3,913 research outputs found
Fine-Grained Complexity Analysis of Two Classic TSP Variants
We analyze two classic variants of the Traveling Salesman Problem using the
toolkit of fine-grained complexity. Our first set of results is motivated by
the Bitonic TSP problem: given a set of points in the plane, compute a
shortest tour consisting of two monotone chains. It is a classic
dynamic-programming exercise to solve this problem in time. While the
near-quadratic dependency of similar dynamic programs for Longest Common
Subsequence and Discrete Frechet Distance has recently been proven to be
essentially optimal under the Strong Exponential Time Hypothesis, we show that
bitonic tours can be found in subquadratic time. More precisely, we present an
algorithm that solves bitonic TSP in time and its bottleneck
version in time. Our second set of results concerns the popular
-OPT heuristic for TSP in the graph setting. More precisely, we study the
-OPT decision problem, which asks whether a given tour can be improved by a
-OPT move that replaces edges in the tour by new edges. A simple
algorithm solves -OPT in time for fixed . For 2-OPT, this is
easily seen to be optimal. For we prove that an algorithm with a runtime
of the form exists if and only if All-Pairs
Shortest Paths in weighted digraphs has such an algorithm. The results for
may suggest that the actual time complexity of -OPT is
. We show that this is not the case, by presenting an algorithm
that finds the best -move in time for
fixed . This implies that 4-OPT can be solved in time,
matching the best-known algorithm for 3-OPT. Finally, we show how to beat the
quadratic barrier for in two important settings, namely for points in the
plane and when we want to solve 2-OPT repeatedly.Comment: Extended abstract appears in the Proceedings of the 43rd
International Colloquium on Automata, Languages, and Programming (ICALP 2016
Keyframe-based monocular SLAM: design, survey, and future directions
Extensive research in the field of monocular SLAM for the past fifteen years
has yielded workable systems that found their way into various applications in
robotics and augmented reality. Although filter-based monocular SLAM systems
were common at some time, the more efficient keyframe-based solutions are
becoming the de facto methodology for building a monocular SLAM system. The
objective of this paper is threefold: first, the paper serves as a guideline
for people seeking to design their own monocular SLAM according to specific
environmental constraints. Second, it presents a survey that covers the various
keyframe-based monocular SLAM systems in the literature, detailing the
components of their implementation, and critically assessing the specific
strategies made in each proposed solution. Third, the paper provides insight
into the direction of future research in this field, to address the major
limitations still facing monocular SLAM; namely, in the issues of illumination
changes, initialization, highly dynamic motion, poorly textured scenes,
repetitive textures, map maintenance, and failure recovery
Epidemic Spreading with External Agents
We study epidemic spreading processes in large networks, when the spread is
assisted by a small number of external agents: infection sources with bounded
spreading power, but whose movement is unrestricted vis-\`a-vis the underlying
network topology. For networks which are `spatially constrained', we show that
the spread of infection can be significantly speeded up even by a few such
external agents infecting randomly. Moreover, for general networks, we derive
upper-bounds on the order of the spreading time achieved by certain simple
(random/greedy) external-spreading policies. Conversely, for certain common
classes of networks such as line graphs, grids and random geometric graphs, we
also derive lower bounds on the order of the spreading time over all
(potentially network-state aware and adversarial) external-spreading policies;
these adversarial lower bounds match (up to logarithmic factors) the spreading
time achieved by an external agent with a random spreading policy. This
demonstrates that random, state-oblivious infection-spreading by an external
agent is in fact order-wise optimal for spreading in such spatially constrained
networks
Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs
A dynamic graph algorithm is a data structure that answers queries about a property of the current graph while supporting graph modifications such as edge insertions and deletions. Prior work has shown strong conditional lower bounds for general dynamic graphs, yet graph families that arise in practice often exhibit structural properties that the existing lower bound constructions do not possess. We study three specific graph families that are ubiquitous, namely constant-degree graphs, power-law graphs, and expander graphs, and give the first conditional lower bounds for them. Our results show that even when restricting our attention to one of these graph classes, any algorithm for fundamental graph problems such as distance computation or approximation or maximum matching, cannot simultaneously achieve a sub-polynomial update time and query time. For example, we show that the same lower bounds as for general graphs hold for maximum matching and (s,t)-distance in constant-degree graphs, power-law graphs or expanders. Namely, in an m-edge graph, there exists no dynamic algorithms with both O(m^{1/2 - ?}) update time and O(m^{1 -?}) query time, for any small ? > 0. Note that for (s,t)-distance the trivial dynamic algorithm achieves an almost matching upper bound of constant update time and O(m) query time. We prove similar bounds for the other graph families and for other fundamental problems such as densest subgraph detection and perfect matching
Parallel Anisotropic Unstructured Grid Adaptation
Computational Fluid Dynamics (CFD) has become critical to the design and analysis of aerospace vehicles. Parallel grid adaptation that resolves multiple scales with anisotropy is identified as one of the challenges in the CFD Vision 2030 Study to increase the capacity and capability of CFD simulation. The Study also cautions that computer architectures are undergoing a radical change and dramatic increases in algorithm concurrency will be required to exploit full performance. This paper reviews four different methods to parallel anisotropic grid generation. They cover both ends of the spectrum: (i) using existing state-of-the-art software optimized for a single core and modifying it for parallel platforms and (ii) designing and implementing scalable software with incomplete, but rapidly maturating functionality. A brief overview for each grid adaptation system is presented in the context of a telescopic approach for multilevel concurrency. These methods employ different approaches to enable parallel execution, which provides a unique opportunity to illustrate the relative behavior of each approach. Qualitative and quantitative metric evaluations are used to draw lessons for future developments in this critical area for parallel CFD simulation
- âŠ