4,691 research outputs found
Decremental Single-Source Reachability in Planar Digraphs
In this paper we show a new algorithm for the decremental single-source
reachability problem in directed planar graphs. It processes any sequence of
edge deletions in total time and explicitly
maintains the set of vertices reachable from a fixed source vertex. Hence, if
all edges are eventually deleted, the amortized time of processing each edge
deletion is only , which improves upon a previously
known solution. We also show an algorithm for decremental
maintenance of strongly connected components in directed planar graphs with the
same total update time. These results constitute the first almost optimal (up
to polylogarithmic factors) algorithms for both problems.
To the best of our knowledge, these are the first dynamic algorithms with
polylogarithmic update times on general directed planar graphs for non-trivial
reachability-type problems, for which only polynomial bounds are known in
general graphs
Improved Algorithms for Decremental Single-Source Reachability on Directed Graphs
Recently we presented the first algorithm for maintaining the set of nodes
reachable from a source node in a directed graph that is modified by edge
deletions with total update time, where is the number of edges and
is the number of nodes in the graph [Henzinger et al. STOC 2014]. The
algorithm is a combination of several different algorithms, each for a
different vs. trade-off. For the case of the
running time is , just barely below . In
this paper we simplify the previous algorithm using new algorithmic ideas and
achieve an improved running time of . This gives,
e.g., for the notorious case . We obtain the
same upper bounds for the problem of maintaining the strongly connected
components of a directed graph undergoing edge deletions. Our algorithms are
correct with high probabililty against an oblivious adversary.Comment: This paper was presented at the International Colloquium on Automata,
Languages and Programming (ICALP) 2015. A full version combining the findings
of this paper and its predecessor [Henzinger et al. STOC 2014] is available
at arXiv:1504.0795
Fully Dynamic Single-Source Reachability in Practice: An Experimental Study
Given a directed graph and a source vertex, the fully dynamic single-source
reachability problem is to maintain the set of vertices that are reachable from
the given vertex, subject to edge deletions and insertions. It is one of the
most fundamental problems on graphs and appears directly or indirectly in many
and varied applications. While there has been theoretical work on this problem,
showing both linear conditional lower bounds for the fully dynamic problem and
insertions-only and deletions-only upper bounds beating these conditional lower
bounds, there has been no experimental study that compares the performance of
fully dynamic reachability algorithms in practice. Previous experimental
studies in this area concentrated only on the more general all-pairs
reachability or transitive closure problem and did not use real-world dynamic
graphs.
In this paper, we bridge this gap by empirically studying an extensive set of
algorithms for the single-source reachability problem in the fully dynamic
setting. In particular, we design several fully dynamic variants of well-known
approaches to obtain and maintain reachability information with respect to a
distinguished source. Moreover, we extend the existing insertions-only or
deletions-only upper bounds into fully dynamic algorithms. Even though the
worst-case time per operation of all the fully dynamic algorithms we evaluate
is at least linear in the number of edges in the graph (as is to be expected
given the conditional lower bounds) we show in our extensive experimental
evaluation that their performance differs greatly, both on generated as well as
on real-world instances
Conditional Lower Bounds for Space/Time Tradeoffs
In recent years much effort has been concentrated towards achieving
polynomial time lower bounds on algorithms for solving various well-known
problems. A useful technique for showing such lower bounds is to prove them
conditionally based on well-studied hardness assumptions such as 3SUM, APSP,
SETH, etc. This line of research helps to obtain a better understanding of the
complexity inside P.
A related question asks to prove conditional space lower bounds on data
structures that are constructed to solve certain algorithmic tasks after an
initial preprocessing stage. This question received little attention in
previous research even though it has potential strong impact.
In this paper we address this question and show that surprisingly many of the
well-studied hard problems that are known to have conditional polynomial time
lower bounds are also hard when concerning space. This hardness is shown as a
tradeoff between the space consumed by the data structure and the time needed
to answer queries. The tradeoff may be either smooth or admit one or more
singularity points.
We reveal interesting connections between different space hardness
conjectures and present matching upper bounds. We also apply these hardness
conjectures to both static and dynamic problems and prove their conditional
space hardness.
We believe that this novel framework of polynomial space conjectures can play
an important role in expressing polynomial space lower bounds of many important
algorithmic problems. Moreover, it seems that it can also help in achieving a
better understanding of the hardness of their corresponding problems in terms
of time
SAT Modulo Monotonic Theories
We define the concept of a monotonic theory and show how to build efficient
SMT (SAT Modulo Theory) solvers, including effective theory propagation and
clause learning, for such theories. We present examples showing that monotonic
theories arise from many common problems, e.g., graph properties such as
reachability, shortest paths, connected components, minimum spanning tree, and
max-flow/min-cut, and then demonstrate our framework by building SMT solvers
for each of these theories. We apply these solvers to procedural content
generation problems, demonstrating major speed-ups over state-of-the-art
approaches based on SAT or Answer Set Programming, and easily solving several
instances that were previously impractical to solve
Improving Reachability and Navigability in Recommender Systems
In this paper, we investigate recommender systems from a network perspective
and investigate recommendation networks, where nodes are items (e.g., movies)
and edges are constructed from top-N recommendations (e.g., related movies). In
particular, we focus on evaluating the reachability and navigability of
recommendation networks and investigate the following questions: (i) How well
do recommendation networks support navigation and exploratory search? (ii) What
is the influence of parameters, in particular different recommendation
algorithms and the number of recommendations shown, on reachability and
navigability? and (iii) How can reachability and navigability be improved in
these networks? We tackle these questions by first evaluating the reachability
of recommendation networks by investigating their structural properties.
Second, we evaluate navigability by simulating three different models of
information seeking scenarios. We find that with standard algorithms,
recommender systems are not well suited to navigation and exploration and
propose methods to modify recommendations to improve this. Our work extends
from one-click-based evaluations of recommender systems towards multi-click
analysis (i.e., sequences of dependent clicks) and presents a general,
comprehensive approach to evaluating navigability of arbitrary recommendation
networks
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