4 research outputs found
Best and worst case permutations for random online domination of the path
We study a randomized algorithm for graph domination, by which, according to
a uniformly chosen permutation, vertices are revealed and added to the
dominating set if not already dominated. We determine the expected size of the
dominating set produced by the algorithm for the path graph and use this
to derive the expected size for some related families of graphs. We then
provide a much-refined analysis of the worst and best cases of this algorithm
on and enumerate the permutations for which the algorithm has the
worst-possible performance and best-possible performance. The case of
dominating the path graph has connections to previous work of Bouwer and Star,
and of Gessel on greedily coloring the path.Comment: 13 pages, 1 figur
On Regularity Lemma and Barriers in Streaming and Dynamic Matching
We present a new approach for finding matchings in dense graphs by building
on Szemer\'edi's celebrated Regularity Lemma. This allows us to obtain
non-trivial albeit slight improvements over longstanding bounds for matchings
in streaming and dynamic graphs. In particular, we establish the following
results for -vertex graphs:
* A deterministic single-pass streaming algorithm that finds
a -approximate matching in bits of space. This constitutes
the first single-pass algorithm for this problem in sublinear space that
improves over the -approximation of the greedy algorithm.
* A randomized fully dynamic algorithm that with high probability maintains a
-approximate matching in worst-case update time per each edge
insertion or deletion. The algorithm works even against an adaptive adversary.
This is the first update-time dynamic algorithm with approximation
guarantee arbitrarily close to one.
Given the use of regularity lemma, the improvement obtained by our algorithms
over trivial bounds is only by some factor.
Nevertheless, in each case, they show that the ``right'' answer to the problem
is not what is dictated by the previous bounds.
Finally, in the streaming model, we also present a randomized
-approximation algorithm whose space can be upper bounded by the
density of certain Ruzsa-Szemer\'edi (RS) graphs. While RS graphs by now have
been used extensively to prove streaming lower bounds, ours is the first to use
them as an upper bound tool for designing improved streaming algorithms
PSI from PaXoS: Fast, Malicious Private Set Intersection
We present a 2-party private set intersection (PSI) protocol which provides security against malicious participants, yet is almost as fast as the fastest known semi-honest PSI protocol of Kolesnikov et al. (CCS 2016).
Our protocol is based on a new approach for two-party PSI, which can be instantiated to provide security against either malicious or semi-honest adversaries. The protocol is unique in that the only difference between the semi-honest and malicious versions is an instantiation with different parameters for a linear error-correction code. It is also the first PSI protocol which is concretely efficient while having linear communication and security against malicious adversaries, while running in the OT-hybrid model (assuming a non-programmable random oracle).
State of the art semi-honest PSI protocols take advantage of cuckoo hashing, but it has proven a challenge to use cuckoo hashing for malicious security. Our protocol is the first to use cuckoo hashing for malicious-secure PSI. We do so via a new data structure, called a probe-and-XOR of strings (PaXoS), which may be of independent interest. This abstraction captures important properties of previous data structures, most notably garbled Bloom filters. While an encoding by a garbled Bloom filter is larger by a factor of than the original data, we describe a significantly improved PaXoS based on cuckoo hashing that achieves constant rate while being no worse in other relevant efficiency measures