# 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

April 1993

### 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