Beyond Level Planarity 24th International Symposium on Graph Drawing - PowerPoint PPT Presentation

Beyond Level Planarity 24th International Symposium on Graph Drawing & Network Visualization 19–21 September, Athens, Greece Patrizio Angelini, Giordano Da Lozzo , Fabrizio Frati, Giuseppe Di Battista, Maurizio Patrignani, Ignaz Rutter UR I NFORMATIK · T ¨ W ILHELM -S CHICKHARD -I NSTITUT F ¨ UBINGEN U NIVERSITY D EPARTMENT OF E NGINEERING · R OMA T RE U NIVERSITY F ACULTY OF I NFORMATICS · K ARLSRUHE I NSTITUTE OF T ECHNOLOGY GD ’16 – Beyond Level Planarity GD ’16 – Beyond Level Planarity GD ’16 – Beyond Level Planarity

Outline • Level Embeddings on Surfaces Problems: ◦ Cyclic and Torus Level Planarity Extensions of Consecutivity constraints: Level Planarity ◦ Radial, Cyclic, and Torus T -Level Planarity • Simultaneous Level Planarity GD ’16 – Beyond Level Planarity

Level Embeddings on Surfaces ( C YCLIC AND T ORUS L EVEL P LANARITY ) “in order to enlarge the class of level graphs that allow for a level embedding (level drawing with no crossings), the notion of Level Planarity has been extended to surfaces different from the plane” GD ’16 – Beyond Level Planarity

Levels on the PLANE ℓ k ℓ 4 ℓ 3 = I × { 3 } ℓ 3 Level Planarity ℓ 2 ℓ 1 P = I × I ( V , E , γ ), γ : V → { 1, 2, ..., k } edges ( u , v ) with γ ( u ) < γ ( v ) are not allowed • Di Battista, G., Nardelli, E.: Hierarchies and planarity theory. IEEE Trans. Systems, Man, and Cybernetics, 1988. • J¨ unger,M.,Leipert,S.,Mutzel,P .: Level planarity testing in linear time. GD, 1998 . GD ’16 – Beyond Level Planarity

Levels on the SPHERE/STANDING CYLINDER axis ℓ k = ℓ 4 ℓ 3 S 2 Radial Level Planarity ℓ 2 2 ℓ 3 = S 1 × k − 1 ℓ 1 S = I × S 1 ( V , E , γ ), γ : V → { 1, 2, ..., k } edges ( u , v ) with γ ( u ) < γ ( v ) are not allowed (we can now draw edges that “wrap around” the cylinder axis) • Bachmaier, C., Brandenburg, F ., Forster, M.: Radial level planarity testing and embedding in linear time . JGAA, 2005. • Fulek, R., Pelsmajer, M., Schaefer, M.: Hanani-Tutte for Radial Planarity II . GD ’16 GD ’16 – Beyond Level Planarity

Levels on the ROLLING CYLINDER ℓ 3 = I × { e 2 π i 2 k } ℓ 3 ℓ 2 ℓ 1 axis ℓ k Cyclic Level Planarity e 2 π i 2 k R = S 1 × I ( V , E , γ ), γ : V → { 1, 2, ..., k } edges ( u , v ) with γ ( u ) < γ ( v ) are allowed • Bachmaier, C., Brunner, W.: Linear time planarity testing and embedding of strongly connected cyclic level graphs. ESA, 2008. • General instances? Bachmaier, Brunner, and K¨ onig [GD’07] claimed that an O ( | V | 6 ) -time algorithm for Cyclic LP can be obtained from the LP testing algorithm by Healy and Kuusik GD ’16 – Beyond Level Planarity

Levels on the ℓ 3 = S 1 × { e 2 π i 2 k } axis e 2 π i 2 k Torus Level ℓ 3 Planarity ℓ 2 ℓ 1 ℓ k T = S 1 × S 1 ( V , E , γ ), γ : V → { 1, 2, ..., k } edges ( u , v ) with γ ( u ) < γ ( v ) are allowed • Bachmaier, C., Brunner, W., K¨ ? onig, C.: Cyclic Level Planarity Testing and Embedding. GD ’07. • Brunner, W.: Cyclic Level Drawings of Directed Graphs. PhD thesis, 2010. open • Hammersen, K.: A Characterization of Radial Graphs. Deutschen problem in: Nationalbibliothek, 2013. GD ’16 – Beyond Level Planarity

Level Planarity Variants Lemma C YCLIC AND R ADIAL L EVEL P LANARITY ≤ L T ORUS L EVEL P LANARITY Level Cyclic Level Radial Level Torus Level ℓ 2 + = ℓ 1 A (planar) level graph that is neither cyclic nor radial level planar , yet it is torus level planar GD ’16 – Beyond Level Planarity

Simultaneous PQ-Ordering: basic concepts 1/2 Lemma Proper Torus Level Planarity ≤ L Simultaneous PQ-Ordering • Orders and Suborders . . . . . . ◦ finite sets A ( ) and S ( ) ◦ injective map φ : S → A ◦ order O on A ◦ order O S on S order O extends order O S when: O ′ = O S = O = a 1 , . . . , a | A | s 1 , . . . , s | S | φ ( s 1 ), . . . , ψ ( s | S | ) O ′ is the O ′ is the image of O S restriction of O on φ [ S ] ⊆ A via φ GD ’16 – Beyond Level Planarity

Simultaneous PQ-Ordering: basic concepts 2/2 • PQ-representable orders : (circular) PQ-trees represent (circular) orders of their leaves with consecutivity constraints ◦ two versions: rooted [Booth&Lueker, ’76] ; unrooted: [Hsu&McConnell, ’01] ◦ two types of internal nodes PQ-tree T with L eaves ( T ) = A P-nodes : permutations Q-nodes : flips GD ’16 – Beyond Level Planarity

Simultaneous PQ-Ordering: problem definition • input: a DAG = ( N , Z ) T 4 T 3 T 7 ◦ each node T i ∈ N is a PQ-tree − → T 1 T 5 T 6 ◦ each arc T i T j ∈ Z is equipped with an injective map T 8 T 2 φ : L eaves ( T j ) → L eaves ( T i ) T 1 arc ( T 1 , T 2 , φ ) is satisfied arc by orders O 1 of T 1 and O 2 of T 2 if ( T 1 , T 2 , φ ) φ O 1 extends O 2 T 2 source target • question: do there exist orders O i for each PQ-tree T i ∈ N that simultaneously satisfy all the arcs in Z ? GD ’16 – Beyond Level Planarity

From k levels to 2 levels ℓ 2 } ℓ 1 ℓ 2 ℓ 1 Torus Level Embedding of Radial Level Embedding of G = ( � k i =1 V i , E , γ ) G 1,2 = ( V i ∪ V i +1 , ( V i × V i +1 ) ∩ E , γ ) Observation: A proper level graph G = ( � k i =1 V i , E , γ ) has a torus level embedding with orders O 1 , . . . , O k on V 1 , . . . , V k along ℓ 1 , . . . , ℓ k if and only if ∃ radial level embedding of level graphs ( V i ∪ V i +1 , ( V i × V i +1 ) ∩ E , γ ) on two levels with orders O 1 ( O i +1 ) on V i ( V i +1 ) along ℓ i ( ℓ i +1 ) GD ’16 – Beyond Level Planarity

Orders in Radial Level Embeddings:vertex ordering Level graph G 1,2 = ( V 1 ∪ V 2 , E 1,2 = E ∩ V 1 × V 2 , γ ) between ℓ 1 and ℓ 2 Radial Level ℓ 2 Embedding Γ ℓ 1 Orders along ℓ 1 Orders along ℓ 2 O − 2 on V − O + 1 on V + O 1 on V 1 O 2 on V 2 1 2 GD ’16 – Beyond Level Planarity

Orders in Radial Level Embeddings: edge ordering Level graph G 1,2 = ( V 1 ∪ V 2 , E 1,2 = E ∩ V 1 × V 2 , γ ) between ℓ 1 and ℓ 2 edge ordering on E 1,2 in Γ circular order in which the edges inter- is sect curve C C ℓ 2 vertex-consecutive order circular order O on E 1,2 s.t. ∀ v ∈ V 1 ∪ ℓ 1 V 2 the edges incident to v are consecutive in O Radial Level Embedding Γ Observation: vertex-consecutive orders (and hence edge orders ) are PQ-representable GD ’16 – Beyond Level Planarity

Radial Level Planarity on 2 Levels Level graph G 1,2 = ( V 1 ∪ V 2 , E 1,2 = E ∩ V 1 × V 2 , γ ) between ℓ 1 and ℓ 2 C ℓ 2 ℓ 2 ℓ 1 ℓ 1 O i := circular order on V i O := circular order on E Lemma ∃ RLE of G 1,2 with edge ordering O in which O 1 and O 2 are the orders on V 1 and V 2 if and only if • order O is vertex-consecutive • orders O 1 and O 2 extend the orders O + 1 and O − 1 and V − 2 on V + induced by O 2 GD ’16 – Beyond Level Planarity

Simultaneous PQ-Ordering: from 2 to k levels layer tree T 1,2 represents exactly the (vertex-consecutive) edge orderings level tree T 1 := level tree T 2 := T 1,2 T 2 T 1 univ. PQ-tree on V 1 univ. PQ-tree on V 2 φ − φ + ι ι 1 2 T − T + 1 2 Instance I ( G 1,2 ) consistency tree T − 2 := consistency tree T + 1 := univ. PQ-tree on V − univ. PQ-tree on V + 2 levels 2 Radial 1 Torus k levels GD ’16 – Beyond Level Planarity

Simultaneous PQ-Ordering: from 2 to k levels layer tree T 1,2 represents exactly the (vertex-consecutive) edge orderings level tree T 1 := level tree T 2 := T 1,2 T 2 T 1 univ. PQ-tree on V 1 univ. PQ-tree on V 2 φ − φ + ι ι 1 2 T − T + 1 2 Instance I ( G 1,2 ) consistency tree T − 2 := consistency tree T + 1 := univ. PQ-tree on V − univ. PQ-tree on V + 2 levels 2 Radial 1 Torus Instance I ( G ) = � k k levels i =1 I ( G i , i +1 ) (where k+1=1) T 1,2 T i − 1, i T i , i +1 T i +1 T i +1, i +2 T k − 1, k T k ,1 T 1 T i T k φ − φ − φ − φ + φ + φ + φ + ι ι ι ι ι ι ι ι . . . . . . . . . . . . 1 i i +1 k i i +1 k T − T − T − T − T + T + T + T + 1 i i i +1 i +1 k 1 k I ( G i , i +1 ) I ( G ) can be tested in P-time ... I ( G ) can be tested in P-time ... GD ’16 – Beyond Level Planarity

Simultaneous PQ-Ordering: Fixedness P-node µ ′ in T ′ (parent) is fixed by node µ in T (child) if T ′ φ ( a ) there exist vertex-disjoint paths: φ ( c ) 1. a → µ , b → µ , c → µ in T and µ ′ 2. φ ( a ) → µ ′ , φ ( b ) → µ ′ , φ ( c ) → µ ′ in T ′ T ′ T ′ T ′ . . . φ ( b ) r 1 2 T P-node φ µ ∈ T T a c T 1 , T 2 , . . . , T ω , T ω +1 , . . . , T k { { µ fixing µ not fixing µ • some nodes ( ω ) in a children might fix a node in b the parent • normalization : we can assume that all nodes in a children fix exactly a node in the parent GD ’16 – Beyond Level Planarity

Simultaneous PQ-Ordering: Fixedness Ti − 1, i Ti , i +1 Ti +1 Ti +1, i +2 Ti . . . . . . . . . . . . T − T − T + T + i i i +1 i +1 fixedness fixed ( µ ) = ω + � r i =1 ( fixed ( µ i ) − 1) P-node µ i ∈ T ′ children fixig µ i fixed by µ source PQ-trees sink PQ-trees (layer and level trees) (consistency trees) ω = 2 and r = 0 ω = 0 and r = 2 2-fixed!! Th. 3.2,3.3 [Bl¨ asius & Rutter, SODA ’13] Sim. PQ-Ordering is solvable in quadratic 2-fixed!! 2-fixed!! time for fixed ( µ ) ≤ 2 instances. Theorem Torus Level Planarity ( Cyclic Level Planarity ) can be decided in O ( | V | 2 ) time for proper level graphs ( O ( | V | 4 ) time for general level graphs) GD ’16 – Beyond Level Planarity