File: minimum-value-to-get-positive-step-by-step-sum/solution.py

Date: 2026-06-06

Time: 18:03

minimum-value-to-get-positive-step-by-step-sum/solution.py

Purpose

Solves LeetCode 1413 — Minimum Value to Get Positive Step by Step Sum. Given an integer array nums, find the minimum positive integer startValue such that the cumulative sum startValue + nums[0] + nums[1] + ... is never less than 1 at any step.

This file owns a single responsibility: the algorithmic solution for this problem, packaged in the standard Solution class convention used across the repo.

Key Components

Solution.maxSideLength(self, nums: List[int]) -> int — The method name is mismatched; it should be minStartValue per the LeetCode problem signature. Despite the name, the implementation is correct for the "minimum start value" problem, not the "maximum side length" problem (which is LeetCode 1292, a completely different problem).

Flow

1. Initialize minsum = 0 and runningsum = 0.

2. Iterate through nums, accumulating a prefix sum in running_sum.

3. At each step, track the minimum prefix sum seen so far in min_sum.

4. Return max(1, 1 - min_sum):

• If the minimum prefix sum is negative (say -5), we need startValue = 6 so that 6 + (-5) = 1 >= 1.
• If the minimum prefix sum is non-negative, the answer is 1 (the smallest positive integer).

The max(1, ...) guard handles the case where all prefix sums are positive — startValue must be at least 1 per the problem constraints.

Concrete trace: nums = [-3, 2, -3, 4, 2]

• Prefix sums: -3, -1, -4, 0, 2
• min_sum = -4
• Result: max(1, 1 - (-4)) = 5
• Verification: 5, 2, 4, 1, 5, 7 — all >= 1.

Patterns

• Single-pass prefix minimum: O(n) time, O(1) space. Avoids materializing the full prefix sum array by tracking only the running minimum. This is the idiomatic approach for this class of problem.
• Standard repo structure: Solution class with a single method, matching the LeetCode submission format used uniformly across all ~500 problem directories.

Dependencies

• Imports: typing.List — standard type hint, no external dependencies.
• Imported by: minimum-value-to-get-positive-step-by-step-sum/test_solution.py directly. The massive "Imported By" list in the prompt is an artifact of the repo's shared test infrastructure — those test files import Solution from their own sibling solution.py, not from this file.

Invariants

• The return value is always >= 1 (enforced by the max(1, ...) expression).
• The algorithm guarantees that for every index i, startValue + sum(nums[0..i]) >= 1.
• min_sum is initialized to 0, not float('inf') — this is intentional. It represents the "do nothing" baseline: if no prefix sum dips below 0, the minimum stays 0 and the answer is 1.

Error Handling

None. The function assumes valid input per LeetCode constraints (non-empty list of integers). No bounds checking, no exception handling. An empty nums list would return 1 correctly (the loop doesn't execute, min_sum stays 0).

Beliefs

• wrong-method-name-min-start-value — Solution.maxSideLength is misnamed; it implements minStartValue (LeetCode 1413), not maxSideLength (LeetCode 1292)
• min-sum-init-zero-is-intentional — min_sum is initialized to 0 (not -inf) so that a non-negative prefix sum sequence correctly yields startValue=1 without a special case
• single-pass-o1-space — The algorithm runs in O(n) time and O(1) auxiliary space, tracking only two scalar accumulators
• empty-input-returns-one — An empty nums list produces a correct result of 1 without any special-case handling