345 research outputs found
Fast distributed almost stable marriages
In their seminal work on the Stable Marriage Problem, Gale and Shapley
describe an algorithm which finds a stable matching in communication
rounds. Their algorithm has a natural interpretation as a distributed algorithm
where each player is represented by a single processor. In this distributed
model, Floreen, Kaski, Polishchuk, and Suomela recently showed that for bounded
preference lists, terminating the Gale-Shapley algorithm after a constant
number of rounds results in an almost stable matching. In this paper, we
describe a new deterministic distributed algorithm which finds an almost stable
matching in communication rounds for arbitrary preferences. We
also present a faster randomized variant which requires rounds.
This run-time can be improved to rounds for "almost regular" (and in
particular complete) preferences. To our knowledge, these are the first
sub-polynomial round distributed algorithms for any variant of the stable
marriage problem with unbounded preferences.Comment: Various improvements in version 2: algorithms for general (not just
"almost regular") preferences; deterministic variant of the algorithm;
streamlined proof of approximation guarante
Packet Forwarding with a Locally Bursty Adversary
We consider packet forwarding in the adversarial queueing theory (AQT) model introduced by Borodin et al. We introduce a refinement of the AQT (?, ?)-bounded adversary, which we call a locally bursty adversary (LBA) that parameterizes injection patterns jointly by edge utilization and packet origin. For constant (O(1)) parameters, the LBA model is strictly more permissive than the (?, ?) model. For example, there are injection patterns in the LBA model with constant parameters that can only be realized as (?, ?)-bounded injection patterns with ? + ? = ?(n) (where n is the network size). We show that the LBA model (unlike the (?, ?) model) is closed under packet bundling and discretization operations. Thus, the LBA model allows one to reduce the study of general (uniform) capacity networks and inhomogenous packet sizes to unit capacity networks with homogeneous packets.
On the algorithmic side, we focus on information gathering networks - i.e., networks in which all packets share a common destination, and the union of packet routes forms a tree. We show that the Odd-Even Downhill (OED) forwarding protocol described independently by Dobrev et al. and Patt-Shamir and Rosenbaum achieves buffer space usage of O(log n) against all LBAs with constant parameters. OED is a local protocol, but we show that the upper bound is tight even when compared to centralized protocols. Our lower bound for the LBA model is in contrast to the (?, ?)-model, where centralized protocols can achieve worst-case buffer space usage O(1) for ?, ? = O(1), while the O(log n) upper bound for OED is optimal only for local protocols
With Great Speed Come Small Buffers: Space-Bandwidth Tradeoffs for Routing
We consider the Adversarial Queuing Theory (AQT) model, where packet arrivals
are subject to a maximum average rate and burstiness
. In this model, we analyze the size of buffers required to avoid
overflows in the basic case of a path. Our main results characterize the space
required by the average rate and the number of distinct destinations: we show
that space suffice, where is the number of distinct
destinations and ; and we show that space is necessary. For directed trees, we describe an algorithm
whose buffer space requirement is at most where is the
maximum number of destinations on any root-leaf path
A Stable Marriage Requires Communication
The Gale-Shapley algorithm for the Stable Marriage Problem is known to take
steps to find a stable marriage in the worst case, but only
steps in the average case (with women and men). In
1976, Knuth asked whether the worst-case running time can be improved in a
model of computation that does not require sequential access to the whole
input. A partial negative answer was given by Ng and Hirschberg, who showed
that queries are required in a model that allows certain natural
random-access queries to the participants' preferences. A significantly more
general - albeit slightly weaker - lower bound follows from Segal's general
analysis of communication complexity, namely that Boolean queries
are required in order to find a stable marriage, regardless of the set of
allowed Boolean queries.
Using a reduction to the communication complexity of the disjointness
problem, we give a far simpler, yet significantly more powerful argument
showing that Boolean queries of any type are indeed required for
finding a stable - or even an approximately stable - marriage. Notably, unlike
Segal's lower bound, our lower bound generalizes also to (A) randomized
algorithms, (B) allowing arbitrary separate preprocessing of the women's
preferences profile and of the men's preferences profile, (C) several variants
of the basic problem, such as whether a given pair is married in every/some
stable marriage, and (D) determining whether a proposed marriage is stable or
far from stable. In order to analyze "approximately stable" marriages, we
introduce the notion of "distance to stability" and provide an efficient
algorithm for its computation
On Sampling Edges Almost Uniformly
We consider the problem of sampling an edge almost uniformly from an unknown graph, G = (V, E). Access to the graph is provided via queries of the following types: (1) uniform vertex queries, (2) degree queries, and (3) neighbor queries. We describe a new simple algorithm that returns a random edge e in E using tilde{O}(n/sqrt{eps m}) queries in expectation, such that each edge e is sampled with probability (1 +/- eps)/m. Here, n = |V| is the number of vertices, and m = |E| is the number of edges. Our algorithm is optimal in the sense that any algorithm that samples an edge from an almost-uniform distribution must perform Omega(n/sqrt{m}) queries
Lower Bounds for Approximating Graph Parameters via Communication Complexity
In a celebrated work, Blais, Brody, and Matulef [Blais et al., 2012] developed a technique for proving property testing lower bounds via reductions from communication complexity. Their work focused on testing properties of functions, and yielded new lower bounds as well as simplified analyses of known lower bounds. Here, we take a further step in generalizing the methodology of [Blais et al., 2012] to analyze the query complexity of graph parameter estimation problems. In particular, our technique decouples the lower bound arguments from the representation of the graph, allowing it to work with any query type.
We illustrate our technique by providing new simpler proofs of previously known tight lower bounds for the query complexity of several graph problems: estimating the number of edges in a graph, sampling edges from an almost-uniform distribution, estimating the number of triangles (and more generally, r-cliques) in a graph, and estimating the moments of the degree distribution of a graph. We also prove new lower bounds for estimating the edge connectivity of a graph and estimating the number of instances of any fixed subgraph in a graph. We show that the lower bounds for estimating the number of triangles and edge connectivity also hold in a strictly stronger computational model that allows access to uniformly random edge samples
Stable Matchings with Restricted Preferences: Structure and Complexity
It is well known that every stable matching instance has a rotation poset
that can be computed efficiently and the downsets of are in
one-to-one correspondence with the stable matchings of . Furthermore, for
every poset , an instance can be constructed efficiently so that the
rotation poset of is isomorphic to . In this case, we say that
realizes . Many researchers exploit the rotation poset of an instance to
develop fast algorithms or to establish the hardness of stable matching
problems.
In order to gain a parameterized understanding of the complexity of sampling
stable matchings, Bhatnagar et al. [SODA 2008] introduced stable matching
instances whose preference lists are restricted but nevertheless model
situations that arise in practice. In this paper, we study four such
parameterized restrictions; our goal is to characterize the rotation posets
that arise from these models: -bounded, -attribute, -list,
-range.
We prove that there is a constant so that every rotation poset is
realized by some instance in the first three models for some fixed constant
. We describe efficient algorithms for constructing such instances given the
Hasse diagram of a poset. As a consequence, the fundamental problem of counting
stable matchings remains BIS-complete even for these restricted instances.
For -range preferences, we show that a poset is realizable if and only
if the Hasse diagram of has pathwidth bounded by functions of . Using
this characterization, we show that the following problems are fixed parameter
tractable when parametrized by the range of the instance: exactly counting and
uniformly sampling stable matchings, finding median, sex-equal, and balanced
stable matchings.Comment: Various updates and improvements in response to reviewer comment
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