What does it mean when we say that an algorithm X is asymptotically more efficient than Y?
(A)
X will be a better choice for all inputs
(B)
X will be a better choice for all inputs except possibly small inputs
(C)
X will be a better choice for all inputs except possibly large inputs
(D)
Y will be a better choice for small inputs
Answer: (B)
Explanation:
When we say that an algorithm X is asymptotically more efficient than Y, we mean that as the size of the input to the algorithm grows larger and larger, the running time of X will eventually become faster than the running time of Y. This is typically expressed using big O notation, which gives an upper bound on the worst-case running time of an algorithm.
Specifically, if we say that X is asymptotically more efficient than Y, we mean that there exists a constant c such that for all sufficiently large inputs, the running time of X is at most c times the running time of Y, i.e., X has a lower big O complexity class than Y.
This means that for large inputs, X will always be a better choice than Y in terms of efficiency. However, for small inputs, it is possible that Y may still be faster than X. Therefore, we cannot say that X will always be a better choice for all inputs, but we can say that X will be a better choice for all inputs except possibly small inputs.
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