Related to exponential in terms of? (nonetheless different from it):
Assume that you have an algorithm (producing gizmos) whose inputs are three numbers t, d, and h, which are interpreted by a human as the number of thingums, doodahs, and hickeys. Suppose the running time of the algorithm is the function g, which depends on t, d, and h. Further, let g(t,d,h) ≤ dᶜ, where c is some expression exceeding 1 and depending on t and h, but not on d.
I would like to use plain English to express this fact (e.g., in an abstract) in a simplified way. Do we write
The algorithm runs in polynomial time in the number of doodahs.
The algorithm runs in polynomial time with the number of doodahs.
My English teacher corrected it to
The algorithm runs in polynomial time in terms of the number of doodahs.
I think this cannot be quite right.
Which version is correct? And how about
The algorithm runs in worst-case polynomial time in/with the number of doodahs.
Any better version?
I welcome answers from mathematicians who are native AmE speakers and have an excellent command of English.