134 research outputs found
Tight bounds for undirected graph exploration with pebbles and multiple agents
We study the problem of deterministically exploring an undirected and
initially unknown graph with vertices either by a single agent equipped
with a set of pebbles, or by a set of collaborating agents. The vertices of the
graph are unlabeled and cannot be distinguished by the agents, but the edges
incident to a vertex have locally distinct labels. The graph is explored when
all vertices have been visited by at least one agent. In this setting, it is
known that for a single agent without pebbles bits of memory
are necessary and sufficient to explore any graph with at most vertices. We
are interested in how the memory requirement decreases as the agent may mark
vertices by dropping and retrieving distinguishable pebbles, or when multiple
agents jointly explore the graph. We give tight results for both questions
showing that for a single agent with constant memory
pebbles are necessary and sufficient for exploration. We further prove that the
same bound holds for the number of collaborating agents needed for exploration.
For the upper bound, we devise an algorithm for a single agent with constant
memory that explores any -vertex graph using
pebbles, even when is unknown. The algorithm terminates after polynomial
time and returns to the starting vertex. Since an additional agent is at least
as powerful as a pebble, this implies that agents
with constant memory can explore any -vertex graph. For the lower bound, we
show that the number of agents needed for exploring any graph of size is
already when we allow each agent to have at most
bits of memory for any .
This also implies that a single agent with sublogarithmic memory needs
pebbles to explore any -vertex graph
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
Lifting query complexity to time-space complexity for two-way finite automata
Time-space tradeoff has been studied in a variety of models, such as Turing
machines, branching programs, and finite automata, etc. While communication
complexity as a technique has been applied to study finite automata, it seems
it has not been used to study time-space tradeoffs of finite automata. We
design a new technique showing that separations of query complexity can be
lifted, via communication complexity, to separations of time-space complexity
of two-way finite automata. As an application, one of our main results exhibits
the first example of a language such that the time-space complexity of
two-way probabilistic finite automata with a bounded error (2PFA) is
, while of exact two-way quantum finite automata with
classical states (2QCFA) is , that is, we demonstrate
for the first time that exact quantum computing has an advantage in time-space
complexity comparing to classical computing
Fine-grained complexity and algorithm engineering of geometric similarity measures
Point sets and sequences are fundamental geometric objects that arise in any application that considers movement data, geometric shapes, and many more. A crucial task on these objects is to measure their similarity. Therefore, this thesis presents results on algorithms, complexity lower bounds, and algorithm engineering of the most important point set and sequence similarity measures like the Fréchet distance, the Fréchet distance under translation, and the Hausdorff distance under translation. As an extension to the mere computation of similarity, also the approximate near neighbor problem for the continuous Fréchet distance on time series is considered and matching upper and lower bounds are shown.Punktmengen und Sequenzen sind fundamentale geometrische Objekte, welche in vielen Anwendungen auftauchen, insbesondere in solchen die Bewegungsdaten, geometrische Formen, und ähnliche Daten verarbeiten. Ein wichtiger Bestandteil dieser Anwendungen ist die Berechnung der Ähnlichkeit von Objekten. Diese Dissertation präsentiert Resultate, genauer gesagt Algorithmen, untere Komplexitätsschranken und Algorithm Engineering der wichtigsten Ähnlichkeitsmaße für Punktmengen und Sequenzen, wie zum Beispiel Fréchetdistanz, Fréchetdistanz unter Translation und Hausdorffdistanz unter Translation. Als eine Erweiterung der bloßen Berechnung von Ähnlichkeit betrachten wir auch das Near Neighbor Problem für die kontinuierliche Fréchetdistanz auf Zeitfolgen und zeigen obere und untere Schranken dafür
Engineering Algorithms for Route Planning in Multimodal Transportation Networks
Practical algorithms for route planning in transportation networks are a showpiece of successful Algorithm Engineering. This has produced many speedup techniques, varying in preprocessing time, space, query performance, simplicity, and ease of implementation. This thesis explores solutions to more realistic scenarios, taking into account, e.g., traffic, user preferences, public transit schedules, and the options offered by the many modalities of modern transportation networks
Route Planning in Transportation Networks
We survey recent advances in algorithms for route planning in transportation
networks. For road networks, we show that one can compute driving directions in
milliseconds or less even at continental scale. A variety of techniques provide
different trade-offs between preprocessing effort, space requirements, and
query time. Some algorithms can answer queries in a fraction of a microsecond,
while others can deal efficiently with real-time traffic. Journey planning on
public transportation systems, although conceptually similar, is a
significantly harder problem due to its inherent time-dependent and
multicriteria nature. Although exact algorithms are fast enough for interactive
queries on metropolitan transit systems, dealing with continent-sized instances
requires simplifications or heavy preprocessing. The multimodal route planning
problem, which seeks journeys combining schedule-based transportation (buses,
trains) with unrestricted modes (walking, driving), is even harder, relying on
approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4,
previously published by Microsoft Research. This work was mostly done while
the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at
Microsoft Research Silicon Valle
Algorithm Engineering for Realistic Journey Planning in Transportation Networks
Diese Dissertation beschäftigt sich mit der Routenplanung in Transportnetzen. Es werden neue, effiziente algorithmische Ansätze zur Berechnung optimaler Verbindungen in öffentlichen Verkehrsnetzen, Straßennetzen und multimodalen Netzen, die verschiedene Transportmodi miteinander verknüpfen, eingeführt. Im Fokus der Arbeit steht dabei die Praktikabilität der Ansätze, was durch eine ausführliche experimentelle Evaluation belegt wird
Learning algorithms with applications to robot navigation and protein folding
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1996.Includes bibliographical references (leaves 109-117).by Mona Singh.Ph.D
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