Lower Bounds on the Area Requirements of Series-Parallel Graphs

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1 Discrete Mthemtics nd Theoreticl Computer Science DMTCS vol. 12:5, 2010, Lower Bounds on the Are Requirements of Series-Prllel Grphs Frizio Frti Università Rom Tre, Itly, nd École Polytechnique Fédérle de Lusnne, Switzerlnd received 2 nd Decemer 2009, revised 8 th Octoer 2010, ccepted 9 th Decemer We show tht there exist series-prllel grphs requiring Ω(n2 log n ) re in ny stright-line or poly-line grid drwing. Such result is chieved in two steps. First, we show tht, in ny stright-line or poly-line drwing of K 2,n, one side of the ounding ox hs length Ω(n), thus nswering two questions posed y Biedl et l. Second, we show fmily of series-prllel grphs requiring Ω(2 log n ) width nd Ω(2 log n ) height in ny stright-line or poly-line grid drwing. Comining the two results, the Ω(n2 log n ) re lower ound is chieved. Keywords: plnr grphs, series-prllel grphs, re requirements, grph drwing, stright-line drwings 1 Introduction A plnr grph is grph tht cn e drwn in the plne so tht no two edges intersect, except, possily, t their common endpoints. Determining symptotic ounds for the re requirements of stright-line nd poly-line drwings of plnr grphs is one of the clssicl topics in the Grph Drwing literture. Groundreking works t the eginning of the nineties y de Frysseix et l. [dpp90] nd y Schnyder [Sch90] hve shown tht every n-vertex plnr grph dmits plnr stright-line drwing on n O(n) O(n) grid. Such ound is worst-cse optiml, even for poly-line drwings [DLT84, dpp90]. Hence, it is nturl to serch for interesting su-clsses of plnr grphs dmitting su-qudrtic re drwings. It turns out tht severl importnt su-clsses of plnr grphs contin grphs requiring qudrtic re in ny grid drwing. Every four-connected plne grph whose outer fce hs t lest four vertices dmits stright-line drwing in n 2 ( n 2 1) re, s shown y Miur et l. in [MNN01], improving upon previous results y He [He97]. Miur et l. lso oserve tht such ound is tight, s shown y the grph in Fig. 1 (). Emil: frti@di.unirom3.it. This work is prtilly supported y the Itlin Ministry of Reserch, Grnt numer RBIP06BZW8, FIRB project Advnced trcking system in intermodl freight trnsporttion nd y the Swiss Ntionl Science Foundtion, Grnt No / c 2010 Discrete Mthemtics nd Theoreticl Computer Science (DMTCS), Nncy, Frnce

2 140 Frizio Frti () () (c) Fig. 1: () A four-connected plne grph requiring n 2 ( n 2 1) re in ny poly-line drwing. () A plne grph with mximum degree three requiring qudrtic re in ny poly-line drwing. (c) A plne grph with outerplnrity two requiring qudrtic re in ny poly-line drwing. Every iprtite plne grph dmits stright-line drwing in n 2 ( n 2 1) re, s shown y Biedl nd Brndenurg in [BB05]. The upper ound of Biedl nd Brndenurg is tight, since iprtite plne grphs exist, very similr to the one shown y Miur et l. [MNN01], requiring n 2 ( n 2 1) re in ny poly-line/stright-line drwing. Plnr grphs of mximum degree three exist requiring qudrtic re in ny poly-line/stright-line grid drwing, s shown in Fig. 1 (). Plnr grphs with outerplnrity two exist requiring qudrtic re in ny poly-line/stright-line grid drwing, s shown y the grph in Fig. 1 (c), tht hs een presented y Biedl in [Bie05]. Plnr grphs re the grphs excluding K 5 nd K 3,3 s minors [Wg37]. Which re the clsses of grphs excluding grphs smller thn K 5 nd K 3,3 s minors? The nswer to the previous question is list of some of the most studied su-clsses of plnr grphs. In fct, trees re the grphs excluding K 3 s minor, outerplnr grphs re the grphs excluding K 4 nd K 2,3 s minors, nd series-prllel grphs re the grphs excluding K 4 s minor. Such grph clsses, prt from hving nice chrcteriztions in terms of excluded minors, prt from hving nice lterntive chrcteriztions ( tree is connected cyclic grph, n outerplnr grph is grph tht dmits plnr emedding in which ll the vertices re incident to the sme fce, nd series-prllel grph is grph tht cn e inductively defined y series nd prllel compositions of smller series-prllel grphs), nd prt from eing of rel interest for pplictions, do dmit grid drwings in su-qudrtic re. Concerning trees, slight modifiction of the h-v drwing lgorithm y Crescenzi et l. [CDP92] constructs drwings in O(n log n) re. Optiml O(n) re ounds re known if the degree of the tree is ounded, s proved y Grg et l. for poly-line drwings [GGT96] nd y Grg nd Rusu for stright-line drwings [GR03]. Concerning outerplnr grphs, Biedl [Bie02] hs shown how to construct poly-line drwings in O(n log n) re; Di Bttist nd the uthor [DF09] presented n lgorithm for otining strightline drwings in O(n 1.48 ) re; the uthor [Fr07] exhiited n lgorithm for constructing strightline drwings in O(dn log n) re, where d is the degree of the grph.

3 Lower Bounds on the Are Requirements of Series-Prllel Grphs 141 Fig. 2: A stright-line drwing of K 2,n with liner re nd liner spect rtio. Both for outerplnr grphs nd for trees, no super-liner re lower ounds re known, neither in the cse of stright-line drwings nor in the one of poly-line drwings. In this pper we del with series-prllel grphs, clss of plnr grphs tht hs een widely investigted in Grph Theory nd Grph Drwing (see, e.g., [VTL82, Epp92, BCD + 94, Di 03, DDLW06]). The min known result on the construction of smll-re grid drwings of series-prllel grphs is tht every series-prllel grph dmits poly-line drwing in O(n 3/2 ) re. Such ound ws proved y Biedl in [Bie05, Bie10]; in these ppers, the uthor provides nice inductive construction of visiility representtions of series-prllel grphs nd shows how such representtions cn e turned into poly-line drwings with symptoticlly the sme re. While poly-line drwings cn e relized in O(n 3/2 ) re, no su-qudrtic re upper ound is known in the cse of stright-line drwings. The est known qudrtic upper ound for stright-line drwings is provided in [ZHN10]. In [Bie05], Biedl lso proved n Ω( n log n log log n ) re lower ound for stright-line drwings of series-prllel grphs. The Ω( n log n log log n ) re lower ound for stright-line drwings of series-prllel grphs is direct consequence of the results in [BCLO03], where Biedl, Chn, nd López-Ortiz, settling in the positive conjecture of Felsner et l. [FLW03], proved tht no liner-re stright-line drwing of K 2,n cn chieve constnt spect rtio. Oserve tht drwing of the complete iprtite grph K 2,n cn e thought of s drwing of n pths tht strt nd end t the sme two vertices, in the following denoted y nd, nd tht do not shre ny other vertex. In the following we will refer such pths s the pths of K 2,n. Fig. 2 shows stright-line drwing of K 2,n with liner re nd liner spect rtio. More precisely, Biedl, Chn, nd López-Ortiz proved the following: Theorem 1 (Biedl et l. [BCLO03]) Every plnr stright-line grid drwing of K 2,n on W H grid with W H stisfies W log H Ω(n). Corollry 1 (Biedl et l. [BCLO03]) Every plnr stright-line grid drwing of K 2,n on W H grid stisfies mx{w, H} Ω(n/ log n). Biedl et l. sk whether the log H fctor in Theorem 1 cn e eliminted nd whether the sme lower ound holds even in the cse of poly-line drwings. In this pper we nswer oth the questions in the ffirmtive. Nmely, we prove the following: Theorem 2 Every plnr stright-line or poly-line grid drwing of K 2,n on W H grid stisfies mx{w, H} Ω(n). Such result is chieved y first exhiiting very simple optiml drwing lgorithm for K 2,n. Tht is, if there exists drwing of K 2,n inside n ritrry convex polygon P in which nd re plced t two specified vertices of P, then our lgorithm constructs one such drwing. Second, we study the drwings constructed y the mentioned lgorithm inside rectngle. Such study revels surprisingly regulr

4 142 Frizio Frti ehvior of the drwing of the pths of K 2,n ; we rgue tht such ehvior hs strong reltionship with the genertion of reltively prime numers s expressed in the Stern-Brocot tree. On the se of such reltionship, we derive some rithmeticl properties of the lines pssing through infinite grid points in the plne, tht might e interesting y their own, nd we chieve the climed lower ound. As consequence of Theorem 2, n Ω(n log n) lower ound on the re requirements of poly-line nd stright-line drwings of series-prllel grphs cn e otined. Nmely, consider n O(n)-node seriesprllel grph contining K 2,n nd n n-node complete ternry tree s sugrphs. Since ny poly-line or stright-line drwing of n n-node complete ternry tree requires Ω(log n) width nd Ω(log n) height (see [FLW03, Sud04]), nd since the width or the height of ny such drwing hs Ω(n) length (y Theorem 2), the lower ound follows. However, we cn chieve etter lower ound y mens of the following: Theorem 3 There exist series-prllel grphs requiring Ω(2 log n ) width nd Ω(2 log n ) height in ny stright-line or poly-line grid drwing. Such result is chieved y crefully constructing grph out of severl copies of K 2,n nd y then strongly exploiting Theorem 2 nd some further geometric considertions. Theorem 3, together with Theorem 2, immeditely implies the following min result: Theorem 4 There exist series-prllel grphs requiring Ω(n2 log n ) re in ny stright-line or polyline grid drwing. We remrk tht the function 2 log n is greter thn ny polylogrithmic function of n nd smller thn ny polynomil function of n; we further remrk tht no super-liner re lower ound ws previously known for poly-line drwings of series-prllel grphs nd tht Ω( n log n log log n ) ws the est known re lower ound for stright-line drwings of series-prllel grphs [Bie05]. The rest of the pper is orgnized s follows. In Section 2 we give some preliminries; in Section 3 we give some geometric lemmt; in Section 4 we prove Theorem 2; in Section 5 we prove Theorem 3; finlly, in Section 6 we conclude nd suggest some open prolems. 2 Preliminries A grid drwing of grph is mpping of ech vertex to distinct point of the plne with integer coordintes nd of ech edge to Jordn curve etween the endpoints of the edge. A plnr drwing is such tht no two edges intersect except, possily, t common endpoints. In the following we lwys refer to plnr grid drwings. A stright-line drwing is such tht ll edges re rectiliner segments. A poly-line drwing is such tht the edges re sequences of rectiliner segments. In poly-line drwing end is point in which n edge chnges its slope, i.e., point common to two consecutive segments in the sequence of segments representing the edge. In grid drwing ends hve integer coordintes. A polygonl pth is poly-line grid drwing of pth. The ounding ox of drwing Γ is the smllest rectngle with sides prllel to the xes tht covers Γ completely. The height (width) of Γ is the height (resp. width) of its ounding ox. The re of Γ is the height of Γ times its width. The spect rtio of Γ is the rtio etween the mximum nd minimum of its height nd width. Throughout the pper, grid line is ny line pssing through n infinite numer of grid points. Two grid lines re consecutive if they re prllel nd no grid point is contined in the open strip delimited y

5 Lower Bounds on the Are Requirements of Series-Prllel Grphs 143 (0,1) (1,0) (1,1) (1,2) (2,1) (1,3) (2,3) (3,2) (3,1) (1,4) (2,5) (3,5) (3,4) (4,3) (5,3) (5,2) (4,1) Fig. 3: The Stern-Brocot tree. the two lines. We denote y pq the segment etween two points p nd q. We lso denote y k the polygonl line composed of segments 0 1, 1 2,..., k 1 k. Let p 1 = (, y 1 ) nd p 2 = (, y 2 ) e ordered pirs of rel numers (such pirs might represent grid points or vectors, for exmple) nd let k e constnt. Throughout the pper, we write kp 1 to denote the pir (k, ky 1 ), we write p 1 + p 2 to denote the pir ( +, y 1 + y 2 ), nd we write p 1 p 2 to denote the pir (, y 1 y 2 ). We denote y v the sclr product etween vector v = (v x, v y ) nd point = ( x, y ) in the plne, tht is, v = v x x + v y y. The Stern-Brocot tree [Ste58, Bro60] is n infinite tree whose nodes re in ijective mpping with the irreducile positive rtionl numers, or equivlently, in ijective mpping with the ordered pirs of reltively prime integers. See Fig. 3. The Stern-Brocot tree hs two nodes (0, 1) nd (1, 0) which re oth connected to the sme node (1, 1). Nodes (0, 1) nd (1, 0) re the left prent nd the right prent of (1, 1), respectively. Further, 1 0 nd 0 1 re the left generting frction nd the right generting frction of 1 1, respectively. An ordered inry tree is then rooted t (1, 1) s follows. Consider node (x, y) of the tree. Such node hs two children. The left child of (x, y) is the node (x, y) + (x, y ), where (x, y ) is the ncestor of (x, y) tht is closer to (x, y) (in terms of grph-theoretic distnce on the tree) nd tht hs (x, y) in its right sutree. Then, y x nd y x re the left generting frction nd the right generting frction of y+y x+x, respectively. Anlogously, the right child of (x, y) is the node (x, y) + (x, y ), where (x, y ) is the ncestor of (x, y) tht is closer to (x, y) nd tht hs (x, y) in its left sutree. Then, y y x nd x re the left generting frction nd the right generting frction of y+y x+x, respectively. The following properties of the Stern-Brocot tree re well-known nd esy to oserve: Property 1 Let (x, y) e node of the Stern-Brocot tree nd let y x nd y x e the left nd right generting frctions of y x. Then, the sutree of the Stern-Brocot tree rooted t the left child of (x, y) contins ll nd only the pirs of reltively prime integers (z, w) such tht y x < w z < y x nd the sutree of the Stern- Brocot tree rooted t the right child of (x, y) contins ll nd only the pirs of reltively prime integers (z, w) such tht y x < w z < y x. Property 2 Let (x, y) e node of the Stern-Brocot tree. Then every node (x, y ) tht is descendnt of (x, y) is such tht x x nd y y nd either x x or y y (or oth). It is useful to visulize the Stern-Brocot tree in the following wy. Nodes (0, 1), (1, 1), nd (1, 0) re

6 144 Frizio Frti p l v () () Fig. 4: () Illustrtion for the proof of Lemm 1. Disk D is the smll shded region. () Illustrtion for the proof of Lemm 2. ordered in this wy from left to right nd three verticl lines re ssocited with these nodes. When node (x, y) is drwn, it is plced in the strip delimited y the verticl lines ssocited with its left nd right generting frctions, nd verticl line is ssocited with (x, y). In such visuliztion, ech node of the tree is close to its generting frctions nd nodes (x, y) re ordered from left to right y decresing vlue of y x. 3 Geometric Lemmt In this section we show some lemmt tht will e used to prove Theorems 2 nd 3. We first del with the geometry of K 2,n nd then with the reltionships etween reltively prime numers nd grid lines in the plne. Recll tht K 2,n cn e considered collection of n pths (the pths of K 2,n ) tht connect two vertices nd nd tht do not shre ny other vertex. 3.1 Lemmt on the Geometry of K 2,n Lemm 1 Consider ny poly-line grid drwing of K 2,n, ny pth π of K 2,n, nd ny vector v = (v 1, v 2 ). There exists grid point p π such tht v p v p, for ny point p π. Proof: If v v p or v v p, for every point p π, the lemm follows. Otherwise, consider the prt π of π strting t nd ending t the first point p in which v p v p, for every point p π (see Fig. 4.). Since ech point p p of π is such tht v p < v p, there exists smll disk D centered t p such tht the prt of π enclosed in D is incresing in the direction determined y v, when π is oriented from to p. Further π, when oriented from to, cn not e incresing in the direction determined y v immeditely fter p, otherwise there would exist point p such tht v p > v p. It follows tht π chnges its slope t p nd, y definition of poly-line grid drwing, p is grid point. Lemm 2 Consider ny drwing of K 2,n. Let l e ny line tht does shre ny point with the open segment. No three pths π 1, π 2, nd π 3 of K 2,n exist such tht: (i) π 1, π 2, nd π 3 do not shre ny point except for nd ; (ii) π 1, π 2, nd π 3 re entirely contined in the closed hlf-plne delimited y l nd contining nd ; nd (iii) ech of π 1, π 2, nd π 3 shres t lest one point different from nd with l.

7 Lower Bounds on the Are Requirements of Series-Prllel Grphs 145 Proof: Suppose, for contrdiction, tht three pths π 1, π 2, nd π 3 of K 2,n with the ove properties exist. Pths π 1 nd π 2 form cycle C. Line l is externl to C nd seprtes from in the exterior of C (see Fig. 4.). Consider ny pth π 3 etween nd. If π 3 is internl to C, then it cn not shre point different from nd with l unless it shres points different from nd with C. If π 3 is externl to C, then it either shres points different from nd with C or it is in prt contined in the open hlf-plne delimited y l nd not contining nd. If π 3 is prt internl nd prt externl to C, then it shres points different from nd with C. In ny cse we hve contrdiction. Let P e ny convex polygon in the plne with vertices hving integer coordintes. Let I e the set of grid points in the interior or on the order of P. Let nd e two distinct vertices of P. Let π 1 nd π 2 e the drwings of the two pths tht connect nd nd tht compose P. At lest one out of π 1 nd π 2, sy π 1, is different from segment. Let M e the mximum numer of pths connecting nd tht cn e drwn s non-crossing polygonl pths inside or on the order of P. Lemm 3 There exist M non-crossing polygonl pths connecting nd such tht: Ech pth is inside or on the order of P ; nd one of such pths is π 1. Proof: Consider ny drwing Γ composed of M non-crossing polygonl pths connecting nd nd contined inside or on the order of P. If pth of Γ is π1, there is nothing to prove. Otherwise, oserve tht no two distinct pths π i nd π j pss through points of π1, s otherwise π i nd π j cross. Hence, Γ hs t most one pth π pssing through points of π1. Remove π from Γ, if π exists, nd drw pth in Γ s π1. Since no pth different from π psses through point of π1, the resulting drwing is plnr, hence proving the lemm. Lemm 4 There exist M non-crossing polygonl pths connecting nd such tht: Ech pth is inside or on the order of P ; nd one of such pths is segment. Proof: We prove the clim y induction on M. If M = 1, then drwing pth s segment proves the clim. Suppose M 2. By Lemm 3, there exists drwing Γ composed of M non-crossing polygonl pths connecting nd such tht ech pth is inside or on the order of P nd one of such pths, sy π, is π1. Remove π from Γ nd ll the grid points π psses through, except for nd, from I. Consider the convex polygon P tht is the convex hull of the resulting grid point-set I. The vertices of P hve integer coordintes. Further, P is such tht M 1 pths cn e drwn s non-crossing polygonl pths connecting nd inside or on the order of P. In fct Γ is drwing hving such property. Hence, the inductive hypothesis pplies nd M 1 polygonl pths exist so tht ech pth is inside or on the order of P nd so tht one of the pths is segment. Considering such M 1 pths together with the drwing of π s π1 proves the lemm. Now ssume tht nd re consecutive vertices of P (see Fig. 5). Let I e the set of grid points in the interior or on the order of P. As efore, let π 1 nd π 2 e the drwings of the two pths tht connect

8 146 Frizio Frti Fig. 5: Drwing the mximum numer of pths in convex polygon with vertices hving integer coordintes. Blck circles re vertices of P nd white circles re grid points inside or on the order of P. nd nd tht compose P, where π 1 is different from segment. Let lso M e the mximum numer of pths connecting nd tht cn e drwn s non-crossing polygonl pths inside or on the order of P. We itertively drw some pths π 1, π 2,, π N connecting nd inside or on the order of P s follows. Pth π i is drwn when the current convex grid polygon is P i contining in its interior or on its order set I i of grid points. At the first step P 1 = P nd I 1 = I. If P i does not coincide with segment, drw pth π i s the polygonl pth tht connects nd, tht lies on P i, nd tht is different from segment. Remove the grid points tht lie on P i, except for nd, from I i, otining new set of grid points I i+1. Then, P i+1 is the convex hull of I i+1. If P i coincides with segment, drw pth π i = π N s segment. We oserve the following: Lemm 5 Pths π 1, π 2,, π N re drwn s non-crossing polygonl pths inside or on the order of P. Further, N = M. Proof: The first prt of the sttement is trivil. We prove tht N = M y induction on M. If M = 2, then the clim trivilly holds, since π 1 is drwn s π1 nd π 2 s. Suppose tht M 3. By Lemm 3, there exists drwing Γ composed of M non-crossing polygonl pths connecting nd such tht ech pth is inside or on the order of P nd one of such pths, sy π 1, is π1. Remove π 1 from Γ nd ll the grid points π 1 psses through from I. Consider the convex polygon P tht is the convex hull of the resulting grid point-set I. Clerly, the vertices of P hve integer coordintes. Further, P is such tht M 1 non-crossing polygonl pths connecting nd exist such tht ech pth is inside or on the order of P. In fct Γ is drwing hving such property. Hence, the inductive hypothesis pplies nd the drwing lgorithm descried efore the sttement of the lemm drws M 1 pths s non-crossing polygonl pths inside or on the order of P. Considering such pths together with the drwing of π 1 s π1 proves the lemm. 3.2 A Lemm on the Arithmetics of Consecutive Grid Lines The im of this section is to prove the following useful lemm. Lemm 6 Let l 1 e grid line with slope y x, where x, y > 0 nd (x, y) is pir of reltively prime numers. Let y x nd y x e the left nd right generting frctions of y x. Consider ny grid point (p x, p y )

9 Lower Bounds on the Are Requirements of Series-Prllel Grphs 147 t (t x, t y ) p 3 (x + x, y + y ) p 1 (x, y) q (q x, q y ) ( q x z, q y z ) p 2 (x, y ) p 0 (0, 0) l 1 l 2 Fig. 6: Illustrtion for the proof of Lemm 6. of l 1. Let l 2 (l 3 ) e the grid line pssing through (p x, p y ) + (x, y ) nd through (p x, p y ) (x, y ) (resp. through (p x, p y ) (x, y ) nd through (p x, p y ) + (x, y )). Then, l 1 nd l 2 (resp. l 1 nd l 3 ) re consecutive grid lines. Proof: Refer to Fig. 6. We prove the sttement for l 1 nd l 2, the proof for l 1 nd l 3 eing nlogous. Suppose, for contrdiction, tht l 1 nd l 2 re not consecutive. First, oserve tht l 1 nd l 2 re prllel, s l 2 hs slope py+y p y+y p x+x p x+x = y +y x +x = y x, where the lst equlity holds y the definition of generting frctions. We cn ssume, without loss of generlity up to simultneous trnsltion of l 1 nd l 2, tht l 1 psses through point p 0 (0, 0). Denote p 1 (x, y). Oserve tht simultneous trnsltion of l 1 nd l 2 does not lter whether the open strip delimited y the two lines contins grid point, s the sme trnsltion moves ny grid point etween the two lines efore the trnsltion to grid point etween the two lines fter the trnsltion. Suppose tht point q (q x, q y ) exists etween l 1 nd l 2. Then, we cn ssume tht q is in the prllelogrm P whose vertices re p 0, p 1, p 2 (x, y ), nd p 3 p 1 + p 2, or on its order. Nmely, if grid point t (t x, t y ) is etween l 1 nd l 2, then every grid point t (t x + mx, t y + my) is etween l 1 nd l 2, for ll m Z. Suppose tht q is inside the closed tringle (p 0, p 1, p 2 ), the cse in which it is inside (p 1, p 2, p 3 ) eing nlogous. We cn ssume tht (q x, q y ) nd (x q x, y q y ) re two pirs of reltively prime numers. Nmely, suppose tht q x nd q y hve common divisor, sy z. Then, ( qx z, qy z ) is grid point. Further, such point is on the order of tringle (p 0, p 1, q), ctully on p 0 q. Then, point q ( qx z, qy z ) cn e considered insted of q (q x, q y ). Anlogously, if x q x nd y q y hve common divisor, sy z, then point q (x x qx z, y y qy z ) cn e considered insted of q (q x, q y ). Oserve tht, whenever the currently

10 148 Frizio Frti h c v d h l, d v v d h Fig. 7: Illustrtion of the nottion for the proof of Theorem 2. considered point q (q x, q y ) is replced y new grid point q ( qx z, qy x qx z ) or q (x z, y y qy z ), the sum of the numer of grid points on the order nd of the numer of grid points in the interior of tringle (p 0, p 1, q) decreses. Hence, eventully fter certin numer of replcements, the coordintes q x nd q y of q (nd simultneously x q x nd y q y ) re reltively prime numers. Oserve tht q does not lie on p 0 p 1 s it hs to lie in the open strip delimited y l 1 nd l 2. Further, it does not lie on p 0 p 2 (on p 1 p 2 ) s otherwise x nd y (resp. x nd y ) would not e reltively prime numers. Now consider the slope qy q x. As q is inside tringle (p 0, p 1, p 2 ), it follows tht y x < qy q x nd tht y x < y qy x q x < y x. By Property 1, the reltively prime pirs (q x, q y ) nd (x q x, y q y ) re contined in the sutree of the Stern-Brocot tree rooted t (x, y). By Property 2, q x x nd q y y hold; further, x q x x nd y q y y hold; hence, q x +x q x 2x nd q y +y q y 2y hold. Such contrdictions prove the lemm. 4 Proof of Theorem 2 By definition, stright-line drwing is lso poly-line drwing. Hence, it suffices to prove Theorem 2 for poly-line drwings. Consider ny poly-line grid drwing of K 2,n. Let R e the smllest xis-prllel rectngle enclosing nd (see Fig. 7). Let l, e the line through nd. Suppose, without loss of generlity, tht y() y(). Suppose lso tht x() < x(), the cse in which x() x() eing nlogous. Let c nd d e the upper left corner nd the lower right corner of R, respectively. Let h nd v e the horizontl nd verticl lines through, respectively. Anlogously, let h nd v e the horizontl nd verticl lines through, respectively. Let d h nd d v e the horizontl nd verticl distnce etween nd, respectively. The width W nd the height H of the drwing re such tht W d h nd H d v. For ny line l, denote y H + (l) (resp. y H (l)) the closed hlf-plne delimited y l nd contining the norml vector of l incresing in the y-direction (resp. decresing in the y-direction). If l is verticl line, then H + (l) (resp. H (l)) denotes the closed hlf-plne delimited y l nd contining the norml vector of l incresing in the x-direction (resp. decresing in the x-direction). For ny non-horizontl line l, we sy tht point p is to the right of l (to the left of l) if p is in the open hlf-plne delimited y l nd contining the norml vector of l incresing in the x-direction (resp. decresing in the x-direction). Consider the hlf-plne H + (h ). By Lemm 1 with v = (0, 1), for ech pth π tht hs non-empty intersection with H + (h ), there exists grid point p π whose y-coordinte is mximum mong the < y x

11 Lower Bounds on the Are Requirements of Series-Prllel Grphs 149 c Fig. 8: Pths π 1, π 2,, π M1 in Π. points of π. Clerly, p elongs to H + (h ). Hence, p elongs to horizontl grid line l tht does not intersect or contin the open segment. By Lemm 2, t most two pths of K 2,n hve their points with gretest y-coordinte elonging to l. It follows tht, if liner numer of pths of K 2,n hs nonempty intersection with H + (h ), then their points with gretest y-coordinte elong to liner numer of distinct horizontl grid lines nd hence H Ω(n). Similr rguments show tht, if liner numer of pths of K 2,n hve non-empty intersection with H (h ), H + (v ), or H (v ), then H Ω(n), W Ω(n), or W Ω(n), respectively. If there exists no liner numer of pths of K 2,n hving non-empty intersection with H + (h ), H (h ), H + (v ), or H (v ), then liner numer of pths of K 2,n is completely inside or on the order of R. We show tht this implies tht mx{d h, d v } Ω(n), nd hence tht mx{w, H} Ω(n). Let M e the mximum numer of pths of K 2,n tht cn e drwn inside or on the order of R. By Lemm 4, there exists drwing of M pths connecting nd, nd completely lying inside or on the order of R, such tht one of the pths is drwn s segment. Since M Ω(n), then either liner numer of pths of K 2,n is contined inside or on the order of the tringle T 1 hving,, nd c s vertices, or liner numer of pths of K 2,n is contined inside or on the order of the tringle T 2 hving,, nd d s vertices. Suppose tht liner numer of pths of K 2,n is contined inside or on the order of T 1, the other cse eing symmetric. Let M 1 Ω(n) e the mximum numer of pths of K 2,n tht cn e drwn inside T 1 nd let I 1 e the set of grid points inside or on the order of T 1. By Lemm 5, sequence of M 1 non-crossing polygonl pths Π = (π 1, π 2,, π M1 ) connecting nd nd completely inside or on the order of T 1 cn e drwn y repeting the following two opertions, for 1 i < M 1 : (1) consider the current convex grid polygon P i (when i = 1 then P 1 = T 1 ); let I i e the set of grid points inside or on the order of P i ; drw pth π i s the prt of P i tht connects nd, nd tht is different from segment ; (2) delete from I i the grid points π i psses through, otining set of grid points I i+1. Convex polygon P i+1 is the convex hull of I i+1. Pth π M1 is drwn s segment. See Fig. 8. In order to prove tht M 1 Ω(n) implies mx{d h, d v } Ω(n), we study pths π 1, π 2,, π M1 nd prove tht they hve very regulr ehvior tht is strongly relted to the genertion of reltively prime numers s in the Stern-Brocot tree. In the following, we first sketch description of the geometry of pths π 1, π 2,, π M1, we then detil such description, we lter prove the geometric clims to e correct, nd we finlly prove tht mx{d h, d v } Ω(n). In the reminder of the section we ssume tht d h, d v > 3. Clerly, if one of d h nd d v is O(1), then the other one must e Ω(M 1 ), nd there is nothing

12 150 Frizio Frti to prove. 4.1 Sketch of the geometry of pths π 1, π 2,, π M1 First, we oserve tht ech pth in Π is composed of two or three segments, i.e., ech pth hs one or two ends. A sequence of pths tht re consecutive in Π nd tht re ech composed of three segments is such tht ll the second segments of the pths hve the sme slope. In sequence of pths such tht the second segments of the pths hve the sme slope, ll the ends lie on two lines, hving slopes one greter nd one smller thn dv d h, the slope of segment. Moreover, the two lines on which such ends lie hve slope y1 nd y2, where (, y 1 ) nd (, y 2 ) re two pirs of reltively prime numers; the slope of the second segments of the pths tht hve such ends is y1+y2 where ( +, y 1 + y 2 ) is pir of reltively prime numers, nd y1 nd y2 re the generting frctions, of y1+y2. The more sequences of three-segments pths tht re consecutive in Π re considered, the more the slopes of the first, of the second, nd of the third segments of the pths pproch to the slope of segment. Nmely, if sequence of pths is such tht their ends lie on two lines with slopes y1 nd y2 nd their second segments hve slope y1+y2, then the next sequence of pths whose second segments hve the sme slope is such tht the ends of such pths lie on two lines with slopes y1 y 2 nd y1+y2, depending on whether y1+y2 second segments of such pths hve slope 2y1+y2 2 < dv d h < y1 or y2 < dv d h or y1+2y2 +2, respectively. nd y1+y2 or with slopes < y1+y2, respectively, nd the In order to nlyze mx{d h, d v } s function of M 1, we sudivide Π into disjoint su-sequences Π 1, Π 2,, Π f nd we rgue tht Π 1 hs t most mx{d h, d v } pths nd tht Π i hs t most mx{d h, d v }/2 i 2 pths, for 2 i f; such ounds led to the conclusion tht, s long s M 1 Ω(n), mx{d h, d v } Ω(n). 4.2 Detils of the geometry of pths π 1, π 2,, π M1 Pth π 1 is c. Let p 1 c + (1, 1). Consider the following two sequences of grid points. See Fig. 9.. Sequence S 0,1 is composed of points p 0,1 k p 1 (k 1)(0, 1), for 1 k i 1, where i 1 is the lrgest integer such tht point p 1 (i 1 1)(0, 1) is contined inside T 1 (recll tht T 1 is the tringle hving,, nd c s vertices). Sequence S 1,0 is composed of points p 1,0 k p 1 + (k 1)(1, 0), for 1 k j 1, where j 1 is the lrgest integer such tht point p 1 + (j 1 1)(1, 0) is contined inside T 1. Notice tht the points of S 0,1 lie on line with slope 1 0 = nd the points of S 1,0 lie on line with slope 0 1 = 0. A su-sequence Π 1 of Π, strting t π 2 nd composed of pths consecutive in Π, uses the points in S 0,1 nd in S 1,0, i.e., ech pth in Π 1 psses through point in S 0,1 or point in S 1,0. Actully, the first pths in Π 1 pss through point in S 0,1 nd point in S 0,1. The pths tht use the points in S 0,1 nd in S 1,0 terminte when pth uses point in S 0,1 nd point in S 1,0 tht re colliner with one of nd. Oserve tht when one of S 0,1 nd S 1,0 is over, it is lwys the cse tht the lst drwn pth uses point in S 0,1 nd point in S 1,0 tht re colliner with one of nd. Then, pth π k+1 is the polygonl pth p 0,1 k p1,0 k, for k = 1, 2,,, where is the smllest index greter thn 1 such tht, p 0,1, nd p 1,0 re colliner or p 0,1, p 1,0, nd re colliner. When one of S 0,1 nd S 1,0 is over, tht is, there exist pths pssing through ll of its points, then, p 0,1, nd p 1,0 re colliner or p 0,1, p 1,0, nd re colliner. Notice tht p 0,1 1 = p 1,0 1 = p 1, hence π 2 is composed of only

13 Lower Bounds on the Are Requirements of Series-Prllel Grphs 151 p 0,1 1 p 1,0 1 p 1,0 2 p 1,0 3 p 0,1 2 p 0,1 3 p 1,0 j 1 p 0,1 i 1 p 0,1 1 p 1,0 1 () π k1+1 () Fig. 9: () Sequences S 1,0 nd S 0,1. () Pths π k+1, with 1 k. two segments. The second segment of pth π k+1, for k = 2, 3,,, hs slope 1 1. Oserve tht 1 0 nd 0 1 re the generting frctions of 1 1. See Fig. 9.. Then, three cses hve to e considered, nmely the one in which, p 0,1, p 1,0, nd re ll colliner, the one in which, p 0,1, nd p 1,0 re colliner (nd is not), nd the one in which p 0,1, p 1,0, nd re colliner (nd is not). In the first cse, pth π k1+1 coincides with segment, hence π k1+1 = π M1. In the second cse (the third cse is nlogous to the second one), sequence S 0,1 is replced y sequence S 1,1 defined s follows. See Fig The points of S 1,1 re the points p 1,1 k p 1,0 +1 (k 1)(1, 1), for 1 k i 2, where i 2 is the lrgest integer such tht point p 1,0 k (i )(1, 1) is contined inside T 1. Some pths in Π 1 use the points in S 1,1 nd the remining points in S 1,0. The pths tht use the points in S 1,1 nd in S 1,0 terminte when pth uses point in S 1,1 nd point in S 1,0 tht re colliner with one of nd. Oserve tht when one of S 1,1 nd S 1,0 is over, it is lwys the cse tht the lst drwn pth uses point in S 1,1 nd point in S 1,0 tht re colliner with one of nd. Then, pth π k1+k+1 is the polygonl pth p 1,1 k p1,0 +k, for k = 1, 2,, k 2, where k 2 is the smllest index such tht, p 1,1 k 2, nd p 1,0 +k 2 re colliner or p 1,1 k 2, p 1,0 +k 2, nd re colliner. When one of S 1,1 nd S 1,0 is over, then, p 1,1 k 2, nd p 1,0 +k 2 re colliner or p 1,1 k 2, p 1,0 +k 2, nd re colliner. Notice tht p 1,1 1 = p 1,0 k, hence π is composed of only two segments. Also, oserve tht the ends of pths π k1+k+1, with k = 1, 2,, k 2, lie on two lines with slope 1 1 = 1 nd 0 1 = 0, while the second segments of such pths lie on lines with slope = 1 2, where 0 1 nd 1 1 re the generting frctions of 1 2. See

14 152 Frizio Frti π k1+1 p 1,0 p 1,1 2 p 1,1 3 p 1,1 1 p 1,0 +1 p 1,0 j 1 p 1,1 i 2 p 1,1 1 p 1,0 +1 () π k1+2 π k1+k 2+1 () Fig. 10: () Sequence S 1,1. () Pths π k1 +k+1, with 1 k k 2. Fig Agin, three cses hve to e considered, nmely the one in which, p 1,1 k 2, p 1,0 +k 2, nd re ll colliner, the one in which, p 1,1 k 2, nd p 1,0 +k 2 re colliner (nd is not), nd the one in which p 1,1 k 2, p 1,0 +k 2, nd re colliner (nd is not). In the first cse, pth π k1+k 2+1 coincides with segment, hence π k1+k 2+1 = π M1. Otherwise,, p 1,1 k 2, nd p 1,0 +k 2 re colliner (nd is not), or p 1,1 k 2, p 1,0 +k 2, nd re colliner (nd is not). Then S 1,1 is. Nmely, such points hve coordinte p 2,1 k = p 1,0 k (k 1)(2, 1), for 1 k i 1+k 2+1 3, where i 3 is the lrgest integer such tht point p 1,0 k 1+k 2+1 (i 3 1)(2, 1) is internl to T 1. See Fig Some pths in Π 1 use the points in S 2,1 nd the remining points in S 1,0, tht is, pth π k1+k 2+k+1, re colliner (nd is not). Suppose tht, p 1,1 k 2, nd p 1,0 +k 2 replced y sequence S 2,1 of points lying on line with slope 1 2 with 1 k k 3, psses through point p 2,1 k nd through point p 1,0 k, where +k 2+k 3 is the smllest index such tht, p 2,1 k 3, nd p 1,0 +k 2+k 3 re colliner or p 2,1 k 3, p 1,0 +k 2+k 3, nd re colliner. Oserve tht when one of S 2,1 nd S 1,0 is over, it is lwys the cse tht the lst drwn pth uses point in S 2,1 nd point in S 1,0 tht re colliner with one of nd. Oserve lso tht the ends of pths π k1+k 2+k+1, with k = 1, 2,, k 3, lie on two lines with slope 1 2 nd 0 1, while the second segments of such pths lie on lines with slope = 1 3, where 1 2 nd 0 1 re the generting frctions of 1 3. See Fig The ove rgument itertes till pth is drwn tht psses through, through point p l,1 k l+1 of the current sequence S l,1, through point p 1,0 +k 2+ +k l +k l+1 of S 1,0, nd through in such wy tht

15 Lower Bounds on the Are Requirements of Series-Prllel Grphs 153 π k1+k 2+1 p 1,0 +k 2 p 2,1 3 p 2,1 2 p 2,1 1 p 1,0 +k 2+1 p 1,0 j 1 p 2,1 i 3 () p 2,1 1 p 1,0 +k 2+1 π k1+k 2+2 π j1+1 = π k1+k 2+k 3+1 () Fig. 11: () Sequence S 2,1. () Pths π k1 +k 2 +k+1, with 1 k k 3. p l,1 k l+1, p 1,0 +k 2+ +k l +k l+1, nd re colliner. If sequence S 1,0 is over, tht is, ll its points hve een trversed y pths in Π, then the lst drwn pth psses through, through point p l,1 k l+1 of S l,1, through the lst point p 1,0 +k 2+ +k l +k l+1 of S 1,0, nd through, where p l,1 k l+1, p 1,0 +k 2+ +k l +k l+1, nd re colliner. Hence, ll the pths tht come fter π 1 in Π 1 pss through distinct points of S 1,0, till pth is drwn tht psses through, through point p l,1 k l+1 of S l,1, through point p 1,0 +k 2+ +k l +k l+1 of S 1,0, nd through in such wy tht p l,1 k l+1, p 1,0 +k 2+ +k l +k l+1, nd re colliner. Thus, Π 1 = (π 2, π 3,, π k1+k 2+ +k l +k l+1 +1) is the desired su-sequence Π 1 of Π. Further, there exists n index l 1 such tht: (1) ll the points p i,1 j re trversed y pths in Π 1, for 0 i l 1 nd 1 j k i+1, nd, p i,1 k i+1, nd p 1,0 +k 2+ +k i+1 re colliner, for 0 i l 1; (2) some points of S l,1 re possily trversed y pths in Π 1, nd p l,1 k l+1, p 1,0 +k 2+ +k l +k l+1, nd re colliner. In the exmple in Figs. 9 11, we hve l = 2; indeed, ll the points p 0,1 j re trversed y pths in Π 1, for 1 j ;, p 0,1, nd p 1,0 re colliner; ll the points p 1,1 j re trversed y pths in Π 1, for 1 j k 2 ;, p 1,1 k 2, nd p 1,0 +k 2 re colliner; some points of S 2,1 re trversed y pth in Π 1 ; p 2,1 k 3, p 1,0 +k 2+k 3, nd re colliner. After drwing pth π k1+k 2+ +k l +k l+1 +1 (tht psses through, p l,1 k l+1, p 1,0 +k 2+ +k l +k l+1, nd in such wy tht p l,1 k l+1, p 1,0 +k 2+ +k l +k l+1, nd re colliner), either is colliner with p l,1 k l+1, p 1,0 +k 2+ +k l +k l+1, nd, or not. In the former cse, pth π k1+k 2+ +k l +k l+1 +1 coincides with segment

16 154 Frizio Frti π j1+1 = π k1+k 2+k 3+1 p 2,1 k 3 p 3,1 p 3,1 3 2 p 2,1 k 3+1 p 3,1 1 () p 2,1 k 3+1 p 3,1 1 () Fig. 12: () Sequence S 3,1. () The pths in Π 2., hence π k1+k 2+ +k l +k l+1 +1 = π M1. In the ltter cse, S l,1 still contins points not trversed y ny pth in Π 1. Then, sequence S 1,0 is now replced y sequence S l+1,1, whose points lie on line with slope l = 1 l+1 pssing through the first point of S l,1 tht is not trversed y pth in Π 1, tht is, point p l,1 k l See Fig. 12., where there exists exctly one point of S 2,1 tht is not trversed y pth in Π 1. The whole rgument is now repeted gin. Nmely, su-sequence Π 2 of Π uses the points in S l,1 not trversed y pths in Π 1 nd the points in S l+1,1, i.e., ech pth in Π 2 psses through point in S l,1 or point in S l+1,1. Actully, the first pths in Π 2 pss through point in S l,1 nd point in S l+1,1. Agin, Π 2 is generlly found in severl steps, where t ech step two sequences S x1,y 1 of grid points re considered, where one etween S x1,y 1 is S l,1 or S l+1,1 (t the first step S x1,y 1 = S l,1 = S l+1,1 oth hold). The points on S x1,y 1 (on S x2,y 2 ) lie on line with slope y 1 y (resp. 2 ), where (, y 1 ) nd (, y 2 ) re pirs of reltively prime numers. The second segments of the pths drwn when S x1,y 1 re considered hve slope y1+y2, where ( +, y 1 + y 2 ) is pir of reltively prime numers, nd y1 nd y2 re the generting frctions of y1+y2. At ech step, pth eventully psses through two points of S x1,y 1 colliner with or with. Then, one etween S x1,y 1 (depending on whether the lst pth drwn in the step psses through two points of S x1,y 1 colliner with or with ) is replced y sequence of points lying on line with slope y1+y2, hence strting new step. After certin numer of steps, oth S l,1 nd S l+1,1 hve een replced y other sequences of points. When the lst pth tht psses through point of S l,1 or of S l+1,1 is drwn (tht is, when the lst pth of Π 2 is drwn), it psses through, through two points in

17 Lower Bounds on the Are Requirements of Series-Prllel Grphs 155 the currently considered sequences S x1,y 1, nd through, so tht either these four points re colliner, or nd two points in S x1,y 1 re colliner nd is not, or nd two points in S x1,y 1 re colliner nd is not. In the first cse, the lst drwn pth is π M1, tht hence coincides with segment. In the second nd the third cse, either S x1,y 1 or S x2,y 2 is replced y sequence S y1+y 2, whose grid points lie on line with slope y1+y2, depending on whether nd two points in the currently considered sequences S x1,y 1 re colliner nd is not, or nd two points in the currently considered sequences S x1,y 1 re colliner nd is not. The whole rgument is then repeted gin, serching for su-sequence Π 3 of Π such tht Π 3 uses the points in S x1,y 1 nd the points in S x1+,y 1+y 2. Clerly, there exists n index f such tht Π = {π 1 } Π 1 Π 2 Π f {}. In the exmple considered in Figs. 9 12, the sequences considered t the first step, when determining Π 2, re S 2,1 nd S 3,1. The slope of the second segments is = 2 5, lthough no pth composed of three segments is drwn. Nmely, the first pth of Π 2 psses through the only point of S 2,1 not trversed y pths in Π 1. The sequences considered t the second step re S 2+3,1+1 = S 5,2 nd S 3,1. Sequence S 5,2 hs only one point p 5,2 1 p 3, The slope of the second segments is 5+3 = 3 8, lthough no pth composed of three segments is drwn. Nmely, the second pth of Π 2 psses through the only point of S 5,2. The sequences considered t the third step re S 5+3,2+1 = S 8,3 nd S 3,1. Sequence S 8,3 hs only one point p 8,3 1 p 3, The slope of the second segments is 8+3 = 4 11, lthough no pth composed of three segments is drwn. Nmely, the third pth of Π 2 psses through the only point of S 8,3 nd the lst point of S 3,1 (the two points ctully coincide). Sequence Π 2 is over, s ll the points in S 2,1 nd in S 3,1 re trversed y pths in Π 1 or in Π 2. Further, since S 8,3 nd S 3,1 end simultneously, the only pth of Π fter Π 2 is segment. 4.3 Proof of correctness of the geometry of pths π 1, π 2,, π M1 We now prove tht pths π 1, π 2,, π M1 hve the geometry descried in Section 4.2. In order to do tht, we descrie five possile sets of geometric fetures (in the following clled Conditions 1 5) tht cn hold fter drwing pth π i, we show tht fter drwing pth π 2 Condition 4 is stisfied, nd we prove tht, if fter drwing pth π i one of Conditions 1 5 is stisfied, then fter drwing pth π i+1 one of Conditions 1 5 is still stisfied (unless we re in specil cse in which we cn directly estimte the numer of pths tht come fter π i in Π). After pths π 1, π 2,..., π i hve een drwn, grid point is occupied if it hs een trversed y pth π j, with j i, nd is free otherwise. After pth π i is drwn, we ssocite with the next pth π i+1 to e drwn two sequences S x1,y 1 of points, such tht the following properties re stisfied: Property S1: nd y 1 re reltively prime numers; nd y 2 re reltively prime numers; Property S2: y1 > dv d h > y2 ; Property S3: y1 nd y2 re the left nd right generting frctions of y1+y2, respectively; Property S4: All the points in (possily empty) initil su-sequence of S x1,y 1 nd ll the points in (possily empty) initil su-sequence of S x2,y 2 re occupied; ll the other points of S x1,y 1 nd S x2,y 2 re free nd lie inside polygon π i ; Property S5: The hlf-line l(, y 1 ) strting t the first point p x1,y1 1 of S x1,y 1, hving slope y1, nd directed towrds decresing y-coordintes intersects the interior of segment in point

18 156 Frizio Frti l 1,2 (π i ) l(x2, y 2 ) q 1 (π i ) π i q 2 (π i ) l(x1, y 1 ) Fig. 13: After drwing π i, Condition 1 is stisfied. In ll the figures of Section 4.3, lck dots represent occupied points of S x1,y 1 nd S x2,y 2, white dots represent free points of S x1,y 1 nd S x2,y 2, nd the shded tringles re T (S x1,y 1, ) nd T (S x2,y 2, ). The slopes of the lines in the figures do not correspond to slopes of grid lines in the plne. This llows us to improve the redility of the drwings. q(s x1,y 1, ); the hlf-line l(, y 2 ) strting t the first point p x2,y2 1 of S x2,y 2, hving slope y2, nd directed towrds incresing x-coordintes intersects the interior of segment in point q(s x2,y 2, ); Property S6: There exists no grid point internl to the tringle T (S x1,y 1, ) hving p x1,y1 1, q(s x1,y 1, ), nd s vertices; there exists no grid point internl to the tringle T (S x2,y 2, ) hving p x2,y2 1, q(s x2,y 2, ), nd s vertices. Conditions 1 5 re s follows: Condition 1. Pth π i is q 1 (π i )q 2 (π i ); q 1 (π i ) nd q 2 (π i ) re the lst occupied points of S x1,y 1 nd S x2,y 2, respectively; segment q 1 (π i )q 2 (π i ) hs slope y1+y2 ; the line l 1,2 (π i ) through q 1 (π i ) nd q 2 (π i ) hs nd to its right; finlly, oth S x1,y 1 hve free points (see Fig. 13). Condition 2. Pth π i is q 1 (π i )q 2 (π i ); q 1 (π i ) nd q 2 (π i ) re the lst occupied points of S x1,y 1 nd S x2,y 2, respectively; segment q 1 (π i )q 2 (π i ) hs slope y1+y2 ; the line l 1,2 (π i ) through q 1 (π i ) nd q 2 (π i ) hs nd to its right; finlly, neither S x1,y 1 nor S x2,y 2 hs free points (see Fig. 14). Condition 3. Pth π i is q 1 (π i ); further, either (i) q 1 (π i ) is the lst occupied point of S x1,y 1 nd ll the points of S x2,y 2 re free; the first free point of S x1,y 1 coincides with the first point of S x2,y 2 ; segment q 1 (π i ) hs slope y2 ; y1 is generting frction of y2 ; the line l 1,2 (π i ) through q 1 (π i ) with slope y1+y2 hs nd to its right (see Fig. 15); or (ii) q 1 (π i ) is the lst occupied point of S x2,y 2 nd ll the points of S x1,y 1 re free; the first free point of S x2,y 2 coincides with the first point of S x1,y 1 ; segment q 1 (π i ) hs slope y1 ; y2 is generting frction of y1 ; the line l 1,2 (π i ) through q 1 (π i ) with slope y1+y2 hs nd to its right. Condition 4. Pth π i is q 1 (π i ); q 1 (π i ) is the lst occupied point of S x1,y 1 nd the lst occupied point of S x2,y 2 ; the line l 1,2 (π i ) through q 1 (π i ) with slope y1+y2 hs nd to its right; oth S x1,y 1 hve free points (see Fig. 16). Condition 5. Pth π i is q 1 (π i ); q 1 (π i ) is the lst occupied point of S x1,y 1 nd the lst occupied point of S x2,y 2 ; the line l 1,2 (π i ) through q 1 (π i ) with slope y1+y2 hs nd to its right; neither S x1,y 1 nor S x2,y 2 hs free points (see Fig. 17).

19 Lower Bounds on the Are Requirements of Series-Prllel Grphs 157 l 1,2 (π i ) l(x2, y 2 ) q 2 (π i ) π i q 1 (π i ) l(x1, y 1 ) Fig. 14: After drwing π i, Condition 2 is stisfied. l 1,2 (π i ) q 1 (π i ) π i l(x2, y 2 ) l(x1, y 1 ) We re now redy to prove the following: Fig. 15: After drwing π i, Condition 3 is stisfied. Lemm 7 Suppose tht fter drwing pth π i one of Conditions 1 5 is stisfied. Then, fter drwing pth π i+1 either one of Conditions 1 5 is stisfied or ll the free points inside polygon π i+1 lie on specific grid line. First we prove tht fter drwing pth π 2 Condition 4 is stisfied. Clerly, such pth is p 0,1 1, where p 0,1 1 = p 1,0 1 c + (1, 1). Let S 0,1 nd S 1,0 e defined s in Section 4.2. Then, S x1,y 1 = S 0,1 nd S x2,y 2 = S 1,0 re ssocited with pth π 3, clerly stisfying Properties S1 S6. Further, p 0,1 1 is the lst occupied point of S 0,1 nd S 1,0 ; moreover, s d h, d v > 3, the line through p 0,1 1 with slope 1 1 hs nd to its right, nd oth S 0,1 nd S 1,0 hve free points. It follows tht, fter drwing π 2, Condition 4 is stisfied, with S 0,1 nd S 1,0 ssocited with pth π 3. Next, suppose tht fter drwing π i one of Conditions 1 5 is stisfied, where sequences S x1,y 1 nd S x2,y 2 re ssocited with π i+1 ; then, we rgue out the drwing of pth π i+1 nd out the sequences to e ssocited with π i+2.

20 158 Frizio Frti l 1,2 (π i ) l(x2, y 2 ) q 1 (π i ) π i l(x1, y 1 ) Fig. 16: After drwing π i, Condition 4 is stisfied. π i l(x2, y 2 ) l 1,2 (π i ) q 1 (π i ) l(x1, y 1 ) Fig. 17: After drwing π i, Condition 5 is stisfied. Suppose tht fter drwing π i Condition 1 is stisfied. Consider the first free point of S x1,y 1, tht is, point q 1 (π i+1 ) q 1 (π i ) (, y 1 ). Also consider the first free point of S x2,y 2, tht is, point q 2 (π i+1 ) q 2 (π i ) + (, y 2 ). Such points exist y the hypotheses of Condition 1. We will prove tht π i+1 psses through q 1 (π i+1 ) nd q 2 (π i+1 ), tht is, either π i+1 is q 1 (π i+1 )q 2 (π i+1 ), or π i+1 is q 1 (π i+1 ) with q 2 (π i+1 ) eing point of q 1 (π i+1 ), or π i+1 is q 2 (π i+1 ) with q 1 (π i+1 ) eing point of q 2 (π i+1 ). Denote y l 1,2 (π i ) nd l 1,2 (π i+1 ) the lines through q 1 (π i ) nd q 2 (π i ) nd through q 1 (π i+1 ) nd q 2 (π i+1 ), respectively. Clim 1 Pth π i+1 psses through q 1 (π i+1 ) nd q 2 (π i+1 ). Proof: Refer to Fig. 18. Since l 1,2 (π i ) hs slope y1+y2, y the hypotheses of Condition 1, the slope of l 1,2 (π i+1 ) is: y(q 2 (π i )) + y 2 (y(q 1 (π i )) y 1 ) x(q 2 (π i )) + (x(q 1 (π i )) ) = y 1 + y 2 + (y(q 2 (π i )) y(q 1 (π i ))) + + (x(q 2 (π i )) x(q 1 (π i ))) = y 1 + y 2 + m(y 1 + y 2 ) + + m( + ) = (m + 1)(y 1 + y 2 ) (m + 1)( + ) = y 1 + y 2, +

21 Lower Bounds on the Are Requirements of Series-Prllel Grphs 159 q 1 (π i ) q 2 (π i ) π i π i+1 l 1,2 (π i ) q 2 (π i+1 ) l 1,2 (π i+1 ) l(x2, y 2 ) q 1 (π i+1 ) l(x1, y 1 ) y 1 Fig. 18: Drwing of π i+1 when Condition 1 holds. By Property S3, nd y2 re the generting frctions of y1+y2, hence, y Lemm 6, l 1,2 (π i ) nd l 1,2 (π i+1 ) re consecutive grid lines. Then no grid point is internl to polygon (q 1 (π i ), q 2 (π i ), q 2 (π i+1 ), q 1 (π i+1 )). As tringles (, q 1 (π i ), q 1 (π i+1 )) nd (, q 2 (π i ), q 2 (π i+1 )) re enclosed in T (S x1,y 1, ) nd in T (S x2,y 2, ), respectively, polygon π i (, q 1 (π i+1 ), q 2 (π i+1 ), ) contins no grid point. Hence, s long s (, q 1 (π i+1 ), q 2 (π i+1 ), ) is convex polygon, we hve π i+1 = (, q 1 (π i+1 ), q 2 (π i+1 ), ). Consider the possile plcements of nd with respect to l 1,2 (π i+1 ). Neither nor is to the left of l 1,2 (π i+1 ), s such vertices re to the right of l 1,2 (π i ), y the hypotheses of Condition 1, nd hence, if they were to the left of l 1,2 (π i+1 ), they would e in the open strip delimited y l 1,2 (π i ) nd l 1,2 (π i+1 ), which re consecutive grid lines, thus contrdicting Lemm 6. Hence, either nd re oth on l 1,2 (π i+1 ), or one of nd is on l 1,2 (π i+1 ) nd the other one is to the right of such line, or oth nd re to the right of l 1,2 (π i+1 ). It follows tht (, q 1 (π i+1 ), q 2 (π i+1 ), ) is convex polygon nd hence tht π i+1 = (, q 1 (π i+1 ), q 2 (π i+1 ), ). Now we discuss which condition is stisfied fter drwing π i+1, discussing the cse in which nd re oth on l 1,2 (π i+1 ), the cse in which one of nd is on l 1,2 (π i+1 ) nd the other one is to the right of such line, nd the cse in which oth nd re to the right of l 1,2 (π i+1 ). First, we prove the following: Clim 2 Vertices nd cn not e oth on l 1,2 (π i+1 ). Proof: Suppose, for contrdiction, tht nd re oth on l 1,2 (π i+1 ). Then, we hve tht q 1 (π i+1 ) nd q 2 (π i+1 ) re oth on segment. However, this implies tht q 1 (π i+1 ) nd q 2 (π i+1 ) re not inside T 1, thus violting Property S4, contrdiction. Second, consider the cse in which nd re oth to the right of l 1,2 (π i+1 ). Suppose tht oth S x1,y 1 hve free points s in Fig. 18. Then we hve the following: Clim 3 After drwing π i+1 Condition 1 is stisfied with S x1,y 1 ssocited with pth π i+2.