running time cg(nf(nn input size figure illustrating the "big-ohnotation the function (nis ( ( ))since (nn example the function is (njustificationby the big-oh definitionwe need to find real constant and an integer constant > such that it is easy to see that possible choice is and indeedthis is one of infinitely many choices available because there is trade-off between and for examplewe could rely on constants and the big-oh notation allows us to say that function (nis "less than or equal toanother function (nup to constant factor and in the asymptotic sense as grows toward infinity this ability comes from the fact that the definition uses "<=to compare (nto (ntimes constantcfor the asymptotic cases when > howeverit is considered poor taste to say ( < ( ( )),since the big-oh already denotes the "less-than-or-equal-toconcept likewisealthough commonit is not fully correct to say (no( ( )),with the usual understanding of the "=relationbecause there is no way to make sense of the symmetric statement" ( ( ) (nit is best to sayf (nis ( ( )alternativelywe can say (nis order of (nfor the more mathematically inclinedit is also correct to sayf (no( ( )),for the big-oh notationtechnically speakingdenotes whole collection of functions in this bookwe will stick to presenting big-oh statements as (nis ( ( )even with this interpretationthere is considerable freedom in how we can use arithmetic operations with the bigoh notationand with this freedom comes certain amount of responsibility