Final answer:
An asymptotically fast algorithm is typically better for large input sizes than an asymptotically slow algorithm, even if running on a slower computer, due to the significance of time complexity and how algorithms scale with input size.
Step-by-step explanation:
When considering the performance of algorithms on computers of varying speeds, it's vital to understand that the raw processing power of a computer does not solely determine computation time for large inputs. For large input sizes, the efficiency of an algorithm plays a much more significant role in determining how quickly it can process data. An asymptotically fast algorithm, which has a better growth rate as the input size increases, generally outperforms an asymptotically slow one, even on a slower machine. For example, consider two algorithms, A (with time complexity O(n log n)) and B (with time complexity O(n^2)), running on two different computers. If Algorithm A is running on a computer that is half as fast as the one running Algorithm B, Algorithm A will still eventually outperform Algorithm B for sufficiently large inputs because its runtime growth is far less steep. The most critical consideration here is how the algorithms scale with increasing input size, which is captured by their time complexity. As the input grows, the constant factor due to computer speed becomes negligible compared to the exponential or factorial growth incurred by inefficient algorithms.