68 research outputs found
Fully Dynamic Algorithms for Minimum Weight Cycle and Related Problems
We consider the directed minimum weight cycle problem in the fully dynamic
setting. To the best of our knowledge, so far no fully dynamic algorithms have
been designed specifically for the minimum weight cycle problem in general
digraphs. One can achieve amortized update time by simply
invoking the fully dynamic APSP algorithm of Demetrescu and Italiano [J.
ACM'04]. This bound, however, yields no improvement over the trivial
recompute-from-scratch algorithm for sparse graphs.
Our first contribution is a very simple deterministic
-approximate algorithm supporting vertex updates (i.e., changing
all edges incident to a specified vertex) in conditionally near-optimal
amortized time for digraphs with real edge
weights in . Using known techniques, the algorithm can be implemented on
planar graphs and also gives some new sublinear fully dynamic algorithms
maintaining approximate cuts and flows in planar digraphs.
Additionally, we show a Monte Carlo randomized exact fully dynamic minimum
weight cycle algorithm with worst-case update that works
for real edge weights. To this end, we generalize the exact fully dynamic APSP
data structure of Abraham et al. [SODA'17] to solve the ``multiple-pairs
shortest paths problem'', where one is interested in computing distances for
some (instead of all ) fixed source-target pairs after each update. We
show that in such a scenario, worst-case update time
is possible.Comment: Full version of an ICALP 2021 pape
Fast and Compact Exact Distance Oracle for Planar Graphs
For a given a graph, a distance oracle is a data structure that answers
distance queries between pairs of vertices. We introduce an -space
distance oracle which answers exact distance queries in time for
-vertex planar edge-weighted digraphs. All previous distance oracles for
planar graphs with truly subquadratic space i.e., space
for some constant ) either required query time polynomial in
or could only answer approximate distance queries.
Furthermore, we show how to trade-off time and space: for any , we show how to obtain an -space distance oracle that answers
queries in time . This is a polynomial
improvement over the previous planar distance oracles with query
time
Near-Optimal Decremental SSSP in Dense Weighted Digraphs
In the decremental Single-Source Shortest Path problem (SSSP), we are given a
weighted directed graph undergoing edge deletions and a source
vertex ; let and be the aspect ratio of the
graph. The goal is to obtain a data structure that maintains shortest paths
from to all vertices in and can answer distance queries in time,
as well as return the corresponding path in time.
This problem was first considered by Even and Shiloach [JACM'81], who
provided an algorithm with total update time for unweighted undirected
graphs; this was later extended to directed weighted graphs [FOCS'95, STOC'99].
There are conditional lower bounds showing that is in fact near-optimal
[ESA'04, FOCS'14, STOC'15, STOC'20]. In a breakthrough result, Forster et al.
showed that it is possible to achieve total update time
if the algorithm is allowed to return -approximate paths,
instead of exact ones [STOC'14, ICALP'15]. No further progress was made until
Probst Gutenberg and Wulff-Nilsen [SODA'20] provided a new approach for the
problem, which yields total time .
Our result builds on this recent approach, but overcomes its limitations by
introducing a significantly more powerful abstraction, as well as a different
core subroutine. Our new framework yields a decremental
-approximate SSSP data structure with total update time
. Our algorithm is thus near-optimal for dense graphs
with polynomial edge-weights. Our framework can also be applied to sparse
graphs to obtain total update time .
Our main technique allows us to convert SSSP algorithms for DAGs to ones for
general graphs, which we believe has significant potential to influence future
work.Comment: Accepted to FOCS'2
Space Efficient Edge-Fault Tolerant Routing
Let G be an undirected weighted graph with n vertices and m edges, and k >= 1 be an integer. We preprocess the graph in O^~(mn) time, constructing a data structure of size O^~ k deg{v}+n^{1/k}) words per vertex v in V, which is then used by our routing scheme to ensure successful routing of packets even in the presence of a single edge fault. The scheme adds only O(k) words of information to the message.
Moreover, the stretch of the routing scheme, i.e., the maximum ratio of the cost of the path along which the packet is routed to the cost of the actual shortest path that avoids the fault, is only O(k^2).
Our results match the best known results for routing schemes that do not consider failures, with only the stretch being larger by a small constant factor of O(k). Moreover, a 1963 girth conjecture of Erdos, known to hold for k=1,2,3 and 5, implies that Omega(n^{1+1/k}) space is required by any routing scheme that has a stretch less than 2k+1.
Hence our data structures are essentially space efficient.
The algorithms are extremely simple, easy to implement, and with minor modifications, can be used under a centralized setting to efficiently answer distance queries in the presence of faults.
An important component of our routing scheme that may be of independent interest is an algorithm to compute the shortest cycle passing through each edge. As an intermediate result, we show that computing this in a distributed model that stores at each vertex the shortest path tree rooted at that node requires Theta(mn) message passings in the worst case
Recommended from our members
Graph Theory
Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures
Recommended from our members
Algorithmic Graph Theory
The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions
- …