04-08-2023, 01:15 AM
Worst cases stretch longer though. Numbers tied to Fibonacci sequences force the most steps. I notice each remainder shrinks slowly like a slow leak in a pipe. You count those steps and they top out around the log of the bigger input. That means time grows slowly even for massive values.
Space stays tiny because you track just two variables at once. I bet you picture it using constant room no matter the size. No extra arrays balloon out here. Perhaps you test it with small examples first. Then you scale them up to watch the pattern hold.
And the average behavior lands close to the worst. Most random pairs still cut down fast. But some outliers drag a bit more. You measure it by counting divisions performed. I find that beats naive subtraction loops by miles.
Now think about proof sketches. You link steps to continued fraction stuff without needing deep theory. Each pass halves the size roughly on average. Or it reduces by a factor tied to golden ratio in bad cases. I watch the count stay under 5 times the digits in the number.
Perhaps you compare it to other gcd tricks. This one wins for simplicity and speed in practice. You avoid heavy multiplications or tables. But it assumes positive integers to begin with. Negative ones need a quick flip first.
Then hardware effects creep in. Modern chips handle the divisions quick enough. I reckon cache misses stay rare due to few variables. You run benchmarks on big ints and see linear growth in bit length. That keeps it viable for crypto keys and such.
Also consider recursive calls versus loops. Recursion eats stack space if you go deep. I prefer the loop version to dodge that. You gain nothing from recursion depth here anyway. Yet both yield same complexity bounds.
Maybe edge cases like zero inputs change nothing much. One step and done. You handle them with simple checks upfront. Or equal numbers wrap in one go too.
The method shines in loops for repeated use. I see libraries embed it everywhere for that reason. You rely on it without thinking about the inner cost. But knowing the log bound helps when numbers hit millions of bits.
Overflow risks vanish in languages with big int support. I track no extra memory spikes during runs. You focus on the input size as the driver.
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