What does the notation T(n) mean?

Question

We learned about big O notation, but I often see T(n) as well. For example,

public static Comparable[] mergeSort(Comparable[] A, int low, int high) {
  if (low < high) { //at least 2 elements?                //cost = c
    int mid = (low + high)/2;                             //cost = d
    Comparable[] A1 = mergeSort(A, low, mid);             //cost = T(n/2) + e
    Comparable[] A2 = mergeSort(A, mid+1, high);          //cost = T(n/2) + f
    return merge(A1,A2);                                  //cost = g n + h
  }
  .... //cost = i

I believe c,d,e,... are meant to be arbitrarily named constants.

What does T(n/2) mean? also how is T notation related to big O?

From wikipedia article on O-notation: "A function T(n) that will express how long the algorithm will take to run (in some arbitrary measurement of time) in terms of the number of elements in the input set." — James, Nov 29 '12 at 03:15
`T(n)`, denoting the *exact* time needed to calculate the data of size `n`. It's very useful when calculate the time needed of a recursive function. — xiaoyi, Nov 29 '12 at 03:27

username tbd · Answer 1 · 2012-11-29T03:20:45.520

15

This notation refers to the maximum amount of time (or, more specifically, steps) that a function takes to run.

T(n) may be much more specific than O(n); for example, let's say you have a program that for any input, requires n^2+n+1 steps to run:

T(n) = n^2+n+1
O(n) = n^2

More information can be found here.

edited Nov 29 '12 at 03:20

answered Nov 29 '12 at 03:13

username tbd

9,152
1
20
35

score 0 · Answer 2 · answered Nov 29 '12 at 03:14

I've personally never seen this notation before, but I suspect it refers to "big-Theta" (Θ), which is both an asymptotic upper-bound (big-O) and an asymptotic lower bound.

Also related: Big-Omega (Ω) is used to denote just an asymptotic lower-bound (mirroring big-O).

What does the notation T(n) mean?

2 Answers2