功能增长大

Question

Suppose we have two algorithms, A and B , for solving some problem (say multiplying two n × n matrices, or sorting an array n of integers). Let n denote the input size of the problem, and let T _A (n) and T _B (n) denote the number of steps taken by algorithms A and B , respectively, on inputs of size n . It is known that T _A (n) = O(n ³ ) and T _B (n) = O(n ⁵ ) . Is it true that for sufficiently large n , algorithm A performs less steps than algorithm B ? Why or why not?

Answer 1

First, I suppose you meant that the bounds are O(n ³ ) and O(n ⁵ ) (ie, powers of 3 and 5, respectively).

Since O is an upper bound , you can't really say anything about the comparisons. For example:

• n ² = O(n ³ ) , and n ^1.5 = O(n ⁵ ) , but, even for large n , the former function is larger than the latter.

• Conversely, n ^1.5 = O(n ³ ) , and n ² = O(n ⁵ ) , and here, for large n , the former function is indeed larger than the latter.

If the question would be about Θ , not O , the answer would be different - in that case, you could state that, for large enough n , the former performs less steps than the latter.