24,206 research outputs found
Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails
The Frechet distance is a well-studied and very popular measure of similarity
of two curves. Many variants and extensions have been studied since Alt and
Godau introduced this measure to computational geometry in 1991. Their original
algorithm to compute the Frechet distance of two polygonal curves with n
vertices has a runtime of O(n^2 log n). More than 20 years later, the state of
the art algorithms for most variants still take time more than O(n^2 / log n),
but no matching lower bounds are known, not even under reasonable complexity
theoretic assumptions.
To obtain a conditional lower bound, in this paper we assume the Strong
Exponential Time Hypothesis or, more precisely, that there is no
O*((2-delta)^N) algorithm for CNF-SAT for any delta > 0. Under this assumption
we show that the Frechet distance cannot be computed in strongly subquadratic
time, i.e., in time O(n^{2-delta}) for any delta > 0. This means that finding
faster algorithms for the Frechet distance is as hard as finding faster CNF-SAT
algorithms, and the existence of a strongly subquadratic algorithm can be
considered unlikely.
Our result holds for both the continuous and the discrete Frechet distance.
We extend the main result in various directions. Based on the same assumption
we (1) show non-existence of a strongly subquadratic 1.001-approximation, (2)
present tight lower bounds in case the numbers of vertices of the two curves
are imbalanced, and (3) examine realistic input assumptions (c-packed curves)
The hardness of routing two pairs on one face
We prove the NP-completeness of the integer multiflow problem in planar
graphs, with the following restrictions: there are only two demand edges, both
lying on the infinite face of the routing graph. This was one of the open
challenges concerning disjoint paths, explicitly asked by M\"uller. It also
strengthens Schw\"arzler's recent proof of one of the open problems of
Schrijver's book, about the complexity of the edge-disjoint paths problem with
terminals on the outer boundary of a planar graph. We also give a directed
acyclic reduction. This proves that the arc-disjoint paths problem is
NP-complete in directed acyclic graphs, even with only two demand arcs
Locally Stable Marriage with Strict Preferences
We study stable matching problems with locality of information and control.
In our model, each agent is a node in a fixed network and strives to be matched
to another agent. An agent has a complete preference list over all other agents
it can be matched with. Agents can match arbitrarily, and they learn about
possible partners dynamically based on their current neighborhood. We consider
convergence of dynamics to locally stable matchings -- states that are stable
with respect to their imposed information structure in the network. In the
two-sided case of stable marriage in which existence is guaranteed, we show
that the existence of a path to stability becomes NP-hard to decide. This holds
even when the network exists only among one partition of agents. In contrast,
if one partition has no network and agents remember a previous match every
round, a path to stability is guaranteed and random dynamics converge with
probability 1. We characterize this positive result in various ways. For
instance, it holds for random memory and for cache memory with the most recent
partner, but not for cache memory with the best partner. Also, it is crucial
which partition of the agents has memory. Finally, we present results for
centralized computation of locally stable matchings, i.e., computing maximum
locally stable matchings in the two-sided case and deciding existence in the
roommates case.Comment: Conference version in ICALP 2013; to appear in SIAM J. Disc Mat
Process-oriented Enterprise Mashups
Mashups, a new Web 2.0 technology provide the ability for easy creation of Web-Based applications by end-users. The uses of the mashups are often consumer related. In this paper we explore how mashups can be used in the enterprise area and hat the criteria for enterprise mashups are. We provide categories for the classification of enterprise mashups, and based upon a motivating example we go further in depth on business process enterprise mashup
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