A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In J. van Leeuwen, editor, Algorithms (Proc. ESA '94), volume 855 of Lecture Notes Comput. Sci., pages 12--23. Springer-Verlag, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that good angular resolution cannot always be achieved while simultaneously guaranteeing straight line edges and small sub exponential drawing area [9] By introducing bends in the edges, however, we can guarantee both good resolution and small drawing area. 1. 2 Previous Work Garg and Tamassia [5] consider the problem of drawing with good angular resolution, and Kant [8] shows how to create drawings with angular resolution of #(1 d(v) in an O(n) O(n) area grid, using edges with at most three bends each. Gutwenger and Mutzel [7] describe an improved algorithm with better constant factors ....

A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In Proc. 2nd Annu. European Sympos. Algorithms, volume 855 of Lecture Notes Computer Science, pages 12--23. Springer-Verlag, 1994.

....resolution) In fact, it has been proven [11] that there exist graphs that always require exponential area for straight line embeddings maintaining good angular resolution. The problem of drawing planar graphs with good angular resolution was addressed by Formann et al. 5] Garg and Tamassia [6], and Kant [9, 10] who showed that one could in fact simultaneously achieve O(n) O(n) area and an angular resolution of (1=d(v) for each vertex v, by drawing a planar graph using piecewise linear polylines with at most three bends each. Gutwenger and Mutzel [8] improved the constant factors ....

A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In Proc. 2nd Annu. European Sympos. Algorithms, volume 855 of Lecture Notes Comput. Sci., pages 12-23. Springer-Verlag, 1994.

.... the illumination of the value of bend minimization in polyline drawings by Tamassia [13] the discussion of aspect ratio by Chan et al. 1] and the identi cation of angular resolution as a sign cant aesthetic criterion by Formann et al. 5] and its further study by Kant [10] Garg and Tamassia [7], and Malitz and Papakostas [11] This paper is directed at continuing this tradition by articulating a uni ed framework for describing aesthetic criteria and complexity measures for drawings of graphs that place vertices at points in the plane and represent edges with smooth curves (that is, ....

A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In Proc. 2nd Annu. European Sympos. Algorithms, volume 855 of Lecture Notes Comput. Sci., pages 12-23. Springer-Verlag, 1994.

....resolution denotes the size of the minimum angle formed by any two edge segments sharing a common point in a drawing. Unfortunately, one can show that a good angular resolution in planar straight line drawings can be achieved only at the cost of the area taken up by the drawing. Garg and Tamassia [18] have shown that a class of planar graphs exists that require exponential area in any planar straight line drawing with optimal Omega (1) angular resolution. Drawings with good angular resolution and small grid size are the so called orthogonal (grid) drawings. Here, only horizontal and ....

A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In Proc. 2nd Annual European Sympos. Algorithms (ESA '94), volume 855 of Lecture Notes in Computer Science, pages 12--23. Springer-Verlag, 1994.

....by convex regions. is O(1=n) This area bound is unlike the area bound for classical graphs which is Omega Gamma n 2 ) see [6] Our angle bound is also different from any angle bound of classical graph drawing conventions in which angle bounds are functions of the degree of the graph (see [8]) Our results indicate important tradeoffs between line straightness and area, and between region convexity and area. The rest of the paper is organized as follows. In Section 2, we present some terminology and some basic results. In Section 3, we describe a straight line convex drawing algorithm ....

....area lower bound and an angle upper bound for C planar straightline convex drawings of clustered graphs. Our bounds are unlike the bounds for classical graphs. Planar drawing of classical graphs requires Omega Gamma n 2 ) area [6] and with angle resolution dependent on the degree of the graph [8]. We present a class of clustered graphs which, in any C planar straightline convex drawing, require exponential area and an angle inversely proportional to the size of the clustered graph. This class of clustered graph is an extension of the graphs defined in [2] see Fig. 4) For each n 0 we ....

Ashim Garg and Roberto Tamassia. Planar drawings and angular resolution: algorithms and bounds. In Lecture Notes in Computer Science: Proc. of European Symposium on Algorithms, 1994.

.... drawing of a triangulated planar graph, whose edge resolution is O(n 3 ) Since then, several researchers have worked on extending and tightening these results in the integer grid model [5, 6, 29] Several researchers have also considered trade offs involving the angular resolution (e.g. see [21, 22, 23]) In addition, Di Battista, Tamassia, and Tollis [15] prove an interesting lower bound, which holds under any reasonable finite resolution rule, that producing a straight line 2 dimensional drawing of a directed acyclic planar graph so that all edges point up requires exponential area. Our ....

....volume to draw as 3 dimensional convex polyhedra with constant angular resolution. We establish this lower bound via a reduction from the problem of drawing a fixed degree 3 connected planar graph under angular resolution in R 2 , which was shown to require exponential area by Garg and Tamassia [23]. The main difficulty in extending their proof to convex drawings in R 3 is that the third dimension allows a tremendous amount of extra drawing freedom. For example, a convex drawing in R 3 can achieve angular resolution and yet have many 2 dimensional projections that do not achieve angular ....

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A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In Graph Drawing '95, 1995.

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A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In J. van Leeuwen, editor, Algorithms (Proc. ESA '94), volume 855 of Lecture Notes Comput. Sci., pages 12--23. Springer-Verlag, 1994.

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A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In J. van Leeuwen, editor, Algorithms (Proc. ESA '94), volume 855 of Lecture Notes in Computer Science, pages 12--23. Springer-Verlag, 1994.

....resolution. However, these drawings are in general not planar. Every planar graph has a planar straight line drawing with angular resolu tion Y2( y) for some constant I [18] but there are planar graphs for which the angular resolution in any planar straight line drawing is bounded by O( lo) [12]. Note that maximizing the angular resolution over all planar V (G) straight line drawings of a planar graph is Af7 hard [11] It is shown in [16] how to obtain planar drawings with asymptotically optimal angular resolution when edges may be represented as sequences of straight lines. This ....

A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. Proc. 2nd European Symp. Algorithms (ESA '9), LNCS vol. 855, pp. 12-23. Springer, 1994.

.... n) 17, 98] degree O(n a ) tree planar polyline grid O(n) Omega Gamma n) 56] degree 4 tree planar orthogonal grid O(n) Omega Gamma n) 114, 81] planar graph planar polyline grid O(n 2 ) Omega Gamma n 2 ) 31, 33, 73] planar graph planar straight line Omega Gamma c aen ) [59] planar graph planar straight line grid O(n 2 ) Omega Gamma n 2 ) 21, 96] triconnected planar graph planar straight line convex grid O(n 2 ) Omega Gamma n 2 ) 73] planar graph planar orthogonal grid O(n 2 ) Omega Gamma n 2 ) 7, 73, 103, 108] planar degree 4 graph orthogonal ....

....such that c 1. Class of Graphs Drawing Type Angular Resolution Source general graph straight line Omega Gamma 1 d 2 ) O( log d d 2 ) 48] planar graph straight line Omega Gamma 1 d ) O( 1 d ) 48] planar graph planar straight line Omega Gamma 1 c d ) O( q log d d 3 ) [59, 87] BOUNDS ON THE NUMBER OF BENDS Table 1.3 summarizes selected universal upper bounds and existential lower bounds on the total and maximum number of bends in orthogonal drawings. Some bounds are stated for n 5 or n 7 because the maximum number of bends is at least 2 for K 4 and at least 3 for ....

A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In Proc. 2nd Annu. European Sympos. Algorithms (ESA '94), volume 855 of Lecture Notes in Computer Science, pages 12--23. Springer-Verlag, 1994.

.... log n) 12, 79] degree O(n a ) tree planar polyline grid Omega Gamma n) O(n) 38] degree 4 tree planar orthogonal grid Omega Gamma n) O(n) 93, 60] planar graph planar polyline grid Omega Gamma n 2 ) O(n 2 ) 27, 28, 56] planar graph planar straight line Omega Gamma c aen ) [40] planar graph planar straight line grid Omega Gamma n 2 ) O(n 2 ) 19, 77] triconnected planar graph planar straight line convex grid Omega Gamma n 2 ) O(n 2 ) 56] planar graph planar orthogonal grid Omega Gamma n 2 ) O(n 2 ) 3, 56, 81, 86] planar degree 4 graph ....

....such that c 1. Class of Graphs Drawing Type Angular Resolution Ref. general graph straight line Omega Gamma 1 d 2 ) O( log d d 2 ) 35] planar graph straight line Omega Gamma 1 d ) O( 1 d ) 35] planar graph planar straight line Omega Gamma 1 c d ) O( q log d d 3 ) [40, 65] Table 3: Orthogonal drawings: universal upper bounds and existential lower bounds on the total and maximum number of bends. Notes: y n 7; z n 5. Class of Graphs Drawing Type Total No. Bends Max No. Bends Ref. degree 4 graphy orthogonal n 2n 2 2 2 [3] planar degree 4 graphy orthogonal ....

A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In Proc. 2nd Annu. European Sympos. Algorithms (ESA '94), volume 855 of Lecture Notes in Computer Science, pages 12--23. Springer-Verlag, 1994.

....They tend to give more aesthetically pleasing drawings in general [31, 62] Minimizing bends is also important in VLSI layouts where bends act as hot points because of their higher current density. In view of the trade off between the angular resolution of a drawing and its area requirement [58, 83], these drawings offer a good compromise: they have large angles (multiples of 90 ffi ) small area, and few bends (every degree 4 graph admits am orthogonal drawing with O(n 2 ) area and O(n) bends [109, 75] Finally, because each angle is a multiple of 90 ffi , orthogonal planar drawings ....

....coloring, flow) the techniques in this chapter appear to be interesting in the way they blend geometric and graph theoretic methods. We believe that our results will stimulate further research in this direction. Most of the results and techniques presented in this chapter can also be found in [55, 58]. The rest of this chapter is organized as follows. Section 4.2 provides basic definitions. The upper bound on the angular resolution is shown in Section 4.3. Section 4.4 contains the tradeoff between area and angular resolution. The algorithms for constructing straight line drawings with large ....

[Article contains additional citation context not shown here]

A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In Proc. 2nd Annu. European Sympos. Algorithms (ESA '94), volume 855 of Lecture Notes in Computer Science, pages 12--23. Springer-Verlag, 1994.

....model [6, 7, 34] We will refer to the method from [14, 15, 8] as the shift method, as it works by successively adding vertices to the drawing and shifting horizontally parts of the existing drawing. Several researchers have also considered tradeoffs involving the angular resolution (e.g. see [23, 24, 39]) For example, Garg and Tamassia [24] show that the problem of drawing a fixed degree 3connected planar graph under angular resolution in R 2 requires exponential area. In addition, Di Battista, Tamassia, and Tollis [17] prove an interesting lower bound, which holds under any reasonable ....

....from [14, 15, 8] as the shift method, as it works by successively adding vertices to the drawing and shifting horizontally parts of the existing drawing. Several researchers have also considered tradeoffs involving the angular resolution (e.g. see [23, 24, 39] For example, Garg and Tamassia [24] show that the problem of drawing a fixed degree 3connected planar graph under angular resolution in R 2 requires exponential area. In addition, Di Battista, Tamassia, and Tollis [17] prove an interesting lower bound, which holds under any reasonable finite resolution rule, that there exist ....

[Article contains additional citation context not shown here]

A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In Proc. 2nd Annu. European Sympos. Algorithms (ESA '94), volume 855 of Lecture Notes in Computer Science, pages 12--23. SpringerVerlag, 1994.

....model [6, 7, 34] We will refer to the method from [14, 15, 8] as the shift method, as it works by successively adding vertices to the drawing and shifting horizontally parts of the existing drawing. Several researchers have also considered trade offs involving the angular resolution (e.g. see [23, 24, 39]) For example, Garg and Tamassia [24] show that the problem of drawing a fixed degree 3 connected planar graph under angular resolution in R 2 requires exponential area. In addition, Di Battista, Tamassia, and Tollis [17] prove an interesting lower bound, which holds under any reasonable ....

....from [14, 15, 8] as the shift method, as it works by successively adding vertices to the drawing and shifting horizontally parts of the existing drawing. Several researchers have also considered trade offs involving the angular resolution (e.g. see [23, 24, 39] For example, Garg and Tamassia [24] show that the problem of drawing a fixed degree 3 connected planar graph under angular resolution in R 2 requires exponential area. In addition, Di Battista, Tamassia, and Tollis [17] prove an interesting lower bound, which holds under any reasonable finite resolution rule, that there exist ....

[Article contains additional citation context not shown here]

A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In Proc. 2nd Annu. European Sympos. Algorithms (ESA '94), volume 855 of Lecture Notes in Computer Science, pages 12--23. Springer-Verlag, 1994.

....aesthetic appeal and visual effectiveness of the drawing. The issue of resolution of a drawing has been extensively studied, motivated by the finite resolution of physical rendering devices. Several papers have been published about the resolution and the area of drawings of graphs (see, e.g. [1, 6, 14, 19]) The resolution of a drawing is defined as the minimum distance between two vertices. The grid based algorithms consider edge bends and edge crossings as dummy vertices for computing the resolution. The layering based algorithms, however, do not consider the edgecrossings as dummy vertices for ....

A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In J. van Leeuwen, editor, Algorithms (Proc. ESA '94), volume 855 of Lecture Notes in Computer Science, pages 12--23. Springer-Verlag, 1994.

No context found.

A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In Proc. 2nd European Symposium on Algorithms, pages 12--23, 1994.

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