My first instinct was to go forward from (a, b) and BFS toward (c, d), but the values can balloon fast and I started second-guessing the bounds.
Recognize that the operations are reversible: from (c, d) we can subtract the smaller from the larger to move backwards. Use the Euclidean algorithm to repeatedly reduce the larger number modulo the smaller, checking if we can reach (a, b). This approach is efficient and handles large numbers.
Pro tip: Mention that the modulo optimization reduces time complexity from O(max(a,b)) to O(log(max(a,b))), which is crucial for large inputs. Also, clarify that the order of the pair matters, so (a, b) and (b, a) are distinct unless a=b.
The operations are (a, b) -> (a+b, b) and (a, b) -> (a, a+b). They are reversible: from (x, y) with x > y, the previous state must be (x-y, y).
Start with (c, d) and repeatedly subtract the smaller from the larger to try to reach (a, b). Use modulo to skip multiple subtractions at once.
While both numbers are positive, if the larger is greater than the smaller, replace the larger with larger % smaller. If the smaller becomes 0, stop.
At each step, check if the current pair equals (a, b) or (b, a) (since operations are symmetric in reverse). If found, return true.
If a or b is 0, only possible if the other equals c or d appropriately. Also, if a > c or b > d, impossible unless the other matches.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
The example they gave (answer = 3 for 4 tasks) helped me sanity check but I still fumbled the pairing logic initially.
Clarify that each task takes 1 unit of time, but two tasks of the same type can run in parallel if their combined memory is within the limit. Then, for each type, pair tasks to maximize parallelism using a greedy two-pointer approach after sorting by memory, and sum the time slots across all types. Finally, discuss the time complexity and potential trade-offs.
Pro tip: Explicitly state that tasks of different types cannot be paired, so the problem decomposes by type. This shows you understand the constraints and avoids overcomplicating the solution.
Confirm that each task takes 1 unit of time, and two tasks of the same type can run in parallel only if their combined memory does not exceed the limit. Also confirm that tasks of different types cannot be paired.
Separate the tasks into groups based on their type. The problem then reduces to solving the minimum time for each type independently.
For each type, sort the memory requirements. Use a two-pointer technique: pair the smallest with the largest if their sum is within the limit; otherwise, the largest runs alone. Count the number of time slots needed.
Add the minimum time slots required for each type to get the total minimum time to complete all tasks.
Discuss the time complexity (O(n log n) due to sorting) and space complexity (O(n) for grouping). Mention that the greedy pairing is optimal for each type.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.