What is the time complexity of Number of Subarrays With AND Value of K?

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

What data structures are used to solve Number of Subarrays With AND Value of K?

This problem uses Array, Binary Search, Bit Manipulation, Segment Tree. Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving Number of Subarrays With AND Value of K?

Insight 1: The bitwise AND operation is monotonically decreasing. That is, as you include more elements in a subarray and perform AND operations, the resulting value can only decrease or stay the same; it can never increase. Insight 2: We can use dynamic programming or a similar iterative approach to keep track of the AND values we've seen so far. Specifically, maintaining a mapping of previous AND values and their frequencies allows us to efficiently calculate AND values for new subarrays.

Which companies ask LeetCode 3209. Number of Subarrays With AND Value of K in interviews?

This problem is frequently asked at DE Shaw.

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Number of Subarrays With AND Value of K - LeetCode 3209 Solution

Number of Subarrays With AND Value of K - Complete Solution Guide

Number of Subarrays With AND Value of K is LeetCode problem 3209, 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

Given an array of integers nums and an integer k , return the number of subarrays of nums where the bitwise AND of the elements of the subarray equals k . Example 1: Input: nums = [1,1,1], k = 1 Output: 6 Explanation: All subarrays contain only 1's. Example 2: Input: nums = [1,1,2], k = 1 Output: 3 Explanation: Subarrays having an AND value of 1 are: [ 1 ,1,2] , [1, 1 ,2] , [ 1,1 ,2] . Example 3: Input: nums = [1,2,3], k = 2 Output: 2 Explanation: Subarrays having an AND value of 2 are: [1, 2 ,3

Detailed Explanation

The problem asks us to find the number of subarrays within a given array `nums` whose bitwise AND of all elements in the subarray is equal to a target value `k`. A subarray is a contiguous sequence of elements within the array. The bitwise AND operation combines the bits of two numbers such that the resulting bit is 1 only if both corresponding bits in the input numbers are 1. For example, the bitwise AND of 5 (101 in binary) and 3 (011 in binary) is 1 (001 in binary). We need to iterate through all possible subarrays, calculate the AND value for each, and count the subarrays where the AND value equals `k`.

Solution Approach

The provided solution uses a dynamic programming-like approach to efficiently count the subarrays. It iterates through the `nums` array, maintaining a map `prev_and_counts` to store the bitwise AND values encountered so far, along with the number of times each value has occurred as the result of a subarray ending at the previous index. For each number in the `nums` array, it creates a new map `curr_and_counts`. It calculates new bitwise AND values by ANDing the current number with all the previous AND values. The count of subarrays with AND value equal to `k` ending at the current index is then added to the `total_count`. `prev_and_counts` is updated for the next iteration.

Step-by-Step Algorithm

Step 1: Initialize `total_count` to 0. This variable will store the final count of subarrays with AND value equal to `k`.
Step 2: Initialize an empty map `prev_and_counts`. This map will store the bitwise AND values encountered in the previous iterations and the number of times they occurred.
Step 3: Iterate through the `nums` array. For each number `num` in `nums`:
Step 4: Create a new map `curr_and_counts`. Initialize this map by adding `num` as a key with a value of 1 (since the subarray containing only `num` has an AND value of `num`).
Step 5: Iterate through the `prev_and_counts` map. For each key-value pair (value, frequency) in `prev_and_counts`:
Step 6: Calculate the bitwise AND of `value` and `num`: `new_and = value & num`.
Step 7: Update `curr_and_counts` by adding `new_and` as a key (if it doesn't exist) or incrementing its value by `frequency` (if it already exists).
Step 8: After iterating through `prev_and_counts`, add the count of subarrays with AND value `k` in `curr_and_counts` to `total_count`. That is: `total_count += curr_and_counts.get(k, 0)`.
Step 9: Update `prev_and_counts` to be equal to `curr_and_counts` for the next iteration.
Step 10: After iterating through all the numbers in `nums`, return `total_count`.

Key Insights

Insight 1: The bitwise AND operation is monotonically decreasing. That is, as you include more elements in a subarray and perform AND operations, the resulting value can only decrease or stay the same; it can never increase.
Insight 2: We can use dynamic programming or a similar iterative approach to keep track of the AND values we've seen so far. Specifically, maintaining a mapping of previous AND values and their frequencies allows us to efficiently calculate AND values for new subarrays.
Insight 3: Since we are dealing with bitwise operations, it's essential to consider the range of possible values and potential integer overflow issues, especially in languages like C where data type sizes are crucial. Using `long long` is often important for accumulating counts.
Insight 4: A naive solution of iterating through all possible subarrays (O(n^2)) and calculating ANDs (O(n)) would lead to O(n^3) complexity, which is not optimal. The provided solution optimizes this to O(n*m) where m is the number of distinct AND values which is capped at the number of elements itself since the number of distinct AND values calculated in an iteration can be at most the length of the input array at that iteration.

Complexity Analysis

Time Complexity: O(n*m)

Space Complexity: O(m)

Topics

This problem involves: Array, Binary Search, Bit Manipulation, Segment Tree.

Companies

Asked at: DE Shaw.

Number of Subarrays With AND Value of K | LeetCode 3209

Hard

Array Binary Search Bit Manipulation Segment Tree

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

Solution Code

Available in:

import java.util.HashMap;
import java.util.Map;

class Solution {
    public long countSubarrays(int[] nums, int k) {
        long total_count = 0;
        Map<Integer, Long> prev_and_counts = new HashMap<>();
        for (int num : nums) {
            Map<Integer, Long> curr_and_counts = new HashMap<>();
            curr_and_counts.put(num, 1L);
            for (Map.Entry<Integer, Long> entry : prev_and_counts.entrySet()) {
                int val = entry.getKey();
                long freq = entry.getValue();
                int new_and = val & num;
                curr_and_counts.put(new_and, curr_and_counts.getOrDefault(new_and, 0L) + freq);
            }

            total_count += curr_and_counts.getOrDefault(k, 0L);
            prev_and_counts = curr_and_counts;
        }

        return total_count;
    }
}

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Problem Info

Description

Problem 3209: Number of Subarrays With AND Value of K - Detailed explanation and solution approaches.

Topics

Array Binary Search Bit Manipulation Segment Tree

Companies

DE Shaw

Stats

Difficulty:Hard

Languages:4

Companies:1

Problem #:3209

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

10 steps

1
Step 1: Initialize `total_count` to 0. This variable will store the final count of subarrays with AND value equal to `k`.
2
Step 2: Initialize an empty map `prev_and_counts`. This map will store the bitwise AND values encountered in the previous iterations and the number of times they occurred.
3
Step 3: Iterate through the `nums` array. For each number `num` in `nums`:
4
Step 4: Create a new map `curr_and_counts`. Initialize this map by adding `num` as a key with a value of 1 (since the subarray containing only `num` has an AND value of `num`).
5
Step 5: Iterate through the `prev_and_counts` map. For each key-value pair (value, frequency) in `prev_and_counts`:
6
Step 6: Calculate the bitwise AND of `value` and `num`: `new_and = value & num`.
7
Step 7: Update `curr_and_counts` by adding `new_and` as a key (if it doesn't exist) or incrementing its value by `frequency` (if it already exists).
8
Step 8: After iterating through `prev_and_counts`, add the count of subarrays with AND value `k` in `curr_and_counts` to `total_count`. That is: `total_count += curr_and_counts.get(k, 0)`.
9
Step 9: Update `prev_and_counts` to be equal to `curr_and_counts` for the next iteration.
10
Step 10: After iterating through all the numbers in `nums`, return `total_count`.

💡

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 outer loop iterates through the `nums` array, which takes O(n) time. Inside the loop, we iterate through the `prev_and_counts` map, which in the worst case can have up to 'm' elements. Therefore, the inner loop takes O(m) time. Consequently, the overall time complexity is O(n*m). The space complexity is determined by the size of the `prev_and_counts` map, which stores at most 'm' key-value pairs, resulting in a space complexity of O(m).

📖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

Iterate through all possible subarrays and calculate the bitwise AND for each. Then, check if the AND value equals `k` and increment a counter.

Time: O(n^3) in the worst case, where n is the length of `nums`. O(n^2) for generating all subarrays, and O(n) for calculating the AND for each subarray.Space: O(1)

Trade-offs: This approach is simple to understand but inefficient for large input arrays due to its cubic time complexity. The provided solution is more efficient.

✨

Key Insights

1Insight 1: The bitwise AND operation is monotonically decreasing. That is, as you include more elements in a subarray and perform AND operations, the resulting value can only decrease or stay the same; it can never increase.
2Insight 2: We can use dynamic programming or a similar iterative approach to keep track of the AND values we've seen so far. Specifically, maintaining a mapping of previous AND values and their frequencies allows us to efficiently calculate AND values for new subarrays.
3Insight 3: Since we are dealing with bitwise operations, it's essential to consider the range of possible values and potential integer overflow issues, especially in languages like C where data type sizes are crucial. Using `long long` is often important for accumulating counts.
4Insight 4: A naive solution of iterating through all possible subarrays (O(n^2)) and calculating ANDs (O(n)) would lead to O(n^3) complexity, which is not optimal. The provided solution optimizes this to O(n*m) where m is the number of distinct AND values which is capped at the number of elements itself since the number of distinct AND values calculated in an iteration can be at most the length of the input array at that iteration.

📚

Key Terms Explained

For Students

Array

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

Binary Search

Halving the search space each step. Like finding a word in a dictionary!

Edge Cases

Edge case 1: `nums` is empty. The solution correctly handles this case because the loop won't execute, and `total_count` will remain 0.
Edge case 2: `k` is 0. The solution correctly accounts for this. A subarray with an AND value of 0 will be counted if it exists.
Edge case 3: `nums` contains large numbers. Using `long long` for counting in Java and C++ avoids potential integer overflow.
Edge case 4: k is larger than any number in nums. The solution handles this correctly as `curr_and_counts.get(k, 0)` will return 0 when k is not present as a key

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Incorrectly calculating the bitwise AND. It's crucial to use the correct operator (`&`) and understand how it works.

✗Common mistake 2: Failing to handle edge cases such as empty `nums` or `k` being 0. These cases should be explicitly considered.

✗Common mistake 3: Integer overflow. Using appropriate data types (e.g., `long long` in C++ and Java) is important for storing potentially large counts.

✗Common mistake 4: Incorrectly iterating through all the values calculating the subarrays. Starting with individual elements is crucial.

✅

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: Network routing. Finding paths or segments of a network where certain bitwise properties (e.g., certain flags being set) are consistent across all nodes in the path. This may be used in determining network policies.

🏦Real-world scenario 2: Data analysis. Identifying subsets of data with specific bitwise characteristics. For example, analyzing system logs to find events with specific error codes that have a certain AND relationship.

💡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

These problems test your ability to optimize. Start with a brute force solution, then improve.

⚠️ 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: October 28, 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

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJul 14

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 7

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

@AlgoMasterJun 30

❓

Frequently Asked Questions

When should I use in-place modification vs creating a new array?

Use in-place modification when space complexity matters (O(1) space requirement) and the original array can be modified. Create a new array when you need to preserve the original data or when the problem involves significant restructuring that would complicate in-place logic.

Can binary search be applied to non-sorted arrays?

Binary search requires a monotonic property, not necessarily sorted data. It can be applied to any search space where you can determine if the answer is in the left or right half. Examples include searching rotated arrays or finding minimum in optimization problems.

Is it okay to not fully solve hard problems in interviews?

Yes! Interviewers care more about your problem-solving process than the final answer. Demonstrate: clear communication, systematic approach, ability to recognize when to pivot strategies, and graceful handling of hints. Partial progress with good reasoning is valued.

What is the importance of time and space complexity analysis?

Complexity analysis is crucial because: 1) Interviewers always ask about it, 2) It helps you choose between approaches, 3) It demonstrates CS fundamentals. Always state both time and space complexity, and be prepared to explain how you derived them.

What is the optimal approach for "Number of Subarrays With AND Value of K"?

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 binary search techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

🤖 Need help with this problem?