157 research outputs found
A Fast Algorithm for Source-Wise Round-Trip Spanners
In this paper, we study the problem of efficiently constructing source-wise
round-trip spanners in weighted directed graphs. For a source vertex set
in a digraph , an -source-wise round-trip spanner of
of stretch is a subgraph of such that for every , the round-trip distance between and in is at most times of
the original distance in . We show that, for a digraph with
vertices, edges and nonnegative edge weights, an -sized source vertex
set and a positive integer , there exists an algorithm, in
time , with high probability constructing an
-source-wise round-trip spanner of stretch and size
. Compared with the state of the art for constructing
source-wise round-trip spanners, our algorithm significantly improves their
construction time (where
and 2.373 is the matrix multiplication exponent) to nearly linear
, while still keeping a spanner stretch and
size , asymptotically similar to their stretch
and size , respectively
Roundtrip Spanners with (2k-1) Stretch
A roundtrip spanner of a directed graph is a subgraph of preserving
roundtrip distances approximately for all pairs of vertices. Despite extensive
research, there is still a small stretch gap between roundtrip spanners in
directed graphs and undirected graphs. For a directed graph with real edge
weights in , we first propose a new deterministic algorithm that
constructs a roundtrip spanner with stretch and edges for every integer , then remove the dependence of size on
to give a roundtrip spanner with stretch and edges. While keeping the edge size small, our result improves the previous
stretch roundtrip spanners in directed graphs [Roditty, Thorup,
Zwick'02; Zhu, Lam'18], and almost matches the undirected -spanner with
edges [Alth\"ofer et al. '93] when is a constant, which is
optimal under Erd\"os conjecture.Comment: 12 page
Improved Roundtrip Spanners, Emulators, and Directed Girth Approximation
Roundtrip spanners are the analog of spanners in directed graphs, where the
roundtrip metric is used as a notion of distance. Recent works have shown
existential results of roundtrip spanners nearly matching the undirected case,
but the time complexity for constructing roundtrip spanners is still widely
open.
This paper focuses on developing fast algorithms for roundtrip spanners and
related problems. For any -vertex directed graph with edges (with
non-negative edge weights), our results are as follows:
- 3-roundtrip spanner faster than APSP: We give an
-time algorithm that constructs a roundtrip spanner of
stretch and optimal size . Previous constructions of roundtrip
spanners of the same size either required time [Roditty, Thorup,
Zwick SODA'02; Cen, Duan, Gu ICALP'20], or had worse stretch [Chechik and
Lifshitz SODA'21].
- Optimal roundtrip emulator in dense graphs: For integer , we give
an -time algorithm that constructs a roundtrip \emph{emulator}
of stretch and size , which is optimal for constant
under Erd\H{o}s' girth conjecture. Previous work of [Thorup and Zwick STOC'01]
implied a roundtrip emulator of the same size and stretch, but it required
construction time. Our improved running time is near-optimal for
dense graphs.
- Faster girth approximation in sparse graphs: We give an
-time algorithm that -approximates the girth of a
directed graph. This can be compared with the previous -approximation
algorithm in time by [Chechik and Lifshitz
SODA'21]. In sparse graphs, our algorithm achieves better running time at the
cost of a larger approximation ratio.Comment: To appear in SODA 202
Path-Fault-Tolerant Approximate Shortest-Path Trees
Let be an -nodes non-negatively real-weighted undirected graph.
In this paper we show how to enrich a {\em single-source shortest-path tree}
(SPT) of with a \emph{sparse} set of \emph{auxiliary} edges selected from
, in order to create a structure which tolerates effectively a \emph{path
failure} in the SPT. This consists of a simultaneous fault of a set of at
most adjacent edges along a shortest path emanating from the source, and it
is recognized as one of the most frequent disruption in an SPT. We show that,
for any integer parameter , it is possible to provide a very sparse
(i.e., of size ) auxiliary structure that carefully
approximates (i.e., within a stretch factor of ) the true
shortest paths from the source during the lifetime of the failure. Moreover, we
show that our construction can be further refined to get a stretch factor of
and a size of for the special case , and that it can be
converted into a very efficient \emph{approximate-distance sensitivity oracle},
that allows to quickly (even in optimal time, if ) reconstruct the
shortest paths (w.r.t. our structure) from the source after a path failure,
thus permitting to perform promptly the needed rerouting operations. Our
structure compares favorably with previous known solutions, as we discuss in
the paper, and moreover it is also very effective in practice, as we assess
through a large set of experiments.Comment: 21 pages, 3 figures, SIROCCO 201
On the Communication Complexity of Distributed Name-Independent Routing Schemes
International audienceWe present a distributed asynchronous algorithm that, for every undirected weighted -node graph , constructs name-independent routing tables for . The size of each table is \tO(\sqrt{n}\,), whereas the length of any route is stretched by a factor of at most~ w.r.t. the shortest path. At any step, the memory space of each node is \tO(\sqrt{n}\,). The algorithm terminates in time , where is the hop-diameter of . In synchronous scenarios and with uniform weights, it consumes \tO(m\sqrt{n} + n^{3/2}\min\set{D,\sqrt{n}\,}) messages, where is the number of edges of . In the realistic case of sparse networks of poly-logarithmic diameter, the communication complexity of our scheme, that is \tO(n^{3/2}), improves by a factor of the communication complexity of \emph{any} shortest-path routing scheme on the same family of networks. This factor is provable thanks to a new lower bound of independent interest
A Gap-{ETH}-Tight Approximation Scheme for Euclidean {TSP}
We revisit the classic task of finding the shortest tour of points in -dimensional Euclidean space, for any fixed constant . We determine the optimal dependence on in the running time of an algorithm that computes a -approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in time. This improves the previously smallest dependence on in the running time of the algorithm by Rao and Smith (STOC 1998). We also show that a algorithm would violate the Gap-Exponential Time Hypothesis (Gap-ETH). Our new algorithm builds upon the celebrated quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a simple new idea that we call \emph{sparsity-sensitive patching}. On a high level this lets the granularity with which we simplify the tour depend on how sparse it is locally. Our approach is (arguably) simpler than the one by Rao and Smith since it can work without geometric spanners. We demonstrate the technique extends easily to other problems, by showing as an example that it also yields a Gap-ETH-tight approximation scheme for Rectilinear Steiner Tree
Minimizing the Maximum Flow Time in the Online Food Delivery Problem
We study a common delivery problem encountered in nowadays online food-ordering platforms: Customers order dishes online, and the restaurant delivers the food after receiving the order. Specifically, we study a problem where k vehicles of capacity c are serving a set of requests ordering food from one restaurant. After a request arrives, it can be served by a vehicle moving from the restaurant to its delivery location. We are interested in serving all requests while minimizing the maximum flow-time, i.e., the maximum time length a customer waits to receive his/her food after submitting the order.
We show that the problem is hard in both offline and online settings even when k = 1 and c = ?: There is a hardness of approximation of ?(n) for the offline problem, and a lower bound of ?(n) on the competitive ratio of any online algorithm, where n is number of points in the metric.
We circumvent the strong negative results in two directions. Our main result is an O(1)-competitive online algorithm for the uncapacitated (i.e, c = ?) food delivery problem on tree metrics; we also have negative result showing that the condition c = ? is needed. Then we explore the speed-augmentation model where our online algorithm is allowed to use vehicles with faster speed. We show that a moderate speeding factor leads to a constant competitive ratio, and we prove a tight trade-off between the speeding factor and the competitive ratio
Are there graphs whose shortest path structure requires large edge weights?
The aspect ratio of a weighted graph is the ratio of its maximum edge
weight to its minimum edge weight. Aspect ratio commonly arises as a complexity
measure in graph algorithms, especially related to the computation of shortest
paths. Popular paradigms are to interpolate between the settings of weighted
and unweighted input graphs by incurring a dependence on aspect ratio, or by
simply restricting attention to input graphs of low aspect ratio.
This paper studies the effects of these paradigms, investigating whether
graphs of low aspect ratio have more structured shortest paths than graphs in
general. In particular, we raise the question of whether one can generally take
a graph of large aspect ratio and \emph{reweight} its edges, to obtain a graph
with bounded aspect ratio while preserving the structure of its shortest paths.
Our findings are:
- Every weighted DAG on nodes has a shortest-paths preserving graph of
aspect ratio . A simple lower bound shows that this is tight.
- The previous result does not extend to general directed or undirected
graphs; in fact, the answer turns out to be exponential in these settings. In
particular, we construct directed and undirected -node graphs for which any
shortest-paths preserving graph has aspect ratio .
We also consider the approximate version of this problem, where the goal is
for shortest paths in to correspond to approximate shortest paths in .
We show that our exponential lower bounds extend even to this setting. We also
show that in a closely related model, where approximate shortest paths in
must also correspond to approximate shortest paths in , even DAGs require
exponential aspect ratio
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