Complexity of a Single Face in an Arrangement of s-Intersecting Curves

Complexit y of a Single F ace in an Arrangemen t of s -In tersecting Curv es ∗ Boris Arono v † Dmitriy Drusvy atskiy ‡ Septem b er 9, 2021 Abstract Consider a face F in an arrangemen t of n Jordan curv es in the plane, no tw o of whic h intersect more than s times. W e pro ve that the com binatorial complexit y of F is O ( λ s ( n )), O ( λ s +1 ( n )), and O ( λ s +2 ( n )), when the curv es are bi-infinite, semi-infinite, or b ounded, resp ectiv ely; λ k ( n ) is the maximum length of a Da v enp ort-Sc hinzel sequence of order k on an alphab et of n sym b ols. Our b ounds asymptotically match the known w orst-case low er b ounds. Our pro of settles the still apparen tly op en case of semi-infinite curv es. Moreov er, it treats the three cases in a fairly uniform fashion. 1 In tro duction In this pap er we study the maximum complexit y of a single face in an arrangement of curv es in the plane, no t wo of which intersect more than s times; see b elo w. W e will do this through an extensiv e use of Da venport-Schinzel sequences, which were first in tro duced by Dav enp ort and Sc hinzel in 1965 [DS65]. They were motiv ated, curiously enough, by a problem in differential equations. Definition. Let n , s b e p ositiv e in tegers. A sequence U = h u 1 , . . . , u m i o ver an alphab et of size n is a Davenp ort-Schinzel se quenc e of order s on an alphab et of n sym b ols, or DS(n,s)-se quenc e , for short, if it satisfies the following conditions: 1. u i 6 = u i +1 , for eac h 1 ≤ i < m . 2. There do not exist s + 2 indices 1 ≤ i 1 < i 2 < . . . < i s +2 ≤ m suc h that u i 1 = u i 3 = u i 5 = . . . = a , u i 2 = u i 4 = u i 6 = . . . = b , for some distinct symbols a and b . W e denote b y λ s ( n ) the length of the longest D S ( n, s )-sequence. ∗ This is the version of the paper from June 1, 2009, with sev eral t yp os corrected. † Departmen t of Computer Science and Engineering, Polytec hnic Institute of NYU, Bro oklyn, New Y ork, USA; aronov@poly.edu . W ork of B.A. on this paper had been partially supp orted by a gran t from the U.S.-Israel Binational Science F oundation and by NSA MSP Grant H98230-06-1-0016. B.A. blames D. Halperin for prov oking him in to w orking on this problem. ‡ Sc ho ol of Op erations Research and Information Engineering, Cornell Universit y , Ithaca, New Y ork, USA; http://people.orie.cornell.edu/dd379 . W ork of Dmitriy Drusvyatskiy on this pap er had b een partially sup- p orted b y the NDSEG grant from the Department of Defense and b y the NSF Computatonal Sustainability Gran t 0832782. 1 Da venport and Sc hinzel w ere able to establish a connection betw een these sequences and low er en velopes of collections of functions [ DS65 , SA96 ]. The next significan t step in studying these sequences w as taken by Szemer´ edi in 1974, who established improv ed upper b ounds on the length of Da venport-Schinzel sequences [ Sze74 ]. In 1983, Atalah’s work was the first step in establishing DS-sequences as a fundamental to ol in computational and combi natorial geometry [ A ta85 ]. The fundamen tal question that w as still unanswered w as determining the asymptotic gro wth rate of the functions λ s ( n ), for an y fixed s . F or s = 1 and s = 2, this is very easy ( λ 1 ( n ) = n and λ 2 ( n ) = 2 n − 1) but already for s = 3, this question is highly non trivial. In 1986, Hart and Sharir sho wed that that the maximum length of any D S ( n, 3)-sequence is O ( nα ( n )), where α ( n ) is the v ery slowly growing inv erse of Ack ermann’s function [ HS86 ]. In 1989, Agarwal, Sharir, and Shor completed this classification by showing nearly tight, nearly linear b ounds for all fixed s [ ASS89 ]. Da venport-Schinzel sequences ha ve prov en to b e v ery useful in pro viding tighter metho ds of analysis for man y problems in discrete and computational geometry [SA96, AS00]. In this pap er, we are interested in the follo wing three types of curv es. An unb ounde d Jor dan curve is the image of an op en unit interv al under a top ological em b edding in to R 2 , such that it separates the plane. A semi-infinite Jor dan curve is the image of a half-op en unit interv al under a topological em b edding in to R 2 , suc h that the image is un b ounded with respect to the standard Euclidean norm. A b ounde d Jor dan curve (or Jor dan ar c ) is the image of a closed unit interv al under a top ological em b edding in to R 2 . Let Γ 0 , Γ 1 , and Γ 2 b e collections of n bi-infinite, semi-infinite, and b ounded Jordan curves in the plane, resp ectiv ely , such that any t wo curves in Γ i in tersect at most s times, for some fixed constant s > 0. (The subscript of Γ i signifies the num ber of finite endp oin ts of eac h curve in this collection.) Definition. [ Ede87 , SA96 ] The arr angement A (Γ i ) of Γ i is the planar sub division induced by the arcs of Γ i . Thus A (Γ i ) is a planar map whose vertic es are the endp oin ts of curves of Γ i , if any , and their pairwise intersection p oints. The e dges are maximal connected p ortions of the curv es that do not con tain a vertex. The fac es are the connected comp onen ts of R 2 − S Γ i . The c ombinatorial c omplexity of a face F of A (Γ i ) is the total num b er of v ertices and edges of A (Γ i ) along its b oundary ∂ F . A feature on ∂ F is coun ted in the complexit y as many times as it app ears. W e are in terested in studying the maxim um combinatorial complexity of a single face of A (Γ i ). Sc hw artz and Sharir sho wed that the combinatorial complexit y of a single face of A (Γ 0 ) is at most λ s ( n ) [ SS90 ]. Sharir et al. sho w ed that the combinatorial complexit y of a single face of A (Γ 2 ) is O ( λ s +2 ( n )) [ GSS88 ]. (See Niv asch [ Niv09 ] for some very recent progress in this sub ject.) Ho wev er, the t wo pro ofs provided are very different, which is somewhat unsatisfying. It has also b een conjectured that the com binatorial complexit y of a single face of A (Γ 1 ) is O ( λ s +1 ( n )). There has b een some work to suggest that this is true. F or instance, Alevizos, Boissonnat, and Preparata sho w ed that the complexit y of a single face in an arrangement of rays is linear [ P A90 ]; this is the case when s = 1. In this pap er, we prov e the follo wing theorem: Theorem 1. The c ombinatorial c omplexity of a fac e F in an arr angement of n bi-infinite, semi- infinite, or b ounde d Jor dan curves, no two of which interse ct mor e than s times, is O ( λ s ( n )) , O ( λ s +1 ( n )) , and O ( λ s +2 ( n )) , r esp e ctively. These upp er b ounds are tight in the worst case. This easily follows from the fact that the complexit y of the low er en v elop e of the collection of functions defined by curves of Γ i (these functions are 2 partially defined for i = 1 and for i = 2), in the sp ecial case where the curves are x -monotone, is a lo wer b ound on the com binatorial complexity of a single face of A (Γ i ) [ A ta85 , SA96 ]; the maxim um complexit y of such an env elop e is Θ( λ s ( n )), Θ( λ s +1 ( n )), and Θ( λ s +2 ( n )), resp ectiv ely . The result is known for bi-infinite curves and for Jordan arcs [ GSS88 , SS90 ], but not, to the b est of our kno wledge, for semi-infinite curv es. The adv an tage of our pro of is that firstly it settles the previously men tioned conjecture and secondly it treats all three cases in a reasonably uniform manner. Our paper is organized as follows. In Section 2 w e pro v e a purely combinatorial auxiliary fact (F act 2). Section 3 contains some preliminary mo difications to the geometric problem. In Section 4 w e prov e b ounds on the maxim um complexity of an un b ounded face in A (Γ i ). Finally , in Section 5 w e transform any b ounded face of A (Γ i ) into an unbounded one without an asymptotic increase in its complexit y . This implies b ounds on the maxim um complexit y of a b ounded face in A (Γ i ) and yields our main theorem. 2 A Com binatorial F act In this section, w e state and prov e a simple combinatorial fact ab out Da venport-Schinzel sequences. It or a close relativ e ha ve b een “in the folklore” of this area of research [ SA96 ], although w e ha ve not b een able to pin do wn a source where it was explicitly stated in this form. F or completeness, w e presen t a pro of. Definition. Giv en a sequence S o ver an alphab et Σ, for Λ ⊆ Σ, S | Λ denotes the sequence obtained b y deleting from S all sym b ols not in Λ. Definition. Let Σ b e an alphab et. Denote by Σ ∗ the set of all finite sequences ov er Σ. W e define an op eration  : Σ ∗ − → Σ ∗ as follo ws. Let X ∈ Σ ∗ . X  is obtained from X b y simply collapsing each subsequence of consecutiv e identical elements to a single elemen t, e.g., h . . . , a, b, b, b, b, c, c, d, . . . i w ould b e collapsed to h . . . , a, b, c, d, . . . i . Definition. Let Σ 1 and Σ 2 b e disjoint alphab ets, let k ≥ 1 b e an integer, and let X = h x 1 , . . . , x m i b e a sequence ov er Σ 1 ∪ Σ 2 . W e sa y that X is k-friend ly under (Σ 1 ,Σ 2 ) if the follo wing condition holds: ( ∗ ) There do not exist k + 1 consecutiv e indices 1 ≤ i, i + 1 , . . . , i + k ≤ m suc h that x i = x i +2 = x i +4 = . . . = a , x i +1 = x i +3 = x i +5 = . . . = b , with a ∈ Σ 1 and b ∈ Σ 2 , or vice versa. F act 2. If a se quenc e X is k-friend ly under (Σ 1 , Σ 2 ) , no two c onse cutive symb ols of X ar e the same, and ( X | Σ 1 )  and ( X | Σ 2 )  ar e b oth D S ( n, s ) -se quenc es, then | X | = O ( k λ s ( n )) . Pr o of. Let L 0 = X | Σ 1 , L = ( L 0 )  , R 0 = X | Σ 2 , and R = ( R 0 )  . It is clear that | X | = | L | + | R | + ( | L 0 | − | L | ) + ( | R 0 | − | R | ). Since | L | , | R | ≤ λ s ( n ), without loss of generality , it is sufficien t to b ound ∆ L = | L 0 | − | L | . ∆ L is the num ber of elements that w ere deleted from L 0 b y the  op eration. Supp ose that a subsequence h a, b, . . . , b, c i of | L 0 | w as collapsed to h a, b, c i in L (collapses at the b eginning and at the end of L are handled similarly). Now the only w a y that this could hav e happ ened is that in X , b et w een every tw o corresp onding consecutiv e elemen ts b , there w as a sequence of one or more elemen ts all from R 0 ; denote such a sequence by ξ i . Let T = h b, ξ 1 , b, ξ 2 , . . . , b i . W e c harge each elemen t b in T to an element of ξ i follo wing it, such that if p ossible it is different from the elemen t 3 of ξ i − 1 that w as charged for the previous o ccurrence of b . If it were alwa ys p ossible to do so, then all the elements of R 0 that hav e been c harged would b e preserv ed when R 0 w ere transformed into R and each one would ha v e b een c harged only once, so we could b ound ∆ L b y | R | . The only time that it is not p ossible is when there is a subsequence of X of the form h . . . , b, r , b, r, . . . i , where r is an elemen t of R . Since X is k-friend ly under (Σ 1 , Σ 2 ), the length of suc h a subsequence is no larger than k . It now easily follows that in the ab ov e charging scheme, an elemen t of R ma y b e c harged up to O ( k ) times, so ∆ L = O ( k λ s ( n )). Therefore | X | = | L | + | R | + ∆ L + ∆ R = O ( k λ s ( n )). 3 Geometric Preliminaries W e no w return to the geometric problem. Recall that we start with a set Γ i of curv es in the plane, no t wo intersecting pairwise more than s times. In order to state our argument, no mo difications will b e required for curves in Γ 0 , since only one side of any curve in Γ 0 can app ear on the b oundary of F . How ev er some mo difications will b e needed for the curves in Γ 1 and Γ 2 , whic h w e describ e b elo w. Let a = a ( γ ) b e the endp oin t of a curve γ ∈ Γ 1 . Let γ + b e the directed curv e that constitutes the “righ t side” of γ orien ted from a to infinit y and let γ − b e the “left” side of γ orien ted from infinity to a . Let a = a ( γ ) and b = b ( γ ) b e the t wo endp oin ts of a curve γ ∈ Γ 2 that are c hosen arbitrarily and fixed. Let γ + (the “righ t” side) be the directed curv e γ orien ted from a to b and let γ − (the “left” side) b e the directed curv e γ orien ted from b to a . 3.1 Asso ciate a sequence with a face Let F i of A (Γ i ) be an un b ounded face and let C i b e a connected comp onen t of ∂ F i . In this subsection, w e show how to asso ciate a sequence of curves with C i . • F or A (Γ 2 ), w e trav erse C 2 , k eeping F 2 on the righ t. Let S 2 = h s 1 , s 2 , . . . , s t i b e the circular sequence of orien ted curv es in Γ 2 in the order in which they app ear along C 2 . If during the tra versal we meet the curve γ with endp oin ts a = a ( γ ) and b = b ( γ ), and follow it from a to b (resp ectiv ely b to a ), we add γ + (resp ectiv ely γ − ) to S 2 . • Observ e that for A (Γ 0 ), C 0 is not closed—it divides the plane in to t wo connected comp onen ts. This means that C 0 naturally corresp onds to a linear sequence of un-oriented curv es. Again, w e tra verse it k eeping F on the right. Denote this sequence by S 0 . S 1 is constructed analogously , as a sequence of oriente d curv es. W e will often abuse the notation sligh tly . Giv en a sequence of curves S i , w e will often isolate an alternating subsequence, sa y A = h ξ j , γ j , ξ j +1 , γ j +1 , . . . i , where all ξ j represen t the app earances of the same curv e ξ and all γ j represen t the app earances of the same curv e γ . W e will often treat ξ j and γ j as aliases for the edges of A (Γ i ) that corresp ond to those en tries in S i . 3.2 Preliminary mo dification for curves in Γ 1 Let Σ L and Σ R b e the alphab ets consisting of the left sym bols and righ t symbols, resp ectiv ely . F or notational purp oses, let S L 1 = ( S 1 | Σ L ) ♦ and S R 1 = ( S 1 | Σ R ) ♦ . In this section, w e prov e a k ey lemma, which is a v ariation of the Circular Consistency Lemma [ GSS88 , SA96 ] b elo w, and state an imp ortan t observ ation. 4 ζ a b π C 1 Figure 1: Pro of of Linear Consistency Lemma. Lemma 3 (Linear Consistency Lemma) . (a) The p ortions of e ach ar c ξ + i app e ar in S R 1 in the same or der as their or der along ξ + i ; analo gous statement holds for S L 1 . (b) The p ortions of e ach ar c ξ i app e ar in S 0 in the same or r everse or der as c omp ar e d to their or der along ξ i . Pr o of. W e only argue (a), since (b) follo ws by an almost identical argumen t. Let a and b b e p ortions of ξ + that o ccur in S R 1 in that order. Assume that along ξ + , a follo ws b ; refer to Fig 1. Denote b y π the p ortion of C 1 connecting a to b . Denote by ζ the p ortion of ξ + from b to a . Now aπ bζ is a closed con tour. It is easy to verify that the infinite “end” of ξ + m ust b e enclosed in this closed con tour, which is a contradiction. W e now mak e the follo wing simple but imp ortan t observ ation. It can easily b e chec k ed that S 1 is k -friendly under (Σ L , Σ R ), for some k = O ( s ). (Indeed, the existence of a contiguous subsequence of the form h ζ − , ξ + , ζ − , ξ + , . . . i of length s + 2, where ζ − ∈ Σ L and ξ + ∈ Σ R , w ould force s + 1 distinct p oin ts of intersection of ζ and ξ —a contradiction). This observ ation will b e critical for the pro of of Theorem 6. 3.3 Preliminary mo dification for curves in Γ 2 W e will also need the follo wing lemma. Lemma 4 (Circular Consistency Lemma [ GSS88 , SA96 ]) . The p ortions of e ach ar c ξ + i (r esp e ctively ξ − i ) app e ar in S 2 in a cir cular or der c onsistent with their or der along the oriente d ξ + i (r esp e ctively ξ − i ). That is, ther e exists a starting p oint in S , which dep ends on ξ i , such that if we r e ad S in a cir cular or der starting fr om that p oint, we enc ounter these p ortions in their or der along ξ i . W e no w perform a cutting of the circular sequence S 2 as in [ GSS88 , SA96 ]. Consider S 2 = h s 1 , . . . , s t i as a linear, rather than a circular sequence by breaking it at an arbitrary vertex. F or each directed arc γ i , consider the linear sequence V i of all app earances of γ i in S 2 , arranged in the order they app ear along γ i . Let µ i and ν i denote, resp ectively , the index in S 2 of the first and of the last elemen t of V i . F or each arc γ i , if µ i > ν i , w e split the symbol γ i in to tw o distinct symbols γ i 1 and γ i 2 , and replace all app earances of γ i in S 2 b et w een µ i and t (resp ectiv ely , b et w een 1 and ν i ) by γ i 1 (resp ectiv ely , by γ i 2 ). Notice that b y Lemma 4, w e are able to split γ i in to t wo subarcs such that γ i 1 represen ts the app earances of the first subarc and γ i 2 represen ts the app earances of the second subarc. This splitting pro duces a sequence, of the same length as S 2 on the alphab et of at most 4 n sym b ols. T o simplify the notation, hereafter we refer to this new linear sequence as S 2 . 5 T o summarize: • W e did not mo dify S 0 . It is a linear sequence of curv es. • S 1 is a linear sequence of oriented curv es, from which we hav e deriv ed tw o subsequences, S L 1 and S R 1 . • After the cutting pro cedure, S 2 is a linear sequence of oriented curves. T o arriv e at our first geometric theorem, we need the following lemma [GSS88, SA96]. Lemma 5 (Quadruple Lemma [ GSS88 , SA96 ]) . Consider a quadruple of c onse cutive elements in a fixe d alternating subse quenc e of S 2 . L et this quadruple b e h ξ 1 , γ 1 , ξ 2 , γ 2 i , such that ξ i and γ j c onstitute p ortions of curves a and b , r esp e ctively. L et π a b e the p ortion of a c onne cting ξ 1 to ξ 2 and let π b b e the p ortion of b c onne cting γ 1 to γ 2 . Then π a and π b must interse ct. F urthermor e, this p oint of interse ction is distinct for e ach such quadruple in this subse quenc e. Although stated for S 2 , the Quadruple Lemma also holds for S 0 , S R 1 , and S L 1 . 4 Complexit y of an Unbounded F ace Theorem 6. The c omplexity of an unb ounde d fac e F in an arr angement of n (0) bi-infinite, (1) semi-infinite, or (2) b ounde d Jor dan curves, no p air of which cr osses mor e than s times, is O ( λ s ( n )) , O ( λ s +1 ( n )) , O ( λ s +2 ( n )) , r esp e ctively. Pr o of. Since λ s ( n ) is at least linear, and no curve can app ear on several connected comp onents of ∂ F , we can consider each comp onent C separately and assume that all of the curves app ear on it. Now consider the sequences S 0 (case 0), S R 1 (case 1), and S 2 (case 2). W e claim that these are D S ( n, s )-, D S (2 n, s + 1)-, and D S (4 n, s + 2)-sequences, resp ectiv ely . Let S b e the sequence in question. W e aim to argue that it is a D S -sequence. By construction, S do es not contain any consecutive iden tical elements. Assume that it has an alternating subsequence of length l . Let this subsequence b e A = h ξ 1 , γ 1 , ξ 2 , γ 2 , . . . i , suc h that ξ i and γ j constitute p ortions of curv es a and b , resp ectively . W e argue that l cannot b e to o large. By Lemma 5, consecutive quadruples of A force l − 3 distinct crossings b et ween a and b . • In case 2, setting l = s + 4 forces l − 3 = s + 1 distinct intersections b etw een a and b . This is a con tradiction. Th us no such subsequence exists, and S 2 is a D S (4 n, s + 2)-sequence, implying that the complexit y of F is O ( λ s +2 ( n )). • In case 1, setting l = s + 3 forces l − 3 = s distinct in tersections b etw een a and b . Now, consider the last quadruple in A . Without loss of generality , let it b e h ξ i , γ i , ξ i +1 , γ i +1 i . Let π b e the p ortion of b connecting γ i to γ i +1 . W e claim that there must b e an additional intersection b et w een a and b at a p oin t on π that w e hav e not accounted for, so far. Let θ ⊃ ξ i +1 b e the p ortion of C connecting γ i to γ i +1 ; π ∪ θ is a closed con tour. Refer to Figure 2. No w trav erse a in the infinite direction starting from ξ i +1 . Since a cannot cross θ , during the trav ersal, a m ust in tersect b at a p oint on π . By the Linear Consistency Lemma, ho wev er this intersection has not b een accoun ted for and thus there are at least s + 1 distinct intersections b etw een a 6 F C a b γ i ξ i +1 ξ i π θ γ i +1 Figure 2: Semi-infinite curves. F C a b ξ 1 ξ 2 ξ 3 γ 1 γ 2 γ 3 θ 1 π 1 π 2 θ 2 Figure 3: Bi-infinite curves. and b . This, of course, is a contradiction and therefore S R 1 is a D S (2 n, s + 1)-sequence. By Theorem 2 and the discussion of Section 3.2, | S 2 | = O ( λ s +1 ( n )), implying that the complexity of F is O ( λ s +1 ( n )). • In case 0, setting l = s + 2 forces ( s + 2) − 3 = s − 1 distinct intersections b etw een a and b . No w, consider the first and last quadruples of A . By the same argumen t as in case 1, there are now tw o additional distinct intersections b et w een a and b , for a total of at least s + 1 distinct intersections—a contradiction. Thus S 0 is a D S ( n, s )-sequence and the combinatorial complexit y of F is λ s ( n ). Refer to Figure 3. 5 Complexit y of an Arbitrary F ace The next step is to generalize our results to b ounded faces. W e will transform the problem while only increasing the complexit y of a face F , to make F un b ounded, and then apply our results from the previous section. More precisely , we will build a tunnel from F to the “outside”, so that after the transformation, the new face will b e part of an unbounded face of the arrangement. Pr o of of The or em 1. If F is unbounded, we are done by Theorem 6. Consider a b ounded face F in an arrangemen t of (a) bi-infinite, (b) semi-infinite, or (c) b ounded Jordan curv es. W e assume that all the curves app ear on ∂ F , otherwise one or more curv es can b e deleted without affecting the complexit y of F . F urthermore since λ s ( n ) is at least linear, it is sufficient to argue the complexit y of just one connected comp onen t of ∂ F . Thus without loss of generality , we can assume that ∂ F is connected. Step 1: Finding the site for the tunnel • In cases (a) and (b), pick an arbitrary infinite edge of the arrangemen t, say of curve a and follow it until it first meets ∂ F , say at p oint p , where it meets curve b . Denote 7 F a b p ζ F a b c ζ p y Figure 4: Finding the initial cut. Γ 1 on the left and Γ 2 on the righ t. this p ortion of a , from infinity to p , by ζ ; ζ is the future site for our tunnel. Refer to Figure 4(left). • In case (c), if an endp oin t of a Jordan arc lies on the b oundary of the unbounded cell, w e start at this p oint. Otherwise, w e pic k an arbitrary edge of the infinite cell of the arrangemen t, cut the curve containing this edge into t wo curves, and mov e them slightly apart; this increases the num b er of curv es by one. In b oth cases, we no w hav e an endp oin t y of a curve a . Now follo w a from y to its first p oin t of intersection p with ∂ F , where it meets curve b . Denote this p ortion of a , from y to p , by ζ ; ζ is the future site for our tunnel. Refer to Figure 4(right). Step 2: Digging the tunnel W e no w “dig a tunnel” along ζ from p to its “infinite end”. Namely , at eac h intersection of a with another curve c of Γ i , we split c in to tw o new curv es, and leav e a small gap b et ween the tw o resulting curv es, for a to pass through (see Figure 5). a tunnel c 1 a tunnel c c 2 Figure 5: Extending the tunnel. The picture b efore (left) and after (right). By construction, during our tra versal of ζ , a did not meet F again. Th us, as a result of our transformation w e hav e only enlarged F , increased its complexit y , and connected it to an infinite face. Notice that no new in tersections are created. Namely , the resulting curves do not self-intersect and if the curv es in the original problem intersected pairwise no more than s times, then none of the newly created curv es will in tersect pairwise more than s times. The num ber of curves in the resulting picture is at most 1 + ( s + 1)( n − 1) = O ( sn ), if we did not ha v e to cut at y ; the remaining case is similar. W e are almost done — the trouble is that in case (a), by splitting an existing curve, we cut a bi-infinite curv e in to semi-infinite curves or even finite sections; similar complications arise in case (b). In case (a), we fix this by extending infinite non-crossing ”tails” along a to infinit y in suc h a w ay that they follow infinitesimally close to a but do not cross pairwise. Case (b) is handled analogously . 8 a a Figure 6: Fixing the tunnel. The picture b efore (left) and after (right) in the case of Γ 0 , magnified. A t eac h split p , w e would add one infinite “tail” to the finite sections, which were created as a result of the original split. Refer to Figure 6. W e note that λ s ( k n ) = O ( λ s ( n )) for any constants s and k . Th us by Theorem 6, the complexity of F in cases (a), (b), and (c) is O ( λ s ( n )), O ( λ s +1 ( n )), and O ( λ s +2 ( n )), resp ectiv ely . References [AS00] P . K. Agarw al and M. Sharir. Dav enport-Schinzel sequences and their geometric applications. In J. Sack and J. Urrutia, editors, Handb o ok of Computational Ge ometry , pages 1–47. North-Holland Publishing Co., Amsterdam, Netherlands, 2000. [ASS89] P . K. Agarwal, M. Sharir, and P . Shor. Sharp upp er and low er b ounds on the length of general Da venport-Schinzel Sequences. J. Comb. The ory Ser. A , 52(2):228–274, 1989. [A ta85] M. A tallah. Some dynamic computational geometry problems. Computers and Mathematics with Applic ations , 11:1171–1181, 1985. [DS65] H. Dav enport and A. Schinzel. A combinatorial problem connected with differen tial equations. A meric an J. Mathematics , 87:684–694, 1965. [Ede87] H. Edelsbrunner. Algorithms in Combinatorial Ge ometry . Springer-V erlag New Y ork, Inc., New Y ork, NY, USA, 1987. [GSS88] L. J. Guibas, M. Sharir, and S. Sifron y . On the general motion planning problem with tw o degrees of freedom. In SCG ’88: Pr o c e e dings of the F ourth Annual Symp osium on Computational Ge ometry , pages 289–298, New Y ork, NY, USA, 1988. ACM. [HS86] S. Hart and M. Sharir. Nonlinearity of Dav eport-Schinzel sequences and of generalized path compression schemes. Combinatoric a , 6(2):151–177, 1986. [Niv09] G. Niv asc h. Impro ved b ounds and new techniques for dav enport–schinzel sequences and their generalizations. In Pr o c e e dings of the Twentieth Annual ACM-SIAM Symp osium on Discr ete A lgorithms, SODA 2009, January 4-6, 2009 , pages 1–10. SIAM, 2009. [P A90] F. Preparata P . Alevizos, J. D. Boissonnat. An optimal algorithm for the b oundary of a cell in a union of rays. Algorithmic a , 5(1–4):573–590, 1990. [SA96] M. Sharir and P . K. Agarwal. Davenp ort-Schinzel Se quenc es and Their Ge ometric Applic ations . Cam bridge Universit y Press, New Y ork, NY, USA, 1996. [SS90] J. T. Sch w artz and M. Sharir. On the tw o-dimensional Da venport-Schinzel problem. J. Symb. Comput. , 10(3-4):371–393, 1990. [Sze74] E. Szeremedi. On a problem of Dav enp ort and Sc hinzel. A cta Arithmetic a , 25:213–224, 1974. 9