Stochastics and Statistics Seminar

Optimal lower bounds for universal relation, and for samplers and finding duplicates in streams

Speaker Name: Jelani Nelson (Harvard University)

Date: September 29, 2017

Time: 11:00am

Location: E18-304


Consider the following problem: we monitor a sequence of edge
insertions and deletions in a graph on n vertices, so there are N = (n
choose 2) possible edges (e.g. monitoring a stream of friend
accepts/removals on Facebook). At any point someone may say "query()",
at which point must output a random edge that exists in the graph at
that time from a distribution that is statistically close to uniform.
More specifically, with probability p our edge should come from a
distribution close to uniform, and with probability 1-p our sampler
can do anything (i.e. it can say "Fail", or output a non-existent
edge). The primary goal is to design an algorithm that uses very
little memory. Whereas the trivial solution maintains N bits, keeping
track of which edges currently exist, we show that the optimal space
complexity for this problem is Theta(min{N, log(1/p) * log^2(2 + N /
log(1/p))}) bits. The upper bound follows by a very minor modification
of previous work, whereas the lower bound is our main novel

This is joint work with Michael Kapralov (EPFL), Jakub Pachocki
(OpenAI), Zhengyu Wang (Harvard), David P. Woodruff (IBM Almaden), and
Mobin Yahyazadeh (Sharif).

Speaker Bio:

Jelani Nelson is Associate Professor of Computer Science at Harvard.
His main research interest is in algorithm design and analysis, with
recent focus on streaming algorithms, dimensionality reduction,
compressed sensing, and randomized linear algebra algorithms. He
completed his Ph.D. in computer science at MIT in 2011, receiving the
George M. Sprowls Award for best computer science doctoral
dissertations at MIT. He is the recipient of an NSF CAREER Award, ONR
Young Investigator Award, ONR Director of Research Early Career Award,
Alfred P. Sloan Research Fellowship, and Presidential Early Career
Award for Scientists and Engineers (PECASE).