Chapter 13 Rectangular Duals 13.1 Intro duction In this chapter we consider the problem of representing a graph G bya rectangu- lar dual. This is applied in the design of o or planning of electronic chips and in architectural design. A rectangular dual is de ned as follows. A rectangular subdi- vision system of a rectangle R is a partition of R into a set = fR ;R ;:::;R g 1 2 n of non-overlapping rectangles such that no four rectangles in meet at the same p oint. A rectangular dual of a planar graph G is a rectangular sub division system and a one-to-one corresp ondence R : V ! suchthattwovertices u and v are adjacentin G if and only if their corresp onding rectangles RuandRv share a common b oundary. In the application of this representation, the vertices of G repre- sent circuit mo dules and the edges represent mo dule adjacencies. A rectangular dual provides a placement of the circuit mo dules that preserves the required adjacencies. Figure 13.1 shows an example of a planar graph and its rectangular dual. This problem was rst studied by Bhasker & Sahni [6, 7] and Ko zminski & Kin- nen [73 ]. Bhasker & Sahni gave a linear time algorithm to construct rectangular duals [7]. The algorithm is fairly complicated and requires manyintriguing pro ce- dures. The co ordinates of the rectangular dual constructed by it are real numb ers and b ear no meaningful relationship to the structure of the graph. This algorithm consists of two ma jor steps: 1 constructing a so-called regular edge labeling REL of G; and 2 constructing the rectangular dual using this lab eling. A simpli cation of step 2 is given in [47]. The co ordinates of the rectangular dual constructed by the algorithm in [47] are integers and carry clear combinatorial meaning. However, step 1 still relies on the complicated algorithm in [7]. A parallel implementation of this algorithm, working in O log n log n time with O n pro cessors, is given by He [46 ]. In this pap er we present a linear time algorithm for step 1: nding a regular edge lab eling. In [71 ] another algorithm is presented. This algorithm extends the canonical ordering of triconnected planar graphs, de ned in Section 2.5 to 4- 179 180 Rectangular Duals connected triangular planar graphs. It turns out that the canonical ordering also gives a reduction of a factor 2 in the width of the visibility representation of 4- connected planar graphs. Moreover, using this ordering it is shown that a visibility representation of any planar graph can b e constructed on a grid of size at most 3 b n 1 n 1 grid. 2 This chapter is organized as follows: in Section 13.2 we present the de nition of the regular edge lab eling REL and we review the algorithm in [47] that computes a rectangular dual from a REL. In Section 13.3, we present the REL algorithm based on the canonical ordering. Section 13.4 discusses the algorithm for the visibilityrep- resentation of 4-connected planar graphs. Section 13.5 brie y outlines the visibility representation algorithm for general planar graphs. 13.2 The Rectangular Dual Algorithm Consider a plane graph H . Let u ;u ;u ;u b e four vertices on the exterior face 0 1 2 3 in counterclo ckwise order. Let P i =0; 1; 2; 3 b e the path on the exterior face i consisting of the vertices b etween u and u addition is mo d 4. We seek a rectan- i i+1 gular dual R of H suchthat u ;u ;u ;u corresp ond to the four corner rectangles H 0 1 2 3 of R and the vertices on P P ;P ;P , resp ectively corresp ond to the rectangles H 0 1 2 3 lo cated on the north west, south, east, resp ectively b oundary of R . In order to H simplify the problem, we mo dify H as follows: Add four new vertices v ;v ;v ;v . N W S E Connect v v ;v ;v , resp ectively to every vertex on P P ;P ;P , resp ectively N W S E 0 1 2 3 and add four new edges v ;v ; v ;v ; v ;v ; v ;v . Let G b e the resulting S W W N N E E S graph. It is easy to see that H has a rectangular dual R if and only if G has a H rectangular dual R with exactly four rectangles on the b oundary of R see Figure G G 13.1a and b. Let a quadrangle b e a cycle of length 4. The following theorem was proved in [6, 73 ]: Theorem 13.2.1 A planar graph G has a rectangular dual R with four rectangles on the boundary of R if and only if 1 every interior face is a triangle and the exterior face is a quadrangle; 2 G has no separating triangles. A graph satisfying the conditions in Theorem 13.2.1 is called a proper triangular planar PTP graph. From nowon,we will discuss only such graphs. Note that condition 2 of Theorem 13.2.1 implies that G is 4-connected. Since G has no separating triangles, the degree of anyinterior vertex v of G is at least 4. If degv = 3, then the triangle induced on the neighb ors of v would b e a separating triangle. The rectangular dual algorithm in [47] heavily dep ends on the concept of regular edge labeling REL de ned as follows [7, 47 ]: De nition 13.2.1 Aregular edge labeling of a PTP graph G isapartition of the interior edges of G into two subsets T ;T of directededges such that: 1 2 13.2 The Rectangular Dual Algorithm 181 vn vn a g j a g f d j f v b v d b v w ve w e i e e c h i c h v s v s a The initial graph G. b A rectangular dual of G. v 6 n n* vn 5 j a g e* j a g f 4 5 w* f d b d v 2 w 3 ve v 1 b 6 2 w 4 ve 2 3 0 i i 5 e c h e c h 7 1 v v s s* s 0 c The graph G . d The graph G . 1 2 Figure 13.1: A PTP graph, its rectangular dual, and the st-graphs G and G 1 2 1. For each interior vertex v , the edges incident to v appear in counterclockwise order around v as fol lows: a set of edges in T leaving v ;asetofedges in T 1 2 entering v ; a set of edges in T entering v ;asetofedges in T leaving v . 1 2 2. Let v ;v ;v ;v be the four exterior vertices in counterclockwise order. All N W S E interior edges incident to v arein T and entering v . Al l interior edges N 1 N incident to v arein T and leaving v .Al l interior edges incident to v are W 2 W S in T and leaving v .Al l interior edges incident to v areinT and entering 1 S E 2 v . E The regular edge lab eling is closely related to planar st-graphs, describ ed in Section 9.3. Let G b e a PTP graph and fT ;T g b e a REL of G.From fT ;T g we construct 1 2 1 2 two planar st-graphs as follows. Let G b e the graph consisting of the edges of T 1 1 182 Rectangular Duals high(F) above(v) left(e)e right(e) left(v)v right(v) v F below(v) GG12 low(F) Figure 13.2: Prop erties of planar st-graphs. plus the four exterior edges directed as v ! v , v ! v , v ! v , v ! v , S W W N S E E N and a new edge v ;v . G is a planar st-graph with source v and sink v .For S N 1 S N eachvertex v , the face of G that separates the incoming edges of v from the outgoing 1 edges of v in the clo ckwise direction is denoted by leftv . The other face of G that 1 separates the incoming and the outgoing edges of v is denoted by rightv . See Figure 13.2. Let G b e the graph consisting of the edges of T plus the four exterior edges 2 2 directed as v ! v , v ! v , v ! v , v ! v , and a new edge v ;v . G W S S E W N N E W E 2 is a planar st-graph with source v and sink v .For eachvertex v , the face of G W E 2 that separates the incoming edges of v from the outgoing edges of v in the clo ckwise direction is denoted by abovev . The other face of G that separates the incoming 2 and the outgoing edges of v is denoted by belowv . See Figure 13.2. The dual graph G of G is de ned as follows. Every face F of G is a no de 1 k 1 1 v in G , and there exists an edge v ;v inG if and only if F and F share F F F i k i k 1 k 1 a common edge in G .We direct the edges of G as follows: if F and F are the 1 l r 1 left and the right face of an edge v; wof G , direct the dual edge from F to F 1 l r is a planar st-graph if v; w 6=v ;v and from F to F if v; w=v ;v .