Incremental Memory Leak

Incremental Memory Leak is a math plus simulation problem where the only tricky part is the allocation rule when both sticks have equal memory. The safe baseline is to simulate second by second, always subtracting the current second from the larger stick, and stop when neither stick can afford the next allocation. In interviews, the useful follow-up is not a fancy data structure, but proving why the number of simulated seconds stays small enough.

Problem Statement

You have two memory sticks with memory1 and memory2 bits available. A faulty process requests more memory every second: 1 bit at second 1, 2 bits at second 2, and so on. On each second, the request is assigned to the stick with more free memory, and if both sticks are tied, the request goes to the first stick.

The program crashes at the first second when the next request cannot fit in either stick. You need to return three values: the second when the crash happens, the remaining memory on the first stick at that moment, and the remaining memory on the second stick. For example, when memory1 = 2 and memory2 = 2, the requests go to stick 1, then stick 2, and the crash result is [3,1,0].

Examples

Example 1

Input: memory1 = 2, memory2 = 2

Output: [3,1,0]

The memory is allocated as follows:

• At the 1st second, 1 bit of memory is allocated to stick 1. The first stick now has 1 bit of available memory.
• At the 2nd second, 2 bits of memory are allocated to stick 2. The second stick now has 0 bits of available memory.
• At the 3rd second, the program crashes. The sticks have 1 and 0 bits available respectively.

Example 2

Input: memory1 = 8, memory2 = 11

Output: [6,0,4]

The memory is allocated as follows:

• At the 1st second, 1 bit of memory is allocated to stick 2. The second stick now has 10 bit of available memory.
• At the 2nd second, 2 bits of memory are allocated to stick 2. The second stick now has 8 bits of available memory.
• At the 3rd second, 3 bits of memory are allocated to stick 1. The first stick now has 5 bits of available memory.
• At the 4th second, 4 bits of memory are allocated to stick 2. The second stick now has 4 bits of available memory.
• At the 5th second, 5 bits of memory are allocated to stick 1. The first stick now has 0 bits of available memory.
• At the 6th second, the program crashes. The sticks have 0 and 4 bits available respectively.

Constraints

• 0 <= memory1, memory2 <= 231 - 1

Solution Approach

Direct simulation with exact tie handling

Start at second 1 and repeat until the next allocation cannot be placed. On each step, compare memory1 and memory2, subtract the current second from the larger one, and use stick 1 when they are equal. This mirrors the problem statement exactly, avoids hidden assumptions, and is the clearest correct approach for Incremental Memory Leak.

Why the simulation is fast enough

The request sizes grow as 1, 2, 3, and so on, so after t successful seconds the program has consumed 1 + 2 + ... + t = t(t + 1)/2 bits in total. That means the number of iterations is bounded by the largest t whose triangular sum fits inside memory1 + memory2. Since t grows roughly like the square root of total memory, even large inputs do not create a long loop.

Math-aware optimization mindset

You can think about the process in two phases: the larger stick absorbs requests until the gap narrows, then the sticks may alternate more often. That observation can support a more math-heavy derivation, but for this problem the main interview win is recognizing that the increasing request size already gives a strong upper bound. A clean simulation is usually the best trade-off between correctness and implementation risk.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

A direct simulation runs in O(t) time and O(1) space, where t is the crash time. Because the total allocated memory after t successful seconds is t(t + 1)/2, we get t = O(sqrt(memory1 + memory2)). That bound is the key math insight for this Math plus Simulation problem.

What Interviewers Usually Probe

• You notice that the tie rule is not cosmetic: sending equal-memory cases to stick 1 changes the final crash state.
• You justify the loop bound with triangular numbers instead of hand-waving that simulation is probably fine.
• You keep the implementation state minimal: current second, memory1, and memory2 are enough.

Common Pitfalls or Variants

Common pitfalls

• Allocating to stick 2 on ties, which breaks the sample with memory1 = 2 and memory2 = 2.
• Stopping when the larger stick cannot fit the next request, even though the smaller stick might still be able to fit it.
• Overengineering the solution and missing the real interview question, which is the crash-time bound from increasing requests.

Follow-up variants

• Change the tie rule to prefer stick 2, which forces different crash states even when the high-level simulation stays the same.
• Add three or more memory sticks, where each second goes to the stick with the most available memory.
• Ask for the full allocation history instead of only [crashTime, memory1crash, memory2crash].

FAQ

What is the main pattern in Incremental Memory Leak?

The core pattern is Math plus Simulation. You simulate the allocation exactly as written, then use the triangular sum 1 + 2 + ... + t to explain why the number of steps is only about the square root of the total memory.

Why is the upper bound for the number of seconds small?

After t successful allocations, the program has used t(t + 1)/2 bits total. Since that total cannot exceed memory1 + memory2 before the crash, t is bounded by a square-root relationship, which keeps the simulation short.

Why does the equal-memory rule matter so much?

Because the next subtraction can change which stick becomes smaller later, the tie rule affects the whole future path. In this problem, equal memory must allocate to stick 1, and ignoring that detail produces the wrong crash state.

Can I solve LeetCode 1860 without simulation?

You can derive parts of the behavior with math, especially the upper bound, but a fully math-only implementation is usually less readable than the direct simulation. For interview code, the clean simulation plus a short proof is the safer choice.

What should I return when the program crashes?

Return the current second as crashTime, along with the remaining memory on stick 1 and stick 2 before any failed allocation is applied. The failed request is not subtracted from either stick.