Find minimum subarray sum
Find minimum subarray sum is a medium arrays & hashing problem solved with the kadane's algorithm pattern.
The best approach, optimal (kadane for minimum), runs in O(n) time and O(1) space.
Below are 2 approaches in Java, from brute force up.
Problem
Given an integer array, find the smallest sum of any non-empty contiguous subarray.
This is the mirror image of the maximum subarray problem, and it is the key step in the circular version.
Examples
Example 1
- Input
arr = [3, -4, 2, -3, -1, 7, -5]- Output
-6- Why
- The subarray [-4, 2, -3, -1] sums to -6.
Example 2
- Input
arr = [2, 6, 8, 1, 4]- Output
1- Why
- All values are positive, so the smallest single element wins.
Constraints
1 <= arr.length <= 10^5-10^4 <= arr[i] <= 10^4
Animated walkthrough
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Concepts first, then the problem and every approach, step by step.
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Solutions
Try it yourself first. Then compare: each approach lists its idea, the steps, its time and space, and the Java code.
| Approach | Time | Space |
|---|---|---|
| Brute force | O(n²) | O(1) |
| Optimal (Kadane for minimum) | O(n) | O(1) |
1Brute force
O(n²)O(1)Sum every subarray and keep the smallest.
- For each start i, extend j to the end with a running sum.
- Track the minimum.
class Solution {
static int smallestSumSubarray(int[] a, int size) {
int best = Integer.MAX_VALUE;
for (int i = 0; i < size; i++) {
int sum = 0;
for (int j = i; j < size; j++) {
sum += a[j];
best = Math.min(best, sum);
}
}
return best;
}
}2Optimal (Kadane for minimum)
O(n)O(1)The smallest subarray ending at i either extends the one ending at i - 1 or starts fresh. Carry the running sum only while it is negative, because a positive carry can only make things bigger.
- cur = best = a[0].
- cur = min(a[i], cur + a[i]).
- best = min(best, cur).
class Solution {
static int smallestSumSubarray(int[] a, int size) {
int cur = a[0], best = a[0];
for (int i = 1; i < size; i++) {
cur = Math.min(a[i], cur + a[i]);
best = Math.min(best, cur);
}
return best;
}
}Edge cases to test
- All positive numbers
- Single element
Hints
Hint 1
This is Kadane's algorithm with min in place of max.
FAQ
What is the best time complexity for Find minimum subarray sum?
Optimal (Kadane for minimum) runs in O(n) time and O(1) extra space.
Which pattern does Find minimum subarray sum use?
It is a arrays & hashing problem that uses the kadane's algorithm pattern. Other problems with the same pattern: Kadane's Algorithm - Simple, Kadane's Algo - Circular Array - No Extra Space.
Is there a brute force solution for Find minimum subarray sum?
Yes. Brute force takes O(n²) time and O(1) space. Sum every subarray and keep the smallest.
Which edge cases should I test for Find minimum subarray sum?
All positive numbers; Single element.