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#3344MediumPremium on LC~35 min

Maximum Sized Array

Time precompute: O(maxs^(1/5) * log(maxs)) runtime: O(log(maxs)) · Space O(maxs^(1/5)) · Official statement on LeetCode

bit manipulationbit manipulationbinary search
cpppython

Solutions

// Time:  precompute: O(max_s^(1/5) * log(max_s))
//        runtime:    O(log(max_s))
// Space: O(max_s^(1/5))

// precompute, bitmasks, combinatorics, binary search
int i = 1;
int64_t area = 0;
vector<int64_t> vol = {0};
class Solution {
public:
    int maxSizedArray(long long s) {
        const auto& bit_length = [](int x) {
            return (x ? std::__lg(x) : -1) + 1;
        };

        for (; vol.back() <= s; ++i) {
            const int l = bit_length(i - 1);
            int line = (i - 1) * i;
            for (int bit = 0; bit < l; ++bit) {
                if (!((i - 1) & (1 << bit))) {
                    line += (((i - 1) >> (bit + 1)) * (1 << bit)) * (1 << bit);
                }
            }
            area += 2 * line - ((i - 1) | (i - 1));
            vol.emplace_back(((0 + (i - 1)) * i / 2) * area);
        }
        return distance(cbegin(vol), upper_bound(cbegin(vol), cend(vol), s)) - 1;
    }
};

Beginner Explanation

What is Maximum Sized Array?

Maximum Sized Array (LeetCode #3344) is a Medium problem that primarily trains bit manipulation.

How to think about it

  1. Restate the goal in your own words before coding.
  2. Work a tiny example by hand so the invariant becomes obvious.
  3. Identify the pattern — this problem aligns with bit manipulation and binary search.
  4. Only then translate the idea into code.

Why this problem matters

It sits in the sweet spot of interview difficulty: multiple valid approaches, clear trade-offs. Official solution notes mention: Precompute, Bitmasks, Combinatorics, Binary Search.

AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.

Interview Walkthrough

Interview approach for Maximum Sized Array

Opening (30–60 seconds)

  • Clarify inputs/outputs and edge cases (empty input, single element, duplicates, overflow).
  • State a brute force so the interviewer knows you can solve it naively.
  • Propose the optimal direction tied to bit manipulation and binary search.

Core solution narrative

  1. Define the state you track (pointers, DP cell, set membership, stack top, etc.).
  2. Explain the transition when you process the next element.
  3. Call out time (precompute: O(maxs^(1/5) * log(maxs)) runtime: O(log(maxs))) and space (O(maxs^(1/5))) before coding.
  4. Code cleanly; narrate variable names.

What interviewers listen for

  • Correctness on edge cases
  • Complexity honesty
  • Ability to discuss trade-offs (e.g., hash map space vs. sort + two pointers)

Follow-up questions they may ask

  • Can you solve it with less memory?
  • What if the input stream is infinite / doesn't fit in RAM?
  • How would tests look for adversarial inputs?

Optimized Approach

Optimized solution notes

The reference solutions on AlgoForge target precompute: O(maxs^(1/5) * log(maxs)) runtime: O(log(maxs)) time and O(maxs^(1/5)) space.

Pattern focus: bit manipulation and binary search

Use the pattern as a checklist:

  • bit manipulation — confirm the invariant holds after each step
  • binary search — confirm the invariant holds after each step

Start from the primary solution, then rewrite from memory to lock it in.

Implementation tips

  • Prefer readable names over micro-optimizations in interviews.
  • Extract helpers only when they clarify (e.g., expand-around-center, DFS visit).
  • After AC-level logic, re-scan for off-by-one and null checks.

Complexity Analysis

Complexity

Measure Bound
Time precompute: O(maxs^(1/5) * log(maxs)) runtime: O(log(maxs))
Space O(maxs^(1/5))

How to justify this in an interview

  • Time: count loops, map/set operations, and recursive branching; state average vs worst case if relevant.
  • Space: include hash maps, recursion stack, and output allocation when the problem asks for it.

If your implementation differs from the reference, re-derive big-O from your code — never memorize a complexity you cannot defend.

Common Mistakes

Common mistakes on Maximum Sized Array

  1. Skipping edge cases — empty collections, single-element inputs, max constraints.
  2. Wrong invariant for bit manipulation and binary search — updating state too early or too late.
  3. Mutating input unexpectedly when the problem forbids it.
  4. Off-by-one in windows, ranges, or binary search bounds.
  5. Ignoring overflow / precision for integer arithmetic problems.
  6. Overengineering — jumping to an advanced structure when a simpler approach works.

Alternative Approaches

AI expand later

Alternatives

Placeholder for multi-approach comparison. Future AI content generation can expand:

  • Brute force baseline
  • Optimal bit manipulation and binary search solution
  • Space-optimized rewrite

Prompt slot: expand alternatives for maximum-sized-array.

Edge Cases

Edge cases checklist

  • Minimum input size
  • Maximum input size / time limits
  • Duplicates and already-sorted input
  • Negative numbers / zeros (if applicable)
  • Disconnected structures (graphs/trees)
  • Single path vs branching recursion depth

Pattern Recognition

Spotting this pattern

Signal phrases that point to bit manipulation and binary search:

  • Sorted input or ability to sort without changing the answer class
  • Need for contiguous subarray / substring → consider sliding window
  • Need for O(1) membership → hash set/map
  • Optimal substructure + overlapping subproblems → DP
  • Connectivity / components → graph DFS/BFS or Union-Find

Primary topics: bit manipulation.

Follow-up Interview Questions

Follow-ups

  1. How does the solution change if the input is a stream?
  2. Can you solve it in-place?
  3. What if duplicates must be handled differently?
  4. How would you parallelize the approach?
  5. Design tests that would break a buggy implementation.

Practice Recommendations

What to practice next

  1. Re-solve Maximum Sized Array in a second language (cpp, python).
  2. Drill 3–5 more problems tagged bit manipulation.
  3. Teach the solution out loud in under 5 minutes.
  4. Add this problem to your revision calendar in 3 days and 14 days.

Visualization

Conceptual diagram for Maximum Sized Array: show input structure (bit manipulation), highlight the moving parts of the bit manipulation and binary search approach, and annotate each step with the maintained invariant and complexity.

Study checklist

  • Read the official problem statement on LeetCode
  • Solve on paper / whiteboard first
  • Implement the bit manipulation and binary search approach
  • Verify edge cases from the checklist
  • State time and space complexity aloud
  • Compare with the AlgoForge reference solution
  • Schedule a revision session

Revision notes

Maximum Sized Array (#3344) — Medium. Pattern: bit manipulation and binary search. Complexity: precompute: O(maxs^(1/5) * log(maxs)) runtime: O(log(maxs)) time / O(maxs^(1/5)) space. Re-derive the invariant before coding.

FAQs

What is the time complexity of Maximum Sized Array?+

The reference solutions aim for precompute: O(maxs^(1/5) * log(maxs)) runtime: O(log(maxs)) time and O(maxs^(1/5)) space. Always re-derive complexity from the code you write in the interview.

What pattern does Maximum Sized Array use?+

It primarily maps to bit manipulation and binary search, within the broader topic of bit manipulation.

Is Maximum Sized Array good for interviews?+

Yes — as a Medium problem it is a solid practice target. Pair it with related problems in the same pattern family for spaced repetition.

Where can I read the official statement?+

Open the official LeetCode page for constraints and examples: https://leetcode.com/problems/maximum-sized-array/