Computing NodeTrix Representations of Clustered Graphs 039-dht-06537-dht-03.ppt 537-dht-03.ppt Computing NodeTrix Representations of Clustered Graphs Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani Roma Tre University 1

NodeTrix Hybrid Representations NodeTrix combines node-link and matrix-based representations [Henry, Fekete, McGuffin, IEEE TVCG, 2007]

Crossings in NodeTrix Drawings Demo available at http://www.aviz.fr/Research/Nodetrix

NodeTrix Literature In order to reduce crossings and improve readability vertices may be allowed to have duplicates in different matrices [Henry, Bezerianos, Fekete, IEEE TVCG, 2008] clusters can be automatically computed so to have dense intra-cluster graphs and a planar inter-cluster graph [Batagelj, Brandenburg, Didimo, Liotta, Palladino, Patrignani, IEEE TVCG 2011]

Flat Clustered Graphs A flat clustered graph (V,E,C) is a graph (V,E) with a partition C of V into sets V1, …, Vk, called clusters An edge (u,v)∈E with u∈Vi and v∈Vj is an intra-cluster edge if i=j an inter-cluster edge if ij 12 1 11 2 10 13 3 9 intra-cluster edge 8 4 inter-cluster edge 6 5 7

NodeTrix Representations In a NodeTrix representation of a flat clustered graph (V,E,C) clusters V1, …, Vk are represented by non-overlapping symmetric adjacency matrices M1, …, Mk matrices M1, …,Mk convey the information about the intra-cluster edges of (V,E,C) each inter-cluster edge (u,v) with u∈Vi and v∈Vj is represented by a curve connecting a point on the border of Mi with a point on the border of Mj such points belong to the column or to the row of Mi and Mj associated with u and v

A NodeTrix Representation

NodeTrix Planarity A NodeTrix representation is planar if no inter-cluster edge e intersects any matrix Mi, except at an end-point of e no pair of inter-cluster edges cross each other, except possibly at a common end-point

Fixed Row/Column Order Complexity Results Complexity of deciding planarity for NodeTrix representations Free sides Fixed sides Free Row/Column Order NP-Complete Fixed Row/Column Order Linear

Fixed Order & Fixed Side Complexity Theorem NodeTrix Planarity with Fixed Order and Fixed Side can be solved in linear time reducible to constrained planarity solvable in linear time with known techniques [Gutwenger, Klein, Mutzel, JGAA 2008] M1 M2 M3 v1 v3 v2

Fixed Side Complexity Theorem Proof NodeTrix Planarity with Fixed Side is NP-complete even for instances with two clusters Proof reduction from Betweenness an instance is a collection of m ordered triplets of items {(a1,b1,c1), (a2,b2,c2),… , (am,bm,cm)} the target is to find a total order of the n items in which, for each of the given triplets, the middle item in the triplet appears somewhere between the other two items

NodeTrix Planarity with Fixed Sides 1 2 M1 M2 3

A More Practical Scenario The user places the matrices Inter-cluster edges have to be drawn in the convex hull of their incident matrices

Monotone NodeTrix Representations A monotone NodeTrix representation is a NodeTrix representation in which the matrices have prescribed positions the inter-cluster edges are represented by xy-monotone curves inside the convex hull of their incident matrices we require that this convex hull does not intersect any other matrix

Monotone Representations & Planarity A monotone NodeTrix representation is locally planar if no pair of inter-cluster edges attached to the same matrix cross allowed crossing forbidden crossing forbidden crossing

Local Planarity Complexity Results Complexity of deciding local planarity for monotone NodeTrix representations Free sides Fixed sides Free Row/Column Order NP-Complete Fixed Row/Column Order Polynomial (if the number of clusters is constant)

Monotone Fixed Order & Fixed Side Theorem Monotone NodeTrix Local Planarity with Fixed Order and Fixed Side can be solved in polynomial time Proof first, we prove that the instance is locally planar if and only if it admits a locally planar straight-line drawing second, we check such drawing for planarity

Monotone Fixed Order & Free Side Theorem Monotone NodeTrix Local Planarity with Fixed Order can be tested in |E|O(|C|2) time, where |C| is the number of clusters S-drawn edges Proof for each pair of adjacent clusters we guess one inter-cluster edge that could be S-drawn (if any) we construct a boolean 2SAT formula to describe feasible choices for the sides

Intuition of the Proof Let e be an S-drawn edge Any other edge admits at most two alternative drawings M1 M2 M1 M2 e e M1 M2 M1 M2 e e

An (unfeasible) polynomial heuristics For each pair of adjacent clusters guess one possible S-drawn edge Construct one instance of 2SAT for each of the |E|O(|C|2) guesses If one of the formulas admits a solution use it to draw the edges Otherwise, search for a solution to a MAX2SAT instance with some heuristics each false clause will correspond to a crossing

A more practical approach Forbid S-drawn edges altogether construct a single 2SAT formula that is satisfiable if and only if the edges can be drawn planarly otherwise, search for a solution of MAX2SAT with a greedy approach Demo available at: http://www.dia.uniroma3.it/~dalozzo/projects/matrix

Equivalent to “Bipartite Book-Embedding with Spine Crossings” Open Problems Monotone NodeTrix Planarity with Free Order and Free Side the case of two clusters Equivalent to “Bipartite Book-Embedding with Spine Crossings” What complexity??

How could we model this problem? Open Problems Monotone NodeTrix Planarity with Free Order and Free Side the case of two clusters How could we model this problem?

Thanks!