3 research outputs found
The Communication Complexity of Set Intersection and Multiple Equality Testing
In this paper we explore fundamental problems in randomized communication
complexity such as computing Set Intersection on sets of size and Equality
Testing between vectors of length . Sa\u{g}lam and Tardos and Brody et al.
showed that for these types of problems, one can achieve optimal communication
volume of bits, with a randomized protocol that takes
rounds. Aside from rounds and communication volume, there is a \emph{third}
parameter of interest, namely the \emph{error probability} .
It is straightforward to show that protocols for Set Intersection or Equality
Testing need to send bits. Is it
possible to simultaneously achieve optimality in all three parameters, namely
communication and rounds? In
this paper we prove that there is no universally optimal algorithm, and
complement the existing round-communication tradeoffs with a new tradeoff
between rounds, communication, and probability of error. In particular:
1. Any protocol for solving Multiple Equality Testing in rounds with
failure probability has communication volume .
2. There exists a protocol for solving Multiple Equality Testing in rounds with communication, thereby essentially
matching our lower bound and that of Sa\u{g}lam and Tardos.
Our original motivation for considering as an independent
parameter came from the problem of enumerating triangles in distributed
() networks having maximum degree . We prove that
this problem can be solved in time with
high probability .Comment: 44 page
Efficient Algorithms for Large Scale Network Problems
In recent years, the growing scale of data has renewed our understanding of what is an efficient algorithm and poses many essential challenges for the algorithm designers. This thesis aims to improve our understanding of many algorithmic problems in this context. These include problems in communication complexity, matching theory, and approximate query processing for database systems.
We first study the fundamental and well-known question of {SetIntersection} in communication complexity. We give a result that incorporates the error probability as an independent parameter into the classical trade-off between round complexity and communication complexity. We show that any -round protocol that errs with error probability requires bits of communication. We also give several almost matching upper bounds.
In matching theory, we first study several generalizations of the ordinary matching problem, namely the -matching and -edge cover problem. We also consider the problem of computing a minimum weight perfect matching in a metric space with moderate expansion. We give almost linear time approximation algorithms for all these problems.
Finally, we study the sample-based join problem in approximate query processing. We present a result that improves our understanding of the effectiveness and limitations in using sampling to approximate join queries and provides a guideline for practitioners in building AQP systems from a theory perspective.PHDComputer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155263/1/hdawei_1.pd