Accelerated Bend Minimization

Cornelsen, Sabine; Karrenbauer, Andreas

doi:10.1007/978-3-642-25878-7_12

Cited by 25 publications

(57 citation statements)

References 31 publications

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“…For the case that deg(s) + deg(t) = 5 the equation H = 2 holds and thus we only have to show convexity on the interval [2,3]. Obviously, cost H (·) is convex on this interval if and only if cost H (2) ≤ cost H (3). As this is the case for both partial cost functions, it is also…”

Section: Lemma 6 If Theorem 5 Holds For Each Principal Split Componementioning

confidence: 97%

“…This is not even true if every edge can have a single bend for free and then has to pay cost 1 for every additional bend, see Figure 1(c). To solve this problem, we essentially show that it is sufficient to compute the cost functions on the small interval [0,3]. We can then show that the cost functions we compute are always convex on this interval.…”

Section: Contribution and Outlinementioning

confidence: 99%

“…With this result we are able to drop the fixed planar embedding (Section 6). We first consider biconnected graphs in Section 6.1 and compute cost functions on the interval [0,3], which can be shown to be convex on that interval, bottom up in the SPQR-tree. In Section 6.2 we extend this result to connected graphs using the BC-tree (see Section 2.2 for a definition).…”

Section: Contribution and Outlinementioning

confidence: 99%

“…Second, as mentioned in the introduction (see Figure 1) the cost function of a split component may already be non-convex on the interval [0,5]. Fortunately, the second reason is not really a problem since there may be at most a single edge with up to five bends, all remaining edges have at most three bends and thus we only need to consider their cost functions on the interval [0,3].…”

Section: Corollary 1 Let G Be a Graph With Positive Flexibility And mentioning

confidence: 99%

“…Then its skeleton has only two embeddings E and E where E is obtained by flipping E. We have to show that the minimum over the two partial cost functions cost E H (·) and cost E H (·) remains convex. For the case that deg(s) + deg(t) = 5 the equation H = 2 holds and thus we only have to show convexity on the interval [2,3]. Obviously, cost H (·) is convex on this interval if and only if cost H (2) ≤ cost H (3).…”

Section: Lemma 6 If Theorem 5 Holds For Each Principal Split Componementioning

confidence: 99%

See 4 more Smart Citations

Optimal Orthogonal Graph Drawing with Convex Bend Costs

Bläsius

Rutter

Wagner

2013

Automata, Languages, and Programming

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Traditionally, the quality of orthogonal planar drawings is quantified by either the total number of bends, or the maximum number of bends per edge. However, this neglects that in typical applications, edges have varying importance. Moreover, as bend minimization over all planar embeddings is N P-hard, most approaches focus on a fixed planar embedding.We consider the problem OptimalFlexDraw that is defined as follows. Given a planar graph G on n vertices with maximum degree 4 and for each edge e a cost function cost e : N 0 −→ R defining costs depending on the number of bends on e, compute an orthogonal drawing of G of minimum cost. Note that this optimizes over all planar embeddings of the input graphs, and the cost functions allow fine-grained control on the bends of edges.In this generality OptimalFlexDraw is N P-hard. We show that it can be solved efficiently if 1) the cost function of each edge is convex and 2) the first bend on each edge does not cause any cost (which is a condition similar to the positive flexibility for the decision problem FlexDraw). Moreover, we show the existence of an optimal solution with at most three bends per edge except for a single edge per block (maximal biconnected component) with up to four bends. For biconnected graphs we obtain a running time of O(n · T flow (n)), where T flow (n) denotes the time necessary to compute a minimum-cost flow in a planar flow network with multiple sources and sinks. For connected graphs that are not biconnected we need an additional factor of O(n).

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Section: Lemma 6 If Theorem 5 Holds For Each Principal Split Componementioning

confidence: 97%

Section: Contribution and Outlinementioning

confidence: 99%

Section: Contribution and Outlinementioning

confidence: 99%

Section: Corollary 1 Let G Be a Graph With Positive Flexibility And mentioning

confidence: 99%

Section: Lemma 6 If Theorem 5 Holds For Each Principal Split Componementioning

confidence: 99%

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Optimal Orthogonal Graph Drawing with Convex Bend Costs

Bläsius

Rutter

Wagner

2013

Automata, Languages, and Programming

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On Turn-Regular Orthogonal Representations

Bekos

Binucci

Battista

et al. 2020

Lecture Notes in Computer Science

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An interesting class of orthogonal representations consists of the so-called turn-regular ones, i.e., those that do not contain any pair of reflex corners that "point to each other" inside a face. For such a representation H it is possible to compute in linear time a minimum-area drawing, i.e., a drawing of minimum area over all possible assignments of vertex and bend coordinates of H. In contrast, finding a minimum-area drawing of H is NP-hard if H is non-turn-regular. This scenario naturally motivates the study of which graphs admit turn-regular orthogonal representations. In this paper we identify notable classes of biconnected planar graphs that always admit such representations, which can be computed in linear time. We also describe a linear-time testing algorithm for trees and provide a polynomial-time algorithm that tests whether a biconnected plane graph with "small" faces has a turn-regular orthogonal representation without bends.

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On Orthogonally Convex Drawings of Plane Graphs

Chang

Yen

2013

Graph Drawing

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Abstract. We investigate the bend minimization problem with respect to a new drawing style called orthogonally convex drawing, which is orthogonal drawing with an additional requirement that each inner face is drawn as an orthogonally convex polygon. For the class of bi-connected plane graphs of maximum degree 3, we give a necessary and sufficient condition for the existence of a no-bend orthogonally convex drawing, which in turn, enables a linear time algorithm to check and construct such a drawing if one exists. We also develop a flow network formulation for bend-minimization in orthogonally convex drawings, yielding a polynomial time solution for the problem. An interesting application of our orthogonally convex drawing is to characterize internally triangulated plane graphs that admit floorplans using only orthogonally convex modules subject to certain boundary constraints.

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Accelerated Bend Minimization

Abstract: √ log n) shown by Garg and Tamassia in 1996 could be improved. To answer this question, we show how to solve the uncapacitated min-cost flow problem on a planar bidirected graph with bounded costs and face sizes in O(n 3/2 ) time.

Cited by 25 publications

References 31 publications

Optimal Orthogonal Graph Drawing with Convex Bend Costs

Optimal Orthogonal Graph Drawing with Convex Bend Costs

On Turn-Regular Orthogonal Representations

On Orthogonally Convex Drawings of Plane Graphs

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