21 research outputs found
Sparse Hopsets in Congested Clique
We give the first Congested Clique algorithm that computes a sparse hopset
with polylogarithmic hopbound in polylogarithmic time. Given a graph ,
a -hopset with "hopbound" , is a set of edges
added to such that for any pair of nodes and in there is a path
with at most hops in with length within of
the shortest path between and in .
Our hopsets are significantly sparser than the recent construction of
Censor-Hillel et al. [6], that constructs a hopset of size
, but with a smaller polylogarithmic hopbound. On the other
hand, the previously known constructions of sparse hopsets with polylogarithmic
hopbound in the Congested Clique model, proposed by Elkin and Neiman
[10],[11],[12], all require polynomial rounds.
One tool that we use is an efficient algorithm that constructs an
-limited neighborhood cover, that may be of independent interest.
Finally, as a side result, we also give a hopset construction in a variant of
the low-memory Massively Parallel Computation model, with improved running time
over existing algorithms
Massively Parallel Approximate Distance Sketches
Data structures that allow efficient distance estimation (distance oracles, distance sketches, etc.) have been extensively studied, and are particularly well studied in centralized models and classical distributed models such as CONGEST. We initiate their study in newer (and arguably more realistic) models of distributed computation: the Congested Clique model and the Massively Parallel Computation (MPC) model. We provide efficient constructions in both of these models, but our core results are for MPC. In MPC we give two main results: an algorithm that constructs stretch/space optimal distance sketches but takes a (small) polynomial number of rounds, and an algorithm that constructs distance sketches with worse stretch but that only takes polylogarithmic rounds.
Along the way, we show that other useful combinatorial structures can also be computed in MPC. In particular, one key component we use to construct distance sketches are an MPC construction of the hopsets of [Elkin and Neiman, 2016]. This result has additional applications such as the first polylogarithmic time algorithm for constant approximate single-source shortest paths for weighted graphs in the low memory MPC setting
Brief Announcement: Massively Parallel Approximate Distance Sketches
Data structures that allow efficient distance estimation have been extensively studied both in centralized models and classical distributed models. We initiate their study in newer (and arguably more realistic) models of distributed computation: the Congested Clique model and the Massively Parallel Computation (MPC) model. In MPC we give two main results: an algorithm that constructs stretch/space optimal distance sketches but takes a (small) polynomial number of rounds, and an algorithm that constructs distance sketches with worse stretch but that only takes polylogarithmic rounds. Along the way, we show that other useful combinatorial structures can also be computed in MPC. In particular, one key component we use is an MPC construction of the hopsets of Elkin and Neiman (2016). This result has additional applications such as the first polylogarithmic time algorithm for constant approximate single-source shortest paths for weighted graphs in the low memory MPC setting
Distributed Distance-Bounded Network Design Through Distributed Convex Programming
Solving linear programs is often a challenging task in distributed settings.
While there are good algorithms for solving packing and covering linear
programs in a distributed manner (Kuhn et al.~2006), this is essentially the
only class of linear programs for which such an algorithm is known. In this
work we provide a distributed algorithm for solving a different class of convex
programs which we call "distance-bounded network design convex programs". These
can be thought of as relaxations of network design problems in which the
connectivity requirement includes a distance constraint (most notably, graph
spanners). Our algorithm runs in rounds in the
model and finds a -approximation to the optimal
LP solution for any , where is the largest distance
constraint. While solving linear programs in a distributed setting is
interesting in its own right, this class of convex programs is particularly
important because solving them is often a crucial step when designing
approximation algorithms. Hence we almost immediately obtain new and improved
distributed approximation algorithms for a variety of network design problems,
including Basic - and -Spanner, Directed -Spanner, Lowest Degree
-Spanner, and Shallow-Light Steiner Network Design with a spanning demand
graph. Our algorithms do not require any "heavy" computation and essentially
match the best-known centralized approximation algorithms, while previous
approaches which do not use heavy computation give approximations which are
worse than the best-known centralized bounds
DISTRIBUTED, PARALLEL AND DYNAMIC DISTANCE STRUCTURES
Many fundamental computational tasks can be modeled by distances on a graph. This has inspired studying various structures that preserve approximate distances, but trade off this approximation factor with size, running time, or the number of hops on the approximate shortest paths.
Our focus is on three important objects involving preservation of graph distances: hopsets, in which our goal is to ensure that small-hop paths also provide approximate shortest paths; distance oracles, in which we build a small data structure that supports efficient distance queries; and spanners, in which we find a sparse subgraph that approximately preserves all distances.
We study efficient constructions and applications of these structures in various models of computation that capture different aspects of computational systems. Specifically, we propose new algorithms for constructing hopsets and distance oracles in two modern distributed models: the Massively Parallel Computation (MPC) and the Congested Clique model. These models have received significant attention recently due to their close connection to present-day big data platforms.
In a different direction, we consider a centralized dynamic model in which the input changes over time. We propose new dynamic algorithms for constructing hopsets and distance oracles that lead to state-of-the-art approximate single-source, multi-source and all-pairs shortest path algorithms with respect to update-time.
Finally, we study the problem of finding optimal spanners in a different distributed model, the LOCAL model. Unlike our other results, for this problem our goal is to find the best solution for a specific input graph rather than giving a general guarantee that holds for all inputs.
One contribution of this work is to emphasize the significance of the tools and the techniques used for these distance problems rather than heavily focusing on a specific model.
In other words, we show that our techniques are broad enough that they can be extended to different models
Epic Fail: Emulators Can Tolerate Polynomially Many Edge Faults for Free
A t-emulator of a graph G is a graph H that approximates its pairwise shortest path distances up to multiplicative t error. We study fault tolerant t-emulators, under the model recently introduced by Bodwin, Dinitz, and Nazari [ITCS 2022] for vertex failures. In this paper we consider the version for edge failures, and show that they exhibit surprisingly different behavior.
In particular, our main result is that, for (2k-1)-emulators with k odd, we can tolerate a polynomial number of edge faults for free. For example: for any n-node input graph, we construct a 5-emulator (k = 3) on O(n^{4/3}) edges that is robust to f = O(n^{2/9}) edge faults. It is well known that ?(n^{4/3}) edges are necessary even if the 5-emulator does not need to tolerate any faults. Thus we pay no extra cost in the size to gain this fault tolerance. We leave open the precise range of free fault tolerance for odd k, and whether a similar phenomenon can be proved for even k
Vertex Fault-Tolerant Emulators
A -spanner of a graph is a sparse subgraph that preserves its shortest
path distances up to a multiplicative stretch factor of , and a -emulator
is similar but not required to be a subgraph of . A classic theorem by
Thorup and Zwick [JACM '05] shows that, despite the extra flexibility available
to emulators, the size/stretch tradeoffs for spanners and emulators are
equivalent. Our main result is that this equivalence in tradeoffs no longer
holds in the commonly-studied setting of graphs with vertex failures. That is:
we introduce a natural definition of vertex fault-tolerant emulators, and then
we show a three-way tradeoff between size, stretch, and fault-tolerance for
these emulators that polynomially surpasses the tradeoff known to be optimal
for spanners.
We complement our emulator upper bound with a lower bound construction that
is essentially tight (within factors of the upper bound) when the
stretch is and is either a fixed odd integer or . We also show
constructions of fault-tolerant emulators with additive error, demonstrating
that these also enjoy significantly improved tradeoffs over those available for
fault-tolerant additive spanners.Comment: To appear in ITCS 202
Massively Parallel Algorithms for Distance Approximation and Spanners
Over the past decade, there has been increasing interest in
distributed/parallel algorithms for processing large-scale graphs. By now, we
have quite fast algorithms -- usually sublogarithmic-time and often
-time, or even faster -- for a number of fundamental graph
problems in the massively parallel computation (MPC) model. This model is a
widely-adopted theoretical abstraction of MapReduce style settings, where a
number of machines communicate in an all-to-all manner to process large-scale
data. Contributing to this line of work on MPC graph algorithms, we present
round MPC algorithms for computing
-spanners in the strongly sublinear regime of local memory. To
the best of our knowledge, these are the first sublogarithmic-time MPC
algorithms for spanner construction. As primary applications of our spanners,
we get two important implications, as follows:
-For the MPC setting, we get an -round algorithm for
approximation of all pairs shortest paths (APSP) in the
near-linear regime of local memory. To the best of our knowledge, this is the
first sublogarithmic-time MPC algorithm for distance approximations.
-Our result above also extends to the Congested Clique model of distributed
computing, with the same round complexity and approximation guarantee. This
gives the first sub-logarithmic algorithm for approximating APSP in weighted
graphs in the Congested Clique model
Deterministic Incremental APSP with Polylogarithmic Update Time and Stretch
We provide the first deterministic data structure that given a weighted
undirected graph undergoing edge insertions, processes each update with
polylogarithmic amortized update time and answers queries for the distance
between any pair of vertices in the current graph with a polylogarithmic
approximation in time.
Prior to this work, no data structure was known for partially dynamic graphs,
i.e., graphs undergoing either edge insertions or deletions, with less than
update time except for dense graphs, even when allowing
randomization against oblivious adversaries or considering only single-source
distances