samuelaplc

2022-10-09

Bounding a logarithmic relation

If I have the following relation $T(n)\le an\lceil \mathrm{lg}(n)\rceil -an+2bn+n$, is it possible to bound $T(n)$ such that it is in the form $T(n)\le an\mathrm{lg}(n)+bn$ for some constants $a,b\ge 0$?

If I have the following relation $T(n)\le an\lceil \mathrm{lg}(n)\rceil -an+2bn+n$, is it possible to bound $T(n)$ such that it is in the form $T(n)\le an\mathrm{lg}(n)+bn$ for some constants $a,b\ge 0$?

vakleraarrc

Beginner2022-10-10Added 6 answers

Note:

$an\lceil \mathrm{log}(n)\rceil -an+2bn+n\ge an\mathrm{log}(n))-an+2bn+n=an\mathrm{log}(n)+(2b-a+1)n$

The RHS is greater than the required expression so long as $2b-a+1>b$, so it is not necessarily possible.

$an\lceil \mathrm{log}(n)\rceil -an+2bn+n\ge an\mathrm{log}(n))-an+2bn+n=an\mathrm{log}(n)+(2b-a+1)n$

The RHS is greater than the required expression so long as $2b-a+1>b$, so it is not necessarily possible.

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