File: k-items-with-the-maximum-sum/solution.py

Date: 2026-06-06

Time: 17:10

K Items With the Maximum Sum

Purpose

This file solves LeetCode 2600. It implements a greedy strategy for picking k items from three bags — one containing 1s, one containing 0s, one containing -1s — to maximize the total sum. It's a pure algorithmic module with no dependencies; its single function is imported by k-items-with-the-maximum-sum/test_solution.py.

Key Components

max_sum(numOnes, numZeros, numNegOnes, k) -> int — The sole exported function. It computes the maximum sum achievable by greedily taking items in value order: all available 1s first, then 0s, then -1s.

The implementation is a closed-form expression rather than a loop:

return min(k, numOnes) - max(0, k - numOnes - numZeros)

Breaking this apart:

• min(k, numOnes) — The number of 1s we actually pick. We take all of them unless k is smaller.
• k - numOnes - numZeros — How many items remain after exhausting all 1s and 0s. If positive, these must come from the -1 bag.
• max(0, ...) — Clamps to zero: if we didn't need to dip into -1s, this contributes nothing.

The sum equals (count of 1s picked) × 1 + (count of 0s picked) × 0 + (count of -1s picked) × (-1), which simplifies to exactly the expression above.

Patterns

• Closed-form greedy: Rather than simulating the pick process, the solution reduces the greedy strategy to arithmetic. This is a common pattern in easy-tier LeetCode problems where the greedy choice is monotonic (always pick the highest-value item available).
• No explicit validation: The function trusts that k <= numOnes + numZeros + numNegOnes (a problem constraint). It doesn't guard against invalid inputs.

Dependencies

• Imports: None. Pure computation with no library dependencies.
• Imported by: k-items-with-the-maximum-sum/testsolution.py directly, plus hundreds of other test files in the repo (the "Imported By" list in the prompt is likely an artifact of a shared test infrastructure, not actual usage of maxsum).

Flow

1. Caller passes counts of each item type and the pick budget k.

2. The function computes how many 1s are picked (min(k, numOnes)), which is the positive contribution.

3. It computes how many -1s are forced (max(0, k - numOnes - numZeros)), which is the negative contribution.

4. Returns the difference — no mutation, no side effects.

Invariants

• The greedy ordering (1 > 0 > -1) is hardcoded into the formula. The function assumes this is optimal, which it is because item values are fixed.
• The implicit precondition is k <= numOnes + numZeros + numNegOnes. Violating this would produce a mathematically valid but semantically meaningless result (the -1 count would exceed numNegOnes).

Error Handling

None. The function has no error paths — no exceptions, no validation, no edge-case guards. It relies entirely on the caller (and LeetCode's problem constraints) to provide valid inputs.

Topics to Explore

• [file] k-items-with-the-maximum-sum/test_solution.py — Verify which edge cases are tested (k=0, k=numOnes, all -1s)
• [file] k-items-with-the-maximum-sum/plan.md — See the original reasoning that led to the closed-form solution
• [general] greedy-closed-form-pattern — Other solutions in this repo that reduce a greedy simulation to a one-liner (e.g., count-of-matches-in-tournament, nim-game)
• [function] maximize-sum-of-array-after-k-negations/solution.py:max_sum — A harder variant where item values aren't fixed and the greedy strategy requires sorting

Beliefs

• max-sum-is-closed-form — max_sum computes the answer in O(1) time with no loops or data structures, using only min/max and arithmetic.
• max-sum-no-input-validation — The function does not validate that k <= numOnes + numZeros + numNegOnes; it assumes the caller respects this constraint.
• max-sum-greedy-correctness — The formula is equivalent to greedily picking all 1s, then all 0s, then -1s, which is provably optimal because item values are strictly ordered.
• max-sum-pure-function — The function has no side effects, no imports, and no mutable state; it is a pure mathematical mapping from inputs to output.