
                i++;
            } else {
                if (length != 0) {
                    length = lps[length - 1];
                } else {
                    lps[i] = 0;
                    i++;
                }
            }
        }

        int count = 0;
        int j = 0;
        for (i = 0; i < n - 1; i++) {
            int current_s_val = (nums[i + 1] > nums[i]) ? 1 : ((nums[i + 1] < nums[i]) ? -1 : 0);

            while (j > 0 && pattern[j] != current_s_val) {
                j = lps[j - 1];
            }

            if (pattern[j] == current_s_val) {
                j++;
            }

            if (j == m) {
                count++;
                j = lps[j - 1];
            }
        }

        return count;
    }
}

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Problem Info

Description

Problem 3036: Number of Subarrays That Match a Pattern II - Detailed explanation and solution approaches.

Topics

Array Rolling Hash String Matching Hash Function

Companies

AutodeskThoughtWorks

Stats

Difficulty:Hard

Languages:4

Companies:2

Problem #:3036

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

6 steps

1
Step 1: Construct the LPS (Longest Proper Prefix Suffix) array for the `pattern` array. The LPS array stores, for each index `i` in `pattern`, the length of the longest proper prefix of `pattern[0...i]` that is also a suffix of `pattern[0...i]`.
2
Step 2: Iterate through the `nums` array, calculating the difference between consecutive elements and converting them into -1, 0, or 1 based on whether `nums[i+1]` is less than, equal to, or greater than `nums[i]`, respectively.
3
Step 3: Use the KMP algorithm to search for the `pattern` in the sequence of differences generated in Step 2. Keep track of the current index `j` in the `pattern`.
4
Step 4: If a mismatch is found during the KMP matching, update `j` using the LPS array to shift the pattern to the right, avoiding unnecessary comparisons.
5
Step 5: If a complete match of the `pattern` is found (i.e., `j` reaches the length of `pattern`), increment the count of matching subarrays and reset `j` using the LPS array to continue searching for further matches.
6
Step 6: Return the final count of matching subarrays.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n+m)

💾

Space Complexity

O(m)

📝

Why?

The KMP algorithm's efficiency stems from its use of the LPS array, which avoids redundant comparisons. The LPS array construction iterates through the pattern array once, and the pattern matching iterates through the modified nums array once. Therefore, the time complexity is linear with respect to the lengths of both arrays. The dominant space complexity comes from the LPS array, which grows linearly with the pattern's length.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Brute-Force Approach

Iterate through all possible subarrays of `nums` with length `m + 1` and, for each subarray, check if it matches the `pattern`. This involves comparing consecutive elements within the subarray according to the rules defined by the pattern.

Time: O(n*m*m), where n is the length of `nums` and m is the length of `pattern`. There are O(n) possible starting positions for subarrays, and for each subarray, we need to perform O(m) comparisons, each consisting of O(m) work.Space: O(1). No extra space is needed besides a few variables.

Trade-offs: The brute-force approach is simpler to implement but significantly less efficient than the KMP-based solution. It's suitable for small input sizes but becomes impractical for larger arrays due to its quadratic time complexity. The KMP approach is preferred for larger inputs because of its linear time complexity.

✨

Key Insights

1Insight 1: The problem can be transformed into a string matching problem where we convert the differences between consecutive numbers in `nums` into a sequence of -1, 0, and 1.
2Insight 2: The KMP (Knuth-Morris-Pratt) algorithm is an efficient way to search for occurrences of the `pattern` within the transformed `nums` sequence.
3Insight 3: The Longest Proper Prefix Suffix (LPS) array helps to avoid unnecessary comparisons during the pattern matching phase of the KMP algorithm.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Edge Cases

Edge case 1: Empty `nums` or `pattern` array. The code implicitly handles an empty pattern, returning 0 as no matches can occur. If `nums` contains fewer elements than the pattern requires, the loop condition `i < n - 1` will prevent out-of-bounds access, and the algorithm will naturally return the correct (zero) count.
Edge case 2: `pattern` array longer than `nums` array. The loop `i < n - 1` ensures that the algorithm does not attempt to find a subarray of length greater than the number of available elements in `nums` after transforming it to differences array. This avoids out-of-bounds errors and incorrect matching.
Edge case 3: All elements in `nums` are identical. The transformation from `nums` to the difference array will result in all 0's. The KMP algorithm will proceed as expected, searching for matches of `pattern` within this sequence of 0's. If the pattern consists of all 0s, the code correctly identifies the matching subarrays.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Incorrectly calculating the LPS array. A wrong LPS array will lead to inaccurate shifts during the KMP search and thus an incorrect count of matching subarrays.

✗Common mistake 2: Forgetting to handle the case when `j` reaches the end of the `pattern` and needs to be reset using `lps[j - 1]`. Failing to do so will cause the algorithm to miss subsequent overlapping matches.

✗Common mistake 3: Not properly converting the relationship between consecutive `nums` elements into the -1, 0, 1 representation according to the problem description. Incorrect comparisons will lead to wrong results.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Pattern recognition in financial time series data. The `nums` array could represent stock prices over time, and the `pattern` could represent a specific trend (e.g., increasing, flat, decreasing). The algorithm can then be used to identify occurrences of this trend in the stock price history.

🏦Real-world scenario 2: Anomaly detection in sensor data. Imagine a sensor stream represented by `nums`, and a known anomalous pattern defined by `pattern`. The algorithm can pinpoint sections of the sensor stream matching that anomalous pattern, flagging potential issues or errors.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

Hard problems often combine multiple techniques. Break the problem into sub-problems.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: November 4, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 14

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 7

❓

Frequently Asked Questions

What is the most efficient way to solve array problems?

Most array problems can be solved efficiently using techniques like two-pointer approach, sliding window, or hash maps. The key is identifying whether the array is sorted (enabling binary search) or if tracking seen elements (using hash maps) can help achieve O(n) time complexity.

How do I approach hard algorithm problems?

Hard problems often require: 1) Breaking down into subproblems, 2) Combining multiple advanced techniques, 3) Careful optimization of brute force. Start with a working solution (even O(n²) or worse), then optimize. Draw examples, identify patterns, and think about what makes this problem unique.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Number of Subarrays That Match a Pattern II"?

For this hard-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether rolling hash techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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