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Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio - - PowerPoint PPT Presentation
Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio - - PowerPoint PPT Presentation
Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani, Vincenzo Roselli Graph G = (V,E) Vertices are points Edges are straight-line segments Planar: edges do not cross each other Metamorphosis I -- M.
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Graph G = (V,E) Vertices are points Edges are straight-line segments Planar: edges do not cross each other
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Metamorphosis I -- M. C. Escher
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Morphing is a special effect in motion pictures and animations that changes (or morphs) one image or shape into another through a seamless transition. (Wikipedia)
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… changes (or morphs) one planar graph drawing into another through a seamless transition
Vertices move at uniform speed along straight-line
segments
Planarity is preserved during the whole morph
(…)
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Some pairs of drawings require several steps The complexity of a morphing algorithm is defined as the number of steps it performs
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Tietze, Rend. Circ. Matematico di Palermo, 38(1):247-304, 1914
We are in the centennial of the Morphing problem!
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Smith, 1917 Veblen, 1917 Alexander, 1923
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Cairns, 1944
A planar morph exists between any two planar drawings of any maximal planar graph (with an exponential number of steps)
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Thomassen, Journal of Combinatorial Theory B, 1983
A planar convex morph exists between any two planar convex drawings of any maximal planar graph, (with an exponential number of steps)
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Aicholzer et al, 2011 Floater, Gotsman, Surazhsky several papers Lubiw Petrick, 2011 Biedl et al., 2006
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Alamdari et al., SODA 2013
A planar morph exists between any two planar drawings of any maximal planar graph, (with a polynomial numer of steps) O(n2) steps for maximal planar O(n4) steps for maximal planar
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Angelini et al., GD 2013 Barrera-Cruz et al., MCDMCG, 2013
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Angelini et al., GD 2013 Barrera-Cruz et al., MCDMCG, 2013
At each morphing step, all the vertices move along the same direction
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Angelini et al., GD 2013 Barrera-Cruz et al., MCDMCG, 2013
At each morphing step, all the vertices move along the same direction Maximal Planar graphs in O(n2) steps Series-parallel graphs in O(n) steps
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Theorem 1: There exist an algorithm to morph
planar graph drawings in O(n) steps
Theorem 2: There exist two drawings of a planar
graph (a path) such that morphing one into the
- ther requires at least Ω(n) steps
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Every planar graph contains at least a vertex of degree at most 5 with no chord between its neighbors By Euler’s formula, |E| <= 3n-6
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Every polygon with at most 5 vertices has at least
- ne vertex on the boundary of its kernel
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Every polygon with at most 5 vertices has at least
- ne vertex on the boundary of its kernel
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Contraction of a low-degree vertex
- n one of its neighbors
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1: Select a vertex v of low degree. It might be contractible
- n different neighbors
in the two drawings v v
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2: Contract v in both the drawings. We obtain different graphs! v v O(1) O(1)
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3: Remove v and compute a drawing in which the polygon of its neighbors is convex v v O(1) O(1)
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In this drawing, passing from one neighbor to the other is always possible! v v O(1) O(1)
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In this drawing, passing from one neighbor to the other is always possible! v v O(1) O(1)
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In this drawing, passing from one neighbor to the
- ther is always possible!
v v O(1) O(1) O(1)
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Recursively morph the two contracted drawings to the convex ones in T(n-1) steps
T(n) = 2 T(n-1) + O(1) = O(2n)
v v T(n-1) T(n-1) O(1) O(1) O(1)
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The morph is not to a specific convex drawing! v v
O(n)
T(n-1) O(1) O(1) O(1)
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The morph is not to a specific convex drawing!
T(n) = T(n-1) + O(n) = O(n2)
v v T(n-1) O(1) O(1)
O(n)
O(1)
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The morph is to a (very) specific convex drawing! v v
O(1)
T(n-1) O(1) O(1) O(1)
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The morph is to a (very) specific convex drawing!
T(n) = T(n-1) + O(1) = O(n)
v v T(n-1) O(1) O(1)
O(1)
O(1)
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Given:
- a triconnected planar graph G = (V,E)
- A set L of parallel lines
- A mapping of the vertices of V to lines of L such that
- rienting the edges of E according to the order of the lines
in L yields an st-orientation of G
st-orientation: each face is composed of two oriented paths
Then, G admits a convex drawing in which each
vertex of V lies on the line of L it is mapped to
Hong, Nagamochi. J. of Discrete Algorithms, 2010
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Hong, Nagamochi. J. of Discrete Algorithms, 2010
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A linear number of steps are necessary to morph
- ne of these drawings into the other
- There exists an edge that must perform a linear number
- f complete rotations
- In a single step, only (a bit less
than) half rotation can be performed
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All the algorithms produce intermediate drawings
- n a very large grid
- What about morphing on a limited-size grid?
- Can we avoid contractions?
Can we produce convexity-preserving morphings
with a polynomial number of steps?
How to morph non-planar graphs?
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