2875. Minimum Size Subarray in Infinite Array

Problem Description

You are given an array called nums, which starts from the index 0. There is also an integer called target. There's a hypothetical array called infinite_nums, which is made by continuously repeating the nums array infinitely. Your task is to find the shortest contiguous subarray within infinite_nums that adds up to the target value. The length of this subarray (the number of elements it includes) is what you need to return. If such a subarray can't be found, you must return -1.

For example, if nums is [1,2,3] and the target is 5, you can use the subarray [2,3] from nums or [3,1,1] from infinite_nums, both sum up to 5 but the shortest length is 2 from the [2,3] subarray in nums.

Intuition

The intuition behind the solution is to use mathematical properties and an efficient data structure to avoid direct iteration through the infinite array, which would be impractical.

First, calculate the sum of all elements in nums, which is referred to as s. Compare s with the target. If the target is larger than s, then you can construct part of the required subarray using whole chunks of nums, each of which contributes s to the overall sum. This means you can repeatedly subtract s from target until target is less than s, and count how many times you've done this as part of the subarray length.

Then, the problem is reduced to two cases within the original nums array:

1. Find the shortest subarray that sums up exactly to target.
2. If it's not possible (because target is now less than s after subtraction), find two subarrays where one is at the start and one is at the end of nums such that their total sum equals s - target. This effectively simulates wrapping around due to the infinite nature of infinite_nums.

To achieve this efficiently, we use a prefix sum array and a hash table. The prefix sum array allows us to calculate the sum of any subarray quickly, and the hash table lets us lookup whether a needed sum to reach the target has been seen before as we iterate through nums. When we find such sums, we can calculate the length of the subarray that reaches the target.

With the above strategy, we ensure that we can get our answer without explicitly dealing with the infinite array, all while maintaining good time complexity.

Learn more about Prefix Sum and Sliding Window patterns.

Solution Approach

The given solution approach leverages a combination of prefix sums and hash tables to solve the problem with efficiency.

Here's a step-by-step breakdown of the implementation:

1. Compute the sum s of the entire nums array. Then check if target is greater than s. If yes, you can cover a significant portion of the target by using whole arrays of nums. Calculate a = n * (target // s), which denotes the array length contributed by the complete nums blocks. After this, subtract (target // s * s) from target, reducing the problem to finding a shorter subarray that sums up to the new target. If the new target matches s, return n since it is the shortest subarray that can be formed by the whole array.

2. Initialize a hash table pos to keep track of the last position of every prefix sum encountered. Set the prefix sum for an empty array to -1 to handle cases where the subarray starts at the beginning of nums.

3. Initialize a variable pre to hold the ongoing prefix sum as we iterate through the array and a variable b as inf (infinity) which will later hold the length of the shortest qualifying subarray found.

4. Iterate through the nums array, updating the pre (prefix sum) by adding each element x from nums to it.

5. Two main checks are performed for each iteration:

   • If the difference t = pre - target exists in pos, it means a subarray sums up to target. Update b to be the minimum of its current value or the distance from the current index to the index stored in pos[t].
   • If the difference t = pre - (s - target) exists in pos, it implies that there's a contiguous subarray from the start and end of nums which together add up to target. Again, update b to be the minimum of its current value or n - (i - pos[t]).

6. After the loop, check if b has changed from inf. If it has, this means a valid subarray was found, and the solution returns a + b. Otherwise, return -1 indicating that a subarray satisfying the conditions does not exist.

This approach uses a hash table to store prefix sums and their corresponding last index. This data structure combined with the prefix sum concept allows us to find contiguous subarrays that add up to a target effectively. By doing this in a single pass, the algorithm avoids the need for nested loops, resulting in an efficient solution.

Example Walkthrough

Let's illustrate the solution approach using a small example. Suppose the given nums array is [3, 1, 2], and the target is 6. We want to find the shortest contiguous subarray within infinite_nums that sums up to 6.

Following the solution approach:

1. First, we compute the sum s of the entire nums array, which in this case is 3 + 1 + 2 = 6. Since target is equal to s, we can directly return the length of nums, which is 3, as the shortest subarray because using the whole nums array once sums to the target. However, for sake of walkthrough, let's proceed as if we were to look for a subarray since often the target might not exactly match s.

2. Initialize a hash table pos with -1 as the prefix sum for an empty array, and a variable pre for ongoing prefix sums starting at 0. Also, initialize b as inf which indicates the shortest found subarray (to be minimized).

3. As we iterate, we build our prefix sum array on the fly, and update pos with the last position we encountered a specific prefix sum.

4. We start iterating nums:

   • At index 0, pre becomes 3. pos does not have pre - target = 3 - 6 = -3, so we continue.
   • At index 1, pre becomes 4. pos does not have pre - target = 4 - 6 = -2, so we continue.
   • At index 2, pre becomes 6. pos does have pre - target = 6 - 6 = 0, which is the prefix sum for an empty array, so we find that subarray [3, 1, 2] sums to target. We update b to current index (2) plus one minus -1 (position of 0 in pos), which equals 3.

5. After the iteration, b is no longer inf; it's 3. So, we can directly return the length 3, indicating that [3, 1, 2] is the shortest subarray that sums up to the target.

In this case, iteration reveals that using the entire array nums itself is the shortest subarray to reach the target of 6. In scenarios where target is not equal to s, the method would identify the shortest contiguous subarray as per the steps described in the solution approach. If no such subarray exists, -1 would be returned.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the function is O(n) where n is the length of the array nums. This is because there is a single loop that goes through each element of nums exactly once. The operations within the loop have constant time complexity, such as the calculation of the running sum (pre), checking for the existence of a value in the hash table (pos), and updating the hash table. As a result, these constant-time operations do not change the overall linear time complexity.

Space Complexity

The space complexity of the function is also O(n), which comes from the use of a hash table pos that keeps track of the indices of the prefix sums. In the worst case, if all prefix sums are unique, the hash table will have as many entries as there are elements in nums. Therefore, the space used by the hash table is directly proportional to the size of the input array, leading to linear space complexity.

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