#1354

Hard

auto_awesomeArray plus Heap (Priority Queue)

LeetCode Problem Workspace

Construct Target Array With Multiple Sums

This problem requires constructing a target array from an array of ones, using multiple sum operations and a priority queue approach.

array heap priority queue

Problem Statement

You are given an array target of n integers. Start with an array arr consisting of n 1's. From here, you can repeatedly select the largest element in arr, reduce it by the sum of all other elements, and update arr. This process can be done until you either reach the target array or find it impossible to do so. Your task is to determine if it's possible to construct the target array from arr using the given procedure.

Return true if it is possible to construct the target array from arr, otherwise return false. You must use a heap (priority queue) to efficiently simulate the process of selecting the largest element and updating the array.

Examples

Example 1

Input: target = [9,3,5]

Output: true

Start with arr = [1, 1, 1] [1, 1, 1], sum = 3 choose index 1 [1, 3, 1], sum = 5 choose index 2 [1, 3, 5], sum = 9 choose index 0 [9, 3, 5] Done

Example 2

Input: target = [1,1,1,2]

Output: false

Impossible to create target array from [1,1,1,1].

Example 3

Input: target = [8,5]

Output: true

Example details omitted.

Constraints

n == target.length
1 <= n <= 5 * 104
1 <= target[i] <= 109

Solution Approach

Heap-based Simulation

Simulate the process using a max heap to always select the largest element in the array. Adjust the largest element by subtracting the sum of the other elements and update the heap. This continues until either the target is reached or the sum condition fails.

Check for Impossibility Conditions

If the largest element in the current array becomes smaller than or equal to any of the elements before it, it implies the target array cannot be formed. Thus, checking for impossible conditions is crucial at each step.

Efficient Time Complexity Handling

Given the constraints on array length, ensure the solution uses heap operations with logarithmic time complexity to maintain efficiency, particularly with large inputs up to 5 * 10^4.

Complexity Analysis

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

Time complexity is mainly influenced by the heap operations, where each insert and remove operation takes O(log n). With potentially n steps, the overall complexity becomes O(n log n). Space complexity depends on storing the heap, which requires O(n) space.

What Interviewers Usually Probe

Check if the candidate optimizes the simulation using a heap for time efficiency.
Look for the candidate's ability to handle edge cases such as arrays with repeated elements.
Evaluate how well the candidate identifies and addresses impossibility conditions.

Common Pitfalls or Variants

Common pitfalls

Forgetting to update the heap after each operation, leading to incorrect simulation.
Not correctly handling cases where the largest element cannot be reduced to a valid target.
Inefficient handling of large arrays, resulting in timeouts or excessive space usage.

Follow-up variants

Modify the problem to allow negative numbers and observe how it impacts the approach.
Consider the impact of different sum operations, such as subtracting a fixed value instead of the total sum.
Change the constraints to allow non-integer values and assess how the algorithm needs to adapt.

FAQ

What is the main strategy for solving the Construct Target Array With Multiple Sums problem?

The main strategy is to simulate the process of updating the array using a max heap, always selecting the largest element and reducing it by the sum of others.

How does the heap help in solving this problem efficiently?

The heap allows you to always extract the largest element in O(log n) time, which is crucial for simulating the transformation efficiently.

What makes this problem challenging to solve?

The challenge lies in ensuring that the process of repeatedly updating the largest element while maintaining the sum condition is handled efficiently, especially with large arrays.

Can the target array be constructed if the elements in the target array are not strictly increasing?

No, the target array must be strictly increasing in order to ensure that the largest element is reduced to its final value in the last step.

What is the time complexity of the solution?

The time complexity of the solution is O(n log n), where n is the length of the array, due to the heap operations for each element.

terminal

Solution

Solution 1: Reverse Construction + Priority Queue (Max Heap)

We observe that if we start constructing the target array $\textit{target}$ from the array $\textit{arr}$ in a forward manner, it is difficult to determine which index $i$ to choose each time, making the problem quite complex. However, if we construct in reverse starting from the array $\textit{target}$, each construction step must select the largest element in the current array, which ensures that each construction is unique, making the problem relatively simple.

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class Solution:
    def isPossible(self, target: List[int]) -> bool:
        s = sum(target)
        pq = [-x for x in target]
        heapify(pq)
        while -pq[0] > 1:
            mx = -heappop(pq)
            t = s - mx
            if t == 0 or mx - t < 1:
                return False
            x = (mx % t) or t
            heappush(pq, -x)
            s = s - mx + x
        return True

Continue Practicing

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