Smallest Color-Spanning Objects

Abellanas, Manuel; Hurtado, Ferrán; Icking, Christian; Klein, Rolf; Langetepe, Elmar; Ma, Lihong; Palop, Belén; Sacristán, Vera

doi:10.1007/3-540-44676-1_23

Cited by 63 publications

(64 citation statements)

References 16 publications

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“…The algorithm runs in O(n log 2 n) time and does not test every minimal candidates (in fact, we show that there may be (kn) minimal color-spanning axis-parallel squares in the worst case). Hence, this result is an improvement to the result proposed by Abellanas et al [1] in case k = ! (log n) 6 .…”

supporting

confidence: 61%

“…Then, we exploit the data structure to show that there is O(n log 2 n) time algorithm to compute the smallest color-spanning square for a set of n points with k colors in the plane. This is a new way to improve O(nk log n) time algorithm presented by Abellanas et al [1] when k = ! (log n).…”

mentioning

confidence: 99%

“…For the axis-parallel case, Abellanas et al [1] showed that there are ((n k) 2 ) minimal rectangles in the worst case and proposed an algorithm of O((n k) 2 log 2 k) running time to solve the problem. The algorithm has been improved to O(n(n k) log k) time by Das et al [10].…”

mentioning

confidence: 99%

“…The other approach mentioned by Abellanas et al [1] is to obtain the smallest color-spanning circle and the smallest colorspanning axis-parallel square in O(kn log n) time using the upper envelope of Voronoi surfaces [13]. In case there are exactly two points of each color, Arkin et al [2] presented O(n log 2 n) time algorithm to compute the smallest color-spanning square.…”

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confidence: 99%

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Effiiently computing the smallest axis-parallel squares spanning all colors

et al. 2017

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Abstract. For a set of colored points, a region is called color-spanning if it contains at least one point of each color. In this paper, we rst consider the problem of maintaining the smallest color-spanning interval for a set of n points with k colors on the real line, such that the insertion and deletion of an arbitrary point takes O(log 2 n) the worst-case time. Then, we exploit the data structure to show that there is O(n log 2 n) time algorithm to compute the smallest color-spanning square for a set of n points with k colors in the plane. This is a new way to improve O(nk log n) time algorithm presented by Abellanas et al. [1] when k = !(log n). We also consider the problem of computing the smallest color-spanning square in a special case in which we have, at most, two points from each color. We present O(n log n) time algorithm to solve the problem which improves the result presented by Arkin et al. [2] by a factor of log n.

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confidence: 61%

mentioning

confidence: 99%

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confidence: 99%

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Effiiently computing the smallest axis-parallel squares spanning all colors

et al. 2017

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“…The bichromatic closest pair (e.g., see Preparata and Shamos [26] Section 5.7, Agarwal et al [2], and Graf and Hinrichs [17]), the chromatic nearest neighbor search (see Mount et al [25]), the problems on finding smallest color-spanning objects (see Abellanas et al [1]), the colored range searching problems (see Agarwal et al [3]), and the group Steiner tree problem where, for a graph with colored vertices, the objective is to find a minimum weight subtree that covers all colors (see Mitchell [23], Section 7.1). Borgelt et al [7] discuss computing planar red-blue minimum spanning trees where edges may connect red and blue points only, and the tree must be planar.…”

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confidence: 99%

Colored Spanning Graphs for Set Visualization

Hurtado

Korman

Kreveld

et al. 2013

Lecture Notes in Computer Science

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Abstract. We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A red-blue-purple spanning graph (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem can be solved in polynomial time using matroid techniques. In addition, we discuss more efficient algorithms for the case in which points are located on a line or a circle, and also describe a fast ( 1 2 ρ + 1)-approximation algorithm, where ρ is the Steiner ratio.In memoriam : Ferran Hurtado (1951-2014 This paper was initiated during a research visit hosted by Ferran and his group in Barcelona. Following tradition, hours of research were complemented by relaxing evenings with great food and wine. The authors would like to thank Ferran for being a supportive mentor, an inspiring colleague, and a great friend.

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