# Big-Theta extension of Brent's Theorem?

Is there an extension or translation of Brent's theorem into asymptotics aside from big-$O$?

Brent's Theorem: source

Running time of a parallel algorithm with $p$ processors (say, $f(n,p)$), $W(n)$ Work complexity, and $S(n)$ Step complexity takes $\leq \frac{W(n)}{p} + S(n)$ time. The $\leq$ lets me use $O$ directly, but not $\Omega$. If it's also true, I'd be able to say something like:

$f(n,p) \in \frac{\Theta(W(n))}{p} + \Theta(S(n))$

It seems like it is true. Is it? I'd love to have a reference.

-

3
You're asking for a lower bound on the conversion of an algorithm into a parallel algorithm ? that is likely to be hard. – Suresh Venkat♦ Feb 14 at 18:29
The algorithm shouldn't need to be converted. I just want to know whether Big-Theta holds. – Kyle Feb 14 at 19:33
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 10, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9483824968338013, "perplexity_flag": "middle"}