523. Continuous Subarray Sum

Problem Description

The problem provides an integer array nums and an integer k. The task is to determine whether there exists at least one subarray within nums that is both of length two or more and whose sum of elements is a multiple of k. A subarray is defined as a contiguous sequence of elements within the parent array. It's important to note that any integer is considered a multiple of k if it can be expressed as n * k for some integer n. Zero is also considered a multiple of k by this definition (since 0 = k * 0).

Intuition

To solve this problem, we can utilize the properties of modular arithmetic. The key observation here is that if the sum of a subarray nums[i:j] (where i < j) is a multiple of k, then the cumulative sums sum[0:i-1] and sum[0:j] will have the same remainder when divided by k. This stems from the fact that if (sum[0:j] - sum[0:i-1]) is a multiple of k, then (sum[0:j] % k) = (sum[0:i-1] % k).

The algorithm proceeds as follows:

Iterate through the array, computing the cumulative sum s as we go.
At each step, calculate the remainder of the sum s divided by k (denoted as r = s % k).
Maintain a dictionary (mp) that maps each remainder to the earliest index where that remainder was seen.
For each calculated remainder r, check if we have seen this remainder before. If we have and the distance between the current index and the index stored in the dictionary mp[r] is at least two, this means we've found a good subarray, and we return True.
If the remainder has not been seen before, store the current index in the dictionary against the remainder r.
If no good subarray is found throughout the iteration, return False after the loop completes.

By using this approach, we are effectively tracking the cumulative sums in such a way that we can efficiently check for subarrays whose sum is a multiple of k. The storage of the earliest index where each remainder occurs is crucial for determining the length of the subarray without having to store all possible subarrays.

Learn more about Math and Prefix Sum patterns.

Solution Approach

The solution approach leverages the concept of prefix sums and modular arithmetic to identify a subarray sum that is a multiple of k. Here is the step-by-step explanation of how the solution is implemented:

Initialize a Variable to Store Cumulative Sum (s): We define a variable s that will hold the cumulative sum of the elements as we iterate through the array.
Create a Dictionary (mp) to Store Remainders and Their Earliest Index: A Python dictionary mp is used to map each encountered remainder when dividing the cumulative sum by k to the lowest index where this remainder occurs. The dictionary is initialized with {0: -1} which handles the edge case wherein the cumulative sum itself is a multiple of k from the beginning of the array (i.e., the subarray starts at index 0).
Iterate Through the Array: Using a for-loop, we iterate through the array while keeping track of the current index i and the element value v.
Update Cumulative Sum: With each iteration, we update the cumulative sum s by adding the current element value v to it: s += v.
Calculate Remainder: We calculate the remainder r of the current cumulative sum s when divided by k: r = s % k.
Check for a Previously Encountered Remainder: If the remainder r has been seen before, and the index difference i - mp[r] is greater than or equal to 2, we have found a "good subarray." This is because the equal remainders signify that the sum of elements in between these two indices is a multiple of k. If such a condition is met, the function returns True.
Store the Remainder and Index If Not Already Present: If the remainder r has not been previously encountered, we store this remainder with its corresponding index i into the dictionary: mp[r] = i.
Return False If No Good Subarray Is Found: If the for-loop completes without returning True, it implies that no "good subarray" has been found. In this case, the function returns False.

By using a hashmap to keep track of the remainders, the algorithm ensures a single-pass solution with O(n) time complexity and O(min(n, k)) space complexity, since the number of possible remainders is bounded by k.

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Example Walkthrough

Let's go through an example to illustrate the solution approach. Suppose we have an array nums = [23, 2, 4, 6, 7] and an integer k = 6. We want to find out if there exists at least one subarray with a sum that is a multiple of k.

Initialize Cumulative Sum and Dictionary: s = 0. Dictionary mp is initialized as {0: -1}.
Iteration 1:
- Index i = 0, Element v = 23.
- Update s: s = 0 + 23 = 23.
- Calculate remainder r: r = 23 % 6 = 5.
- Remainder 5 is not in mp, so we add it: mp = {0: -1, 5: 0}.
Iteration 2:
- Index i = 1, Element v = 2.
- Update s: s = 23 + 2 = 25.
- Calculate remainder r: r = 25 % 6 = 1.
- Remainder 1 is not in mp, so we add it: mp = {0: -1, 5: 0, 1: 1}.
Iteration 3:
- Index i = 2, Element v = 4.
- Update s: s = 25 + 4 = 29.
- Calculate remainder r: r = 29 % 6 = 5.
- Remainder 5 is already in mp, and i - mp[5] = 2 - 0 = 2 which is equal to or greater than 2, hence we have found a "good subarray" [23, 2, 4] with sum 29 which is a multiple of k (since 29 - 23 = 6 which is 6*1).
- Return True.

In this example walkthrough, we found a "good subarray" in the third iteration and therefore returned True. This means at least one subarray meets the criteria, thus the function would terminate early with a positive result.

Solution Implementation

1class Solution:
2    def checkSubarraySum(self, nums: List[int], k: int) -> bool:
3        # Initialize the prefix sum as zero
4        prefix_sum = 0
5      
6        # A dictionary to keep track of the earliest index where 
7        # a particular modulus (prefix_sum % k) is found.
8        mod_index_map = {0: -1}  # The modulus 0 is at the "imaginary" index -1
9      
10        # Iterate over the list of numbers
11        for index, value in enumerate(nums):
12            # Update the prefix sum with the current value
13            prefix_sum += value
14          
15            # Get the modulus of the prefix sum with 'k'
16            modulus = prefix_sum % k
17          
18            # If the modulus has been seen before and the distance between 
19            # the current index and the earlier index of the same modulus 
20            # is at least 2, we found a subarray sum that's multiple of k
21            if modulus in mod_index_map and index - mod_index_map[modulus] >= 2:
22                return True
23          
24            # Store the index of this modulus if it's not seen before
25            if modulus not in mod_index_map:
26                mod_index_map[modulus] = index
27      
28        # No subarray found that sums up to a multiple of k
29        return False
30

1class Solution {
2    public boolean checkSubarraySum(int[] nums, int k) {
3        // HashMap to store the remainder of the sum encountered so far and its index
4        Map<Integer, Integer> remainderIndexMap = new HashMap<>();
5        // To handle the case when subarray starts from index 0
6        remainderIndexMap.put(0, -1);
7        // Initialize the sum to 0
8        int sum = 0;
9      
10        // Iterate through the array
11        for (int i = 0; i < nums.length; ++i) {
12            // Add current number to the sum
13            sum += nums[i];
14            // Calculate the remainder of the sum w.r.t k
15            int remainder = sum % k;
16            // If the remainder is already in the map and the subarray is of size at least 2
17            if (remainderIndexMap.containsKey(remainder) && i - remainderIndexMap.get(remainder) >= 2) {
18                // We found a subarray with a sum that is a multiple of k
19                return true;
20            }
21            // Put the remainder and index in the map if not already present
22            remainderIndexMap.putIfAbsent(remainder, i);
23        }
24        // If we reach here, no valid subarray was found
25        return false;
26    }
27}
28

1#include <vector>
2#include <unordered_map>
3using namespace std;
4
5class Solution {
6public:
7    // Function to check if the array has a contiguous subarray of size at least 2 
8    // that sums up to a multiple of k
9    bool checkSubarraySum(vector<int>& nums, int k) {
10        // Create a map to store the modulus occurrence with their index
11        unordered_map<int, int> modIndexMap;
12        modIndexMap[0] = -1; // Initialize with a special case to handle edge case
13        int sum = 0; // Accumulated sum
14      
15        // Iterate through the numbers in the vector
16        for (int i = 0; i < nums.size(); ++i) {
17            sum += nums[i]; // Add current number to sum
18            int mod = sum % k; // Current modulus of sum by k
19          
20            // Check if the modulus has been seen before
21            if (modIndexMap.count(mod)) {
22                // If the distance between two same modulus is at least 2,
23                // it indicates a subarray sum that is a multiple of k
24                if (i - modIndexMap[mod] >= 2) return true;
25            } else {
26                // If this modulus hasn't been seen before, record its index
27                modIndexMap[mod] = i;
28            }
29        }
30      
31        // If no qualifying subarray is found, return false
32        return false;
33    }
34};
35

1// Importing necessary utilities
2import { HashMap } from 'hashmap';
3
4// Function to check if the array has a contiguous subarray of size at least 2 
5// that sums up to a multiple of k
6function checkSubarraySum(nums: number[], k: number): boolean {
7    // Create a map to store the modulus occurrence with their index
8    let modIndexMap: Map<number, number> = new Map();
9    modIndexMap.set(0, -1); // Initialize with a special case to handle edge case
10    let accumulatedSum = 0; // Accumulated sum
11
12    // Iterate through the numbers in the array
13    for (let i = 0; i < nums.length; ++i) {
14        accumulatedSum += nums[i]; // Add current number to the accumulated sum
15        let mod = accumulatedSum % k; // Current modulus of the accumulated sum by k
16
17        // Check if the modulus has been seen before
18        if (modIndexMap.has(mod)) {
19            // If the distance between two same modulus is at least 2,
20            // it indicates a subarray sum that is a multiple of k
21            if (i - modIndexMap.get(mod)! >= 2) return true;
22        } else {
23            // If this modulus hasn't been seen before, record its index
24            modIndexMap.set(mod, i);
25        }
26    }
27
28    // If no qualifying subarray is found, return false
29    return false;
30}
31

Time and Space Complexity

Time Complexity

The provided code consists of a single loop that iterates over the list nums once. For each element of nums, it performs constant-time operations involving addition, modulus, and dictionary access (both lookup and insert). Therefore, the time complexity is determined by the loop and is O(n), where n is the number of elements in nums.

Space Complexity

The space complexity of the code is primarily dependent on the dictionary mp that is used to store the remainders and their respective indices. In the worst case, each element could result in a unique remainder when taken modulo k. Therefore, the maximum size of mp could be n (where n is the number of elements in nums). Thus, the space complexity is also O(n).

Learn more about how to find time and space complexity quickly using problem constraints.

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Question 1 out of 10

Consider the classic dynamic programming of fibonacci numbers, what is the recurrence relation?

dp[i] = dp[i - 1] + dp[i - 2]

dp[i] = dp[i - 1] + 1

dp[i] = dp[i - 1] + dp[i - 2] + dp[i - 3]

dp[i] = dp[i - 1]

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