15 research outputs found
Computing pseudotriangulations via branched coverings
We describe an efficient algorithm to compute a pseudotriangulation of a
finite planar family of pairwise disjoint convex bodies presented by its
chirotope. The design of the algorithm relies on a deepening of the theory of
visibility complexes and on the extension of that theory to the setting of
branched coverings. The problem of computing a pseudotriangulation that
contains a given set of bitangent line segments is also examined.Comment: 66 pages, 39 figure
Multitriangulations, pseudotriangulations and primitive sorting networks
We study the set of all pseudoline arrangements with contact points which
cover a given support. We define a natural notion of flip between these
arrangements and study the graph of these flips. In particular, we provide an
enumeration algorithm for arrangements with a given support, based on the
properties of certain greedy pseudoline arrangements and on their connection
with sorting networks. Both the running time per arrangement and the working
space of our algorithm are polynomial.
As the motivation for this work, we provide in this paper a new
interpretation of both pseudotriangulations and multitriangulations in terms of
pseudoline arrangements on specific supports. This interpretation explains
their common properties and leads to a natural definition of
multipseudotriangulations, which generalizes both. We study elementary
properties of multipseudotriangulations and compare them to iterations of
pseudotriangulations.Comment: 60 pages, 40 figures; minor corrections and improvements of
presentatio
The greedy flip tree of a subword complex
We describe a canonical spanning tree of the ridge graph of a subword complex
on a finite Coxeter group. It is based on properties of greedy facets in
subword complexes, defined and studied in this paper. Searching this tree
yields an enumeration scheme for the facets of the subword complex. This
algorithm extends the greedy flip algorithm for pointed pseudotriangulations of
points or convex bodies in the plane.Comment: 14 pages, 10 figures; various corrections (in particular deletion of
Section 4 which contained a serious mistake pointed out by an anonymous
referee). This paper is subsumed by our joint results with Christian Stump on
"EL-labelings and canonical spanning trees for subword complexes"
(http://arxiv.org/abs/1210.1435) and will therefore not be publishe
The brick polytope of a sorting network
The associahedron is a polytope whose graph is the graph of flips on
triangulations of a convex polygon. Pseudotriangulations and
multitriangulations generalize triangulations in two different ways, which have
been unified by Pilaud and Pocchiola in their study of flip graphs on
pseudoline arrangements with contacts supported by a given sorting network.
In this paper, we construct the brick polytope of a sorting network, obtained
as the convex hull of the brick vectors associated to each pseudoline
arrangement supported by the network. We combinatorially characterize the
vertices of this polytope, describe its faces, and decompose it as a Minkowski
sum of matroid polytopes.
Our brick polytopes include Hohlweg and Lange's many realizations of the
associahedron, which arise as brick polytopes for certain well-chosen sorting
networks. We furthermore discuss the brick polytopes of sorting networks
supporting pseudoline arrangements which correspond to multitriangulations of
convex polygons: our polytopes only realize subgraphs of the flip graphs on
multitriangulations and they cannot appear as projections of a hypothetical
multiassociahedron.Comment: 36 pages, 25 figures; Version 2 refers to the recent generalization
of our results to spherical subword complexes on finite Coxeter groups
(http://arxiv.org/abs/1111.3349
Brick polytopes of spherical subword complexes and generalized associahedra
International audienceWe generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description of generalized associahedra, a Minkowski sum decomposition into Coxeter matroid polytopes, and a combinatorial description of the exchange matrix of any cluster in a finite type cluster algebra
Algorithms for Optimizing Search Schedules in a Polygon
In the area of motion planning, considerable work has been done on guarding
problems, where "guards", modelled as points, must guard a polygonal
space from "intruders". Different variants
of this problem involve varying a number of factors. The guards performing
the search may vary in terms of their number, their mobility, and their
range of vision. The model of intruders may or may not allow them to
move. The polygon being searched may have a specified starting point,
a specified ending point, or neither of these. The typical question asked
about one of these problems is whether or not certain polygons can be
searched under a particular guarding paradigm defined by the types
of guards and intruders.
In this thesis, we focus on two cases of a chain of guards searching
a room (polygon with a specific starting point) for mobile intruders.
The intruders must never be allowed to escape through the door undetected.
In the case of the two guard problem, the guards must start at the door
point and move in opposite directions along the boundary of the
polygon, never crossing the door point. At all times, the
guards must be able to see each other. The search is complete once both
guards occupy the same spot elsewhere on the polygon. In the case of
a chain of three guards, consecutive guards in the chain must always
be visible. Again, the search starts at the door point, and the outer
guards of the chain must move from the door in opposite directions.
These outer guards must always remain on the boundary of the polygon.
The search is complete once the chain lies entirely on a portion of
the polygon boundary not containing the door point.
Determining whether a polygon can be searched is a problem in the area
of visibility in polygons; further to that, our work is related
to the area of planning algorithms. We look for ways to find optimal schedules that minimize
the distance or time required to complete the search. This is done
by finding shortest paths in visibility diagrams that indicate valid
positions for the guards. In the case of
the two-guard room search, we are able to find the shortest distance
schedule and the quickest schedule. The shortest distance schedule
is found in O(n^2) time by solving an L_1 shortest path problem
among curved obstacles in two dimensions. The quickest search schedule is
found in O(n^4) time by solving an L_infinity shortest path
problem among curved obstacles in two dimensions.
For the chain of three guards, a search schedule minimizing the total
distance travelled by the outer guards is found in O(n^6) time by
solving an L_1 shortest path problem among curved obstacles in two dimensions
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by Erd˝os
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n ≥ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version
Vektori- ja rasteriaineistojen yhdistäminen kustannuspinnaksi reitinetsintää varten
Alhaisimman kustannuksen reitin etsintä on paikkatietoanalyysi, jolla pyritään selvittämään edullisin mahdollinen kustannus kustannuspinnan kohteiden välillä. Perinteisesti paikkatieto-ohjelmissa analyysi on toteutettu siten, että kustannuspinta mallinnetaan rasterina, jossa jokainen solun arvo kuvaa kustannusta liikkua kyseisen solun alueella.
Kustannuspinnan mallintamiselle on rasterin lisäksi olemassa useita muitakin vaihtoehtoja, kuten kustannusviivat ja polygonit. Eri tavat mallintaa kustannuspintaa soveltuvat käytettäväksi erilaisten aineistojen kanssa, mutta mikään tavoista ei sovellu hyvin käytettäväksi kaiken tyyppisten aineistojen kanssa.
Tässä työssä esitetään menetelmä, jolla eri aineistotyyppejä voidaan yhdistää yhdeksi kustannuspinnaksi ja muodostaa siitä yhtenäinen verkko reitinetsintää varten. Esitelty menetelmä hyödyntää kuudentoista naapurin menetelmää rasteriaineistoissa ja näkyvyysverkkoon perustuvaa menetelmää aluemaisten vektorikohteiden osalta. Työssä esitellään kuinka erityyppisten aineistojen yhdistäminen yhdeksi verkoksi tehdään teoriatasolla, selvitetään kuinka tämä verkonmuodostus saadaan tehtyä tehokkaasti käytännön sovelluksessa ja testataan algoritmin toimintaa käytännön tapaustutkimuksen kautta.
Kehitetty menetelmä todettiin tapaustutkimuksessa käyttökelpoiseksi ja sen havaittiin mahdollistavan entistä monimutkaisempien kulkukustannukseen vaikuttavien ilmiöiden mallintamisen käyttäen tavanomaisia paikkatietoaineistoja. Kehityskohteita havaittiin liittyen algoritmin tehokkuuteen, tarkkuuteen ja algoritmin vaatiman kustannuspinnan muodostamisen helppouteen. Useisiin havaituista kehityskohteista esitetään mahdollisia ratkaisuvaihtoehtoja, joiden käytännön toteutus jätettiin kuitenkin jatkotutkimuksen kohteeksi