23 research outputs found
Spanners and Reachability Oracles for Directed Transmission Graphs
Let P be a set of n points in d dimensions, each with an associated radius r_p > 0. The transmission graph G for P has vertex set P and an edge from p to q if and only if q lies in the ball with radius r_p around p. Let t > 1. A t-spanner H for G is a sparse subgraph of G such that for any two vertices p, q connected by a path of length l in G, there is a p-q-path of length at most tl in H. We show how to compute a t-spanner for G if d=2. The running time is O(n (log n + log Psi)), where Psi is the ratio of the largest and smallest radius of two points in P. We extend this construction to be independent of Psi at the expense of a polylogarithmic overhead in the running time. As a first application, we prove a property of the t-spanner that allows us to find a BFS tree in G for any given start vertex s of P in the same time.
After that, we deal with reachability oracles for G. These are data structures that answer reachability queries: given two vertices, is there a directed path between them? The quality of a reachability oracle is measured by the space S(n), the query time Q(n), and the preproccesing time. For d=1, we show how to compute an oracle with Q(n) = O(1) and S(n) = O(n) in time O(n log n). For d=2, the radius ratio Psi again turns out to be an important measure for the complexity of the problem. We present three different data structures whose quality depends on Psi: (i) if Psi = sqrt(3), we get Q(n) = O(Psi^3 sqrt(n)) and S(n) = O(Psi^5 n^(3/2)); and (iii) if Psi is polynomially bounded in n, we use probabilistic methods to obtain an oracle with Q(n) = O(n^(2/3)log n) and S(n) = O(n^(5/3) log n) that answers queries correctly with high probability. We employ our t-spanner to achieve a fast preproccesing time of O(Psi^5 n^(3/2)) and O(n^(5/3) log^2 n) in case (ii) and (iii), respectively
Shortest Paths in Geometric Intersection Graphs
This thesis studies shortest paths in geometric intersection graphs, which can model, among others, ad-hoc communication and transportation networks. First, we consider two classical problems in the field of algorithms, namely Single-Source Shortest Paths (SSSP) and All-Pairs Shortest Paths (APSP). In SSSP we want to compute the shortest paths from one vertex of a graph to all other vertices, while in APSP we aim to find the shortest path between every pair of vertices. Although there is a vast literature for these problems in many graph classes, the case of geometric intersection graphs has been only partially addressed.
In unweighted unit-disk graphs, we show that we can solve SSSP in linear time, after presorting the disk centers with respect to their coordinates. Furthermore, we give the first (slightly) subquadratic-time APSP algorithm by using our new SSSP result, bit tricks, and a shifted-grid-based decomposition technique.
In unweighted, undirected geometric intersection graphs, we present a simple and general technique that reduces APSP to static, offline intersection detection. Consequently, we give fast APSP algorithms for intersection graphs of arbitrary disks, axis-aligned line segments, arbitrary line segments, d-dimensional axis-aligned boxes, and d-dimensional axis-aligned unit hypercubes. We also provide a near-linear-time SSSP algorithm for intersection graphs of axis-aligned line segments by a reduction to dynamic orthogonal point location.
Then, we study two problems that have received considerable attention lately. The first is that of computing the diameter of a graph, i.e., the longest shortest-path distance between any two vertices. In the second, we want to preprocess a graph into a data structure, called distance oracle, such that the shortest path (or its length) between any two query vertices can be found quickly. Since these problems are often too costly to solve exactly, we study their approximate versions.
Following a long line of research, we employ Voronoi diagrams to compute a (1+epsilon)-approximation of the diameter of an undirected, non-negatively-weighted planar graph in time near linear in the input size and polynomial in 1/epsilon. The previously best solution had exponential dependency on the latter. Using similar techniques, we can also construct the first (1+epsilon)-approximate distance oracles with similar preprocessing time and space and only O(log(1/\epsilon)) query time.
In weighted unit-disk graphs, we present the first near-linear-time (1+epsilon)-approximation algorithm for the diameter and for other related problems, such as the radius and the bichromatic closest pair. To do so, we combine techniques from computational geometry and planar graphs, namely well-separated pair decompositions and shortest-path separators. We also show how to extend our approach to obtain O(1)-query-time (1+epsilon)-approximate distance oracles with near linear preprocessing time and space. Then, we apply these oracles, along with additional ideas, to build a data structure for the (1+epsilon)-approximate All-Pairs Bounded-Leg Shortest Paths (apBLSP) problem in truly subcubic time
Greedy routing and virtual coordinates for future networks
At the core of the Internet, routers are continuously struggling with
ever-growing routing and forwarding tables. Although hardware advances
do accommodate such a growth, we anticipate new requirements e.g. in
data-oriented networking where each content piece has to be referenced
instead of hosts, such that current approaches relying on global
information will not be viable anymore, no matter the hardware
progress. In this thesis, we investigate greedy routing methods that
can achieve similar routing performance as today but use much less
resources and which rely on local information only. To this end, we
add specially crafted name spaces to the network in which virtual
coordinates represent the addressable entities. Our scheme enables participating
routers to make forwarding decisions using only neighbourhood information,
as the overarching pseudo-geometric name space structure already
organizes and incorporates "vicinity" at a global level.
A first challenge to the application of greedy routing on virtual
coordinates to future networks is that of "routing dead-ends"
that are local minima due to the difficulty of consistent coordinates
attribution. In this context, we propose a routing recovery scheme
based on a multi-resolution embedding of the network in low-dimensional Euclidean spaces.
The recovery is performed by routing greedily on a blurrier view of the network. The
different network detail-levels are obtained though the embedding of
clustering-levels of the graph. When compared with
higher-dimensional embeddings of a given network, our method shows a
significant diminution of routing failures for similar header and
control-state sizes.
A second challenge to the application of virtual coordinates and
greedy routing to future networks is the support of
"customer-provider" as well as "peering" relationships between
participants, resulting in a differentiated services
environment. Although an application of greedy routing within such a
setting would combine two very common fields of today's networking
literature, such a scenario has, surprisingly, not been studied so
far. In this context we propose two approaches to address this scenario.
In a first approach we implement a path-vector protocol similar to
that of BGP on top of a greedy embedding of the network. This allows
each node to build a spatial map associated with each of its
neighbours indicating the accessible regions. Routing is then
performed through the use of a decision-tree classifier taking the
destination coordinates as input. When applied on a real-world dataset
(the CAIDA 2004 AS graph) we demonstrate an up to 40% compression ratio of
the routing control information at the network's core as well as a computationally efficient
decision process comparable to methods such as binary trees and tries.
In a second approach, we take inspiration from consensus-finding in social
sciences and transform the three-dimensional distance data structure
(where the third dimension encodes the service differentiation) into a
two-dimensional matrix on which classical embedding tools can be used.
This transformation is achieved by agreeing on a set of
constraints on the inter-node distances guaranteeing an
administratively-correct greedy routing. The computed distances are
also enhanced to encode multipath support. We demonstrate a good
greedy routing performance as well as an above 90% satisfaction of multipath constraints
when relying on the non-embedded obtained distances on synthetic datasets.
As various embeddings of the consensus distances do not fully exploit their multipath potential, the use of compression techniques such as transform coding to
approximate the obtained distance allows for better routing performances
Distributed Approximation Algorithms for Weighted Shortest Paths
A distributed network is modeled by a graph having nodes (processors) and
diameter . We study the time complexity of approximating {\em weighted}
(undirected) shortest paths on distributed networks with a {\em
bandwidth restriction} on edges (the standard synchronous \congest model). The
question whether approximation algorithms help speed up the shortest paths
(more precisely distance computation) was raised since at least 2004 by Elkin
(SIGACT News 2004). The unweighted case of this problem is well-understood
while its weighted counterpart is fundamental problem in the area of
distributed approximation algorithms and remains widely open. We present new
algorithms for computing both single-source shortest paths (\sssp) and
all-pairs shortest paths (\apsp) in the weighted case.
Our main result is an algorithm for \sssp. Previous results are the classic
-time Bellman-Ford algorithm and an -time
-approximation algorithm, for any integer
, which follows from the result of Lenzen and Patt-Shamir (STOC 2013).
(Note that Lenzen and Patt-Shamir in fact solve a harder problem, and we use
to hide the O(\poly\log n) term.) We present an -time -approximation algorithm for \sssp. This
algorithm is {\em sublinear-time} as long as is sublinear, thus yielding a
sublinear-time algorithm with almost optimal solution. When is small, our
running time matches the lower bound of by Das Sarma
et al. (SICOMP 2012), which holds even when , up to a
\poly\log n factor.Comment: Full version of STOC 201
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum