PLANE 3-TREES: EMBEDDABILITY & APPROXIMATION

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1 PLANE 3-TREES: EMBEDDABILITY & APPROXIMATION STEPHANE DUROCHER AND DEBAJYOTI MONDAL Astrt. We give n O(nlog 3 n)-time liner-spe lgorithm tht, given plne 3-tree G ith n verties nd set S of n points in the plne, determines hether G hs point-set emedding on S (i.e., plnr stright-line dring of G here eh verte is mpped to distint point of S), improving the O(n 4/3+ε )-time O(n 4/3 )-spe lgorithm of Moos nd Rhmn (2011). Given n ritrr plne grph G nd point set S, Kufmnn nd Wiese (2002) gve n lgorithm to ompute 2-end point-set emeddings of G on S. Lter, Di Giomo nd Liott (2010) shoed ho suh dring n e omputed using O(W 3 ) re, here W is the length of the longest edge of the ounding o of S. Their lgorithm uses O(W 3 ) re even hen the input grphs re restrited to plne 3-trees. We introdue ne tehniques for omputing 2-end point-set emeddings of plne 3-trees tht tkes onl O(W 2 ) re. We lso give pproimtion lgorithms for point-set emeddings of plne 3-trees. Our results on 2-end point-set emeddings nd pproimte point-set emeddings hold for prtil plne 3-trees (e.g., series-prllel grphs nd Hlin grphs). Ke ords. grph dring, point-set emedding, plne 3-tree, pproimtion AMS sujet lssifitions. 05C10, 68R10 1. Introdution. A plnr dring of grph G is n emedding (i.e., mpping) of G onto the Euliden plne R 2, here eh verte in G is ssigned unique point in R 2 ndehedgein Gissimpleurvein R 2 joiningthe points orresponding to its endverties suh tht no to urves interset eept possil t their endpoints. A grph is plnr if it hs plnr dring. A stright-line dring of plnr grph is plnr dring, here eh edge is drn s stright line segment. The stright-line dring stle is populr sine it nturll produes drings tht re esier to red nd to displ on smller sreens [13, 33, 36]. To meet the requirements of different prtil pplitions, reserhers hve emined grph dring prolems under vrious onstrints, e.g., hen the verties re onstrined to e pled on set of pre-speified lotions [4, 9], or hen ijetion eteen the verties nd lotions re given [32]. If the pre-speified lotions for pling the verties of the input grph re points on the Euliden plne, then e ll the prolem point-set emedding prolem. Suh prolems hve pplitions in VLSI iruit lout, here different iruits need to e mpped onto fied printed iruit ord [25], simultneous displ of different soil nd iologil netorks [7], nd onstrution of desired netork mong set of fied lotions. Formll, point-set emedding of plne grph G (i.e., fied omintoril plnr emedding of G) ith n verties on set S of n points is stright-line dring of G, here the verties re pled on distint points of S. Figures 1(), () nd () illustrte plne grphg, point set S nd point-set emedding of G on S, respetivel. Point-Set Emeddings. In 1994, Ikee et l. [24] gve n O(n 2 )-time lgorithm to emed n tree ith n verties on n set of n points in generl position, i.e., no three A preliminr version of the pper ppered in Dejoti Mondl s M.S. thesis [28] nd in the Proeedings of the 13th Interntionl Smposium on Algorithms nd Dt Strutures (WADS 2013) [20]. Deprtment of Computer Siene, Universit of Mnito, Cnd, duroher@s.umnito.. Work of the uthor is supported in prt the Nturl Sienes nd Engineering Reserh Counil of Cnd (NSERC). Deprtment of Computer Siene, Universit of Mnito, Cnd, joti@s.umnito.. Work of the uthor is supported in prt Universit of Mnito Grdute Felloship. 1

2 2 Plne 3-trees: Emeddilit & Approimtion d e g f f f g g d f e g d e () () () (d) (e) (f) e d Fig. 1. () A plne grph G. () A point set S. () A point-set emedding of G on S. (d) A point set S. (e) A 2-end point set emedding of G on S. (f) A stright-line emedding of G, here S (Γ) = 5. The verties tht re not mpped to n points of S re shon in lk squre, nd the points of S tht does not ontin n verte of G re shon in gr. points re olliner. Lter, Bose et l. [5] devised divide nd onquer lgorithm tht runs in O(nlogn) time. In 1996, Cstñed nd Urruti [10] gve n O(n 2 )- time lgorithm to onstrut point-set emeddings of miml outerplnr grphs. Lter, Bose [4] improved the running time of their lgorithm to O(nlog 3 n) using dnmi onve hull dt struture. In the sme pper Bose posed n open prolem tht sks to determine the time ompleit of testing the point-set emeddilit for plnr grphs. In 2006, Cello [9] proved the prolem to e NP-omplete for grphs tht re 2-onneted nd 2-outerplnr. The prolem remins NP-omplete for 3-onneted plnr grphs [19], even hen the treeidth is onstnt [3]. In the lst fe ers reserhers hve emined the point-set emeddilit prolem restrited to plne 3-trees (lso knon s stked poltopes, Apollonin netorks, nd miml plnr grphs ith treeidth three) euse of their ide rnge of pplitions in mn theoretil nd pplied fields [1, 2, 14], e.g., in struture nlsis of poldisperse grnulr pkings, s model for porous medi nd for the nlsis of eletril suppl sstems. Nisht et l. [30] first gve n O(n 2 )-time lgorithm for deiding point-set emeddilit of plne 3-trees, nd proved n Ω(n log n)-time loer ound. Lter, Duroher et l. [21] nd Moos nd Rhmn [29] independentl improved the running time to O(n 4/3+ε ), for n ε > 0. Sine Ω(n 4/3 ) is loer ound on the orst-se time ompleit for solving vrious geometri prolems [22], it m e nturl to ept the possiilit tht the O(n 4/3+ε )-time lgorithm ould e smptotill optiml. In ft, Moos nd Rhmn mention tht n o(n 4/3 )-time lgorithm seems unlikel using urrentl knon tehniques. Hoever, in this pper e prove tht the Ω(nlogn) loer ound is nerl tight, giving n O(nlog 3 n)-time lgorithm for deiding point-set emeddilit of plne 3-trees. Theorem 2.4 Let G e plne 3-tree ith n verties nd let S e set of n points in generl position in R 2. We n deide in O(nlog 3 n) time nd liner spe, hether G dmits point-set emedding on S nd ompute suh n emedding if it eists. Universl Point Set. Oserve tht plnr grph m not ls dmit point-set emedding on given point set. Attempts hve een mde t onstruting set S of k n points suh tht ever plnr grph ith n verties dmits point-set emedding on suset of S [6, 12, 18, 23, 27]. Suh point set tht supports ll plnr grphs ith n verties is lled universl point set of sie n. A long stnding open question in grph dring sks to design set of O(n) points tht is universl for ll plnr grphs ith n verties [6]. The est knon upper ound on the sie of universl point set is O(n 2 ), hih is implied eisting lgorithms for dring plnr grphs on n O(n) O(n) integer grid [13, 33]. Reentl, Everett et l. [23] hve designed 1-end universl point set S n for plnr grphs ith n verties, i.e.,

3 ever plnr grph ith n verties dmits stright-line dring on S n suh tht eh verte is mpped to distint point nd eh edge is drn s hin of t most to stright line segments. The point-set emeddilit prolem seems to hve lose reltion ith the universl point set prolem. Cstñed nd Urruti [10] proved tht n set of n points in generl position is universl for ll outerplnr grphs ith n verties. Lter, Kufmnn nd Wiese [26] proved tht n set S of n points is 2-end universl for n (i.e., ever plnr grph ith n verties dmits stright-line dring on S suh tht eh verteis mpped to distint point nd eh edge is drns hin of t most three stright line segments, s shon in Figures 1(d) (e)). Hoever, the re required for the dring ould e eponentil in W, here W is the length of the side of the smllest is-prllel squre tht enloses S. Di Giomo nd Liott [16, Theorem 7] shoed tht using the onept of monotone topologil ook emedding [15] one n redue the re requirement to O(W 3 ). Even hen restrited to simpler lsses of grphs (e.g., series prllel grphs or plne 3-trees), the tehnique of Di Giomo nd Liottis the est knon, hih still requireso(w 3 ) re. In this pper, eontriute ne tehnique tht uses onl O(W 2 ) re to ompute 2-end point set emeddings of plne 3-trees, nd hene lso for prtil plne 3-trees (e.g., series-prllel grphs nd Hlin grphs). Theorem 3.1 Given plne 3-tree G ith n verties nd point set S of n points in generl position, e n ompute 2-end point-set emedding of G in O(nlog 3 n) time ith O(W 2 ) re, here W is the length of the side of the smllest is-prllel squre tht enloses S. Approimte Point-Set Emeddings. Although n set of n points in generl position is universl for n-verte outerplnr grphs [10], plne 3-tree ith n verties m not dmit point-set emedding on given set of n points [30]. On the other hnd, hile lloing to ends per edge, n set of n points in generl position is 2-end universl for plne 3-trees. Due to this pprent diffiult of defining lgorithms tht simultneousl minimie re, the numer of ends, nd running time, e onsider lgorithms tht provide pproimte solutions, tht is, t lest frtion ρ of the verties of the input grph re mpped to distint points of the given point set. Speifill, let G e plne grph nd let Γ e stright-line dring of G. Let S e set of n points in generl position. Then S(Γ) denotes the numer of verties in Γ tht re mpped to distint points of S, e.g., see Figure 1(f). The optiml point-set emedding of G is stright-line dring Γ suh tht S(Γ ) S(Γ ) for n strightline dring Γ of G. A ρ-pproimtion point-set emedding lgorithm omputes stright-line dring Γ of G suh tht S(Γ)/S(Γ ) ρ. As opposed to emedding onl sugrph of G (hih ould e highl disonneted nd, therefore, signifintl esier to emed) the output is stright-line plnr emedding of the entire grph G for hih the frtion of the verties of G drn on points of S is optimied. This distintion is importnt, sine otherise it ould suffie to mp the verties of n independent set of sie n/4 onto S to hieve onstnt-ftor pproimtion. We sho tht given plne 3-tree G ith n verties, e n onstrut strightline dringγof G suh tht S(Γ) is Ω( n), nd hene the point-set emeddilit is pproimle ith ftor Ω(1/ n) for plne 3-trees. Speifill, if the input points re in generl position, then e prove tht the point-set emeddilit of plne 3-trees is pproimle ith ftor Ω(1/ n). Theorem 4.3 If the input points re in generl position, then the point-set emeddilit of plne 3-trees is pproimle ith ftor Ω(1/ n). 3

4 4 Plne 3-trees: Emeddilit & Approimtion 2. Fster Point-Set Emeddings of Plne 3-Trees. Nisht et l. [30] gve n O(n 2 ) lgorithm for deiding point-set emeddilit of plne 3-trees. Reentl, Duroher et l. [21] nd Moos et l. [29] proposed different tehniques to improve the running time of the lgorithm of Nisht et l. to O(n 4/3+ε ), for n ε > 0, using tringulr rnge serh dt struture. We use the si tehniques of Nisht et l. s lgorithm, ut e do not use n rnge serh dt struture. Insted, e mke some simple ut ruil oservtions tht help eploit dnmi onve hull dt struture to hieve even fster lgorithm. Our lgorithm tkes O(nlog 3 n) time to deide point-set emeddilit of plne 3-trees, nd onstruts the emedding ithin the sme time if suh n emedding eists. Before going into detils, e revie fe definitions. A plne 3-tree G ith n 3 verties is tringulted plne grph suh tht if n > 3, then G ontins verte hose deletion ields plne 3-tree ith n 1 verties. Let r,s,t e le of three verties in G. B G rst e denote the sugrph indued r,s,t nd the verties tht lie interior to the le. Ever plne 3-tree G ith n > 3 verties ontins verte tht is the ommon neighor of ll the three outer verties of G. We ll this verte the representtive verte of G. Let p e the representtive verte of G nd let,, e the three outer verties of G in lokise order. Then eh of the sugrphs G p,g p nd G p is plne 3-tree. Let S e set of n points in the plne. Let p,q nd r e three points tht do not neessril elong to S. Then S(pqr) onsists of the points of S tht lie either on the oundr or in the interior of the tringle pqr. Overvie of Knon Algorithms. Let G e plne 3-tree ith n verties, nd let,,ndpe the threeoutervertiesndthe representtiveverteofg, respetivel. Nisht et l. [30] used the folloing steps to ompute point-set emedding of G on S. Step 1. If the numer of points on the oundr of the onve hull C of S is not etl three, then G does not dmit point-set emedding on S. Otherise, let,, e the points on C. Step 2. For eh of the possile si different mppings of the outer verties,, to the points,,, eeute Step 3. Step 3. Let n 1,n 2 nd n 3 e the numer of verties of G p,g p nd G p, respetivel. Without loss of generlit ssume tht the urrent mpping of, nd is to, nd, respetivel. Find the unique mpping of the representtive verte p of G to point S suh tht the tringles, nd ontin etl n 1,n 2 nd n 3 points, respetivel. If no suh mpping of p eists, then G does not dmit point-set emedding on S for the urrent mpping of,,; hene go to Step 2 for the net mpping. Otherise, reursivel ompute point-set emeddings of G p,g p nd G p on S(), S() nd S(), respetivel. See Figures 2() (d). The time ompleit of n lgorithm tht uses Steps 1 3 is dominted the ost of Step 3 nd the ottlenek of this step is the reursive omputtion of the vlid mppings. Oserve tht the reurrene reltion for the time tken in Step 3 is T(n) = T(n 1 ) + T(n 2 ) + T(n 3 ) + T, here T denotes the time required to find the mpping of the representtive verte. The lgorithm of Nisht et l. [30] preproesses the set S in O(n 2 ) time so tht the omputtion for the mpping of representtive verte tkes O(n) time. Hene T = O(n) nd the overll time ompleit eomes O(n 2 ). Moos nd Rhmn [29] used inr serh tehnique

5 5 f p d h e p f p d e h () () () (d) Fig. 2. () A plne 3-tree G. () A point set S. () (d) A vlid mpping of the representtive verte of G, nd the reursive omputtion of the three suprolems. ith the help of tringulr rnge serh dt struture of Chelle et l. [11] to otin T = min{n 1,n 2,n 3 } n 1/3+ε nd T(n) = O(n 4/3+ε ). Duroher et l. [21] use the sme ide, ut insted of inr serh the use rndomied serh. Emedding Plne 3-trees in O(nlog 3 n) time. We speed up the mpping of the representtive verte s follos. We first selet O(min{n 1,n 2,n 3 }) points interior to the tringle in O(min{n 1 + n 2,n 2 + n 3,n 1 + n 3 } log 2 n) time using dnmi onve hull dt struture. We prove tht these re the onl ndidtes for the mpping of the representtive verte. We then mke some non-trivil oservtions to test nd ompute mpping for the representtive verte in O(min{n 1,n 2,n 3 }) time. Hene e otin T = O(min{n 1 + n 2,n 2 + n 3,n 1 + n 3 }log 2 n) nd running time of T(n) = O(nlog 3 n), hih domintes the O(nlog 2 n) time for uilding the initil dnmi onve hull dt struture. In the folloing e use three lemms to otin our min result. Lemm 2.1 selets region R ontining the ndidte points inside the tringle. Lemm 2.2 redues the prolem of finding mpping inside the tringle to the prolem of finding point stisfing speifi riteri inside R. Lemm 2.3 gives n effiient tehnique to find suh point. Finll, e use these lemms to otin mpping for the representtive verte in O(min{n 1 +n 2,n 2 +n 3,n 1 +n 3 }log 2 n) time. Without loss of generlit ssume tht n 3 n 2 n 1. Oservetht n 1 +n 2 +n 3 5 = n. Let S e set of n points in generl position suh tht the onve hull of S ontins etl three points,, on its oundr. Without loss of generlit ssume tht the verties outer verties,, re mpped to the points,,, respetivel. Let u nd v e to points on the stright line segment suh tht S(u) = n 1 1nd S(v) = n 2 1,sshoninFigure3(). Notethtnoneofundv elong to S. It is strightforrd to verif tht if vlid mpping for the representtive verte eists (i.e, there eists point S suh tht S() = n 1, S() = n 2 nd S() = n 3 ), then the orresponding point (i.e., the point ) must lie inside S(uv). Let r nd s e to points on the stright line segments u nd v, respetivel, suh tht S(ru) = S(sv) = n 3 1. We ll the region defined the simple polgon,u,v,,s,,r, the region of interest. An emple is shon in Figures 3(). The folloing lemm shos tht the region of interest must ontin the edges orresponding to vlid mpping. Lemm 2.1. If there eists point S tht orresponds to vlid mpping for the representtive verte of G, then the stright line segments, nd lie inside the region of interest R. Moreover, the numer of points in R tht elong to S is O(n 3 ), nd the folloing properties hold. () If the points s,, (respetivel, points r,,) re distint, then S(s) = n 2 n 3 +2 (respetivel, S(r) = n 1 n 3 +2).

6 6 Plne 3-trees: Emeddilit & Approimtion n 3 1 n 1 1 u n 3 1 n 2 1 v r s u v n 3 1 r s () () () (d) Fig. 3. () () Illustrtion for the lines u,v,r nd s. The region of interest is shon in gr. () (d) Illustrtion for the proof of Lemm 2.1. () Otherise, point s (respetivel, point r) oinides ith (respetivel, ) nd S(s) = 2 (respetivel, S(r) = 2). Proof. The point must e in S(uv). Otherise, either S() < n 1 or S() < n 2 holds, hih implies tht does not orrespond to vlid mpping. We no lim tht the stright line segments, nd lie interior to R, s shon in Figure 3(). Sine S(uv), the stright line segment must lie inside R. Suppose for ontrdition tht either one of, or oth properl rosses the oundr of R. In this sitution S() must ontin one of S(ru) nd S(sv) impling tht S() > n 3, s shon in Figure 3(d). Consequentl,, nd must lie interior to R. B the onstrution of R, the numer of points tht lie on the oundr nd the interior of R is t most (n 3 1)+2(n 3 1) = O(n 3 ). We no determine the vlue of S(s). If the points s,, re distint, then the tringle s ontins (n 2 1) (n 3 1) 1 points of S in its proper interior nd three points (i.e.,, nd the other point on line s) of S on its oundr. Therefore, S(s) = n 2 n Otherise, if s oinides ith, then n 3 = n 2 nd S(s) onsists of etl to points of S, i.e., nd. Sine nd re distint, e re onl left ith the se hen s oinides ith. But this se does not rise sine s is point of the segment v, nd v does not oinide ith sine n 4 nd n 3 3. We n determine the vlue of S(r) in similr. Let S S e the set tht onsists of the points ling on the oundr of R nd the points ling in the proper interior of R. We ll S the set of interest. B Lemm 2.1, S = O(n 3 ). We redue the prolem of finding vlid mpping in S to the prolem of finding point ith ertin properties in S, s shon in the folloing lemm. Lemm 2.2. There eists vlid mpping for the representtive verte of G in S if nd onl if there eists point S suh tht S ( ) = n 2 S(s) +3, S ( ) = n 1 S(r) +3 nd S ( ) = n 3. Proof. Assume tht there eists vlid mpping for the representtive verte of G in S nd the point orresponding to the vlid mpping is. B Lemm 2.1, S. We no prove tht if e hoose =, then S ( ), S ( ) nd S ( ) must hve the required numer of points. Oserve tht the numer of points in the proper interior of tringle s is S(s) 3. All the points on the oundr ofsre lsoon the oundrof R. Sine orrespondstovlidmppingins, S() = n 2. Consequentl, S ( ) = S() S(s) +3 = n 2 S(s) +3. We n prove tht S ( ) = n 1 S(r) +3 nd S ( ) = n 3 in similr. Assume no tht there eists point S suh tht S ( ) = n 2 S(s) +3, S ( ) = n 1 S(r) +3 nd S ( ) = n 3. We no prove tht S() = n 2. Sine the numer of points in the proper interior of tringle s is

7 7 m m m m m m () () () (d) Fig. 4. Illustrtion for the proof of Lemm 2.3, here {m,m } S=Ø nd {,,,} S. S(s) 3, S() = S ( ) + S(s) 3 = n 2. Similrl, e n verif tht S() = n 1 nd S() = n 3. We n use the point to define prtition of the set S into three susets S ( ), S ( ) nd S ( ), here the vlues S ( ), S ( ) nd S ( ) re fied. In other ords, ts s vlid mpping for S ith respet to the vlues S ( ), S ( ) nd S ( ). Sine vlid mpping is unique [30], the point must lso e unique. We ll the point the prinipl point of S. Oserve tht this prinipl point orresponds to the vlid mpping of the representtive verte of G in S. We ill use the folloing lemm to effiientl find vlid mpping. Lemm 2.3. Let S e set of t 4 points in generl position suh tht the onve hull of S is tringle. Let i,j,k e three non-negtive integers, here i 3,j 3 nd k = t+5 i j. Then e n deide in O(t) time hether there eists point S suh tht S() = i, S() = j nd S() = k, nd ompute suh point if it eists. Proof. Consider first vrition of the prolem, here e nt to onstrut point m S interior to suh tht S(m) = i + 1, S(m) = j 1 nd S(m) = k 1. Steiger nd Streinu [35] proved the eistene of m nd gve n O(t)-time lgorithm to find m. If there eists point S suh tht S() = i, S() = j nd S() = k, then it is strightforrd to oserve tht there eists point m S interior to suh tht S(m) = i+1, S(m) = j 1 nd S(m) = k 1. We no prove tht the eistene of m implies unique prtition of S. Hene e n effiientl test hether eists. Welimthtifthereeistspointm m, herem S,suhtht S(m ) = i+1, S(m ) = j 1 nd S(m ) = k 1, then the sets S(m ),S(m ) nd S(m ) must oinide ith the sets S(m),S(m) nd S(m). To verif the lim ssume ithout loss of generlit tht m S(m). Sine the tringle m lies interior to the tringle m, the sets S(m ) nd S(m) must e identil. On theotherhnd,eitherthetringlem liesinteriortothetringlem, orthetringle m lies interior to the tringle m, s shon in Figures 4() (). Therefore, either the sets S(m) nd S(m ), or the sets S(m) nd S(m ) must e identil. Consequentl, the remining pir of sets must lso e identil. Oserve tht if the point S e re looking for eists, then must lie interior to S(m), s shon in Figure 4(). Otherise, if S(m) (respetivel, S(m)), then S(m) S() = j (respetivel, S(m) S() = k), hih ontrdits our initil ssumption tht S(m) = j 1 (respetivel, S(m) = k 1). Figure 4(d) depits suh senrio. If eists, then the onve hull of S(m) must e tringle m, here m S(m). If S(m ) = i, S(m ) = j nd S(m ) = k, then m is the required point.

8 8 Plne 3-trees: Emeddilit & Approimtion Otherise, no suh eists. We n test hether the onve hull of S(m) is tringle in O(t) time (e.g., find the leftmost point, the rightmost point nd the point ith the lrgest perpendiulr distne to the line determined the line segment, nd then test hether tringle ontins ll the points). It is lso strightforrd to ompute the vlues S(m ), S(m ) nd S(m ) in O(t) time. Given the set of interest S S, e use Lemms 2.2 nd 2.3 to find the prinipl point S in O(n 3 ) time. Oserve tht this prinipl point orresponds to the vlid mpping of the representtive verte of G in S. We no sho ho to ompute the set S in O((n 2 + n 3 )log 2 n) time using the dnmi plnr onve hull dt struture of Overmrs nd vn Leeuen [31], hih supports single updte (i.e., single insertion or deletion) in O(log 2 n) time. We refer the reder to Figures 3() () to rell the definition of the region of interest. Step A. Assume tht the points of S re pled in dnmi onve hull dt struture D. We repetedl delete the neighor of on the oundr of the onve hull of S strting from in ntilokise order. After deleting n 2 2 points, e insert ll the deleted points into ne dnmi onve hull dt struture D. We then insert op of into D. Rell u nd v from Figure 3(). Oserve tht ll the points of S(v) re pled in D. In similr e onstrut nother dnmi onvehull dt struture D tht mintins ll the points of S(uv). Consequentl, D no onl mintins the points of S(u). Sine single insertion or deletion tkes O(log 2 n) time, ll the ove O(n 2 +n 3 ) insertions nd deletions tke O((n 2 +n 3 )log 2 n) time in totl. Step B. We no onstrut to other dnmi onve hull dt strutures D 1 nd D 2 using D nd D suh tht the mintin the points of S(ru) nd S(sv), respetivel. Sine S(ru) + S(sv) = O(n 3 ), this tkes O(n 3 log 2 n) time. Step C. We onstrut the point set S using the points mintined in D,D 1 nd D 2, hih lso tkes O(n 3 log 2 n) time. In similr e n restore the originl point set S nd the initil dt struture D in O((n 2 + n 3 )log 2 n) time. The time for the onstrution of S using Steps A C is O((n 2 + n 3 )log 2 n), hih domintes the time required for the omputtion of the vlid mpping of the representtive verte p. Let e the point tht orrespondsto the vlid mpping. We no need to onstrut the point sets S(), S() nd S() for reursivel testing the point-set emeddilit of G p,g p nd G p, respetivel. We n onstrut S(), S() nd S() nd their orresponding dnmi onve hull dt strutures in O((n 2 +n 3 )log 2 n) time s follos. Let l e the point of intersetion of the infinite stright lines determined the line segments nd. First onstrut the set S(l) nd then modif it to otin the sets S() nd S(l), hih tkes O((n 2 +n 3 )log 2 n) time. No modif the set S(l) to onstrut the set S(l), nd then use the sets S(l) nd S(l) to onstrut S(), hih tkes O(n 3 log 2 n) time. Oserve tht fter the modifition of the set S(l), e re left ith the set S(). We no sho tht the totl time tken is T(n) dnlog 3 n, for some onstnt d, s follos. There eists > 0 suh tht for ll d,

9 9 T(n) = T(n 1 )+T(n 2 )+T(n 3 )+O((n 2 +n 3 )log 2 n) dn 1 log 3 n 1 +dn 2 log 3 n 2 +dn 3 log 3 n 3 +(n 2 +n 3 )log 2 n dn 1 log 3 n+dn 2 log 2 nlog n 2 +dn 3log 2 nlog n 2 +(n 2 +n 3 )log 2 n = dn 1 log 3 n+dn 2 log 2 n(logn 1)+dn 3 log 2 n(logn 1)+(n 2 +n 3 )log 2 n = d(n 1 +n 2 +n 3 )log 3 n (d )(n 2 +n 3 )log 2 n dnlog 3 n. Oserve tht the onstrution of the initil dt struture D tkes O(nlog 2 n) time, hih is dominted T(n). The dnmi plnr onve hull of Brodl nd Jo [8] tkes mortied O(log n) time per updte. Therefore, using their dt struture insted of Overmrs nd vn Leeuen s dt struture [31] e n improve the epeted running time of our lgorithm. Sine the lgorithms of [31, 35] tke liner spe, the spe ompleit of our lgorithm is O(n). Theorem 2.4. Given plne 3-tree G ith n verties nd set S of n points in generl position in R 2, e n deide the point-set emeddilit of G on S in O(nlog 3 n) time nd O(n) spe, nd ompute suh n emedding if it eists. Our lgorithm ould e esil dpted to the se hen the input points re not in generl position, provided tht the lgorithms of Overmrs nd vn Leeuen [31] nd Steiger nd Streinu [35] n hndle degenerte ses. 3. Universl Point Set for Plne 3-Trees. In this setion e give n lgorithm to ompute 2-end point-set emeddings of plne 3-trees on set of n points in generl position in O(W 2 ) re, here W is the length of the side of the smllest is-prllel squre tht enloses S. Wedesrienoutlineofthe lgorithm. Givenplne3-treeGndsetofpoints S in generl position, e first onstrut stright-line dring Γ of G suh tht ever point of S other thn pir of points on the onve hull of S lies in the proper interior of some distint inner fe in Γ, s shon in Figure 5(). While onstruting Γ, e ompute ijetive funtion φ from the verties of Γ to the points of S. We then etend eh edge (u,v) in Γ using to ends to ple the verties u nd v onto the points φ(u) nd φ(v), respetivel, s shon in Figure 5(d). We prove tht Γ nd φ mintin ertin properties so tht the resulting dring Γ remins plnr. In the folloing e desrie the lgorithm in detil. Let H e the onve hull of S. Construt tringle ith O(W 2 ) re suh tht enloses H nd the side psses through pir of onseutive points, on the oundr of H. Assume tht is loser to thn. Set φ() = nd φ() =. Set φ() =, here is the point on the onve hull of S() for hih the ngle is smllest. Figure 5(e) illustrtes the tringle nd the funtion φ. We ll the stright line segments φ(), φ(), φ() the ings of. Oserve tht onl φ() mong the three ings of lie in the proper interior of. We use this invrint throughout the lgorithm, i.e., ever fe f in the dring ill ontin t most one ing tht is in the proper interior of f. We ll suh ing the mjor ing of f. Let,, e the outer verties of G in ntilokise order nd let p e the representtive verte of G. Mp the verties,, to the points,,. Let S \ {,, } e the point set S. Let ˆn 1,ˆn 2 nd ˆn 3 e the numer of inner verties of G p,g p nd G p, respetivel. Sine the mjor ing of is inident to, e ompute point S suh tht S () = ˆn 1,S () = ˆn 2 +1 nd S () = ˆn 3,

10 10 Plne 3-trees: Emeddilit & Approimtion s shon in Figure 5(e). Steiger nd Streinu [35] proved tht suh point ls eists nd gve n O( S )-time lgorithm to find. Sine the ngle φ() is the smllest, if or intersets φ(), then ontinuit there must eist nother point on the line suh tht S ( ) = ˆn 1,S ( ) = ˆn 2 + 1, S ( ) = ˆn 3 hold, nd e hoose s the point. Figures 5(f) (g) depit suh senrios. Note tht φ() no lies either in the tringle or. Set φ() =, here is the point on the onve hull of S () for hih the ngle is smllest. Sine does not ontin φ(), the mpping e ompute mintins the invrint tht ever fe ontins t most one mjor ing. d f e p () () d f p () e d (d) f p e ( )= ( )= ( )= (e) (f) (g) e 3 e 2 e 1 u E (h) C v m ( v) et e 3 e 2 e 1 P v ( v) (i) e t Fig. 5. () A plne 3-tree G. () A set of points S. () Γ nd φ, here φ is illustrted ith dshed lines. (d) A 2-end point-set emedding of G on S. (e) Illustrtion for the tringle. (f) (g) Constrution of nd φ(), here φ() = is shon in hite nd the onve hull of S() is shon in gr. (h) The region R nd ellipse E, here R is shon in gr. (i) Illustrtion for P v. We no reursivel onstrut the drings of G p,g p nd G p ith the point sets S (), S () \ nd S (), respetivel. Note tht hile reursivel onstruting point for the representtive verte inside some tringle, then the tringle m not hve n mjor ing. Also in this se, it suffies to ompute suh tht S () = ˆn 1,S () = ˆn 2 +1 nd S () = ˆn 3 holds. One e omplete the reursive omputtion, e otin stright-line dring Γ of G, nd ijetive funtion φ from the verties of Γ to the points of S. The ide no is to etend eh edge (u,v) in Γ using to ends to ple the verties u nd v onto the points φ(u) nd φ(v), respetivel. We use φ nd the propert tht ever fe in Γ ontins t most one mjor ing, to mintin plnrit. We no desrie the onstrution detils. For eh verte v in Γ e do the folloing. Let e 1,e 2,...,e t e the edges djent to v in lokise order suh tht e 1 nd e t form the smllest ngle tht ontins φ(v). Construt stritl onve polgon P v = φ(v),v 1,v 2,...,v t, here v i,1 i t, is point on e i nd no point of S other thn φ(v) lie inside the polgon. No delete the stright line segments vv i nd dr the segments φ(v)v i, s shon in Figure 5(i). Sine ever fe in Γ ontins t most one mjor ing, e n onstrut the P v s suh tht for to different verties v 1 nd v 2 in Γ, the orresponding onve polgons P v1 nd P v2 re disjoint. Lter in this setion, e preisel desrie suh onstrution

11 for P v s. We lim tht the resulting dring Γ is 2-end point-set emedding of G on S. Sine ever edge in Γ hs onl to endpoints, the orresponding edge e in Γ hs etl to ends. Sine φ is ijetive funtion, e does not rete n loop in Γ. It no suffies to prove tht Γ is plnr dring of G. Oserve tht Γ is plnr stright-line dring. Therefore, if Γ is not plnr dring, then either to of the nel dded segments properl interset (Cse 1), or nel dded segment intersets n old segment tht originll elongs to Γ (Cse 2). Cse 1 nnot hppen sine ll P v s re disjoint. Cse 2 nnot pper sine ever nel dded segment lie in some onve polgon P v tht does not ontin n old segment in Γ. We n onstrut Γ in O(nlog 3 n) time ith similr tehnique s in Setion 2. We no desrie ho to onstrut Γ from Γ in O(nlogn) time. Let S e set tht onsists of the points orresponding to the verties in Γ nd the points tht elong to S. Let η e the Euliden distne eteen the losest pir of points in S. We n ompute η in O(nlogn) time [34]. For eh verte v in Γ, e no onstrut the onve polgon P v in O(deg(v)) time s follos. Let C e the irle entered t v ith rdius η/2. Assume tht e 1 = (u,v), nd m is the intersetion point of edge e t nd the line determined u,φ(v). Let R e the region determined the union of C nd the tringle uvm. Consider no n ellipse E ith foi v nd φ(v) suh tht the hlf of the ellipse (determined the minor is of E) tht ontins v lies interior to R. We n ls find suh n ellipse sine stright line segment n e vieed s degenerte se of n ellipse. We no define v 1,v 2,...,v t s the intersetion points of E ith e 1,e 2,...,e t, respetivel, s shon in Figures 5(h) (i). Sine R does not ontin n point of S other thn φ(v), the polgon P v = φ(v),v 1,v 2,...,v t does not ontin n point of S other thn φ(v). Sine ever fe in Γ ontins t most one mjor ing nd C is irle ith rdius η/2, n to P v s must e disjoint. Consequentl, the lgorithm tkes O(nlog 3 n) + O(nlogn) + v O(deg(v)) = O(nlog 3 n) time. The folloing theorem summries the results of this setion. Theorem 3.1. Given plne 3-tree G ith n verties nd point set S of n points in generl position, e n ompute 2-end point-set emedding of G in O(nlog 3 n) time ith O(W 2 ) re, here W is the length of the side of the smllest is-prllel squre tht enloses S. 4. Approimte Point-Set Emeddings. Let Γ e stright-line dring of G. Then S(Γ) denotes the numer of verties in Γ tht re mpped to distint points of S. The optiml point-set emedding of G is stright-line dring Γ suh tht S(Γ ) S(Γ ) for n stright-line dring Γ of G. A ρ-pproimtion point-set emedding lgorithm omputes stright-line dring Γ of G suh tht S(Γ)/S(Γ ) ρ. In this setion e sho tht given plne 3-tree G ith n verties, e n onstrut stright-line dring Γ of G suh tht S(Γ) = Ω( n), nd hene point-set emeddilit is pproimle ith ftor Ω(1/ n) for plne 3-trees. We first introdue fe more definitions. Let G e plne 3-tree ith the outer verties,, nd representtive verte p, nd let the numer of verties of G e n. Then the representtive tree T n 3 of G stisfies the folloing onditions [30]. () If n = 3, then T n 3 is empt. () If n = 4, then T n 3 onsists of single verte. () If n > 4, then the root p of T n 3 is the representtive verte of G nd the sutrees rooted t the three ounter-lokise ordered hildren p 1, p 2 nd p 3 of p in T n 3 re the representtive trees of G p, G p nd G p, respetivel. 11

12 12 Plne 3-trees: Emeddilit & Approimtion Sine rooted tree ith n nodes is prtill ordered set under the suessor reltion, Dilorth s theorem [17], either the height or the numer of leves in the tree is t lest n. Let G e the input plne 3-tree ith n verties nd let T e its representtive tree ith n 3 verties [30]. If T hs Ω( n) leves, then e use the tehnique of Theorem 3.1 to hve stright-line dring Γ of G suh tht S(Γ) = Ω( n) s follos. Find stright-line dring of G nd the ijetive funtion φ, s shon in Figure 5(). Oserve tht the leves of T (i.e., the inner verties of degree three in G) orrespond to distint K 4 s in the dring, nd hene e n ple eh lef l to the point φ(l) voiding n edge rossing. Otherise, the height of T is Ω( n). In this se e prove tht G hs nonil ordering tree (lso, lled Shnder s relier [33]) ith height Ω( n), s shon in Lemm 4.1. We then sho (in Lemm 4.2) simple to ompute stright-line dring Γ of G suh tht S(Γ) = Ω( n). Before proving Lemms 4.1 nd 4.2 e rell the definition of nonil ordering. Let G e tringulted plne grph ith the outer verties, nd in lokise order on the outer fe. Let π = (u 1 (= ),u 2 (= ),...,u n (= )) e n ordering of ll verties of G. B G k, 3 k n, e denote the sugrph of G indued {u 1,u 2,...,u k } nd C k the outer le (i.e., the oundr of the outer fe) of G k. We ll π nonil ordering of G ith respet to the outer edge (,) if for eh inde k, 3 k n, the folloing onditions re stisfied [13]. () G k is 2-onneted nd internll tringulted. () If k+1 n, then u k+1 is n outer verte of G k+1 nd the neighors of u k+1 in G k ppers onseutivel on C k. Assumethtforsomek 3,theouterleC k is 1 (= ),..., p, q (= u k ), r..., t (= ), herethe vertiespperin lokiseorderonc k. Then ellthe edges( p,u k ) nd (u k, r ) the left-edge nd the right-edge of u k, respetivel. Let E e the set of edges tht does not elong to n C k, 3 k n. Then the grph indued the edges in E is tree. The grph indued the right-edges (respetivel, left-edges) of the verties u k,3 k n 1, is lso tree. These three trees form the Shnder s relier of G, nd eh of them is knon s nonil ordering tree of G. Lemm 4.1. Let G e plne 3-tree nd let T e its representtive tree. If the height of the representtive tree is Ω( n), then G hs nonil ordering tree ith height Ω( n). Proof. Let P = (v 1,v 2,...,v k ), k = Ω( n), e the longest pth from the root v 1 of T to some lef v k. Without loss of generlit ssume tht k is even. B G i, here 1 i k, e denote the plne 3-tree indued the outer verties of G nd the verties v 1,v 2,...,v i. We no inrementll onstrut G k. First onstrut tringle, ple the verte v 1 interior to nd dd the segments v 1,v 1,v 1. Sine v 2 is hild of v 1, v 2 must e pled interior to one of the tringles inident to v 1. Sine v i+1, here i + 1 k, is hild of v i, this ondition holds throughout the onstrution. Let T,T,T e the trees of the Shnder s relier rooted t,,, respetivel. Figure 6() illustrtes the relier of G 2, here the heights of T,T nd T re one, one nd to, respetivel. B A nd B e denote the rooted trees isomorphi to T nd T in G 2, respetivel. The nodes of T, {,,}, here the relier gros hile dding v i+1 to G i, i 2, re lled the onnetors of T in G i. Figures 6(e) (g) illustrte ll possile s to insert v 3, nd sho the verties here the relier T n gro in gr. Here the relier gros onl t the verties nd v 2, nd hene these re the onnetors for T hile dding v 3 to G 2. Figure 6() illustrtes the onnetors for ever T in gr.

13 Consider no the steps hen e otin the grphs G 2,G 4,G 6,...,G k. Oserve tht eh time some tree of the form A (or B) gets onneted ith some T, {,,}, of G i, the onnetors of A (or B) eome the onl onnetors of T in G i+2. We desrie this senrioith n emple of G 4, s shon in Figure 6(). Figure 6(d) shos the onnetors of T,T nd T in gr. The tree T onsists of to trees: one is of the form A nd the other is of the form B, here the root of B oinides ith some onnetor of A (i.e., v 2 ). Sine the susequent verte v 5 must lie inside the tringle v 1 v 3, the onnetors of B eome the onl onnetors of T in G 4. Similrl, e n verif this ondition for T nd T. Consequentl, eh time some tree of the form B gets onneted ith some T, {,,}, of G i, the height of T inreses one in G i+2. Sine e need k/2 steps efore e otin G k, one of T,T or T must hve height t lest k/6 = Ω( n). Sine eh tree of the Shnder s relier of G k is sutree of distint tree of the Shnder s relier of G, the proof is omplete. 13 v 1 v v v v v v v v v 1 v 2 v v v v1 v4 v v v v v v 3 v2 v2 A A B B A A () () () (d) A A B v 3 v 3 v 2 v 1 v 2 v 1 v 2 v 1 v 3 (e) (f) (g) Fig. 6. () Illustrtion for G 2, here eh edge of T,T nd T is shon in one, to nd three prllel lines, respetivel. () Illustrtion for the onnetors, shon in gr. () (d) Emple of onnetion of A,A,B ith B,A,A, respetivel. (e) (g) Illustrtion for the onnetors of T onsidering the possile insertions of v 3 in G 2. Lemm 4.2. Let G e plne 3-tree ith n verties nd let S e set of n or more points in generl position. If G ontins nonil ordering tree T ith height k, then G dmits stright-line dring Γ suh tht S(Γ) = k. Proof. We prove the lemm using n indution on the numer of verties n of G. The se hen n 4 is strightforrd. We ssume tht the lemm holds for ll plne 3-trees ith feer thn n verties, nd no onsider the se hen G hs n verties. Let P = (v 1,v 2,...,v k ), e the longest pth from the root v 1 of T to some lef v k. Let p 1,p 2,...,p k e k points of S in deresing order of their -oordintes. Let C e polgonl hin tht onsists of the stright-line segments p i,p i+1, 1 i k 1. We no onstrut tringle suh tht = p 1, nd ll the points of the hin C re visile to oth nd, i.e., the stright-lines from the points of C to the point or do not ross n segment of C, s illustrted in Figure 7. Let,, e the outerverties of G nd let p e the representtive verte of G. Without loss of generlit ssume tht T is rooted t. If the edge (,p) is not ontined in the pth P, then ithout loss of generlit

14 14 Plne 3-trees: Emeddilit & Approimtion ssume tht (,p) is n inner edge of G p. We no find point interior to the tringle suh tht the distne eteen nd is ver smll nd ll the points of C revisileto. We then onstrutstright-linedringofg p ndg p interior to the tringles nd. B indution, G p dmits stright-line dring Γ interior to suh tht S(Γ ) = k. See Figures 7() (). Otherise, the pth P ontins the edge (,p). In this se e hoose = p 2. We then onstrut stright-line dring of G p nd G p interior to the tringles nd. B indution, G p dmits stright-line dring Γ interior to suh tht S(Γ ) = k 1. See Figures 7(d) (e). () p () () p (d) (e) Fig. 7. Illustrtion for the proof of Lemm 4.2. The points of S re shon in gr. The folloing theorem summries the result of this setion. Theorem 4.3. Given plne 3-tree G ith n verties nd point set S of n points in generl position in R 2, e n ompute stright-line dring Γ of G in polnomil time suh tht the numer of verties in Γ tht re mpped to distint points of S is 1/(6 n) times to the optiml. Hene the point-set emeddilit of plne 3-trees is pproimle ith ftor Ω(1/ n). Oserve tht e n use the tehnique of Theorem 3.1 to hve 1-end strightline dring of G tht uses n/4 distint points of S (e.g., hoose n independent set of n/4 verties nd ple those verties on n/4 distint points of S determined φ using t most one end per edge). Consequentl, 1-end point-set emeddilit is pproimle ith t lest ftor 0.25 for plne 3-trees. 5. Conlusion. Using tehniques tht re ompletel different from those used in the previousl est knon pprohes for testing point-set emeddilit of plne 3-trees (hieving O(n 4/3+ε ) time nd O(n 4/3 ) spe), in Setion 2 e desried n lgorithm tht solves the prolem for given plne 3-tree in O(nlog 3 n) time using O(n) spe. As suggested n nonmous revieer, one possiilit for potentill reduing the running time further might e to ppl the lgorithm of Moos nd Rhmn [29], here n orthogonl rnge serh ould e used insted of tringulr rnge serh. Speifill, given points nd nd n integer k, tringle tht ontins k points n e found enoding eh point using to vlues: the slopes of nd. The tringle is then mpped to to-sided is-ligned orthogonl rnge quer. It is not ovious, hoever, ho this tehnique ould e pplied in reursive levels. One possiilit might e to use dnmi orthogonl rnge ounting dt struture. A nturl optimition question in this diretion is s follos.

15 Open Question 1. Given plne 3-tree G ith n verties nd set S of n points in generl position, ho fst n e ompute stright-line emedding of G suh tht the numer of verties pled on the points of S is mimied? In Setion 3 e proved tht ever plne 3-tree dmits 2-end point-set emedding on n set of n points in generl position in O(W 2 ) re. An importnt issue here is to emine the mount of sle up required to ensure tht the verties nd end points of the drings produed in Setion 3 lie on integer oordintes, i.e., the re requirement under minimum resolution ssumption. While our result holds for prtil plne 3-trees, one m tr to hrterie the grphs tht dmit 2-end point-set emeddings in smll re. Open Question 2. Chrterie the plnr grphs ith n verties tht dmit 2-end point-set emeddings on n set of n points in generl position in O(W 2 ) re, here W is the length of the side of the smllest is prllel squre tht enloses the given point set. In Setion 4 e proved tht the point-set emeddilit prolem(respetivel, the 1-endpoint-setemeddilitprolem)ispproimleithftortlestΩ(1/ n) (respetivel, 0.25) for plne 3-trees ithin O(nlog 3 n) time, hih motivtes us to sk the folloing question. Open Question 3. Design n o(n 2 )-time lgorithm tht n pproimte point-set emeddilit for plne 3-trees ith onstnt ftor, or prove tht no suh lgorithm eists. Aknoledgement. We thnk Vlentin Polishhuk nd the other nonmous revieers for mn onstrutive nd helpful omments. 15 REFERENCES [1] José S. Andrde, Jr., Hns J. Herrmnn, Roerto F. S. Andrde, nd Luino R. d Silv, Apollonin netorks: Simultneousl sle-free, smll orld, euliden, spe filling, nd ith mthing grphs, Phsil Revie Letters, 94 (2005). [2] Re Mhmoodi Brm, Poldisperse Grnulr Pkings nd Berings, PhD thesis, Universit of Stuttgrt, Germn, [3] Therese Biedl nd Mrtin Vtshelle, The point-set emeddilit prolem for plne grphs, in Pro. of the 28th Annul Smposium on Computtionl geometr (SoCG), ACM, 2012, pp [4] Prosenjit Bose, On emedding n outer-plnr grph in point set, Comput. Geom., 23 (2002), pp [5] Prosenjit Bose, Mihel MAllister, nd Jk Snoeink, Optiml lgorithms to emed trees in point set, J. Grph Algorithms Appl., 1 (1997), pp [6] Frn J. Brndenurg, Dvid Eppstein, Mihel T. Goodrih, Stephen G. Koourov, Giuseppe Liott, nd Petr Mutel, Seleted open prolems in grph dring, in Pro. of the 11th Interntionl Smposium on Grph Dring (GD), vol of LNCS, Springer, 2004, pp [7] Peter Brss, Eon Cenek, Christin A. Dunn, Alon Efrt, Cesim Erten, Dn Ismilesu, Stephen G. Koourov, Ann Lui, nd Joseph S. B. Mithell, On simultneous plnr grph emeddings, Comput. Geom., 36 (2007), pp [8] Gerth Stølting Brodl nd Riko Jo, Dnmi plnr onve hull, in Pro. of the 43rd Annul Smposium on Foundtions of Computer Siene (FOCS), IEEE, 2002, pp [9] Sergio Cello, Plnr emeddilit of the verties of grph using fied point set is NP-hrd, J. Grph Algorithms Appl., 10 (2006), pp [10] Nethulootl Cstñed nd Jorge Urruti, Stright line emeddings of plnr grphs on point sets, in Pro. of CCCG, 1996, pp [11] Bernrd Chelle, Mih Shrir, nd Emo Well:, Qusi-optiml upper ounds for simple rnge serhing nd ne one theorems, Algorithmi, 8 (1992), pp

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