26 research outputs found
Improved approximation for Fr\'echet distance on c-packed curves matching conditional lower bounds
The Fr\'echet distance is a well-studied and very popular measure of
similarity of two curves. The best known algorithms have quadratic time
complexity, which has recently been shown to be optimal assuming the Strong
Exponential Time Hypothesis (SETH) [Bringmann FOCS'14].
To overcome the worst-case quadratic time barrier, restricted classes of
curves have been studied that attempt to capture realistic input curves. The
most popular such class are c-packed curves, for which the Fr\'echet distance
has a -approximation in time [Driemel
et al. DCG'12]. In dimension this cannot be improved to
for any unless SETH fails
[Bringmann FOCS'14].
In this paper, exploiting properties that prevent stronger lower bounds, we
present an improved algorithm with runtime .
This is optimal in high dimensions apart from lower order factors unless SETH
fails. Our main new ingredients are as follows: For filling the classical
free-space diagram we project short subcurves onto a line, which yields
one-dimensional separated curves with roughly the same pairwise distances
between vertices. Then we tackle this special case in near-linear time by
carefully extending a greedy algorithm for the Fr\'echet distance of
one-dimensional separated curves
Improved Protocols and Hardness Results for the Two-Player Cryptogenography Problem
The cryptogenography problem, introduced by Brody, Jakobsen, Scheder, and
Winkler (ITCS 2014), is to collaboratively leak a piece of information known to
only one member of a group (i)~without revealing who was the origin of this
information and (ii)~without any private communication, neither during the
process nor before. Despite several deep structural results, even the smallest
case of leaking one bit of information present at one of two players is not
well understood. Brody et al.\ gave a 2-round protocol enabling the two players
to succeed with probability and showed the hardness result that no
protocol can give a success probability of more than~.
In this work, we show that neither bound is tight. Our new hardness result,
obtained by a different application of the concavity method used also in the
previous work, states that a success probability better than 0.3672 is not
possible. Using both theoretical and numerical approaches, we improve the lower
bound to , that is, give a protocol leading to this success
probability. To ease the design of new protocols, we prove an equivalent
formulation of the cryptogenography problem as solitaire vector splitting game.
Via an automated game tree search, we find good strategies for this game. We
then translate the splits that occurred in this strategy into inequalities
relating position values and use an LP solver to find an optimal solution for
these inequalities. This gives slightly better game values, but more
importantly, it gives a more compact representation of the protocol and a way
to easily verify the claimed quality of the protocol.
These improved bounds, as well as the large sizes and depths of the improved
protocols we find, suggests that finding good protocols for the
cryptogenography problem as well as understanding their structure are harder
than what the simple problem formulation suggests
Multivariate Fine-Grained Complexity of Longest Common Subsequence
We revisit the classic combinatorial pattern matching problem of finding a
longest common subsequence (LCS). For strings and of length , a
textbook algorithm solves LCS in time , but although much effort has
been spent, no -time algorithm is known. Recent work
indeed shows that such an algorithm would refute the Strong Exponential Time
Hypothesis (SETH) [Abboud, Backurs, Vassilevska Williams + Bringmann,
K\"unnemann FOCS'15].
Despite the quadratic-time barrier, for over 40 years an enduring scientific
interest continued to produce fast algorithms for LCS and its variations.
Particular attention was put into identifying and exploiting input parameters
that yield strongly subquadratic time algorithms for special cases of interest,
e.g., differential file comparison. This line of research was successfully
pursued until 1990, at which time significant improvements came to a halt. In
this paper, using the lens of fine-grained complexity, our goal is to (1)
justify the lack of further improvements and (2) determine whether some special
cases of LCS admit faster algorithms than currently known.
To this end, we provide a systematic study of the multivariate complexity of
LCS, taking into account all parameters previously discussed in the literature:
the input size , the length of the shorter string
, the length of an LCS of and , the numbers of
deletions and , the alphabet size, as well as
the numbers of matching pairs and dominant pairs . For any class of
instances defined by fixing each parameter individually to a polynomial in
terms of the input size, we prove a SETH-based lower bound matching one of
three known algorithms. Specifically, we determine the optimal running time for
LCS under SETH as .
[...]Comment: Presented at SODA'18. Full Version. 66 page
Smoothed approximation ratio of the 2-opt heuristic for the TSP
The 2-Opt heuristic is a simple, easy-to-implement local search heuristic for the traveling salesman problem. While it usually provides good approximations to the optimal tour in experiments, its worst-case performance is poor. In an attempt to explain the approximation performance of 2-Opt, we prove an upper bound of exp(O(sqrt(log(1/sigma))) for the smoothed approximation ratio of 2-Opt. As a lower bound, we prove that the worst-case lower bound of Omega(log n/log log n) for the approximation ratio holds for sigma = O(1/ sqrt(n)).\ud
Our main technical novelty is that, different from existing smoothed analyses, we do not separately analyze objective values of the global and the local optimum on all inputs, but simultaneously bound them on the same input
Quasirandom Rumor Spreading: An Experimental Analysis
We empirically analyze two versions of the well-known "randomized rumor
spreading" protocol to disseminate a piece of information in networks. In the
classical model, in each round each informed node informs a random neighbor. In
the recently proposed quasirandom variant, each node has a (cyclic) list of its
neighbors. Once informed, it starts at a random position of the list, but from
then on informs its neighbors in the order of the list. While for sparse random
graphs a better performance of the quasirandom model could be proven, all other
results show that, independent of the structure of the lists, the same
asymptotic performance guarantees hold as for the classical model. In this
work, we compare the two models experimentally. This not only shows that the
quasirandom model generally is faster, but also that the runtime is more
concentrated around the mean. This is surprising given that much fewer random
bits are used in the quasirandom process. These advantages are also observed in
a lossy communication model, where each transmission does not reach its target
with a certain probability, and in an asynchronous model, where nodes send at
random times drawn from an exponential distribution. We also show that
typically the particular structure of the lists has little influence on the
efficiency.Comment: 14 pages, appeared in ALENEX'0
Tight(er) bounds for similarity measures, smoothed approximation and broadcasting
In this thesis, we prove upper and lower bounds on the complexity of sequence similarity measures, the approximability of geometric problems on realistic inputs, and the performance of randomized broadcasting protocols.
The first part approaches the question why a number of fundamental polynomial-time problems - specifically, Dynamic Time Warping, Longest Common Subsequence (LCS), and the Levenshtein distance - resists decades-long attempts to obtain polynomial improvements over their simple dynamic programming solutions. We prove that any (strongly) subquadratic algorithm for these and related sequence similarity measures would refute the Strong Exponential Time Hypothesis (SETH). Focusing particularly on LCS, we determine a tight running time bound (up to lower order factors and conditional on SETH) when the running time is expressed in terms of all input parameters that have been previously exploited in the extensive literature.
In the second part, we investigate the approximation performance of the popular 2-Opt heuristic for the Traveling Salesperson Problem using the smoothed analysis paradigm. For the Fréchet distance, we design an improved approximation algorithm for the natural input class of c-packed curves, matching a conditional lower bound.
Finally, in the third part we prove tighter performance bounds for processes that disseminate a piece of information, either as quickly as possible (rumor spreading) or as anonymously as possible (cryptogenography).Die vorliegende Dissertation beweist obere und untere Schranken an die Komplexität von Sequenzähnlichkeitsmaßen, an die Approximierbarkeit geometrischer Probleme auf realistischen Eingaben und an die Effektivität randomisierter Kommunikationsprotokolle.
Der erste Teil befasst sich mit der Frage, warum für eine Vielzahl fundamentaler Probleme im Polynomialzeitbereich - insbesondere für das Dynamic-Time-Warping, die längste gemeinsame Teilfolge (LCS) und die Levenshtein-Distanz - seit Jahrzehnten keine Algorithmen gefunden werden konnten, die polynomiell schneller sind als ihre einfachen Lösungen mittels dynamischer Programmierung. Wir zeigen, dass ein (im strengen Sinne) subquadratischer Algorithmus für diese und verwandte Ähnlichkeitsmaße die starke Exponentialzeithypothese (SETH) widerlegen würde. Für LCS zeigen wir eine scharfe Schranke an die optimale Laufzeit (unter der SETH und bis auf Faktoren niedrigerer Ordnung) in Abhängigkeit aller bisher untersuchten Eingabeparameter.
Im zweiten Teil untersuchen wir die Approximationsgüte der klassischen 2-Opt-Heuristik für das Problem des Handlungsreisenden anhand des Smoothed-Analysis-Paradigmas. Weiterhin entwickeln wir einen verbesserten Approximationsalgorithmus für die Fréchet-Distanz auf einer Klasse natürlicher Eingaben.
Der letzte Teil beweist neue Schranken für die Effektivität von Prozessen, die Informationen entweder so schnell wie möglich (Rumor-Spreading) oder so anonym wie möglich (Kryptogenografie) verbreiten