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

Peter S. Gemmell and Mor Harchol

EECS Department, University of California, Berkeley

Technical Report No. UCB/CSD-93-737

, 1993

http://www2.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},
    Year= {1993},
    Month= {Apr},
    Url= {http://www2.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://www2.eecs.berkeley.edu/Pubs/TechRpts/1993/6024.html
%F Gemmell:CSD-93-737