What is the brute force approach and complexity for Sum of Subarray Minimums?

Fix the start, extend the end, add the running minimum each time with time complexity O(N²) and space complexity O(1).

What is the optimal approach and complexity for Sum of Subarray Minimums?

arr[i] is the min of left[i] × right[i] subarrays; find both counts with a monotonic stack with optimal time complexity O(N) and space complexity O(N).

How do you identify the pattern for Sum of Subarray Minimums?

Look for key signals in the prompt: "sum over all subarrays" indicates n²/2 subarrays is too many to visit → count each element's share instead.; "minimum of each subarray" indicates An element is the min until a smaller one appears → previous smaller / next smaller walls.; "modulo 10^9 + 7" indicates The total is big → long long for the products, then mod..

Sum of Subarray Minimums

Medium·LeetCode #907 Pro OnlyBasic Reinforcement

LeetCode

Problem Statement

You are given an integer array arr.

Look at every contiguous subarray of arr. Take the minimum of each one and add all those minimums together. The total can be huge, so return it modulo 10^9 + 7.

Input

An integer array arr.

Output

The sum of min(subarray) over all subarrays, modulo 10^9 + 7.

Example 1

Input: arr = [3, 1, 2, 4]

Output: 17

The subarrays are [3], [1], [2], [4], [3,1], [1,2], [2,4], [3,1,2], [1,2,4], [3,1,2,4]. Their minimums are 3, 1, 2, 4, 1, 1, 2, 1, 1, 1, which add up to 17.

Example 2

Input: arr = [11, 81, 94, 43, 3]

Output: 444

Constraints

• 1 ≤ arr.length ≤ 3 · 10^4
• 1 ≤ arr[i] ≤ 3 · 10^4

Pattern Recognition Clues

Recognising these key signals in the problem description immediately identifies the pattern:

“sum over all subarrays”

→ n²/2 subarrays is too many to visit → count each element's share instead.

“minimum of each subarray”

→ An element is the min until a smaller one appears → previous smaller / next smaller walls.

“modulo 10^9 + 7”

→ The total is big → long long for the products, then mod.

Approach Complexity Ladder

Brute Force — every subarray with a running minO(N²)O(1)

Fix the start, extend the end, add the running minimum each time

Optimal — contribution of each elementO(N)O(N)

arr[i] is the min of left[i] × right[i] subarrays; find both counts with a monotonic stack

Brute Force Intuition

Fix the start of the subarray. Move the end one step at a time and keep the running minimum. Each (start, end) pair is one subarray, so add the current minimum to the total.

Recognition rule for this sub-pattern:

For each element, find how far it stretches left and right as the min (or max). Histograms: area = height × width. Sums over subarrays: value × (left span) × (right span), with one side strict and the other ≤ so ties count once.

Remember this · Monotonic Stack

Trigger

Next/previous greater or smaller element, spans, histogram areas, "for each element, how far does it dominate?"

Think

When a new element arrives, which older elements can never be the answer again?

Key question

Is my stack increasing or decreasing, and what do I record when I pop?

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