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On Upward Point-Set Embedding of Upward Planar Digraphs. Md. Emran Chowdhury. Muhammad Jawaherul Alam. Md. Saidur Rahman. Department of Computer Science & Engineering Bangladesh University of Engineering & Technology (BUET). Upward Point-Set Embedding. d. d. f. f. e. e. a. a. G. - PowerPoint PPT Presentation
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Md. Emran ChowdhuryMd. Emran Chowdhury
Muhammad Jawaherul AlamMuhammad Jawaherul Alam Md. Saidur RahmanMd. Saidur Rahman
Department of Computer Science & EngineeringBangladesh University of Engineering & Technology (BUET)
Department of Computer Science & EngineeringBangladesh University of Engineering & Technology (BUET)
Upward Point-Set EmbeddingUpward Point-Set Embedding
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Sa
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Each vertex is placed at a distinct pointEach vertex is placed at a distinct point
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Upward Point-Set EmbeddingUpward Point-Set Embedding
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Each vertex is placed at a distinct pointEach vertex is placed at a distinct point
Each edge is drawn upwardEach edge is drawn upward
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Upward Point-Set EmbeddingUpward Point-Set Embedding
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Each vertex is placed at a distinct pointEach vertex is placed at a distinct point
Each edge is drawn upwardEach edge is drawn upward
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G’ S
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G S
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Upward Point-Set EmbeddingUpward Point-Set Embedding
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There is an Upward Point-set Embedding of G onS if and only if Gis acyclic
There is an Upward Point-set Embeddingof G on S if and only if G is acyclic There is an Upward Point-set Embedding
ofG on S if and only if Gis upward planar
There is an Upward Point-set Embeddingof G on S if and only if G is upward planar
Upward Point-Set EmbeddingUpward Point-Set Embedding
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Upward Point-Set EmbeddingUpward Point-Set Embedding
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Giordano et. al.
Upward Point-Set Embedding ofany upward planar digraph with on
any point set withat most two bends per edge
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Upward Point-Set Embedding with mappingUpward Point-Set Embedding with mapping
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φ
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bφ
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φ’
No upward point-setembedding with this mapping
No upward point-setembedding with this mapping
Upward Point-Set Embedding with mappingUpward Point-Set Embedding with mapping
Giordano et. al.
• O(n3)-time testing algorithm
• O(n2)-time drawing algorithm* (2n-3) bends per edge
Ours
• O(n2)-time drawing algorithm* (n-3) bends per edge
Upward TopologicalBook Embedding with
a given ordering
Upward Point-Set Embedding with mappingUpward Point-Set Embedding with mapping
Upward Topological Book EmbeddingUpward Topological Book Embedding
Variant of Upward Point-Set Embedding
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SG
Spine
LeftPage
RightPage
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The vertices on the spine
The edges on the pages
Only ordering of the verticesare important, not their positions
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Our AlgorithmOur Algorithm
G contains directed hamiltonian pathG contains directed hamiltonian path
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A directed path containingall the vertices
A directed path containingall the vertices
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Our AlgorithmOur Algorithm
G contains directed hamiltonian pathG contains directed hamiltonian path
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7The drawing …..
• has no crossings sinceit has the same embeddingas the original graph
• has no spine crossings• has 1 bend per edge
Our AlgorithmOur Algorithm
G contains no directed hamiltonian path
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Our AlgorithmOur Algorithm
G contains no directed hamiltonian path
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Our AlgorithmOur Algorithm
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Our AlgorithmOur Algorithm
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Our AlgorithmOur Algorithm
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Each spine crossingcorresponds to a dummy vertex
Each spine crossingcorresponds to a dummy vertex
Number of Bends per edgeNumber of Bends per edge
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Spine crossing from ito j is at most j-i-3
Spine crossing from ito j is at most j-i-3
The edge (1, n) has no crossingsThe edge (1, n) has no crossings
Spine Crossings per edgeis at most n-4
Spine Crossings per edgeis at most n-4
Bends per edge is at most n-3Bends per edge is at most n-3
Algorithm for Points in General PositionAlgorithm for Points in General Position
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a’
d’c’
b’e’
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Ordering inducedby φ on L
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Giordano et. al. showed that
G admits an upward point-set embedding onS of points with the mapping φ with t bends
if and only if there is a line L such thatG admits an upward topological book embeddingwith the ordering induced by φ on L with t bends
Open ProblemsOpen Problems
Find the minimum number of total bends in all edges
To give an o(n3)-time testing algorithm
Thank You