#1191

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

K-Concatenation Maximum Sum

Solve the K-Concatenation Maximum Sum problem using dynamic programming and optimal sub-array sum calculation.

Problem Statement

Given an integer array arr and an integer k, repeat the array k times and compute the maximum sub-array sum in the modified array. The length of the sub-array can be zero, with its sum being zero in that case.

For instance, if arr = [1, 2] and k = 3, the modified array will be [1, 2, 1, 2, 1, 2]. You are tasked with finding the maximum possible sum of any sub-array within this array. The approach can be optimized by first solving for k=1 and then adjusting for larger k values.

Examples

Example 1

Input: arr = [1,2], k = 3

Output: 9

Example details omitted.

Example 2

Input: arr = [1,-2,1], k = 5

Output: 2

Example details omitted.

Example 3

Input: arr = [-1,-2], k = 7

Output: 0

Example details omitted.

Constraints

1 <= arr.length <= 105
1 <= k <= 105
-104 <= arr[i] <= 104

Solution Approach

Handle k=1 case

First, solve the problem for k=1, which is equivalent to finding the maximum sub-array sum in the original array. This can be done using Kadane's algorithm or a dynamic programming approach. It provides a baseline solution for smaller k values.

Use prefix and suffix sums for large k

When k > 1, consider prefix and suffix sums. Calculate the maximum prefix sum, suffix sum, and the total sum of the array. Use these values to construct the maximum possible sub-array sum by considering various combinations of the prefix, suffix, and middle segments of the repeated arrays.

Final dynamic programming solution

Extend the dynamic programming approach to handle larger values of k by incorporating the calculated prefix and suffix sums into the overall sum. Combine these values effectively to ensure the solution scales efficiently without recomputing redundant parts of the array.

Complexity Analysis

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

The time complexity for this approach is O(n) for calculating prefix and suffix sums, plus O(1) for combining them for larger k values. The space complexity is O(1) if we only store prefix, suffix, and total sums. This solution is optimal for large values of k.

What Interviewers Usually Probe

Can the candidate handle the k=1 base case effectively?
Does the candidate recognize the importance of prefix and suffix sums for larger k?
How efficiently does the candidate scale the solution for large k values?

Common Pitfalls or Variants

Common pitfalls

Forgetting to handle the case where the sub-array sum is zero.
Not properly combining the prefix and suffix sums when k > 1.
Inefficient solutions that do not scale for large values of k.

Follow-up variants

What happens if we reduce k to 1? How does the solution change?
Can this approach be used for negative numbers in the array?
How would you handle cases where all elements in the array are negative?

FAQ

What is the key pattern in solving the K-Concatenation Maximum Sum?

The key pattern involves dynamic programming with state transitions, utilizing prefix and suffix sums for larger values of k.

How do we solve for k=1 in the K-Concatenation Maximum Sum problem?

For k=1, we find the maximum sub-array sum in the original array using algorithms like Kadane's or dynamic programming.

Why do we need to handle both prefix and suffix sums?

Prefix and suffix sums help maximize the sub-array sum when the array is repeated k times, especially for large k values.

What is the expected time complexity for large k in this problem?

The time complexity is O(n) for calculating prefix and suffix sums, and O(1) for combining them to handle large k values efficiently.

How does GhostInterview help with solving dynamic programming problems like this?

GhostInterview walks you through key dynamic programming patterns and scaling techniques, enhancing problem-solving efficiency.

terminal

Solution

Solution 1: Prefix Sum + Case Discussion

We denote the sum of all elements in the array $arr$ as $s$, the maximum prefix sum as $mxPre$, the minimum prefix sum as $miPre$, and the maximum subarray sum as $mxSub$.

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class Solution:
    def kConcatenationMaxSum(self, arr: List[int], k: int) -> int:
        s = mx_pre = mi_pre = mx_sub = 0
        for x in arr:
            s += x
            mx_pre = max(mx_pre, s)
            mi_pre = min(mi_pre, s)
            mx_sub = max(mx_sub, s - mi_pre)
        ans = mx_sub
        mod = 10**9 + 7
        if k == 1:
            return ans % mod
        mx_suf = s - mi_pre
        ans = max(ans, mx_pre + mx_suf)
        if s > 0:
            ans = max(ans, (k - 2) * s + mx_pre + mx_suf)
        return ans % mod

Continue Practicing

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