# Tight Bounds on Expected Time to Add Correctly and Add Mostly Correctly

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/CSD-93-737.pdf

We consider the problem of adding two n-bit numbers which are chosen independently and uniformly at random where the adder is circuit of AND, OR, and NOT gates of fanin two.

The fastest currently known worst-case adder has running time log n + O(sqrt of log n).

We first present a circuit which adds at least 1 - epsilon fraction of pairs of numbers correctly and has running time log log (n\epsilon) + O(sqrt of log log (n\epsilon)).

We then prove that this running time is optimal.

Next we present a circuit which always produces the correct answer. We show this circuit adds two n-bit numbers from the uniform distribution in expected (1\2) log n + O(sqrt of log n) time, a speed up factor of two over the best possible running time of a worst-case adder.

We prove that this expected running time is optimal.

BibTeX citation:

@techreport{Gemmell:CSD-93-737,
Author = {Gemmell, Peter S. and Harchol, Mor},
Title = {Tight Bounds on Expected Time to Add Correctly and Add Mostly Correctly},
Institution = {EECS Department, University of California, Berkeley},
Year = {1993},
Month = {Apr},
URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/6024.html},
Number = {UCB/CSD-93-737},
Abstract = {We consider the problem of adding two <i>n</i>-bit numbers which are chosen independently and uniformly at random where the adder is circuit of AND, OR, and NOT gates of fanin two. <p>The fastest currently known worst-case adder has running time log <i>n</i> + <i>O</i>(sqrt of log <i>n</i>). <p>We first present a circuit which adds at least 1 - epsilon fraction of pairs of numbers correctly and has running time log log (<i>n</i>\epsilon) + <i>O</i>(sqrt of log log (<i>n</i>\epsilon)). <p>We then prove that this running time is optimal. <p>Next we present a circuit which always produces the correct answer. We show this circuit adds two <i>n</i>-bit numbers from the uniform distribution in expected (1\2) log <i>n</i> + <i>O</i>(sqrt of log <i>n</i>) time, a speed up factor of two over the best possible running time of a worst-case adder. <p>We prove that this expected running time is optimal.}
}

EndNote citation:

%0 Report
%A Gemmell, Peter S.
%A Harchol, Mor
%T Tight Bounds on Expected Time to Add Correctly and Add Mostly Correctly
%I EECS Department, University of California, Berkeley
%D 1993
%@ UCB/CSD-93-737
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/6024.html
%F Gemmell:CSD-93-737