83 research outputs found
Dynamic Planar Orthogonal Point Location in Sublogarithmic Time
We study a longstanding problem in computational geometry: dynamic 2-d orthogonal point location, i.e., vertical ray shooting among n horizontal line segments. We present a data structure achieving O(log n / log log n) optimal expected query time and O(log^{1/2+epsilon} n) update time (amortized) in the word-RAM model for any constant epsilon>0, under the assumption that the x-coordinates are integers bounded polynomially in n. This substantially improves previous results of Giyora and Kaplan [SODA 2007] and Blelloch [SODA 2008] with O(log n) query and update time, and of Nekrich (2010) with O(log n / log log n) query time and O(log^{1+epsilon} n) update time. Our result matches the best known upper bound for simpler problems such as dynamic 2-d dominance range searching.
We also obtain similar bounds for orthogonal line segment intersection reporting queries, vertical ray stabbing, and vertical stabbing-max, improving previous bounds, respectively, of Blelloch [SODA 2008] and Mortensen [SODA 2003], of Tao (2014), and of Agarwal, Arge, and Yi [SODA 2005] and Nekrich [ISAAC 2011]
External Memory Planar Point Location with Fast Updates
We study dynamic planar point location in the External Memory Model or Disk Access Model (DAM). Previous work in this model achieves polylog query and polylog amortized update time. We present a data structure with O(log_B^2 N) query time and O(1/B^(1-epsilon) log_B N) amortized update time, where N is the number of segments, B the block size and epsilon is a small positive constant, under the assumption that all faces have constant size. This is a B^(1-epsilon) factor faster for updates than the fastest previous structure, and brings the cost of insertion and deletion down to subconstant amortized time for reasonable choices of N and B. Our structure solves the problem of vertical ray-shooting queries among a dynamic set of interior-disjoint line segments; this is well-known to solve dynamic planar point location for a connected subdivision of the plane with faces of constant size
Computational geometry through the information lens
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.Includes bibliographical references (p. 111-117).This thesis revisits classic problems in computational geometry from the modern algorithmic perspective of exploiting the bounded precision of the input. In one dimension, this viewpoint has taken over as the standard model of computation, and has led to a powerful suite of techniques that constitute a mature field of research. In two or more dimensions, we have seen great success in understanding orthogonal problems, which decompose naturally into one dimensional problems. However, problems of a nonorthogonal nature, the core of computational geometry, have remained uncracked for many years despite extensive effort. For example, Willard asked in SODA'92 for a o(nlg n) algorithm for Voronoi diagrams. Despite growing interest in the problem, it was not successfully solved until this thesis. Formally, let w be the number of bits in a computer word, and consider n points with O(w)-bit rational coordinates. This thesis describes: * a data structure for 2-d point location with O(n) space, and 0( ... )query time. * randomized algorithms with running time 9 ... ) for 3-d convex hull, 2-d Voronoi diagram, 2-d line segment intersection, and a variety of related problems. * a data structure for 2-d dynamic convex hull, with O ( ... )query time, and O ( ... ) update time. More generally, this thesis develops a suite of techniques for exploiting bounded precision in geometric problems, hopefully laying the foundations for a rejuvenated research direction.by Mihai Pǎtraşcu.S.M
Hardness and approximation for the geodetic set problem in some graph classes
In this paper, we study the computational complexity of finding the
\emph{geodetic number} of graphs. A set of vertices of a graph is a
\emph{geodetic set} if any vertex of lies in some shortest path between
some pair of vertices from . The \textsc{Minimum Geodetic Set (MGS)} problem
is to find a geodetic set with minimum cardinality. In this paper, we prove
that solving the \textsc{MGS} problem is NP-hard on planar graphs with a
maximum degree six and line graphs. We also show that unless , there is
no polynomial time algorithm to solve the \textsc{MGS} problem with
sublogarithmic approximation factor (in terms of the number of vertices) even
on graphs with diameter . On the positive side, we give an
-approximation algorithm for the \textsc{MGS}
problem on general graphs of order . We also give a -approximation
algorithm for the \textsc{MGS} problem on the family of solid grid graphs which
is a subclass of planar graphs
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
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