Chandan Saha

Simpler algorithm for estimating frequency moments of data streams

The problem of estimating the <i>k</i>^<i>th</i> frequency moment <i>F</i><inf><i>k</i></inf> over a data stream by looking at the items exactly once as they arrive was posed in [1, 2]. A succession of algorithms have been proposed for this problem [1, 2, 6, 8, 7]. Recently, Indyk and Woodruff [11] have presented the first algorithm for estimating <i>F</i><inf><i>k</i></inf>, for <i>k</i> > 2, using space <i>Õ</i>(<i>n</i>^1-2/<i>k</i>), matching the space lower bound (up to poly-logarithmic factors) for this problem [1, 2, 3, 4, 13] (<i>n</i> is the number of distinct items occurring in the stream.) In this paper, we present a simpler 1-pass algorithm for estimating <i>F</i><inf><i>k</i>.</inf>

Fast integer multiplication using modular arithmetic

We give an O(N • log N • 2O(log*N)) algorithm for multiplying two N-bit integers that improves the O(N • log N • log log N) algorithm by Schönhage-Strassen. Both these algorithms use modular arithmetic. Recently, Fürer gave an O(N • log N • 2O(log*N)) algorithm which however uses arithmetic over complex numbers as opposed to modular arithmetic. In this paper, we use multivariate polynomial multiplication along with ideas from Fürers algorithm to achieve this improvement in the modular setting. Our algorithm can also be viewed as a p-adic version of Fürers algorithm. Thus, we show that the two seemingly different approaches to integer multiplication, modular and complex arithmetic, are similar.

Fast Integer Multiplication Using Modular Arithmetic

We give an

Practical algorithms for tracking database join sizes

N\cdot \log N\cdot 2^{O(\log^*N)}

Separation Between Read-once Oblivious Algebraic Branching Programs (ROABPs) and Multilinear Depth Three Circuits

time algorithm to multiply two

The Power of Depth 2 Circuits over Algebras

N

FACTORING POLYNOMIALS OVER FINITE FIELDS USING BALANCE TEST

-bit integers that uses modular arithmetic for intermediate computations instead of arithmetic over complex numbers as in Furers algorithm, which also has the same and so far the best known complexity. The previous best algorithm using modular arithmetic (by Schonhage and Strassen) has complexity

Covering a set of points in a plane using two parallel rectangles

O(N \cdot \log N \cdot \log\log N)

Covering a Set of Points in a Plane Using Two Parallel Rectangles

. The advantage of using modular arithmetic as opposed to complex number arithmetic is that we can completely evade the task of bounding the truncation error due to finite approximations of complex numbers, which makes the analysis relatively simple. Our algorithm is based upon Furers algorithm, but uses fast Fourier transform over multivariate polynomials along with an estimate of the least prime in an arithmetic progression to achieve this improvement in the modular setting. It can also be viewed as a

Jacobian Hits Circuits: Hitting Sets, Lower Bounds for Depth-

p