#1982

Hard

auto_awesomeArray plus Divide and Conquer

LeetCode Problem Workspace

Find Array Given Subset Sums

Reconstruct an array from given subset sums using divide and conquer approach and leveraging array properties.

Problem Statement

You are given an integer n representing the length of an unknown array. You are also given an array sums containing the values of all 2n subset sums of the unknown array in no particular order.

Your task is to return the array ans of length n representing the unknown array. If multiple answers exist, return any of them. The sum of an empty array is considered to be 0.

Examples

Example 1

Input: n = 3, sums = [-3,-2,-1,0,0,1,2,3]

Output: [1,2,-3]

[1,2,-3] is able to achieve the given subset sums:

[]: sum is 0
[1]: sum is 1
[2]: sum is 2
[1,2]: sum is 3
[-3]: sum is -3
[1,-3]: sum is -2
[2,-3]: sum is -1
[1,2,-3]: sum is 0 Note that any permutation of [1,2,-3] and also any permutation of [-1,-2,3] will also be accepted.

Example 2

Input: n = 2, sums = [0,0,0,0]

Output: [0,0]

The only correct answer is [0,0].

Example 3

Input: n = 4, sums = [0,0,5,5,4,-1,4,9,9,-1,4,3,4,8,3,8]

Output: [0,-1,4,5]

[0,-1,4,5] is able to achieve the given subset sums.

Constraints

1 <= n <= 15
sums.length == 2n
-104 <= sums[i] <= 104

Solution Approach

Initial Observations

The problem presents a set of subset sums for an unknown array. A key insight is that the two largest values in the subset sums give information about the largest elements of the unknown array. Sorting these sums helps identify these values, and from there, you can begin reconstructing the array.

Divide and Conquer Strategy

Use divide and conquer by progressively breaking down the remaining subset sums. After isolating the largest elements, the task becomes recursively finding smaller subsets until the entire array is reconstructed. This is a direct application of the divide and conquer approach, essential for breaking down the problem efficiently.

Handling Multiple Solutions

There can be multiple valid arrays that produce the same subset sums. Since the order of the elements doesn't matter, any permutation of the array is acceptable. This flexibility allows for various correct answers, as long as the given subset sums are matched.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the method chosen to solve the problem, with divide and conquer often leading to a more efficient approach. Space complexity is similarly influenced by the chosen algorithm, but recursive solutions generally require more memory.

What Interviewers Usually Probe

Candidate demonstrates an understanding of subset sum problems.
Candidate leverages sorting to help identify key elements.
Candidate successfully applies divide and conquer techniques to reduce the problem size.

Common Pitfalls or Variants

Common pitfalls

Confusing the order of subset sums and the actual array elements.
Failing to account for multiple possible solutions, leading to incorrect assumptions about uniqueness.
Not applying divide and conquer efficiently, leading to unnecessary recomputation.

Follow-up variants

Working with larger arrays, where the number of subset sums increases exponentially.
Optimizing for time and space complexity, especially in recursive solutions.
Handling edge cases where all array elements are zero, or negative values are involved.

FAQ

How does divide and conquer apply to the 'Find Array Given Subset Sums' problem?

Divide and conquer helps break down the problem by progressively isolating subsets and solving for smaller parts of the unknown array.

Can there be multiple correct answers to this problem?

Yes, any permutation of the elements that result in the same subset sums is valid.

What should the largest subset sums indicate?

The two largest subset sums give clues about the largest elements in the unknown array.

How do you efficiently reduce the problem size in this problem?

By isolating the largest elements first and recursively solving for smaller subsets, you reduce the problem size step by step.

What are some common mistakes in solving 'Find Array Given Subset Sums'?

A common mistake is not correctly handling the multiple valid solutions or misunderstanding the role of the subset sums.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def recoverArray(self, n: int, sums: List[int]) -> List[int]:
        m = -min(sums)
        sl = SortedList(x + m for x in sums)
        sl.remove(0)
        ans = [sl[0]]
        for i in range(1, n):
            for j in range(1 << i):
                if j >> (i - 1) & 1:
                    s = sum(ans[k] for k in range(i) if j >> k & 1)
                    sl.remove(s)
            ans.append(sl[0])
        for i in range(1 << n):
            s = sum(ans[j] for j in range(n) if i >> j & 1)
            if s == m:
                for j in range(n):
                    if i >> j & 1:
                        ans[j] *= -1
                break
        return ans

Solution 2

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class Solution:
    def recoverArray(self, n: int, sums: List[int]) -> List[int]:
        m = -min(sums)
        sl = SortedList(x + m for x in sums)
        sl.remove(0)
        ans = [sl[0]]
        for i in range(1, n):
            for j in range(1 << i):
                if j >> (i - 1) & 1:
                    s = sum(ans[k] for k in range(i) if j >> k & 1)
                    sl.remove(s)
            ans.append(sl[0])
        for i in range(1 << n):
            s = sum(ans[j] for j in range(n) if i >> j & 1)
            if s == m:
                for j in range(n):
                    if i >> j & 1:
                        ans[j] *= -1
                break
        return ans

Continue Practicing

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