Compact Monotone Drawing of Trees

He, Xin; He, Dayu

doi:10.1007/978-3-319-21398-9_36

Cited by 7 publications

(10 citation statements)

References 11 publications

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“…We first establish the "slope" condition. 7 For the case where arctan( 1 2 ) > θ 1 the identified point is (2,1). We note that the slope of the edge e from the origin (0, 0) to (2, 1) is slope(e) = arctan( 1 2 ).…”

Section: Locatingmentioning

confidence: 95%

“…4.5]) that used a grid of size O(n 1.5 ) × O(n 1.5 ). He and He [7] gave an algorithm based on Farey sequence and reduced the required grid size to O(n 1.205 ) × O(n 1.205 ), which was the first result that used less than O(n 3 ) area. Recently, He and He [5] firstly reduced the grid size for a monotone tree drawing to O(n log(n)) × O(n log(n)) and, in a sequel paper, to O(n) × O(n) [6].…”

Section: Introductionmentioning

confidence: 99%

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Simple Compact Monotone Tree Drawings

Oikonomou

Symvonis

2018

Lecture Notes in Computer Science

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A monotone drawing of a graph G is a straight-line drawing of G such that every pair of vertices is connected by a path that is monotone with respect to some direction.Trees, as a special class of graphs, have been the focus of several papers and, recently, He and He [6] showed how to produce a monotone drawing of an arbitrary n-vertex tree that is contained in a 12n × 12n grid.All monotone tree drawing algorithms that have appeared in the literature consider rooted ordered trees and they draw them so that (i) the root of the tree is drawn at the origin of the drawing, (ii) the drawing is confined in the first quadrant, and (iii) the ordering/embedding of the tree is respected. In this paper, we provide a simple algorithm that has the exact same characteristics and, given an n-vertex rooted tree T , it outputs a monotone drawing of T that fits on a n × n grid.For unrooted ordered trees, we present an algorithms that produces monotone drawings that respect the ordering and fit in an (n + 1) × ( n 2 + 1) grid, while, for unrooted non-ordered trees we produce monotone drawings of good aspect ratio which fit on a grid of size at most 3 4 (n + 2) × 3 4 (n + 2) .

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Section: Locatingmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Simple Compact Monotone Tree Drawings

Oikonomou

Symvonis

2018

Lecture Notes in Computer Science

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“…In order to assign a vector V β ∈ V to b l , we may have to skip at most (f + 1) K−j − 1 vectors in V by Lemma 2. Also counting the vector V β assigned to b l , we consume at most (f + 1) K−j vectors in V. Thus the total number of vectors needed by the vector assignment procedure is bounded by: [1,9]. Next, we analyze the size of the drawing.…”

Section: Draw the Vertices Of T As In Step 3 Of Algorithmmentioning

confidence: 99%

Optimal Monotone Drawings of Trees

He¹,

He²

2017

SIAM J. Discrete Math.

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A monotone drawing of a graph G is a straight-line drawing of G such that, for every pair of vertices u, w in G, there exists a path P uw in G that is monotone in some direction l uw . (Namely, the order of the orthogonal projections of the vertices of P uw on l uw is the same as the order they appear in P uw .)The problem of finding monotone drawings for trees has been studied in several recent papers. The main focus is to reduce the size of the drawing. Currently, the smallest drawing size is O(n 1.205 ) × O(n 1.205 ). In this paper, we present an algorithm for constructing monotone drawings of trees on a grid of size at most 12n × 12n. The smaller drawing size is achieved by a new simple Path Draw algorithm, and a procedure that carefully assigns primitive vectors to the paths of the input tree T .We also show that there exists a tree T 0 such that any monotone drawing of T 0 must use a grid of size Ω(n) × Ω(n). So the size of our monotone drawing of trees is asymptotically optimal.

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“…The algorithm in [15], uses a set of more compact primitive vectors and reduces the drawing size to O(n 1.205 ) × O(n 1.205 ).…”

Section: 2mentioning

confidence: 99%

“…Recently, X. He and D. He in [15] reduced the drawing size to O(n 1.205 ) × O(n 1.205 ) by using a set of more compact primitive vectors.…”

Section: Introductionmentioning

confidence: 99%

Nearly optimal monotone drawing of trees

2016

Theoretical Computer Science

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A monotone drawing of a graph G is a straight-line drawing of G such that, for every pair of vertices u, w in G, there exists a path P uw in G that is monotone in some direction l. (Namely, the order of the orthogonal projections of the vertices of P uw on l is the same as the order they appear in P uw .)The problem of finding monotone drawings for trees has been studied in several recent papers. The main focus is to reduce the size of the drawing. Currently, the smallest drawing size is O(n 1.205

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Compact Monotone Drawing of Trees

Cited by 7 publications

References 11 publications

Simple Compact Monotone Tree Drawings

Simple Compact Monotone Tree Drawings

Optimal Monotone Drawings of Trees

Nearly optimal monotone drawing of trees

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