2,327 research outputs found
Latency Optimal Broadcasting in Noisy Wireless Mesh Networks
In this paper, we adopt a new noisy wireless network model introduced very
recently by Censor-Hillel et al. in [ACM PODC 2017, CHHZ17]. More specifically,
for a given noise parameter any sender has a probability of
of transmitting noise or any receiver of a single transmission in its
neighborhood has a probability of receiving noise.
In this paper, we first propose a new asymptotically latency-optimal
approximation algorithm (under faultless model) that can complete
single-message broadcasting task in time units/rounds in any
WMN of size and diameter . We then show this diameter-linear
broadcasting algorithm remains robust under the noisy wireless network model
and also improves the currently best known result in CHHZ17 by a
factor.
In this paper, we also further extend our robust single-message broadcasting
algorithm to multi-message broadcasting scenario and show it can broadcast
messages in time rounds. This new robust
multi-message broadcasting scheme is not only asymptotically optimal but also
answers affirmatively the problem left open in CHHZ17 on the existence of an
algorithm that is robust to sender and receiver faults and can broadcast
messages in time rounds.Comment: arXiv admin note: text overlap with arXiv:1705.07369 by other author
An FPT Algorithm for Directed Spanning k-Leaf
An out-branching of a directed graph is a rooted spanning tree with all arcs
directed outwards from the root. We consider the problem of deciding whether a
given directed graph D has an out-branching with at least k leaves (Directed
Spanning k-Leaf). We prove that this problem is fixed parameter tractable, when
k is chosen as the parameter. Previously this was only known for restricted
classes of directed graphs.
The main new ingredient in our approach is a lemma that shows that given a
locally optimal out-branching of a directed graph in which every arc is part of
at least one out-branching, either an out-branching with at least k leaves
exists, or a path decomposition with width O(k^3) can be found. This enables a
dynamic programming based algorithm of running time 2^{O(k^3 \log k)} n^{O(1)},
where n=|V(D)|.Comment: 17 pages, 8 figure
Near Optimal Parallel Algorithms for Dynamic DFS in Undirected Graphs
Depth first search (DFS) tree is a fundamental data structure for solving
graph problems. The classical algorithm [SiComp74] for building a DFS tree
requires time for a given graph having vertices and edges.
Recently, Baswana et al. [SODA16] presented a simple algorithm for updating DFS
tree of an undirected graph after an edge/vertex update in time.
However, their algorithm is strictly sequential. We present an algorithm
achieving similar bounds, that can be adopted easily to the parallel
environment.
In the parallel model, a DFS tree can be computed from scratch using
processors in expected time [SiComp90] on an EREW PRAM, whereas
the best deterministic algorithm takes time
[SiComp90,JAlg93] on a CRCW PRAM. Our algorithm can be used to develop optimal
(upto polylog n factors deterministic algorithms for maintaining fully dynamic
DFS and fault tolerant DFS, of an undirected graph.
1- Parallel Fully Dynamic DFS:
Given an arbitrary online sequence of vertex/edge updates, we can maintain a
DFS tree of an undirected graph in time per update using
processors on an EREW PRAM.
2- Parallel Fault tolerant DFS:
An undirected graph can be preprocessed to build a data structure of size
O(m) such that for a set of updates (where is constant) in the graph,
the updated DFS tree can be computed in time using
processors on an EREW PRAM.
Moreover, our fully dynamic DFS algorithm provides, in a seamless manner,
nearly optimal (upto polylog n factors) algorithms for maintaining a DFS tree
in semi-streaming model and a restricted distributed model. These are the first
parallel, semi-streaming and distributed algorithms for maintaining a DFS tree
in the dynamic setting.Comment: Accepted to appear in SPAA'17, 32 Pages, 5 Figure
On Efficient Distributed Construction of Near Optimal Routing Schemes
Given a distributed network represented by a weighted undirected graph
on vertices, and a parameter , we devise a distributed
algorithm that computes a routing scheme in
rounds, where is the hop-diameter of the network. The running time matches
the lower bound of rounds (which holds for any
scheme with polynomial stretch), up to lower order terms. The routing tables
are of size , the labels are of size , and
every packet is routed on a path suffering stretch at most . Our
construction nearly matches the state-of-the-art for routing schemes built in a
centralized sequential manner. The previous best algorithms for building
routing tables in a distributed small messages model were by \cite[STOC
2013]{LP13} and \cite[PODC 2015]{LP15}. The former has similar properties but
suffers from substantially larger routing tables of size ,
while the latter has sub-optimal running time of
Completely Independent Spanning Trees in Line Graphs
Completely independent spanning trees in a graph are spanning trees of
such that for any two distinct vertices of , the paths between them in
the spanning trees are pairwise edge-disjoint and internally vertex-disjoint.
In this paper, we present a tight lower bound on the maximum number of
completely independent spanning trees in , where denotes the line
graph of a graph . Based on a new characterization of a graph with
completely independent spanning trees, we also show that for any complete graph
of order , there are completely
independent spanning trees in where the number is optimal, such that completely
independent spanning trees still exist in the graph obtained from by
deleting any vertex (respectively, any induced path of order at most
) for or odd (respectively, even ).
Concerning the connectivity and the number of completely independent spanning
trees, we moreover show the following, where denotes the minimum
degree of . Every -connected line graph has
completely independent spanning trees if is not super edge-connected or
. Every -connected line graph
has completely independent spanning trees if is regular.
Every -connected line graph with has
completely independent spanning trees.Comment: 20 pages with 5 figure
Constant-Time Algorithms for Minimum Spanning Tree and Related Problems on Processor Array with Reconfigurable Bus Systems
[[abstract]]A processor array with a reconfigurable bus system is a parallel computation model that consists of a processor array and a reconfigurable bus system. In this paper, a constant-time algorithm is proposed on this model for finding the cycles in an undirected graph. We can use this algorithm to decide whether a specified edge belongs to the minimum spanning tree of the graph or not. This cycle-finding algorithm is designed on a two-dimensional processor array with a reconfigurable bus system, where is the number of vertices in the graph. Based on this cycle-finding algorithm, the minimum spanning tree problem and the spanning tree problem can be solved in O(1) time by using fewer processors than before, O() and O() processors respectively. This is a substantial improvement over previous known results. Moreover, we also propose two constant-time algorithms for solving the minimum spanning tree verification problem and spanning tree verification problem by using O() and O() processors, respectively.
On Temporal Graph Exploration
A temporal graph is a graph in which the edge set can change from step to
step. The temporal graph exploration problem TEXP is the problem of computing a
foremost exploration schedule for a temporal graph, i.e., a temporal walk that
starts at a given start node, visits all nodes of the graph, and has the
smallest arrival time. In the first part of the paper, we consider only
temporal graphs that are connected at each step. For such temporal graphs with
nodes, we show that it is NP-hard to approximate TEXP with ratio
for any . We also provide an explicit
construction of temporal graphs that require steps to be
explored. We then consider TEXP under the assumption that the underlying graph
(i.e. the graph that contains all edges that are present in the temporal graph
in at least one step) belongs to a specific class of graphs. Among other
results, we show that temporal graphs can be explored in steps if the underlying graph has treewidth and in
steps if the underlying graph is a grid. In the second part of the
paper, we replace the connectedness assumption by a weaker assumption and show
that -edge temporal graphs with regularly present edges and with random
edges can always be explored in steps and steps with high
probability, respectively. We finally show that the latter result can be used
to obtain a distributed algorithm for the gossiping problem.Comment: This is an extended version of an ICALP 2015 pape
A New Perspective on Vertex Connectivity
Edge connectivity and vertex connectivity are two fundamental concepts in
graph theory. Although by now there is a good understanding of the structure of
graphs based on their edge connectivity, our knowledge in the case of vertex
connectivity is much more limited. An essential tool in capturing edge
connectivity are edge-disjoint spanning trees. The famous results of Tutte and
Nash-Williams show that a graph with edge connectivity contains
\floor{\lambda/2} edge-disjoint spanning trees.
We present connected dominating set (CDS) partition and packing as tools that
are analogous to edge-disjoint spanning trees and that help us to better grasp
the structure of graphs based on their vertex connectivity. The objective of
the CDS partition problem is to partition the nodes of a graph into as many
connected dominating sets as possible. The CDS packing problem is the
corresponding fractional relaxation, where CDSs are allowed to overlap as long
as this is compensated by assigning appropriate weights. CDS partition and CDS
packing can be viewed as the counterparts of the well-studied edge-disjoint
spanning trees, focusing on vertex disjointedness rather than edge
disjointness.
We constructively show that every -vertex-connected graph with nodes
has a CDS packing of size and a CDS partition of size
. We prove that the CDS packing bound is
existentially optimal.
Using CDS packing, we show that if vertices of a -vertex-connected graph
are independently sampled with probability , then the graph induced by the
sampled vertices has vertex connectivity . Moreover,
using our CDS packing, we get a store-and-forward broadcast
algorithm with optimal throughput in the networking model where in each round,
each node can send one bounded-size message to all its neighbors
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