13 research outputs found
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
Being Fast Means Being Chatty: The Local Information Cost of Graph Spanners
We introduce a new measure for quantifying the amount of information that the
nodes in a network need to learn to jointly solve a graph problem. We show that
the local information cost () presents a natural lower bound on
the communication complexity of distributed algorithms. For the synchronous
CONGEST-KT1 model, where each node has initial knowledge of its neighbors' IDs,
we prove that bits are
required for solving a graph problem with a -round algorithm that
errs with probability at most . Our result is the first lower bound
that yields a general trade-off between communication and time for graph
problems in the CONGEST-KT1 model.
We demonstrate how to apply the local information cost by deriving a lower
bound on the communication complexity of computing a -spanner that
consists of at most edges, where . Our main result is that any -time
algorithm must send at least bits in the
CONGEST model under the KT1 assumption. Previously, only a trivial lower bound
of bits was known for this problem.
A consequence of our lower bound is that achieving both time- and
communication-optimality is impossible when designing a distributed spanner
algorithm. In light of the work of King, Kutten, and Thorup (PODC 2015), this
shows that computing a minimum spanning tree can be done significantly faster
than finding a spanner when considering algorithms with
communication complexity. Our result also implies time complexity lower bounds
for constructing a spanner in the node-congested clique of Augustine et al.
(2019) and in the push-pull gossip model with limited bandwidth
Total Completion Time Minimization for Scheduling with Incompatibility Cliques
This paper considers parallel machine scheduling with incompatibilities
between jobs. The jobs form a graph and no two jobs connected by an edge are
allowed to be assigned to the same machine. In particular, we study the case
where the graph is a collection of disjoint cliques. Scheduling with
incompatibilities between jobs represents a well-established line of research
in scheduling theory and the case of disjoint cliques has received increasing
attention in recent years. While the research up to this point has been focused
on the makespan objective, we broaden the scope and study the classical total
completion time criterion. In the setting without incompatibilities, this
objective is well known to admit polynomial time algorithms even for unrelated
machines via matching techniques. We show that the introduction of
incompatibility cliques results in a richer, more interesting picture.
Scheduling on identical machines remains solvable in polynomial time, while
scheduling on unrelated machines becomes APX-hard. Furthermore, we study the
problem under the paradigm of fixed-parameter tractable algorithms (FPT). In
particular, we consider a problem variant with assignment restrictions for the
cliques rather than the jobs. We prove that it is NP-hard and can be solved in
FPT time with respect to the number of cliques. Moreover, we show that the
problem on unrelated machines can be solved in FPT time for reasonable
parameters, e.g., the parameter pair: number of machines and maximum processing
time. The latter result is a natural extension of known results for the case
without incompatibilities and can even be extended to the case of total
weighted completion time. All of the FPT results make use of n-fold Integer
Programs that recently have received great attention by proving their
usefulness for scheduling problems
Parallel Algorithms for Small Subgraph Counting
Subgraph counting is a fundamental problem in analyzing massive graphs, often
studied in the context of social and complex networks. There is a rich
literature on designing efficient, accurate, and scalable algorithms for this
problem. In this work, we tackle this challenge and design several new
algorithms for subgraph counting in the Massively Parallel Computation (MPC)
model:
Given a graph over vertices, edges and triangles, our first
main result is an algorithm that, with high probability, outputs a
-approximation to , with optimal round and space complexity
provided any space per machine, assuming
.
Our second main result is an -rounds
algorithm for exactly counting the number of triangles, parametrized by the
arboricity of the input graph. The space per machine is
for any constant , and the total space is ,
which matches the time complexity of (combinatorial) triangle counting in the
sequential model. We also prove that this result can be extended to exactly
counting -cliques for any constant , with the same round complexity and
total space . Alternatively, allowing space per
machine, the total space requirement reduces to .
Finally, we prove that a recent result of Bera, Pashanasangi and Seshadhri
(ITCS 2020) for exactly counting all subgraphs of size at most , can be
implemented in the MPC model in rounds,
space per machine and total space. Therefore,
this result also exhibits the phenomenon that a time bound in the sequential
model translates to a space bound in the MPC model