1. Rectangle Visibility Graphs: Characterization, Construction, Compaction Ileana Streinu (Smith) Sue Whitesides (McGill U.)

2. Rectangle Visibility Graphs We study horizontal and vertical visibilities of non- overlapping, axis aligned rectangles in 2D. A B C D A B C D a) a set of rectanglesb) their visibility graph Rectangles are open; visibility lines are “thick” but may have 0 length.

3. Rectangle Visibility Graph Recognition Problem given: graph G = (V,E) question: Can G be realized as the visibility graph of rectangles? A B C D A B C D ? NP-complete (Shermer)

4. Missing Information … horizontal vs. vertical visibility direction info. (e.g, north, east) A B D.. B sees A vertically, to the north; B sees D horizontally, to the east

5. … missing information … multiple edges A B C D A sees C on both sides of B. ….

6. …. missing information … the cyclic ordering of visibilities around a rectangle A B C D Traversing the boundary of B clockwise, starting at the upper left corner, the bug sees A to the north, then D to the east, then C to the south. ….

7. …. missing information which rectangles can see infinitely far in some direction A B C D e.g., B sees infinitely far to the west; D sees infinitely far in 4 directions.

8. A Frame for the Rectangles A B C D N S E W We add 4 new rectangles to capture seeing infinitely far.

9. New Notion of Visibility Graph We redefine the visibility graph of a set of rectangles to capture the missing topological information. Given a set of rectangles, we frame the set define a graph D V which captures vertical visibilities define a graph D V which captures vertical visibilities define a graph D H which captures horizontal visibilities define a graph D H which captures horizontal visibilities as follows …

10. The Vertical Visibility Graph D V of a set of given rectangles A B C D N S s,t graph : D V is an s,t graph : planar DAG embedded 2-connected underlying graph single source, single sink source and sink on outside face

11. The Horizontal Visibility Graph D H of a given set of rectangles E W A B C D s,t graph D H is an s,t graph

12. Topological Rectangle Visibility Graphs ( TRVG s) Definition: A (pseudo) topological rectangle visibility graph is a pair ( D V, D H ) of s,t graphs ; i.e., it is a combinatorial, topological structure that might arise from the visibilities of some set of rectangles. Notation: A set R of rectangles gives rise to a topological rectangle visibility graph, denoted ( D V ( ), D H ( ) ). R R

13. New Problem: Given a topological rectangle visibility graph ( D V, D H ), does there exist a set of rectangles R such that ( D V, D H ) = ( D V ( ), D H ( ) ) ? RR This is the TRVG Recognition problem.

14. Our Results a combinatorial, topological characterization of TRVG’s a polynomial time algorithm that tests whether a (pseudo) TRVG arises from some set or rectangles, and if so, constructs such a set In fact, if rectangles are required to have all corners at grid points, our construction gives a set of rectangles whose bounding box has minimum possible width and minimum possible height. R

15. Possible Application: Compaction Given a set of rectangles whose visibilities we would like to preserve, we can in linear time : compute ( D V ( ), D H ( ) ) (no need to apply the recognition algorithm) apply our construction algorithm to produce a new, optimally compact set R RR R’R’

16. Prerequisites for stating the characterization Since D V and D H are embedded, planar graphs, they have duals. notation: The dual of D V is D V * ; the dual of D H is D H * Watch out ! The dual of D V is not equal to D H. D V * = D H ; D H * = D V / / fact: The dual of an s,t graph is also an s,t graph; hence D V, D H, D V *, and D H * are all s,t graphs.

17. Another Prequisite: Fact: At any node of an s,t graph, the incoming and outgoing arcs are separated. u the right face of u the left face of u

18. Notation for the Dual Graphs u the right face of u the left face of u node L V *(u) node R V *(u) DV* DV* Arcs of D V * cross arcs of D V left to right. D H * similarly has L H *(u) and R H *(u) (but dual arcs cross primal ones right to left)

19. Our Characterization of TRVG’s Theorem. A pair ( D V, D H ) of s,t graphs can be realized as the TRVG of some set of rectangles if and only if 1) AND 2) hold: Weaves together properties of 4 distinct graphs D V, D H, D V *, D H *. u v in D V L H *(u) R H *(v) AND in D H * u v in D H R V *(u) L V *(v) in D V * 1) for all u, v in D V IF THEN 2) for all u, v in D H IF THEN

20. Intuition for the Necessity of the Conditions E A B C D W L H *( C) R H *( A) There is a path of vertical visibilities from C to A. Use the vertical visibility segments to cross the edges of D H to get a path from L H *( C ) to R H *( A ) in D H *.

21. The Construction of the Rectangle Set when ( D V, D H ) passes conditions 1) and 2) Compute D V * and D H * Assign 0 to the source node in each of the two dual graphs D V * and D H * Number all other nodes of D V * and D H * by the length of a longest path from the respective source Make a rectangle for each node u : u # L V *(u) # R V *(u) # L H *(u) # R H *(u)

22. A Tool for the Proof of Correctness of Construction Lemma (Tamassia and Tollis): In an s,t graph such as D V, for each pair of nodes u,v, exactly one of the following holds: 1.D V has a directed path from u to v OR 2.D V has a directed path from v to u OR 3.D V * has a directed path from R V * ( u ) to L V *( v ) OR 4.D V * has a directed path from R V *( v ) to L V *( u ). This lemma appeared in their characterization of bar visibility graphs. Part of our motivation was to generalize their results (obtained also by Wismath).

23. Conclusion Extensions to higher dimensions? Not by these techniques – 3D vertical visibility graphs of floating rectangles need not be planar Faster algorithm for testing conditions 1) and 2) ?

26. A B C D N S E A D B C N S E W A example at A : Horizontal visibilities at A, in E and W sectors Record all the topological visibility information. A New Notion of Visibility Graph

27. Missing Information horizontal vs. vertical visibility direction info. (e.g, east, north) multiple edges cyclic ordering of visibilities around a vertex which rectangles can see infinitely far to the north, east, south, west

28. intuition A B C D N S

29. Intuition E A W B C D N S