71 research outputs found
Flow Computations on Imprecise Terrains
We study the computation of the flow of water on imprecise terrains. We
consider two approaches to modeling flow on a terrain: one where water flows
across the surface of a polyhedral terrain in the direction of steepest
descent, and one where water only flows along the edges of a predefined graph,
for example a grid or a triangulation. In both cases each vertex has an
imprecise elevation, given by an interval of possible values, while its
(x,y)-coordinates are fixed. For the first model, we show that the problem of
deciding whether one vertex may be contained in the watershed of another is
NP-hard. In contrast, for the second model we give a simple O(n log n) time
algorithm to compute the minimal and the maximal watershed of a vertex, where n
is the number of edges of the graph. On a grid model, we can compute the same
in O(n) time
Flow computations on imprecise terrains
We study water flow computation on imprecise terrains. We
consider two approaches to modeling flow on a terrain: one where water
flows across the surface of a polyhedral terrain in the direction of steepest
descent, and one where water only flows along the edges of a predefined
graph, for example a grid or a triangulation. In both cases each vertex has
an imprecise elevation, given by an interval of possible values, while its
(x, y)-coordinates are fixed. For the first model, we show that the problem
of deciding whether one vertex may be contained in the watershed of
another is NP-hard. In contrast, for the second model we give a simple
O(n log n) time algorithm to compute the minimal and the maximal
watershed of a vertex, where n is the number of edges of the graph.
On a grid model, we can compute the same in O(n) time.Peer ReviewedPostprint (published version
Optimal topological simplification of discrete functions on surfaces
We solve the problem of minimizing the number of critical points among all
functions on a surface within a prescribed distance {\delta} from a given input
function. The result is achieved by establishing a connection between discrete
Morse theory and persistent homology. Our method completely removes homological
noise with persistence less than 2{\delta}, constructively proving the
tightness of a lower bound on the number of critical points given by the
stability theorem of persistent homology in dimension two for any input
function. We also show that an optimal solution can be computed in linear time
after persistence pairs have been computed.Comment: 27 pages, 8 figure
Computing Realistic Terrains from Imprecise Elevations
It is ideal for triangulated terrains to have characteristics or properties that are realistic. In the imprecise terrain model, each vertex of a triangulated terrain has an imprecise eleva- tion value only known to lie within some interval. Under some objective function, the goal is to compute a precise terrain by assigning a single elevation value to each point, so that the objective function is optimized.
This thesis examines various objectives, such as minimizing the number of local extrema and minimizing the terrain’s surface area. We give algorithms in some cases, hardness results in other cases. Specifically, we consider four objectives: (1) minimizing the number of local extrema; (2) optimizing coplanar features; (3) minimizing the surface area; (4) minimizing the maximum steepness.
Problem (1) is known to be NP-hard, but we give an algorithm for a special case. For problem (2) we give an NP-hardness proof for the general case and a positive result for a special case. Meanwhile, problems (3) and (4) can be approximated using Second Order Cone Programming. We also consider versions of these problems for terrains one dimension down, where the output is a polyline. Here we give very efficient algorithms for all objective functions considered.
Finally, we go beyond terrains and briefly consider the distant representatives problem, where the goal is to choose precise points from segments to be as far from each other as possible. For this problem, we give a parameterized algorithm for vertical segments, prove NP-hardness for unit horizontal segments, and show hardness of approximation for vertical and horizontal segments
Connecting Terminals and 2-Disjoint Connected Subgraphs
Given a graph and a set of terminal vertices we say that a
superset of is -connecting if induces a connected graph, and
is minimal if no strict subset of is -connecting. In this paper we prove
that there are at most minimal -connecting sets when and that
these can be enumerated within a polynomial factor of this bound. This
generalizes the algorithm for enumerating all induced paths between a pair of
vertices, corresponding to the case . We apply our enumeration algorithm
to solve the {\sc 2-Disjoint Connected Subgraphs} problem in time
, improving on the recent algorithm of Cygan et
al. 2012 LATIN paper.Comment: 13 pages, 1 figur
Flow computations on imprecise terrains
Abstract. We study water flow computation on imprecise terrains. We consider two approaches to modeling flow on a terrain: one where water flows across the surface of a polyhedral terrain in the direction of steepest descent, and one where water only flows along the edges of a predefined graph, for example a grid or a triangulation. In both cases each vertex has an imprecise elevation, given by an interval of possible values, while its (x, y)-coordinates are fixed. For the first model, we show that the problem of deciding whether one vertex may be contained in the watershed of another is NP-hard. In contrast, for the second model we give a simple O(n log n) time algorithm to compute the minimal and the maximal watershed of a vertex, or a set of vertices, where n is the number of edges of the graph. On a grid model, we can compute the same in O(n) time. Rose knew almost everything that water can do, there are an awful lot when you think what. Gertrude Stein, The World is Round
Estudi de la coautoria de publicacions científiques entre UPC i cinc universitats dels Estats Units : Caltech, Stanford University, UC Davis, UC Irvine i UCLA
S'analitza la coautoria de la UPC amb autors vinculats a institucions acadèmiques dels Estats Units, per totes les àrees temàtiques, de gener de 2009 a juny de 2014.Postprint (published version
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
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