18 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
A Unified Framework for Hopsets
Given an undirected graph G = (V,E), an (?,?)-hopset is a graph H = (V,E\u27), so that adding its edges to G guarantees every pair has an ?-approximate shortest path that has at most ? edges (hops), that is, d_G(u,v) ? d_{G?H}^(?)(u,v) ? ?? d_G(u,v). Given the usefulness of hopsets for fundamental algorithmic tasks, several different algorithms and techniques were developed for their construction, for various regimes of the stretch parameter ?.
In this work we devise a single algorithm that can attain all state-of-the-art hopsets for general graphs, by choosing the appropriate input parameters. In fact, in some cases it also improves upon the previous best results. We also show a lower bound on our algorithm.
In [Ben-Levy and Parter, 2020], given a parameter k, a (O(k^?),O(k^{1-?}))-hopset of size O?(n^{1+1/k}) was shown for any n-vertex graph and parameter 0 < ? < 1, and they asked whether this result is best possible. We resolve this open problem, showing that any (?,?)-hopset of size O(n^{1+1/k}) must have ??? ? ?(k)
Almost Shortest Paths with Near-Additive Error in Weighted Graphs
Let be a weighted undirected graph with vertices and
edges, and fix a set of sources . We study the problem of
computing {\em almost shortest paths} (ASP) for all pairs in in
both classical centralized and parallel (PRAM) models of computation. Consider
the regime of multiplicative approximation of , for an arbitrarily
small constant . In this regime existing centralized algorithms
require time, where is the
matrix multiplication exponent. Existing PRAM algorithms with polylogarithmic
depth (aka time) require work .
Our centralized algorithm has running time , and its PRAM
counterpart has polylogarithmic depth and work , for an
arbitrarily small constant . For a pair , it
provides a path of length that satisfies , where is the weight of the
heaviest edge on some shortest path. Hence our additive term depends
linearly on a {\em local} maximum edge weight, as opposed to the global maximum
edge weight in previous works. Finally, our .
We also extend a centralized algorithm of Dor et al. \cite{DHZ00}. For a
parameter , this algorithm provides for {\em unweighted}
graphs a purely additive approximation of for {\em all pairs
shortest paths} (APASP) in time . Within the same
running time, our algorithm for {\em weighted} graphs provides a purely
additive error of , for every vertex pair , with defined as above.
On the way to these results we devise a suit of novel constructions of
spanners, emulators and hopsets
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
Closing the Gap Between Directed Hopsets and Shortcut Sets
For an n-vertex directed graph , a -\emph{shortcut set}
is a set of additional edges such that has
the same transitive closure as , and for every pair , there is a
-path in with at most edges. A natural generalization of
shortcut sets to distances is a -\emph{hopset} , where the requirement is that and have the same
shortest-path distances, and for every , there is a
-approximate shortest path in with at most
edges.
There is a large literature on the tradeoff between the size of a shortcut
set / hopset and the value of . We highlight the most natural point on
this tradeoff: what is the minimum value of , such that for any graph
, there exists a -shortcut set (or a -hopset) with
edges? Not only is this a natural structural question in its own right,
but shortcuts sets / hopsets form the core of many distributed, parallel, and
dynamic algorithms for reachability / shortest paths. Until very recently the
best known upper bound was a folklore construction showing , but in a breakthrough result Kogan and Parter [SODA 2022] improve
this to for shortcut sets and
for hopsets.
Our result is to close the gap between shortcut sets and hopsets. That is, we
show that for any graph and any fixed there is a
hopset with edges. More generally, we
achieve a smooth tradeoff between hopset size and which exactly matches
the tradeoff of Kogan and Parter for shortcut sets (up to polylog factors).
Using a very recent black-box reduction of Kogan and Parter, our new hopset
implies improved bounds for approximate distance preservers.Comment: Abstract shortened to meet arXiv requirements, v2: fixed a typ
Centralized, Parallel, and Distributed Multi-Source Shortest Paths via Hopsets and Rectangular Matrix Multiplication
Consider an undirected weighted graph . We study the problem of
computing -approximate shortest paths for , for a
subset of sources, for some . We
devise a significantly improved algorithm for this problem in the entire range
of parameter , in both the classical centralized and the parallel (PRAM)
models of computation, and in a wide range of in the distributed (Congested
Clique) model. Specifically, our centralized algorithm for this problem
requires time , where
is the time required to multiply an matrix by an
one. Our PRAM algorithm has polylogarithmic time , and its work complexity is , for any arbitrarily small constant .
In particular, for , our centralized algorithm computes -approximate shortest paths in time. Our
PRAM polylogarithmic-time algorithm has work complexity , for any arbitrarily small constant . Previously existing
solutions either require centralized time/parallel work of
or provide much weaker approximation guarantees.
In the Congested Clique model, our algorithm solves the problem in
polylogarithmic time for sources, for , while previous
state-of-the-art algorithms did so only for . Moreover, it improves
previous bounds for all . For unweighted graphs, the running time is
improved further to
Path-Reporting Distance Oracles with Near-Logarithmic Stretch and Linear Size
Given an -vertex undirected graph , and a parameter , a
path-reporting distance oracle (or PRDO) is a data structure of size ,
that given a query , returns an -approximate shortest
path in within time . Here , and are
arbitrary functions.
A landmark PRDO due to Thorup and Zwick, with an improvement of Wulff-Nilsen,
has , and . The
size of this oracle is for all . Elkin and Pettie and
Neiman and Shabat devised much sparser PRDOs, but their stretch was
polynomially larger than the optimal . On the other hand, for
non-path-reporting distance oracles, Chechik devised a result with
, and .
In this paper we make a dramatic progress in bridging the gap between
path-reporting and non-path-reporting distance oracles. We devise a PRDO with
size ,
stretch and query time . We can also have size , stretch
and query time
.
Our results on PRDOs are based on novel constructions of approximate distance
preservers, that we devise in this paper. Specifically, we show that for any
, any , and any graph and a collection
of vertex pairs, there exists a -approximate preserver with
edges, where
. These new
preservers are significantly sparser than the previous state-of-the-art
approximate preservers due to Kogan and Parter.Comment: 61 pages, 3 figure
A Local-To-Global Theorem for Congested Shortest Paths
P_k such that each P_i is a shortest path from s_i to t_i, and every node in the graph is on at most c paths P_i, or reporting that no such collection of paths exists. When c = k, there are no congestion constraints, and the problem can be solved easily by running a shortest path algorithm for each pair (s_i,t_i) independently. At the other extreme, when c = 1, the (k,c)-SPC problem is equivalent to the k-Disjoint Shortest Paths (k-DSP) problem, where the collection of shortest paths must be node-disjoint. For fixed k, k-DSP is polynomial-time solvable on DAGs and undirected graphs. Amiri and Wargalla interpolated between these two extreme values of c, to obtain an algorithm for (k,c)-SPC on DAGs that runs in polynomial time when k-c is constant.
In the same way, we prove that (k,c)-SPC can be solved in polynomial time on undirected graphs whenever k-c is constant. For directed graphs, because of our counterexample to the original theorem statement, our roundtrip local-to-global result does not imply such an algorithm (k,c)-SPC. Even without an algorithmic application, our proof for directed graphs may be of broader interest because it characterizes intriguing structural properties of shortest paths in directed graphs