Big O Notation - Multiple Variables

Multiple Variables

Big O (and little o, and Ω...) can also be used with multiple variables. To define Big O formally for multiple variables, suppose and are two functions defined on some subset of . We say

if and only if

For example, the statement

asserts that there exist constants C and M such that

where g(n,m) is defined by

Note that this definition allows all of the coordinates of to increase to infinity. In particular, the statement

(i.e., ) is quite different from

(i.e., ).

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