CCCG 207, Ottawa, Ontaio, July 26 28, 207 Upwad ode-peseving 8-gid-dawings of binay tees Theese Biedl Abstact This pape concens upwad ode-peseving staightline dawings of binay tees with the additional constaint that all edges must be outed along edges of the 8-gid (i.e., hoizontal, vetical, diagonal) o some subset theeof. We give an algoithm that daws n-node tees with width O(log 2 n), while the pevious best esult wee dawings of width O(n 0.48 ). If only some of the gid-lines ae allowed to be used, then the algoithm gives (with mino modifications) the same uppe bounds fo the {,, }-gid and the {,, }-gid. On the othe hand, in the {,, }-gid sometimes Ω( n/ log n) width is equied. Intoduction Thee ae many algoithms to daw tees, especially ooted tees, because diffeent applications impose diffeent equiements on the dawing. In this pape, the dawing should satisfy the following constaints: It is plana, i.e., no vetices o edges ovelap unless the coesponding gaph-elements do. It is staight-line, i.e., evey edge is dawn as a staight-line segment that connects the coesponding vetices. It is stictly-upwad, i.e., paents have lage y- coodinates that childen. (Fo some of the esults this is elaxed to upwad dawings whee edges may be hoizontal.) It is ode-peseving, i.e., a given left-to-ight ode of childen at each node must be espected in the dawing. Such dawings wee called ideal dawings peviously [3]. All dawings in this pape must be plana and staightline, and this will not always be mentioned. Futhe, vetices must always be placed at gid-points, i.e., with intege coodinates. Any dawing is assumed (afte possible tanslation) to eside within the [, W ] [, H]-gid David R. Cheiton School of Compute Science, Univesity of Wateloo, Wateloo, Ontaio N2L A2, Canada. biedl@uwateloo.ca Suppoted by NSERC. The autho would like to thank Timothy Chan and Stephanie Lee fo inspiing discussions. whee W and H ae the width and height. Column i consists of all gid-points with x-coodinate i; ow j consists of all gid-points with y-coodinate j. Since the height may have to be Ω(n) in a stictly-upwad dawing, the objective of this pape is to find ideal dawings of binay tees that have small width. Thee ae many esults concening how to daw ooted tees; see fo example [5] fo an oveview, [2] fo some ecent esults, and Table fo the esults especially elevant to this pape. This pape focuses on giddawings, which means that the edges must be dawn along the lines of a gid. This is well-studied fo socalled othogonal dawings, whee the gid is the ectangula gid (also called the 4-gid) and hence all edges ae hoizontal o vetical. Ceszenci et al. [4] showed that evey binay n-node tee has an upwad staight-line 4-gid-dawing in an O(log n) O(n)-gid (the dawing need not be ode-peseving). Fo complete binay tees as well as fo Fibonacci tees, they achieve an O( n) O( n)-gid. Fo ode-peseving dawings, significantly moe aea may be needed: Fati [6] showed that Ω(n) width and height is necessay fo some binay tees in an upwad staight-line 4-gid dawing. The focus of this pape is the octagonal gid o 8- gid that has hoizontal, vetical and diagonal lines in both diections. Dawings in the 8-gid could also be called ASCII-dawings, since they could easily be done in ASCII using chaactes / _ \. Ceszenci et al. [4] ague that thei upwad 4-gid-dawings can easily be conveted into stictly-upwad 8-gid-dawings via a downwad shea. This peseves the same width and gives asymptotically the same height, hence any binay tee has an (unodeed) stictly-upwad dawing in an O(log n) O(n)-gid. Fo ode-peseving dawings, only much weake bounds ae known. Chan [3] studied ideal dawings of binay tees (not necessaily with edges along the gid). As he points out, the fist and second of his fou algoithms adapt easily to ceate ASCII-dawings of binay tees. The width of these depends much on the chosen spine (a concept that will be used in Theoem as well); with a suitable choice Chan achieves ideal 8-gid-dawings of width O(n 0.48 ) and height O(n). Results of this pape: In this pape, we show how to ceate ideal 8-gid-dawings of binay tees. The pevious best known bounds hee ae dawings of width 232
29 th Canadian Confeence on Computational Geomety, 207 Gid-lines Upwad? Ode-peseving width uppe bound width lowe bound {, } upwad no O(log n) [4] Ω(log n) [4] {, } upwad yes O(n) (folkloe) Ω(n) [6] {, } stictly-upwad no O(log n) [4] Ω(log n) [4] {, } stictly upwad yes O(n) (folkloe) Ω(n) (Thm.2) {, } stictly upwad yes O(n) (folkloe) Ω(n) (Thm.2) {,, } stictly upwad yes O(n 0.48 ) [3] Ω(log n) [4] O(log 2 n) (Thm.) {,, } upwad yes O(log 2 n) (Thm.3) Ω(log n) [4] {,, } upwad no O(n) (folkloe) Ω( n/ log n) (Thm.4) Table : Results fo plana, upwad, staight-line gid-dawings of binay tees. Some moe esults can be deived in the obvious way, e.g. the uppe bound fo the {,, }-gid also holds fo the 8-gid. O(n 0.48 ) [3]. This pape impoves this to ceate dawings that have width O(log 2 n). In fact, the width is pw(t ) 2, whee the ooted pathwidth pw(t ) is a lowebound on the width of any upwad dawing of a tee T (even if it need not be ode-peseving o staightline). Since pw(t ) log(n+) [2], ou algoithm can be viewed as an (log(n+))-appoximation algoithm fo the width of ideal 8-dawings. We also study what happens if one set of the paallel gid-lines is emoved; depending on which set is emoved we can eithe achieve the same width-bound o ague that a lowe bound of Ω( n/ log n) holds on the width. See Table. 2 Backgound Let T be a ooted tee with n nodes that is binay, i.e., evey node has at most two childen. Fo any node v, use T v to denote the subtee ooted at v. The ooted pathwidth pw(t ) of a ooted tee is defined as follows [2]: If 3 evey node of T has at most one child, then pw(t ) =. (In othe wods, T 3 is a path fom the oot to the unique 2 3 leaf.) Othewise, set pw(t ) := + min P max T T P pw(t ). Hee the minimum is taken ove all paths P fo which one end is at the oot of T, and the maximum is taken ove all subtees that esult when deleting all nodes of P fom T. A path P whee the minimum is achieved is sometimes called an pw-main-path, though this pape uses the tem spine to mimic the notations of [7]. The figue above shows a tee with the ooted pathwidth indicated fo all nodes; one possible spine is bold. 3 Ideal 8-gid dawings of binay tees Theoem Let T be a ooted binay tee. Then T has an ideal 8-gid-dawing of width at most (pw(t )) 2. 2 3 2 2 Poof. The poof is stongly inspied by the algoithm of Gag and Rusu [7] that give ideal dawings of binay tees of width O(log n). (Thei dawings ae not necessaily in the 8-gid.) Thei key idea was to use dawings that ae stetchable in the sense that fo any given α 0 one can pescibe the contents of the top α ows of the dawing. This then allows to mege dawings of subtees in a ecusive constuction. Fo gid-dawings we need a slightly modified definition as follows: Definition Let T be a ooted binay tee with pw(t ) =, and let α 0 be given. An 8-gid-dawing of T is called a left-α-dawing if within the fist α ows, all points in columns + and futhe to the ight ae unused. Put diffeently, within the top α ows, only the leftmost columns may be used fo placing vetices and edges. Note that (in contast to [7]) this definition of a left-α-dawing makes no estictions whee the oot must be placed (othe than that it must be within the leftmost columns). Define symmetically a ight-αdawing to be one whee within the fist α ows only the ightmost columns may be used. We need two moe types of dawings. Define a leftcone-dawing and a ight-cone-dawing to be a dawing of T whee the oot is at the top-left (top-ight) cone. The main claim, to be poved by induction on pw(t ), is the following: Claim Fix an abitay α 0. Then T has a left-αdawing, a ight-α-dawing, a left-cone-dawing and a ight-cone-dawing, and all fou dawings have width at most (pw(t )) 2. To pove this claim, conside the base case whee pw(t ) =. This implies that T is a path fom the oot to a single leaf. Such a path can easily be dawn Inspection of the constuction given below eveals that the oot is always in column o, but we will not make use of this. 233
CCCG 207, Ottawa, Ontaio, July 26 28, 207 with width = 2, and this satisfies the conditions fo all fou dawings. Now assume that := pw(t ) 2. Fom the definition of ooted pathwidth, we know that T has a spine P such that all subtees T of T P satisfy pw(t ) pw(t ). Let the vetices of P be v 0 v... v m whee v 0 is the oot. 2 Fo simplicity of desciption, assume that evey spine-node v i v 0 has a sibling; if it does not then simply add a sibling and delete it in the obtained dawing late. Adding a sibling that is a leaf does not affect the ooted pathwidth since pw(t ) 2, so this does not affect the width-bound. Thus fom now on evey spine-node except v m has a left and a ight child. We explain hee only how to ceate the left-α-dawing and the left-cone-dawing; the othe two dawings ae obtained in a symmetic fashion. Thee ae thee cases, depending on whethe v is the left o ight child of v 0, and which type of dawing is desied. Case : v is the left child of v 0. In this case, the same constuction woks fo both a left-α-dawing and a left-cone dawing (fo the latte, use α := below). Place the oot v 0 at the top-left cone. Let s 0 be the ight child of v 0, and ecusively obtain a left-α - dawing D(T ) of T, whee α = α. Place D(T ), flush left with column 2 and sufficiently fa below such that the -diagonal fom v 0 ends exactly at s 0. Next obtain ecusively a left-cone-dawing D(T v ) of T v, and place it below D(T ), flush left with column. Connect (v 0, v ) vetically (both ae in column ). This finishes the constuction. Note that pw(t ) pw(t ) = by definition of ooted pathwidth and the spine. Theefoe D(T ) uses only the leftmost columns within the fist α ows. So this gives a left-α-dawing with the oot in the top left cone, as desied. As fo the width, D(T ) has width at most ( ) 2 while D(T v ) has width at most 2 ; theefoe the width is at most max{+( ) 2, 2 } = 2 as desied. Case 2: v is the ight child of v 0, and we want a left-cone-dawing. Let s 0 be the left child of v. Recusively find a left-cone-dawing D(T ) of T, say that it has height H. D(T ) has width at most ( ) 2 since pw(t ) < pw(t ). Recusively find a ight-h - dawing D(T v ) of T v of width 2. (If its width is smalle than 2, then pad it with empty columns on the left.) Thus within the topmost H ows of D(T v ), the leftmost 2 > ( ) 2 columns ae empty. D(T ) fits within this empty space; place it flush left with column. Finally place v 0 vetically above s 0 (i.e., in column 2 The notation hee is the same as in [7], though thei spine is chosen diffeently as to always use the heaviest child, athe than the one that has the lagest ooted pathwidth. ) and high enough so that the -diagonal fom v 0 ends exactly at v. This gives a left-cone-dawing of width 2 as desied. v D(T ) D(Tv ) α D(T ) D(Tv ) Figue : The constuction in Case and Case 2. Case 3: v is the ight child of v 0, and we want an α- dawing. This is the most complicated case whee a longe section of the spine may get dawn befoe ecusing. Figue 2 illustates the constuction. Recall that evey spine-vetex v i v 0 has a sibling by assumption; as in [7] let s i be the sibling of v i. Let k be the smallest intege such that v k is eithe v m o s k is the left child of v k. Fist place vetices v,..., v k of the spine; vetex v 0 will be added late. Thus, place v in column. Now epeat fo i k : ecusively find a left-conedawing D(T si ) of T si, place it flush left with column + and one ow below v i, then place v i+ in column and in the last ow used by D(T si ). This ends with vetex v k having been placed in column. Extend a -diagonal fom v k ; this will late be used to complete edge (v k, v k+ ). Next, ecusively obtain a left-conedawing D(T sk ) of T sk, and place it, flush left with column and ( ) 2 ows below the ow of v k. Note that D(T sk ) has width at most ( ) 2. Theefoe (in the dawing of width 2 that is being ceated) thee ae 2 ( ) 2 ( ) = columns fee to the ight of D(T sk ). These will be used fo T vk+ late. Also note that in the topmost ow of D(T sk ), the -diagonal fom v k is within the ightmost ows, and hence it does not intefee with D(T sk ). Let H be the total numbe of ows that ae used thus fa, i.e., fom the ow of v to the bottommost ow of D(T sk ). Note that columns,..., ae (thus fa) entiely fee. Recusively find a left-α -dawing of T, whee α = H + α. Place it, stating α ows above v and flush left with column. Within the top α ows this uses only columns,..., by pw(t ) < pw(t ), and hence this does not intesect the peviously placed subtees. Place v 0 vetically above v (i.e., in column ) and high enough so that the -diagonal fom v 0 ends exactly v 234
29 th Canadian Confeence on Computational Geomety, 207 at s. Let H be the numbe of ows fom the ow of s k to the bottommost ow of D(T ). Recusively find a ight-h -dawing D(T vk ) of T vk. Place it, flush ight with the ightmost column, and in the ow of s k o below such that the -diagonal extending fom v k exactly meets the point containing v k. Within the topmost H ows, dawing D(T ) uses only the ightmost columns. Recall that columns emained fee next to D(T sk ), and also columns ae fee next to D(T ) since this dawing has width at most ( ) 2. Thus dawing D(T vk ) does not intefee with peviously placed dawings. This gives the desied left-α-dawing of width 2. This ends the constuction fo all cases and poves Theoem. α s 0 v v 0 One can easily ague that the height is at most O(n (pw(t )) 2 ), because (as one can see) any ow without vetex in it intesects a diagonal edge, any such diagonal edge intesects at most pw(t ) 2 ows, and these ows can be assigned to the uppe endpoint of the diagonal edge. If one follows the constuction exactly as descibed, then Ω(n 2 ) height (fo = pw(t )) may esult. (Fo example, conside a tee whee the spine has length Ω(n) and nealy all siblings of spine-vetices ae leaves, but the last few siblings have big enough subtees to foce ooted pathwidth.) Howeve, thee ae some obvious possible impovements to the height. To give just one, in Case 2 the dawing D(T ) could be moved much highe, diectly unde the -diagonal, because due to the stict-upwadness of the dawing, the ith ow of it is empty in column i+ and fathe ight. This alone is not enough to ensue a smalle height, but we suspect that combining this with dawing the spine moe caefully when some siblings have vey small size may lead to a dawing of width O(log 2 n) and height O(n). This emains fo futue wok. s D(T s ) 4 Ideal 6-gid dawings of binay tees H v 2 v k D(T ) s 2 D(T s2 ) s k v k+ D(T sk ) ( ) 2 D(T vk+ ) ( ) 2 + Figue 2: The constuction in Case 3. H Now we tun to the 6-gid, which has gid-lines with angles of 60 between them. Fequently it is easie to think of it instead as a gid that has thee of the fou sets of gid-lines of the 8-gid (e.g., hoizontal, ightwad, and -diagonals). Bachmeie et al. [] studied 6-gid dawings of tees. Thei dawing wee not (necessaily) upwad, and as such, it was ielevant which of the gidlines of the 8-gid ae used fo the 6-gid, since they ae all the same afte 90 otation and/o a shea, and a shea does not affect the asymptotic aea. In contast to this, we study hee upwad dawings of binay tees in the 6-gid, and as befoe, focus on keeping the width small. As will be seen, hee it makes a diffeence exactly which gid-lines ae used to epesent the 6-gid. The following gids will be studied: The {,, }-gid: gid-lines ae vetical o along a 45 diagonal in eithe diection. The {,, }-gid: gid-lines ae hoizontal o along a 45 diagonal in eithe diection. Height-consideation: In most pevious tee-dawing papes, the height is easily shown to be O(n), because all ows (o nealy all ows) intesect at least one vetex. In contast to this, the constuction hee has many ows (e.g. most of the ows,..., 2 in the constuction fo Case 2) that intesect only edges. The {,, }-gid: gid-lines ae hoizontal o vetical o along one of the 45 diagonals. (Fo the {,, }-gid a symmetic set of esults is obtained by using a hoizontal flip.) 235
CCCG 207, Ottawa, Ontaio, July 26 28, 207 We have thus fa mostly studied ideal dawings, which must be stictly-upwad and hence hoizontal lines ae disallowed. In paticula, Theoem ceated stictlyupwad dawings in the 8-gid, which hence ae automatically dawings in the {,, }-gid. Theefoe evey binay tee T has an ideal dawing that is an embedding in the {,, }-gid and has width at most (pw(t )) 2. Now we tun to othe types of 6-gids. Again, having an ideal dawing means being stictly-upwad, so no hoizontal lines can be used. We show that then no small width is possible. v α D(T ) D(Tv ) α D(T ) D(Tv ) v Theoem 2 Thee exists a binay tee T such that any ideal dawing of T in the {, }-gid o the {, }-gid equies width and height Ω(n). Poof. Let T consist of a path of length n/2 and attach at each node a left child that is a leaf. Fo an odepeseving and stictly-upwad dawing, the path must be dawn following the -diagonals. This gives a width and height of at least n/2. D(T ) v v D(Tv ) D(Ts2 ) D(T ) Theefoe, the emaining dawing-esults will be in a elaxed model of ideal dawings whee hoizontal edges ae allowed, hence the dawing is upwad athe than stictly-upwad. Call these weakly-ideal dawings. (As befoe all dawings must be plana, staight-line and ode-peseving.) Theoem 3 Evey binay tee T has a weakly-ideal dawing that is an embedding in the {,, }-gid and has width at most (pw(t )) 2. Poof. The poof is vey simila to the poof of Theoem. As befoe, define (left/ight) cone-dawings and (left/ight) α-dawings. Additionally now demand fo all these dawings that in the topmost ow no point to the ight of the oot is occupied (we say that the oot is ight-fee). Ceate left-cone-dawings and left-α-dawings almost exactly as befoe. The only diffeence is that at the places whee a -diagonal was used befoe, we now use a hoizontal edges instead; this is feasible because the oot of coesponding subtee is ight-fee. Fo ightcone and ight-α-dawings, the constuctions ae not entiely symmetic anymoe, but again, by using hoizontal edges athe than diagonal ones, dawings can be constucted. Figue 3 illustates the constuctions in all cases; the details ae left to the eade. Finally conside the {,, }-gid, which is the same as the 8-gid whee no vetical edges ae allowed. Theoem 2 showed that ideal dawings have to have lage width. We show hee that even weakly-ideal dawings may equie lage with. In fact, the following bound holds fo any plana staight-line dawing in the α D(Tv ) v v2 vk sk D(T ) s D(Ts ) s2 D(Ts2 ) D(Tsk ) ( ) 2 D(Tvk+ ) vk+ α vk+ D(Ts ) s v D(Ts2 ) s2 v2 D(Tsk ) ( ) 2 D(Tvk+ ) vk sk D(T ) Figue 3: The constuction fo the {,, }-gid. {,, }-gid, even if it is not upwad o not odepeseving. Theoem 4 The complete binay tee must have width O( n/ log n) in any staight-line dawing in the {,, }-gid. Poof. The poof is vey simila to the simplest method fo obtaining a width-lowe-bound fo weaklyideal dawings of the complete tenay tee, see [8]. We 236
29 th Canadian Confeence on Computational Geomety, 207 epeat the agument hee fo completeness. Fix an abitay staight-line dawing of the complete binay tee in the {,, }-gid, say it has w columns. So any edge spans a hoizontal distance of at most w. Since only hoizontal and diagonal edges ae allowed, theefoe any edge spans a vetical distance of at most w. Obseve that T has height h := log ( ) n+ 2, i.e., the path fom the oot to each leaf contains h edges. In consequence, any node has vetical distance at most h(w ) fom the oot. Theefoe the entie dawing is contained within a ectangle that has w columns and up to h(w ) ows above and below the oot, hence 2h(w )+ ows in total. Theefoe the dawing esides in a gid with at most 2wh(w )+w gid points. Since all n nodes ae placed on these gid-points, necessaily n 2wh(w ) + w O(w 2 log n), which implies w Ω( n/ log n). We stongly suspect that a lowe bound of Ω( n) on the width holds, but this emains fo futue wok. Fo the complete binay tee, it is easy to find a constuction that has width and height O( n), and in fact, no hoizontal edges ae used. Theoem 5 (based on [4]) The complete binay tee has an ideal dawing in the {, }-gid of gid-size O( n) O( n). Poof. Cescenzi et al. gave a simple ecusive constuction that daws the complete binay tee in a 4- gid of size O( n) O( n) [4]. Moeove, all edges go ightwad o downwad. Scale this dawing by 2 and then otate it by 45 clockwise. Due to the scaling, this maps all vetices to gid-points, and all edges ae now diagonal and downwad as desied. 5 Remaks This pape developed algoithms fo weakly-ideal 8- gid-dawings of binay tees, i.e., plana upwad staight-line ode-peseving dawings with edges dawn along gid-lines fo the 8-gid (o some subset theefoe). We gave constuctions of width O(log 2 n) fo a numbe of such gids. The height is athe lage (O(n log 2 n)), and impoving this emains an open poblem. We also showed that width O( n/ log n) is equied fo the gid whee no vetical lines ae allowed. A natual question is whethe simila bounds could be poved fo tenay tees. Fo unodeed dawings, Bachmeie et al. [] gave simple ecusive constuctions that achieve width O(n log 3 2 ) O(n 0.63 ). In wok done simultaneously with the cuent pape, Lee studied odeed dawings of tenay tees and poved that evey tenay tee has such a weakly-ideal 8-gid dawing of width Ω(n 0.68 ) [8]. Futhemoe, the complete tenay tee equies width Ω(n 0.4 ) in any upwad octagonalgid-dawing [8]. Both the constuctions and the lowe bounds in Lee s thesis ae significantly moe complicated than the ones given hee, and will be published sepaately. As fo open poblems, the obvious one is to close the gap between the width O(log 2 n) achieved with ou algoithm and the lowe bound of Ω(log n) fo the complete binay tee. Ae thee binay tees that equie ω(log n) width in ideal 8-gid dawings? The othe emaining gap concens dawings in the {,, }-gid-gid. Can we achieve a width of O( n) not just fo complete binay tees but fo all tees? Refeences [] Chistian Bachmaie, Fanz-Josef Bandenbug, Wolfgang Bunne, Andeas Hofmeie, Maco Matzede, and Thomas Unfied. Tee dawings on the hexagonal gid. In Ioannis G. Tollis and Mauizio Patignani, editos, Gaph Dawing (GD 2008), volume 547 of Lectue Notes in Compute Science, pages 372 383. Spinge, 2009. [2] Theese Biedl. Ideal tee-dawings of appoximately optimal width (and small height). Jounal of Gaph Algoithms and Applications, 2(4):63 648, 207. [3] Timothy M. Chan. A nea-linea aea bound fo dawing binay tees. Algoithmica, 34(): 3, 2002. [4] Pieluigi Cescenzi, Giuseppe Di Battista, and Adolfo Pipeno. A note on optimal aea algoithms fo upwad dawings of binay tees. Comput. Geom., 2:87 200, 992. [5] Giuseppe Di Battista and Fabizio Fati. A suvey on small-aea plana gaph dawing, 204. CoRR epot 40.006. [6] Fabizio Fati. Staight-line othogonal dawings of binay and tenay tees. In Seok-hee Hong, Takao Nishizeki and Wu Quan, editos, Gaph Dawing (GD 2007), volume 4875 of Lectue Notes in Compute Science, pages 76 87, Spinge, 2007. [7] Ashim Gag and Adian Rusu. Aea-efficient odepeseving plana staight-line dawings of odeed tees. Int. J. Comput. Geomety Appl., 3(6):487 505, 2003. [8] Stephanie Lee. Upwad octagonal dawings of tenay tees. Maste s thesis, Univesity of Wateloo, August 206. (Supevisos: T. Biedl and T. Chan.) Available at https://uwspace.uwateloo.ca/handle/002/0832. 237