1. Computing NodeTrix Representations of Clustered Graphs
039-dht 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

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

3. Crossings in NodeTrix Drawings
Demo available at

4. 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]

5. 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

6. 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

7. A NodeTrix Representation

8. 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

9. 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

10. 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

11. 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

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

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

14. 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

15. 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

16. 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)

17. 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

18. 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

19. 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

20. 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

21. 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:

22. 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??

23. 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?

24. Thanks!