Maximum Sum of a Circular Subarray
Problem
The array is circular: after the last element comes the first one again. Find the biggest possible sum of a non-empty continuous part. The part may wrap around the end (it must not use any element twice). Print the sum. Hint: the best wrapping part is the total minus the smallest normal part.
Constraints
- 1 <= n <= 10^5
- |value| <= 10^4
Examples
Input
4 1 -2 3 -2
Output
3
Input
3 5 -3 5
Output
10
Solutions
A part that wraps around is the whole array without one normal part in the middle. To make it as big as possible we remove the smallest normal part. The all-negative case needs care because removing everything would leave an empty part, which is not allowed.
/*
* Problem : Best circular sum.
* Input : n, then n integers.
* Output : The biggest sum.
* Approach: Two Kadane runs.
* Time : O(n)
* Space : O(1)
* Java : 8 or newer.
*/
import java.util.Scanner;
public class Main {
// This method solves the task. It receives the input and returns the answer.
static long solve(int[] a) {
// Step 1: Run Kadane twice at the same time: once for the biggest sum and once for the smallest sum of a normal part.
long total = a[0];
long bestMax = a[0];
long currentMax = a[0];
long bestMin = a[0];
long currentMin = a[0];
for (int i = 1; i < a.length; i++) {
total += a[i];
currentMax = Math.max(currentMax + a[i], a[i]);
bestMax = Math.max(bestMax, currentMax);
currentMin = Math.min(currentMin + a[i], a[i]);
bestMin = Math.min(bestMin, currentMin);
}
// Step 2: If all numbers are negative, the "total minus smallest part" would be empty, so use the normal answer.
if (bestMax < 0) {
return bestMax;
}
// Step 3: A wrapping part is everything except the smallest normal part.
return Math.max(bestMax, total - bestMin);
}
public static void main(String[] args) {
Scanner scanner = new Scanner(System.in);
int size = scanner.nextInt();
int[] a = new int[size];
for (int i = 0; i < size; i++) {
a[i] = scanner.nextInt();
}
System.out.println(solve(a));
}
}
Hints (1)
- Why is the all-negative case special?