Is Sum Of Special Evenly-Spaced Elements In Array easy or hard?

The problem is rated Hard because it requires recognizing a sqrt decomposition optimization. A naive solution is straightforward, but designing the preprocessing strategy and balancing time vs space for multiple queries requires deeper algorithmic insight.

Sum Of Special Evenly-Spaced Elements In Array Python/Java solution

Implement a DP table for step sizes up to √n and compute values backward: dp[y][i] = nums[i] + dp[y][i+y]. During queries, return dp[y][x] if the step size is small; otherwise iterate through the array. This approach works efficiently in Python, Java, C++, Go, and TypeScript.

How to solve Sum Of Special Evenly-Spaced Elements In Array in O(n√n)?

Use sqrt decomposition. Let B = √n and precompute dp[y][i] = nums[i] + dp[y][i+y] for every step size y ≤ B. This preprocessing takes O(n√n). When processing queries, return dp[y][x] if y ≤ B; otherwise iterate through indices x, x+y, x+2y until the array ends.

What is the best approach for Sum Of Special Evenly-Spaced Elements In Array?

The most efficient method uses block (sqrt) decomposition with dynamic programming. Precompute results for small step sizes y ≤ √n using a DP table where dp[y][i] stores the sum starting from index i with step y. For larger step sizes, simulate the traversal since at most √n elements are visited. This keeps total complexity around O(n√n + q√n).

Is Sum Of Special Evenly-Spaced Elements In Array asked at Google/Amazon/Meta?

Problems involving query optimization, sqrt decomposition, and dynamic programming over arrays commonly appear in interviews at companies like Google, Amazon, and Meta. This specific problem reflects the type of reasoning used in system queries and competitive programming style interview questions.

What data structure is used in Sum Of Special Evenly-Spaced Elements In Array?

The solution primarily uses arrays and a 2D dynamic programming table for precomputed step sizes. The technique combines array traversal with sqrt decomposition to split queries into precomputed and simulated cases.

What is the time complexity of Sum Of Special Evenly-Spaced Elements In Array?

The optimized solution runs in O(n√n + q√n) time with O(n√n) preprocessing space. Precomputation builds DP values for all step sizes up to √n, and each query either returns a precomputed value in O(1) or iterates at most √n elements when the step size is large.

Description Hints 3 Solution Video 24:47

1714. Sum Of Special Evenly-Spaced Elements In Array

Hard49.9% acceptance1 companyPremium elsewhere, free here

You are given a 0-indexed integer array nums consisting of n non-negative integers.

You are also given an array queries, where queries[i] = [x_i, y_i]. The answer to the i^th query is the sum of all nums[j] where x_i <= j < n and (j - x_i) is divisible by y_i.

Return an array answer where answer.length == queries.length and answer[i] is the answer to the i^th query modulo 10⁹+ 7.

Example 1:

Input: nums = [0,1,2,3,4,5,6,7], queries = [[0,3],[5,1],[4,2]]
Output: [9,18,10]
Explanation: The answers of the queries are as follows:
1) The j indices that satisfy this query are 0, 3, and 6. nums[0] + nums[3] + nums[6] = 9
2) The j indices that satisfy this query are 5, 6, and 7. nums[5] + nums[6] + nums[7] = 18
3) The j indices that satisfy this query are 4 and 6. nums[4] + nums[6] = 10

Example 2:

Input: nums = [100,200,101,201,102,202,103,203], queries = [[0,7]]
Output: [303]

Constraints:

n == nums.length
1 <= n <= 5 * 10⁴
0 <= nums[i] <= 10⁹
1 <= queries.length <= 1.5 * 10⁵
0 <= x_i < n
1 <= y_i <= 5 * 10⁴

Asked at 1 company

MakeMyTrip

Topics

Array Dynamic Programming

Hints open one at a time

Hint 1

Think if y cannot be small. You can solve a query in O(n/y), so if y is large enough, it won't be a problem.

Hint 2

If y is small, like less than B, you can preprocess the answers for all such ys in O(n * B), then answer each such query in O(1).

Hint 3

As you might have already guessed, the optimal value for B is ~sqrt(n).

24:47

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Examples

Case 1Case 2

Input

[0,1,2,3,4,5,6,7]
[[0,3],[5,1],[4,2]]

Input

[100,200,101,201,102,202,103,203]
[[0,7]]

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