Morphing Planar Graph Drawings

Transcription

1 Morphing Planar Graph Drawings Giuseppe Di Battista Università degli Studi Roma Tre The 12th International Conference and Workshops on Algorithms and Computation WALCOM 2018

2 Basic definitions Graph drawing A drawing of a graph G V, E maps vertices in V to distinct points of the plane and edges in E to open curves connecting their end-vertices 2 V = 1,2,3,4,5 E = 1,2, 1,4, 2,3, 3,4, 2,5, (4,5)

3 Basic definitions Planar drawing A drawing of a graph G V, E is planar if the curves representing the edges do not intersect; a graph admitting a planar drawing is a planar graph 2 V = 1,2,3,4,5 E = 1,2, 1,4, 2,3, 3,4, 2,5, (4,5)

4 Basic definitions Face A planar drawing partitions the plane into topological connected regions called faces, the unbounded face is the outer face 2 V = 1,2,3,4,5 E = 1,2, 1,4, 2,3, 3,4, 2,5, (4,5)

5 Basic definitions Straight-line drawing A drawing of a graph is straight-line if the curves representing edges are straight-line segments 2 V = 1,2,3,4,5 E = 1,2, 1,4, 2,3, 3,4, 2,5, (4,5)

6 The problem of morphing graph drawings input: two drawings Γ 0 and Γ 1 of the same graph G A morph between Γ 0 and Γ 1 is a continuously changing family of drawings of G indexed by time t 0,1, such that the drawing at time t = 0 is Γ 0 and the drawing at time t = 1 is Γ 1

7 A morph Γ 0 Γ 1

8 A morph Γ 0 Γ 1

9 A morph Γ 0 Γ 1

10 A morph Γ 0 Γ 1

11 A morph Γ 0 Γ 1

12 A morph Γ 0 Γ 1

13 A morph Γ 0 Γ 1

14 A morph Γ 0 Γ 1

15 Morphing preserving one or more properties suppose that both Γ 0 and Γ 1 have a certain geometric property, e.g. they are planar drawings they are straight-line drawings their edges are polygonal lines composed of horizontal and vertical segments their faces are covex polygons. it is interesting that all the drawings of the morph preserve that property

16 A morph that does not preserve planarity

17 A morph that does not preserve planarity

18 A morph that does not preserve planarity

19 A morph that does not preserve planarity

20 A morph that does not preserve planarity

21 100 Years of morphing planar straight-line graph drawings

22 Morphs of planar graphs From now on, for a while, we discuss morphs of planar straight-line graph drawings that preserve: planarity straight-line drawing of the edges

23 1914: Squares Courtesy of Vincenzo Roselli

24 1917: Squares Courtesy of Vincenzo Roselli

25 1917: Polygons Courtesy of Vincenzo Roselli

26 1923: Polygons Courtesy of Vincenzo Roselli

27 1944: Triangulations Cairns, first algorithmic proof Courtesy of Vincenzo Roselli

28 Basic definitions Triangulation A triangulation is a straight-line planar drawing of a maximal planar graph

29 Cairns theorem Theorem - Given two triangulations of the same maximal planar graph and with the same outer face, a morph preserving planarity and straight-line edges exists between them complexity of morph? trajectory of vertices?

30 Morphing steps A morph is composed of one or more morphing steps In a morphing step a subset of the vertices and their incident edges move to a new location vertices can move along different types of trajectories

31 Cairn s double recursion approach Γ 0 Γ 1

32 Cairn s double recursion approach Γ 0 Γ 1

33 Cairn s double recursion approach Γ 0 Γ 1

34 Cairn s double recursion approach contraction expansion recursion position change recursion

35 A closer look at the position change position change

36 A closer look at the position change position change

37 A closer look at the position change position change

38 A closer look at the position change position change

39 A closer look at the position change position change

40 A closer look at the position change position change

41 A closer look at the position change position change

42 Cairn s double recursion approach number S(n) of morphing steps S n = 2S n 1 + O(1) contraction O(1) S(n) O 2 n expansion O(1) recursion position change recursion S(n 1) O(1) S(n 1)

43 Basic definitions Equivalent drawings Two planar drawings of a connected planar graph are equivalent if they have the same circular ordering of the edges around vertices and have the same outer face

44 Basic definitions Embedding and plane graph A planar embedding is an equivalence class of planar drawings of the same graph; a plane graph is a planar graph with a given planar embedding

45 1983: Plane graphs Thomassen Courtesy of Vincenzo Roselli

46 Cairns and Thomassen Thomassen extended the result of Cairns to all straight-line drawings of plane graphs Augmentation of both Γ 0 and Γ 1 to isomorphic (compatible) triangulations which reduces the general case to Cairns's result The idea of compatible triangulations was rediscovered and explored (in 1993) by Aronov et al., who showed, that two drawings of a connected graph on n vertices have a compatible triangulation of size O n 2 ; tight in the worst case

47 The last 6 years

48 Results from S. Alamdari, P. Angelini, T.M. Chan, gdb, F. Frati, A. Lubiw, M. Patrignani, V. Roselli, S. Singla, and B. T. Wilkinson, Morphing planar graph drawings with a polynomial number of steps, SODA 2013 P. Angelini, F. Frati, M. Patrignani, and V. Roselli, Morphing planar graph drawings Efficiently, GD 2013 P. Angelini, G. Da Lozzo, gdb, F. Frati, M. Patrignani, and V. Roselli, Morphing planar graph drawings optimally, ICALP 2014 S. Alamdari, P. Angelini, F. Barrera-Cruz, T.M. Chan, G. Da Lozzo, gdb, F. Frati, P. Haxell, A. Lubiw, M. Patrignani, V. Roselli, S. Singla, B.T. Wilkinson. How to morph planar graph drawings, SICOMP 2017

49 Linear morphs and morphing steps In a linear morphing step every vertex moves along a straight-line segment at uniform speed Vertices may move at different speeds, and some vertices may remain stationary A linear morph consists only of linear morphing steps

50 Unidirectional morphing steps A unidirectional morphing step is a linear morphing step in which every vertex moves parallel to the same line i.e. there is a line with unit direction vector l such that each vertex moves linearly from an initial position v 0 to a final position v 0 + k v l for some k v R Different vertices may move different amounts, and k v can be positive or negative

51 Unidirectional morphing steps A unidirectional morphing step is a linear morphing step in which every vertex moves parallel to the same line i.e. there is a line with unit direction vector l such that each vertex moves linearly from an initial position v 0 to a final position v 0 + k v l for some k v R Different vertices may move different amounts, and k v can be positive or negative

52 A unidirectional morphing lemma Lemma - Let x, y, z be the clockwise-ordered vertices of the triangular outer face of a triangulation. Define any point p inside the triangle as a convex combination of x, y, z. In this way the motion of x, y, z determines the motion of p. Suppose that x, y, z move linearly in the direction of a vector l in such a way that their clockwise order is preserved. The result is a unidirectional morph of the straight-line planar drawing where planarity is preserved.

53 Proof Unidirectionality of morph is trivial The fact that planarity is preserved follows from a more general result The transformation of points x, y, x determines an affine transformation of the plane, that by hypothesis preserves the orientation of triangle x, y, x Affine transformations preserve convex combinations; thus our definition of the movement of any interior point p is the same as applying the affine transformation to p An affine transformation that preserves the orientation of one triangle preserves the orientations of all triangles This implies that the drawing is planar at all times of the morph

54 Morphing triangulations with a few steps Theorem - Let Γ 0 and Γ 1 be two triangulations that are topologically equivalent drawings of an n-vertex maximal planar graph. There is a morph from Γ 0 to Γ 1 that is composed of O n unidirectional morphing steps. The morph can be constructed in O n 3 time.

55 A combinatorial tool Every planar graph contains at least one internal vertex of degree at most 5 with no chord between its neighbors By Euler s formula, E 3n 6

56 A geometric tool Every polygon with at most 5 vertices has a non-empty kernel (the set of points from which all the polygon is visible) and at least one vertex of the polygon is on the boundary of the kernel

57 A geometric tool Every polygon with at most 5 vertices has a non-empty kernel and at least one vertex of the polygon lies on the boundary of the kernel

58 Using the combinatorial and the geometric tools In any planar straight-line drawing of a graph there exists a vertex with low degree that can be contracted on one of its neighbors without introducing any crossing

59 Convexification Lemma - Given an n-vertex triangulation Γ and given a quadrilateral (pentagon) abcd (abcde) in Γ such abcd (abcde) does not have external chords, a unidirectional morph of Γ exits so that abcd (abcde) becomes convex. Such a morph can be found in O n time. a b a b d c d c

60 Basic definitions Triconnected graph A connected graph is triconnected if deleting any two vertices (and incident edges) results in a graph that is still connected

61 Basic definitions st-orientation An st-orientation of a plane graph orients the edges so that each face is composed of two oriented paths t s

62 A convexification tool Given: a triconnected plane graph G = (V, E) a set L of parallel lines a mapping of the vertices of V to lines of L such that orienting the edges of E according to the order of the lines in L yields an storientation of G Then G admits a convex drawing (all faces are convex polygons) in which each vertex of V lies on the line of L it is mapped to Hong, Nagamochi, JDA, 2010

63 A convexification tool Hong, Nagamochi, JDA, 2010 Courtesy of Patrizio Angelini

64 Fast Convexifier Courtesy of Patrizio Angelini

65 Fast Convexifier

66 Fast Convexifier

67 Fast Convexifier

68 Fast Convexifier

69 Fast Convexifier

70 Fast Convexifier

71 Fast Convexifier

72 Morphing triangulations with a few steps number S(n) of unidirectional morphing steps S n = S n 1 + O(1) contraction O(1) S(n) O n expansion O(1) convexification position change recursion O(1) O(1) S(n 1)

73 Plane graphs To generalize from triangulations to planar straight-line drawings of connected plane graphs Thomassen uses compatible triangulations of both drawings, which increases the size of the graph to O n 2 This can be improved by making use of the freedom to morph the drawings use a sequence of O n unidirectional morphs to triangulate Γ 0 and Γ 1 with the same edges hence, the general problem is reduced to the case of triangulations with the same input size

74 Plane graphs Theorem - Given a planar graph G on n vertices and two straight-line planar drawings Γ 0 and Γ 1 of G with the same faces and the same outer face, including the same nesting of connected components, there is a morph between Γ 0 and Γ 1 that preserves straight-line planarity and consists of O n unidirectional morphing steps. The morph can be found in time O n 3.

75 Morphing to find a compatible triangulation Assume, for semplicity, that G is connected This assumption can be removed with extra work Suppose there is a face f with 4 or more vertices We find two consecutive edges (u, v) and (v, w) of f so that (u, w) can be added to G, i.e., such that u w and (u, w) is not an edge of G

76 Looking for an edge to be inserted Suppose that f has two consecutive edges (x 1, x 2 ) and (x 2, x 3 ) such that x 2 is a cutvertex (its removal disconnects G) Then x 1 x 3 and (x 1, x 3 ) does not belong to G, hence it can be added inside f If such edges do not exist, then f is a simple cycle then consider four consecutive vertices x 1, x 2, x 3, and x 4 along f by planarity, either edge (x 1, x 3 ) or edge (x 2, x 4 ) does not belong to the G, hence it can be added inside f

77 Looking for an edge to be inserted At this point we have two consecutive edges (u, v) and (v, w) of f so that (u, w) can be added to G preserving simplicity and (topological) planarity w v u

78 Making room for the edge We morph Γ 0 and Γ 1 so that (u, w) can be added as a straight-line segment preserving planarity; the argument is the same for both drawings w v u

79 Making room for the edge Add a new vertex r and edges (r, v), (r, u), and (r, w) placing r close enough to v so that the resulting drawing is straight-line planar w w v r v u u

80 Making room for the edge We morph the resulting drawing to make the quadrilateral urwv convex To do this, we temporarily triangulate the drawing and apply Convexification to urwv w w w w v r v r v v u u u u

81 It might happen. It might happen that when we triangulate the drawing, we add the edge (u, w), which would make it impossible to convexify the quadrilateral urwv In this case, we remove (u, w) and retriangulate the resulting quadrilateral by adding a new vertex p and adding straight-line edges from p to the four vertices of the quadrilateral

82 Dealing with an internal edge If (u, w) was an internal edge of the triangulation, then p can be placed at any internal point of the segment u, w w w r v p r v u u

83 Dealing with an external edge If (u, w) was an external edge of the triangulation, then p can be placed outside w p w v r v r u u

84 Pseudomorph and morph The algorithms described so far allow a vertex v to become coincident with another vertex, so they do not build a true morph, but a pseudomorph How to convert a pseudomorph to a morph? Cairns (and Thomassen) solved this issue by keeping a contracted vertex v at the centroid of its surrounding polygon, but this results in a non-linear motion for v

85 From pseudomorph to morph Theorem - Let Γ 0 and Γ 1 be two triangulations that are topologically equivalent drawings of an n-vertex maximal planar graph G. Suppose that there is a pseudomorph from Γ 0 to Γ 1 in which we contract an internal vertex v of degree at most 5, perform k unidirectional morphs, and then uncontract v. There is a morph M from Γ 0 to Γ 1 that consists of k + 2 unidirectional morphs. Furthermore, given the sequence of k + 1 drawings that define the pseudomorph, we can modify them to obtain the sequence of drawings that define M in O(k + n) time.

86 Case analysis Consider polygon P formed by the neighbors of v contracted to a Three cases, depending on deg v that can be 3, 4, or 5 If P is a triangle then we can place v at a fixed convex combination of the triangle vertices in all the drawings If P is a quadrilateral abcd then the segment ac is in the kernel of P because vertex a is in the kernel of P we can place v at a fixed convex combination of a and c in all the drawings The coordinates of v in each drawing can be computed in constant time the total time bound is O(k)

87 Case analysis If P is a pentagon we place v very close to a v is placed at distance ε to a, with ε small enough so that at any time instant during morph Γ 1,., Γ k+1 the intersection between disk D centered at a with radius ε and the kernel of P is a nonempty sector of D Since the morph consists of k linear morphs, we can compute a value for ε as follows For 1 i k let ε i be the minimum distance from a to any of (b, c); (c, d); and (d, e) during the unidirectional morph Γ i,γ i+1 ε i can be computed in constant time on a real RAM model of computation ε is set to the smallest ε i

88 Can we do better? Is it possible to compute morphs with less than linear morphing steps?

89 A linear lower bound Theorem - There exist two straight-line planar drawings of an n-vertex path such that any straight-line planarity preserving morph between them that consists of k linear morphing steps is such that k Ω(n)

90 A linear lower bound Edges are oriented to clarify the drawing v 3 e 3 v 6 e 1 e 2 e 3 e 4 e 5 e 6 v 7 e 6 e 5 e 2 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 4 v 5 e 4 e 1 v 1 v 2

91 Convex faces If the faces are convex polygons in both Γ 0 and Γ 1 can the morph preserve this? Yes, proved in Angelini, Da Lozzo, Frati, Lubiw, Patrignani, Roselli, Optimal Morphs of Convex Drawings, SoCG 2015 And if the convexity is strict, then it is preserved in all steps

92 More morphing Tutte-based

93 Tutte s Algorithm Algorithm by Tutte (1963) for building a straight-line drawing of a 3-connected plane graph draw the outer face as a convex polygon and put each interior vertex at the barycenter of its neighbors' positions The position of the interior vertices is found as the solution of a system of linear equations The drawing is always planar and the faces are convex polygons

94 Floater, Gotsman, and Surazhsky Floater and Gotsman (1999) gave a way to morph straightline planar triangulations based on Tutte's graph drawing algorithm Gotsman and Surazhsky (1999, 2001, 2003) extended to all straight-line planar graph drawings using the idea of compatible triangulations

95 Floater, Gotsman, and Surazhsky These algorithms do not produce explicit vertex trajectories They compute the intermediate drawing (a snapshot ) at any requested time point There are no guarantees about the number of time points required to approximate continuous motion while preserving planarity

96 More morphing Schnyder-based

97 A resolution open problem The algorithms described above use a real RAM the intermediate drawings produced by our morph may have an exponential ratio of the distances between the closest and farthest pairs of vertices If the vertices in Γ 0 and Γ 1 have integer coordinates can we build a morph so that each intermediate drawing has integer coordinates and limited area?

98 Integer morphs for Schnyder drawings A Schnyder (1990) planar drawing of a triangulation G is built by giving a positive integer weight to all the faces of G, by assigning to each vertex v a certain 3-partition of the faces of G, by using the weights of the 3 sets of the 3-partition to assign to v the coordinates in a 3D drawing, and by projecting such drawing on the plane The algorithm by Barrera-Cruz, Haxell, and Lubiw (2014) solves the problem when Γ 0 and Γ 1 are Schnyder drawings The general problem is still open

99 More morphing Orthogonal drawings

100 Basic definitions Orthogonal drawing A drawing of a graph is orthogonal if the curves representing the edges are a sequence of alternately horizontal and vertical line segments

101 Morphing planar orthogonal drawings Biedl, Lubiw, Petrick, and Spriggs (2006, 2013) gave an algorithm to morph between two planar orthogonal drawings of a graph, preserving planarity and orthogonality Quadratic number of linear morphing steps The algorithm is also able to maintain edge directions Is it possible to achieve a linear bound? van Goethem and Verbeek; announced at SoCG 2018

102 More morphing Topological morphing

103 Two different embeddings Suppose that Γ 0 and Γ 1 are planar but have different embeddings, then it is not possible to morph between them without crossings How to morph, with a limited number of elementary changes? Studied by Angelini, Cortese, and gdb (2008, 2013) for biconnected planar graphs

104 Flip and skip flip skip

105 A sequence of three skip operations moving the external face to the face marked by a diamond

106 A sequence of three skip operations moving the external face to the face marked by a diamond

107 A sequence of three skip operations moving the external face to the face marked by a diamond

108 A sequence of flips and skips transforming a planar embedding

109 A sequence of flips and skips transforming a planar embedding

110 A sequence of flips and skips transforming a planar embedding

111 A complexity theorem Theorem - Let G be a biconnected planar graph and let Γ 0 and Γ 1 be two planar embeddings of G. The problem of computing the minimum number of flip/skip that transform Γ 0 into Γ 1 is NP-complete.

112 Simpler versions The problem becomes polynomial-time solvable if only skips are allowed if no separation pair disconnects more than two triconnected components Open general connected case other meaningful topology transformation operations

113 Conclusions

114 An intriguing field Several available methodologies recursive contractions convexification pseudomorphing and morphing Tutte based Schnyder based

115 An intriguing field Theory vs practice most of the theory-based techniques require the production of drawings with exponential precision the techniques used in practice do not give any guarantee on the result

116 Open Problems

117 Open The Graph Drawing and the Computational Geometry communities defined many ways to draw graphs, for each of them a morphing problem can be stated and studied

118 Morphing Planar Graph Drawings Giuseppe Di Battista Università degli Studi Roma Tre The 12th International Conference and Workshops on Algorithms and Computation WALCOM 2018

119 Extras Applications Many areas of Computer Science Computer Graphics Effects in motion pictures and animations Graph visualization When the graph changes over time Linkage reconfiguration

120 Extras More for From pseudomorph to morph Suppose that the given pseudomorph consists of the contraction of an internal vertex v with deg v 5 to a vertex a, followed by a morph M = Γ 1,., Γ k+1 of the reduced graph, and then the uncontraction of v from a Suppose that each Γ i, Γ i+1 is an unidirectional morph It is possible to update the sequence of drawings of M to those of M in time O(n + k)

121 From pseudomorph to morph We show how to add v and its incident edges back into each drawing of M keeping each step unidirectional Planarity is preserved by placing v at an interior point of the kernel of the polygon (v) formed by the neighbors of v Call the resulting morph M We modify M into M in time O(k)

122 From pseudomorph to morph To obtain the final morph M, we replace the original contraction of v to a by a unidirectional morph that moves v from its initial position to its position at the start of M, then follow the steps of M, and then replace the uncontraction of v by a unidirectional morph that moves v from its position at the end of M to its final position The result is a true morph that consists of k + 2 unidirectional morphing steps It takes O(n) time to add the two extra morphs to the sequence, since we must add 2 drawings of an n-vertex graph

123 From pseudomorph to morph Now the problem is to modify the morph M by adding vertex v and its incident edges back into each drawing of the morph sequence in constant time per drawing, preserving planarity and maintaining the property that each morphing step of the sequence is unidirectional Everything outside the polygon P = (v) can be ignored Note that, as vertex a is adjacent to all the vertices of P, it remains in the kernel of P throughout M

124 Extras - More on the lower bound Rotation Let e i j be the drawing of e i = (v i, v j ) in Γ j Define the rotation ρ i j of e i around v i during morphing step Γ j, Γ j+1 assume that v i does not move ρ i j is the angle between e i j and e i j+1 v i e i j e i j+1 v i e i j+1 v i ρ i j e i j

125 A rotation lemma Lemma for each i = 1,., n and j = 1,., k we have ρ j i < π