Rectangle Visibility Graphs:

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

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

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

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

… missing information … • multiple edges A Asees C on both sides ofB . B D C ….

…. missing information … • the cyclic ordering of visibilities around a rectangle 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. A B D C ….

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

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

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 DV which captures vertical visibilities • define a graph DH which captures horizontal visibilities as follows …

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

The Horizontal Visibility GraphDHof a given set of rectangles A W DH is ans,t graph E B D C

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

New Problem: Given a topological rectangle visibility graph ( DV , DH ) , does there exist a set of rectangles R such that ( DV , DH ) = ( DV ( ) , DH ( ) ) ? R R This is the TRVG Recognitionproblem .

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

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

Prerequisites for stating the characterization Since DV and DH are embedded, planar graphs, they have duals. notation: The dual of DV is DV* ; the dual of DH is DH* Watch out ! The dual ofDV isnot equal to DH . DV* = DH ; DH* = DV / / fact: The dual of an s,t graph is also an s,t graph; hence DV, DH , DV*, and DH* are all s,t graphs.

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

Notation for the Dual Graphs DV* u node RV*(u) the left face of u the right face of u node LV*(u) Arcs of DV* cross arcs of DV left to right. DH* similarly has LH*(u) and RH*(u) (but dual arcs cross primal ones right to left)

Our Characterization of TRVG’s Weaves together properties of 4 distinct graphs DV , DH , DV*, DH* . Theorem. A pair ( DV, DH ) of s,t graphs can be realized as the TRVG of some set of rectangles if and only if 1) AND 2) hold: 1) for all u, v in DV v RH*(v) IF in DV THEN in DH* u LH*(u) AND 2) for all u, v in DH LV*(v) v IF in DH THEN u in DV* RV*(u)

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

The Construction of the Rectangle Setwhen ( DV, DH ) passes conditions 1) and 2) • Compute DV* and DH* • Assign 0 to the source node in each of the two dual graphs DV* and DH* • Number all other nodes of DV* and DH* by the length of a longest path from the respective source • Make a rectangle for each node u : # LH*(u) u # RH*(u) # LV*(u) # RV*(u)

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

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) ?

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

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

intuition N A B D C S

Intuition N A W E B D C S

Rectangle Visibility Graphs: