#2829

Medium

auto_awesomeGreedy choice plus invariant validation

LeetCode Problem Workspace

Determine the Minimum Sum of a k-avoiding Array

Determine the minimum sum of a k-avoiding array by choosing distinct integers such that no pair sums to a given k.

Problem Statement

You are given two integers, n and k. You need to construct an array of distinct positive integers of length n, called a k-avoiding array, where no two distinct elements sum to k.

Return the minimum possible sum of such an array, making sure to carefully avoid combinations that would violate the k-avoiding condition.

Examples

Example 1

Input: n = 5, k = 4

Output: 18

Consider the k-avoiding array [1,2,4,5,6], which has a sum of 18. It can be proven that there is no k-avoiding array with a sum less than 18.

Example 2

Input: n = 2, k = 6

Output: 3

We can construct the array [1,2], which has a sum of 3. It can be proven that there is no k-avoiding array with a sum less than 3.

Constraints

1 <= n, k <= 50

Solution Approach

Greedy Choice with Validation

Start by picking the smallest integers from 1 upwards and continuously check for any pair that would sum to k. If a pair is found, skip the current number and continue with the next available number.

Validate the Invariant

Ensure that after adding each number, no pair of numbers already in the array sums to k. This invariant guarantees the k-avoiding property while minimizing the sum.

Iterative Construction

Build the array incrementally by validating the choices as you go. By adding numbers progressively and checking for pairs that sum to k, you can ensure that you reach the minimum sum with minimal computation.

Complexity Analysis

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

The time and space complexity depend on the specific approach. Generally, a greedy method with validation will have a time complexity of O(n) for checking each number, with O(n) space for the array construction.

What Interviewers Usually Probe

Look for the ability to implement a greedy solution that minimizes the sum while maintaining correctness.
Check for validation of the k-avoiding property after each addition.
The interviewer should be able to reason through the process of constructing the array and handling edge cases.

Common Pitfalls or Variants

Common pitfalls

Not checking the sum of pairs during array construction.
Failing to skip numbers that would create an invalid pair.
Rushing through the greedy process and missing an optimal choice.

Follow-up variants

Consider a variation where k is larger or smaller to test scalability.
Test with different n values to explore the boundary conditions.
Explore the possibility of optimizing the space complexity or using different data structures.

FAQ

How do I construct the k-avoiding array for this problem?

Start with the smallest integers, skipping any number that would create a pair summing to k.

What does the greedy approach refer to in this problem?

It refers to always choosing the smallest available integer that does not create a sum of k with any previous element in the array.

What is the main pattern for solving the 'Determine the Minimum Sum of a k-avoiding Array' problem?

The main pattern is greedy choice combined with invariant validation to ensure no two numbers sum to k.

How does the greedy approach minimize the sum in this problem?

By choosing the smallest valid integers, the greedy approach ensures that the sum is as low as possible while still maintaining the k-avoiding condition.

Are there any edge cases to consider when solving the k-avoiding array problem?

Yes, cases where n is small or k is large can lead to tricky choices, and these should be handled by properly checking each number before adding it.

terminal

Solution

Solution 1: Greedy + Simulation

Starting from the positive integer $i = 1$, we sequentially determine if $i$ can be added to the array. If it can be added, we add $i$ to the array, accumulate it to the answer, and then mark $k - i$ as visited, indicating that $k-i$ cannot be added to the array. We continue this process until the array's length reaches $n$.

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class Solution:
    def minimumSum(self, n: int, k: int) -> int:
        s, i = 0, 1
        vis = set()
        for _ in range(n):
            while i in vis:
                i += 1
            vis.add(k - i)
            s += i
            i += 1
        return s

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