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Rectangular Layouts and Contact Graphs
, 2007
"... Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding rectangular layouts is a key problem. We study the areaoptimization problem and show that it is NPhard t ..."
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Cited by 25 (3 self)
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Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding rectangular layouts is a key problem. We study the areaoptimization problem and show that it is NPhard to find a minimumarea rectangular layout of a given contact graph. We present O(n)time algorithms that construct O(n2)area rectangular layouts for general contact graphs and O(n log n)area rectangular layouts for trees. (For trees, this is an O(log n)approximation algorithm.) We also present an infinite family of graphs (rsp., trees) that require Ω(n2) (rsp., Ω(n log n)) area. We derive these results by presenting a new characterization of graphs that admit rectangular layouts using the related concept of rectangular duals. A corollary to our results relates the class of graphs that admit rectangular layouts to rectangle of influence drawings.
MultiDimensional Orthogonal Graph Drawing with Small Boxes
 Proc. 7th International Symp. on Graph Drawing (GD '99
, 1999
"... In this paper we investigate the general position model for the drawing of arbitrary degree graphs in the Ddimensional (D >= 2) orthogonal grid. In this model no two vertices lie in the same grid hyperplane. ..."
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Cited by 14 (6 self)
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In this paper we investigate the general position model for the drawing of arbitrary degree graphs in the Ddimensional (D >= 2) orthogonal grid. In this model no two vertices lie in the same grid hyperplane.
Improved Compact Routing Tables for Planar Networks via Orderly Spanning Trees
 In: 8 th Annual International Computing & Combinatorics Conference (COCOON). Volume 2387 of LNCS
, 2002
"... We address the problem of designing compact routing tables for an unlabeled connected nnode planar network G. For each node r of G, the designer is given a routing spanning tree Tr of G rooted at r, which speci es the routes for sending packets from r to the rest of G. ..."
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Cited by 12 (3 self)
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We address the problem of designing compact routing tables for an unlabeled connected nnode planar network G. For each node r of G, the designer is given a routing spanning tree Tr of G rooted at r, which speci es the routes for sending packets from r to the rest of G.
IMPROVED COMPACT VISIBILITY REPRESENTATION OF Planar Graph via Schnyder’s Realizer
 SIAM J. DISCRETE MATH. C ○ 2004 SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS VOL. 18, NO. 1, PP. 19–29
, 2004
"... Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility repre ..."
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Cited by 8 (1 self)
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Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility representation of G. Given a canonical ordering of the triangulated G, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained �from Schnyder’s � realizer for the triangulated G yields a visibility representation of G no wider than 22n−40. Our easytoimplement O(n)time algorithm bypasses the complicated subroutines for 15 fourconnected components and fourblock trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant’s open question about whether � � 3n−6 is a 2 worstcase lower bound on the required width. Also, if G has no degreethree (respectively, degreefive) internal node, then our visibility representation for G is no wider than � �
A Simple Linear Time Algorithm for Proper Box Rectangular Drawing of Plane Graphs
 Journal of Algorithms
, 2000
"... In this paper we introduce a new drawing style of a plane graph G, called proper box rectangular (PBR ) drawing. It is defined to be a drawing of G such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal or a vertical line segment, and each face is dra ..."
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Cited by 6 (0 self)
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In this paper we introduce a new drawing style of a plane graph G, called proper box rectangular (PBR ) drawing. It is defined to be a drawing of G such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal or a vertical line segment, and each face is drawn as a rectangle. We establish necessary and sufficient conditions for G to have a PBR drawing. We also give a simple linear time algorithm for finding such drawings. The PBR drawing is closely related to the box rectangular (BR ) drawing defined by Rahman, Nakano and Nishizeki [17]. Our method can be adapted to provide a new simpler algorithm for solving the BR drawing problem. 1 Introduction The problem of "nicely" drawing a graph G has received increasing attention [5]. Typically, we want to draw the edges and the vertices of G on the plane so that certain aesthetic quality conditions and/or optimization measures are met. Such drawings are very useful in visualizing planar graphs and fi...
Orthogonal Drawings of Plane Graphs without Bends
, 2003
"... In an orthogonal drawing of a plane graph each vertex is drawn as a point and each edge is drawn as a sequence of vertical and horizontal line segments. A bend is a point at which the drawing of an edge changes its direction. Every plane graph of the maximum degree at most four has an orthogonal dra ..."
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Cited by 5 (2 self)
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In an orthogonal drawing of a plane graph each vertex is drawn as a point and each edge is drawn as a sequence of vertical and horizontal line segments. A bend is a point at which the drawing of an edge changes its direction. Every plane graph of the maximum degree at most four has an orthogonal drawing, but may need bends. A simple necessary and sufficient condition has not been known for a plane graph to have an orthogonal drawing without bends. In this paper we obtain a necessary and sufficient condition for a plane graph G of the maximum degree three to have an orthogonal drawing without bends. We also give a lineartime algorithm to find such a drawing of G if it exists.
Some Applications of Orderly Spanning Trees in Graph Drawing
 IN PROCEEDINGS OF THE 10TH INTERNATIONAL SYMPOSIUM ON GRAPH DRAWING, LNCS 2528
, 2002
"... Orderly spanning trees seem to have the potential of becoming a new and promising technique capable of unifying known results as well as deriving new results in graph drawing. Our exploration in this paper provides new evidence to demonstrate such a potential. Two applications of the orderly spa ..."
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Cited by 2 (2 self)
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Orderly spanning trees seem to have the potential of becoming a new and promising technique capable of unifying known results as well as deriving new results in graph drawing. Our exploration in this paper provides new evidence to demonstrate such a potential. Two applications of the orderly spanning trees of plane graphs are investigated. Our first
ORTHOGONAL DRAWINGS OF SERIESPARALLEL GRAPHS WITH MINIMUM BENDS
, 2007
"... In an orthogonal drawing of a planar graph G, each vertex is drawn as a point, each edge is drawn as a sequence of alternate horizontal and vertical line segments, and any two edges do not cross except at their common end. A bend is a point where an edge changes its direction. A drawing of G is ca ..."
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Cited by 2 (0 self)
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In an orthogonal drawing of a planar graph G, each vertex is drawn as a point, each edge is drawn as a sequence of alternate horizontal and vertical line segments, and any two edges do not cross except at their common end. A bend is a point where an edge changes its direction. A drawing of G is called an optimal orthogonal drawing if the number of bends is minimum among all orthogonal drawings of G. In this paper we give an algorithm to find an optimal orthogonal drawing of any given seriesparallel graph of the maximum degree at most three. Our algorithm takes linear time, while the previously known best algorithm takes cubic time. Furthermore, our algorithm is much simpler than the previous one. We also obtain a best possible upper bound on the number of bends in an optimal drawing.
Sorting Based Data Centric Storage
"... Abstract — Datacentric storage is a very important concept for sensor networks, where data of the same type are aggregated and stored in the same set of nodes. It is essential for many sensornet applications because it supports efficient innetwork query and processing. Multiple approaches have bee ..."
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Cited by 1 (1 self)
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Abstract — Datacentric storage is a very important concept for sensor networks, where data of the same type are aggregated and stored in the same set of nodes. It is essential for many sensornet applications because it supports efficient innetwork query and processing. Multiple approaches have been proposed so far. Their main technique is the hashing technique, where a hashing function is used to map data with the same key value to the same geometric location, and sensors closest to the location are made to store the data. Such solutions are elegant and efficient for implementation. However, two difficulties still remain: load balancing and the support for range queries. When the data of some key values are more abundant than data of other key values, or when sensors are not uniformly placed in the geometric space, some sensors can store substantially more data than other sensors. Since hashing functions map data with similar key values to independent locations, to query a range of data, multiple query messages need to be sent, even if the data of some key value in the range do not exist. In addition to the above two difficulties, obtaining the locations of sensors is also a nontrivial task. In this paper, we propose a new datacentric storage method based on sorting. Our method is robust for different network models and works for unlocalized homogeneous sensor networks, i.e., it requires no location information and no super nodes that have significantly more resources than other nodes. The idea is to sort the data in the network based on their key values, so that queries – including range queries – can be easily answered. The sorting method balances the storage load very well, and we present a sorting algorithm that is both decentralized and very efficient. We present both rigorous theoretical analysis and extensive simulations for analyzing its performance. They show that the sortingbased method has excellent performance for both communication and storage. I.