1,073 research outputs found
Space-Time Tradeoffs for Distributed Verification
Verifying that a network configuration satisfies a given boolean predicate is
a fundamental problem in distributed computing. Many variations of this problem
have been studied, for example, in the context of proof labeling schemes (PLS),
locally checkable proofs (LCP), and non-deterministic local decision (NLD). In
all of these contexts, verification time is assumed to be constant. Korman,
Kutten and Masuzawa [PODC 2011] presented a proof-labeling scheme for MST, with
poly-logarithmic verification time, and logarithmic memory at each vertex.
In this paper we introduce the notion of a -PLS, which allows the
verification procedure to run for super-constant time. Our work analyzes the
tradeoffs of -PLS between time, label size, message length, and computation
space. We construct a universal -PLS and prove that it uses the same amount
of total communication as a known one-round universal PLS, and factor
smaller labels. In addition, we provide a general technique to prove lower
bounds for space-time tradeoffs of -PLS. We use this technique to show an
optimal tradeoff for testing that a network is acyclic (cycle free). Our
optimal -PLS for acyclicity uses label size and computation space . We further describe a recursive space verifier for
acyclicity which does not assume previous knowledge of the run-time .Comment: Pre-proceedings version of paper presented at the 24th International
Colloquium on Structural Information and Communication Complexity (SIROCCO
2017
Silent MST approximation for tiny memory
In network distributed computing, minimum spanning tree (MST) is one of the
key problems, and silent self-stabilization one of the most demanding
fault-tolerance properties. For this problem and this model, a polynomial-time
algorithm with memory is known for the state model. This is
memory optimal for weights in the classic range (where
is the size of the network). In this paper, we go below this
memory, using approximation and parametrized complexity.
More specifically, our contributions are two-fold. We introduce a second
parameter~, which is the space needed to encode a weight, and we design a
silent polynomial-time self-stabilizing algorithm, with space . In turn, this allows us to get an approximation algorithm for the problem,
with a trade-off between the approximation ratio of the solution and the space
used. For polynomial weights, this trade-off goes smoothly from memory for an -approximation, to memory for exact solutions,
with for example memory for a 2-approximation
Covering Points by Disjoint Boxes with Outliers
For a set of n points in the plane, we consider the axis--aligned (p,k)-Box
Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together
contain n-k points. In this paper, we consider the boxes to be either squares
or rectangles, and we want to minimize the area of the largest box. For general
p we show that the problem is NP-hard for both squares and rectangles. For a
small, fixed number p, we give algorithms that find the solution in the
following running times:
For squares we have O(n+k log k) time for p=1, and O(n log n+k^p log^p k time
for p = 2,3. For rectangles we get O(n + k^3) for p = 1 and O(n log n+k^{2+p}
log^{p-1} k) time for p = 2,3.
In all cases, our algorithms use O(n) space.Comment: updated version: - changed problem from 'cover exactly n-k points' to
'cover at least n-k points' to avoid having non-feasible solutions. Results
are unchanged. - added Proof to Lemma 11, clarified some sections - corrected
typos and small errors - updated affiliations of two author
Circumscribing Polygons and Polygonizations for Disjoint Line Segments
Given a planar straight-line graph G=(V,E) in R^2, a circumscribing polygon of G is a simple polygon P whose vertex set is V, and every edge in E is either an edge or an internal diagonal of P. A circumscribing polygon is a polygonization for G if every edge in E is an edge of P.
We prove that every arrangement of n disjoint line segments in the plane has a subset of size Omega(sqrt{n}) that admits a circumscribing polygon, which is the first improvement on this bound in 20 years. We explore relations between circumscribing polygons and other problems in combinatorial geometry, and generalizations to R^3.
We show that it is NP-complete to decide whether a given graph G admits a circumscribing polygon, even if G is 2-regular. Settling a 30-year old conjecture by Rappaport, we also show that it is NP-complete to determine whether a geometric matching admits a polygonization
A Time-Space Tradeoff for Triangulations of Points in the Plane
In this paper, we consider time-space trade-offs for reporting a triangulation of points in the plane. The goal is to minimize the amount of working space while keeping the total running time small. We present the first multi-pass algorithm on the problem that returns the edges of a triangulation with their adjacency information. This even improves the previously best known random-access algorithm
Labeling Schemes for Bounded Degree Graphs
We investigate adjacency labeling schemes for graphs of bounded degree
. In particular, we present an optimal (up to an additive
constant) adjacency labeling scheme for bounded degree trees.
The latter scheme is derived from a labeling scheme for bounded degree
outerplanar graphs. Our results complement a similar bound recently obtained
for bounded depth trees [Fraigniaud and Korman, SODA 10], and may provide new
insights for closing the long standing gap for adjacency in trees [Alstrup and
Rauhe, FOCS 02]. We also provide improved labeling schemes for bounded degree
planar graphs. Finally, we use combinatorial number systems and present an
improved adjacency labeling schemes for graphs of bounded degree with
How Long It Takes for an Ordinary Node with an Ordinary ID to Output?
In the context of distributed synchronous computing, processors perform in
rounds, and the time-complexity of a distributed algorithm is classically
defined as the number of rounds before all computing nodes have output. Hence,
this complexity measure captures the running time of the slowest node(s). In
this paper, we are interested in the running time of the ordinary nodes, to be
compared with the running time of the slowest nodes. The node-averaged
time-complexity of a distributed algorithm on a given instance is defined as
the average, taken over every node of the instance, of the number of rounds
before that node output. We compare the node-averaged time-complexity with the
classical one in the standard LOCAL model for distributed network computing. We
show that there can be an exponential gap between the node-averaged
time-complexity and the classical time-complexity, as witnessed by, e.g.,
leader election. Our first main result is a positive one, stating that, in
fact, the two time-complexities behave the same for a large class of problems
on very sparse graphs. In particular, we show that, for LCL problems on cycles,
the node-averaged time complexity is of the same order of magnitude as the
slowest node time-complexity.
In addition, in the LOCAL model, the time-complexity is computed as a worst
case over all possible identity assignments to the nodes of the network. In
this paper, we also investigate the ID-averaged time-complexity, when the
number of rounds is averaged over all possible identity assignments. Our second
main result is that the ID-averaged time-complexity is essentially the same as
the expected time-complexity of randomized algorithms (where the expectation is
taken over all possible random bits used by the nodes, and the number of rounds
is measured for the worst-case identity assignment).
Finally, we study the node-averaged ID-averaged time-complexity.Comment: (Submitted) Journal versio
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