1,722,052 research outputs found
Improved Map Construction using Subtrajectory Clustering
No full textMap construction is the problem of reconstructing a travel network based on trajectory data of entities travelling on the network. While many map construction algorithms reconstruct the global structure of a network well, local features such as location of crossings and turns are generally harder to reconstruct correctly, in particular for noisy and irregularly sampled data. We demonstrate how subtrajectory clustering can be used to construct maps that capture both the global structure and local features well
An Experimental Evaluation of Grouping Definitions for Moving Entities
No full textOne important pattern analysis task for trajectory data is to find a group: a set of entities that travel together over a period of time. In this paper, we compare four definitions of groups by conducting extensive experiments using various data sets. The grouping definitions are different by one or more of three different characteristics: whether they use the measured sample points or the continuous movement, how distance is used to decide if entities are in the same group, and whether the duration of the group is measured cumulatively or as one contiguous time interval. We are interested in the differences between the definitions and comparisons to human annotated data, if available. We concentrate on pedestrian data and on different crowd densities. Furthermore, we analyze the robustness of the definitions and their dependence on different sampling rates. We use two different types of trajectory data sets: synthetic trajectories from a crowd simulation model, and real-life trajectories extracted from video surveillance. We present the results of the quantitative evaluations. For experiments with real-life trajectories, we augment them with a qualitative evaluation using videos that show groups in the trajectories with a color coding
Efficient Fréchet distance queries for segments
Full text linkWe study the problem of constructing a data structure that can store a two-dimensional polygonal curve P, such that for any query segment ab one can efficiently compute the Fréchet distance between P and ab. First we present a data structure of size O(n log n) that can compute the Fréchet distance between P and a horizontal query segment ab in O(log n) time, where n is the number of vertices of P. In comparison to prior work, this significantly reduces the required space. We extend the type of queries allowed, as we allow a query to be a horizontal segment ab together with two points s, t ∈ P (not necessarily vertices), and ask for the Fréchet distance between ab and the curve of P in between s and t. Using O(n log2 n) storage, such queries take O(log3 n) time, simplifying and significantly improving previous results. We then generalize our results to query segments of arbitrary orientation. We present an O(nk3+ϵ + n2) size data structure, where k ∈ [1, n] is a parameter the user can choose, and ϵ > 0 is an arbitrarily small constant, such that given any segment ab and two points s, t ∈ P we can compute the Fréchet distance between ab and the curve of P in between s and t in O((n/k) log2 n + log4 n) time. This is the first result that allows efficient exact Fréchet distance queries for arbitrarily oriented segments. We also present two applications of our data structure. First, we show that our data structure allows us to compute a local δ-simplification (with respect to the Fréchet distance) of a polygonal curve in O(n5/2+ϵ) time, improving a previous O(n3) time algorithm. Second, we show that we can efficiently find a translation of an arbitrary query segment ab that minimizes the Fréchet distance with respect to a subcurve of P
An Experimental Comparison of Two Definitions for Groups of Moving Entities (Short Paper)
No full textTwo of the grouping definitions for trajectories that have been developed in recent years allow a continuous motion model and allow varying shape groups. One of these definitions was suggested as a refinement of the other. In this paper we perform an experimental comparison to highlight the differences in these two definitions on various data sets
Convexity-Increasing Morphs of Planar Graphs
Full text linkWe study the problem of convexifying drawings of planar graphs. Given any planar straight-line drawing of an internally 3-connected graph, we show how to morph the drawing to one with strictly convex faces while maintaining planarity at all times. Our morph is convexity-increasing, meaning that once an angle is convex, it remains convex. We give an efficient algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines. Moreover, we show that a linear number of steps is worst-case optimal. To obtain our result, we use a well-known technique by Hong and Nagamochi for finding redrawings with convex faces while preserving y-coordinates. Using a variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and Nagamochi's result which comes with a better running time. This is of independent interest, as Hong and Nagamochi's technique serves as a building block in existing morphing algorithms
Competitive searching for a line on a line arrangement
Full text linkWe discuss the problem of searching for an unknown line on a known or unknown line arrangement by a searcher S, and show that a search strategy exists that finds the line competitively, that is, with detour factor at most a constant when compared to the situation where S has all knowledge. In the case where S knows all lines but not which one is sought, the strategy is 79-competitive. We also show that it may be necessary to travel on Ω(n) lines to realize a constant competitive ratio. In the case where initially, S does not know any line, but learns about the ones it encounters during the search, we give a 414.2-competitive search strategy
Mapping Multiple Regions to the Grid with Bounded Hausdorff Distance
Full text linkWe study a problem motivated by digital geometry: given a set of disjoint geometric regions, assign each region Ri a set of grid cells Pi, so that Pi is connected, similar to Ri, and does not touch any grid cell assigned to another region. Similarity is measured using the Hausdorff distance. We analyze the achievable Hausdorff distance in terms of the number of input regions, and prove asymptotically tight bounds for several classes of input regions
Geometric Computing with CGAL and LEDA
No full textLEDA and CGAL are platforms for combinatorial and geometric computing. We discuss the use of LEDA and CGAL for geometric computing and show that they provide a unique framework for exact, efficient and convenient geometric computing
Token Swapping on Trees
Full text linkThe input to the token swapping problem is a graph with vertices , and tokens with labels , one on each vertex.
The goal is to get token to vertex for all using a
minimum number of swaps, where a swap exchanges the tokens on the endpoints of
an edge. We present some results about token swapping on a tree, also known as
"sorting with a transposition tree":
1. An optimum swap sequence may need to perform a swap on a leaf vertex that
has the correct token (a "happy leaf"), disproving a conjecture of Vaughan.
2. Any algorithm that fixes happy leaves -- as all known approximation
algorithms for the problem do -- has approximation factor at least .
Furthermore, the two best-known 2-approximation algorithms have approximation
factor exactly 2.
3. A generalized problem -- weighted coloured token swapping -- is
NP-complete on trees, even when they are restricted to be subdivided stars, but
solvable in polynomial time on paths and stars. In this version, tokens and
vertices have colours, and colours have weights. The goal is to get every token
to a vertex of the same colour, and the cost of a swap is the sum of the
weights of the two tokens involved.Comment: 37 pages, Discrete Mathematics and Theoretical Computer Science,
DMTCS vol. 24:2, 2022, #
Chasing Puppies: Mobile Beacon Routing on Closed Curves
Full text linkWe solve an open problem posed by Michael Biro at CCCG 2013 that was inspired by his and others’ work on beacon-based routing. Consider a human and a puppy on a simple closed curve in the plane. The human can walk along the curve at bounded speed and change direction as desired. The puppy runs with unbounded speed along the curve as long as the Euclidean straight-line distance to the human is decreasing, so that it is always at a point on the curve where the distance is locally minimal. Assuming that the curve is smooth (with some mild genericity constraints) or a simple polygon, we prove that the human can always catch the puppy in finite time
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