Realizing Site Permutations

Transcription

1 CCCG 2011, Toronto ON, August 10 12, 2011 Relizing Site Permuttions Stephne Durocher Seed Mehrbi Debjyoti Mondl Mtthew Skl Abstrct Gien n fixed sites on the plne, there re seerl wys to determine permuttion of the sites s function of unit ector u or ntge point. Gien such scheme nd permuttion π, we cn sk whether there is ny unit ector or ntge point for which the permuttion is π. We gie liner-time lgorithms for this reliztion problem under three schemes for determining permuttions: sweeping line cross the sites in direction u; expnding circle strting from ntge point ; nd sweeping ry from to gie cyclic permuttion. 1 Introduction Gien n rrngement of points clled sites on the plne, there re seerl wys to choose permuttion of the sites. For instnce, we could sweep line cross the rrngement nd enumerte the sites in the order the line touches them. We could strt from some ntge point nd consider the sites in order of incresing distnce from the ntge point. We could insted sweep ry from the ntge point rdilly through ll possible ngles nd consider the circulr ordering of the sites it encounters. Other rules re lso possible. Gien set of sites S nd geometric rule for defining permuttion of S s function of sweep direction or ntge point, some permuttions cn be relized by some choice of sweep direction or ntge point, nd other permuttions cnnot be relized. In this work we consider the lgorithmic problem of recognizing relizble permuttions, nd describe liner-time lgorithms for this problem under three different geometric rules. Problems of this type he pplictions in settings tht inole computing the position of n obserer such s robot [8] within its enironment reltie to sequence of obsertions mde using directionl sensor (such s sonr, rdr, or cmer). 2 Definitions nd nottion Let S = {, s 2,..., s n } be set of points on the Eucliden plne, clled the sites. Let S 1 represent the set of directions, or unit ectors, in the plne. Assume tht Work supported in prt by the Nturl Sciences nd Engineering Reserch Council of Cnd (NSERC). Deprtment of Computer Science, Uniersity of Mnitob, {durocher,mehrbi,jyoti,mskl}@cs.umnitob.c ll points nd directions in the problem re in generl position: tht is, no two points re coincident; no three points re colliner; no point is equidistnt from two others; no four points w, x, y, nd z he the reltionship tht the line wx is prllel to the line yz; nd (in the cse of sweep-line permuttions) the gien sweep direction u is not orthogonl to the line connecting ny two points. For ny unit ector u S 1 in generl position reltie to S, let the sweep-line permuttion of u be the permuttion of sites determined by sweeping line orthogonl to u cross the sites in the direction u nd enumerting the sites in the order encountered. It would be equilent to sy tht we project ll the sites onto directed line prllel to u nd define the permuttion by the order of the projected sites long the line. Insted of sweeping line in direction, we might strt from point clled the ntge point nd enumerte the sites in order of incresing distnce from to form distnce permuttion. This derition cn be imgined s expnding circle centred on nd enumerting the sites in the order encountered; or s sending out sonr ping nd recording the order of the echoes receied. Another wy of determining permuttion would be by tking ry strting from nd sweeping it counterclockwise through complete rottion of 360, enumerting the sites in the order the ry encounters them. 1 Then we obtin cyclic permuttion; tht is, n equilence clss of permuttions up to rottion. This rdil permuttion is nlogous to scnning the sites with rotting serch light or rdr bem, nd recording the order in which we see them without regrd for the ngles or the strting orienttion of the sweep. Figure 1 illustrtes the three kinds of site permuttions we consider. In the figure, line swept in the direction u encounters the sites in the order dcb. An expnding circle strting t encounters the sites in the order bdc; nd ry originting t nd swept counterclockwise encounters them in the order cdb, up to rottion tht depends on the strting orienttion of the sweep. For ny of these schemes, gien permuttion or cyclic permuttion π nd set of sites, unit ector u or ntge point is sid to relize π if π is the permuttion determined by u or for the gien 1 We describe ngles using degrees to oid confusion with the symbol π used for generic permuttion.

2 23 d Cndin Conference on Computtionl Geometry, 2011 b B Figure 1: A. sweep-line permuttion in direction u: dcb. B. distnce permuttion centred t : bdc. C. rdil permuttion centred t : cdb. scheme nd sites. Then π is sid to be relizble if nd only if there is ntge point or unit ector relizing it. In this work we consider the problem of deciding whether permuttion π is relizble nd, if so, computing corresponding unit ector u or ntge point tht relizes π. If the ntge point is sufficiently fr from the sites in the direction opposite to u, then the expnding circle centred on when it psses oer the sites is equilent to line orthogonl to u nd sweeping in the direction u. Similrly, if the ntge point is sufficiently fr from the sites in direction 90 counterclockwise from u, then the sweeping ry from when it psses oer the sites is equilent line sweeping in the direction u. We cn thus mke the following obsertion. Obsertion 1 Eery sweep-line permuttion for n rrngement of sites is lso relized s distnce permuttion nd rdil permuttion. Throughout our lgorithmic results we ssume rel RAM model of computtion, in which we cn perform bsic rithmetic opertions in unit time. This is stndrd ssumption for computtionl geometry lgorithms in generl; nd in prticulr, the liner-time liner progrmming lgorithm of Megiddo [5], which we use, is only liner-time under the ssumption it cn complete in constnt time the multipliction nd diision opertions needed to find the intersections of lines gien s input. Anlysing the lgorithms under some other model to force superliner result would be primrily n explortion of the complexity of rithmetic in generl without giing specific insight into these lgorithms. C c u d A 3 Preious work The cyclic sequence of sweep-line permuttions formed by site rrngement s we rotte the sweep direction through full circle is clled n llowble sequence, nd llowble sequences re well-studied. Goodmn nd Pollck pioneered the use of llowble sequences in chrcterizing the order type of the sites [4]. The llowble sequence for site rrngement is closely connected to the oriented mtroid ssocited with the site rrngement, nd tht connection leds to mny combintoril insights [2]. Cháez, Figuero, nd Nrro introduced distnce permuttions in dtbse context, s wy of clssifying points in high-dimensionl generl metric spces to support efficient proximity queries [3]. Note tht this kind of permuttion (possibly with tiebreking ssumption dded to hndle degenerte cses) is defined for ny spce with rel distnce function it need not een be metric. Skl proed bounds on the number of distinct distnce permuttions tht cn occur s function of the number of sites in rious spces, including n exct count for Eucliden spces [6]. Bieri nd Schmidt studied rdil permuttions s well s sweep-line permuttions nd rition on rdil permuttions in which line is swept insted of ry [1]. Noting tht the number of rdil permuttions relized by site rrngement is Θ(n 4 ) (which follows from the number of bisectors nd the fct tht k lines in generl position on the plne diide the plne into Θ(k 2 ) cells), they gie n lgorithm to generte ll the permuttions in Θ(n 4 ) time interesting becuse the nie size of the output would be Θ(n 5 ). To chiee the fster running time, they order the permuttions in such wy tht ech except the first differs from some preious permuttion by one swp of djcent elements; then the swps cn be found in O(n 4 ) time. Tor, Fred, nd LVlle studied rdil permuttions in the context of robot nigtion; ssuming robot with sensor tht detects the rdil permuttion of lndmrks s seen from its current loction, they show how the robot cn chiee nigtionl gols [8]. 4 Bisectors nd Voronoi digrms The sweep-line method of finding permuttion implicitly diides the set of possible directions into interls corresponding to the relizble permuttions. Similrly, the distnce nd rdil permuttions correspond to cells of Voronoi-like digrm in the plne. These diisions re shown in Figure 2. Note tht the unbounded cells for distnce nd rdil permuttions correspond to the permuttions relized by points t infinity, nd thus to the sweep-line permuttions (Obsertion 1). Eery pir of sites s i nd s j determines bisector: set of points where the ordering of s i nd s j is not

3 CCCG 2011, Toronto ON, August 10 12, 2011 cb cb cb bc bc cb cb bc bc cb b c b c cb b bc cb bc () (b) (c) c bc cb Figure 2: Diision of spce by permuttion schemes: () sweep-line, (b) distnce, (c) rdil. uniquely defined. If we imgine point wndering continuously through the spce (like Tor, Fred, nd LVlle s robot [8]), the permuttion it obseres will chnge by swp of djcent elements ech time it crosses bisector. For sweep-line permuttions, the bisector of s i nd s j consists of the two unit ectors prllel to the line between s i nd s j. For distnce permuttions, it is the set of ll points equidistnt from s i nd s j, which is the line orthogonlly bisecting the segment tht connects the two sites. For rdil permuttions, it is the line connecting s i nd s j, with the segment between them remoed. The rdil bisector is unusul becuse it cn be sid to cut the plne into just one piece: with two sites, only one permuttion exists up to rottion, so there is only one cell. Rdil bisectors become more meningful once there re three or more sites. Exmintion of these diisions of spce leds to simple counts or bounds on the number of permuttions relized. For sweep-line permuttions, the bisectors ech consist of two points, nd distinct bisectors neer coincide when sites re in generl position, so it is triil tht the number of interls nd thus permuttions for n sites is 2 ( n 2). For distnce permuttions, ( n 2) bisectors nd the qudrtic bound on number of cells formed by lines in generl position gies n upper bound of O(n 4 ) permuttions; Skl notes tht bisectors re not in generl position becuse of trnsitiity, nd gies n exct recurrence for the number of permuttions, s well s generlizing the question to higher dimensions of Eucliden spce; in d dimensions the number of permuttions is shown to be Θ(n 2d ) [6]. For rdil permuttions, the sme kind of rgument gies n obious O(n 4 ) upper bound, but the possibility for permuttion s cell to be non-conex or een disconnected (s in Figure 2(c) ) complictes mtters. Bieri nd Schmidt stte s theorem (without detiled proof) tht the upper bound is chieed by some rrngement of n sites for eery n [1]. 5 Rdil permuttions in the dul spce For ech site s i in, s 2,..., s n, define line s i s follows: let (x i, y i ) be the coordintes of s i, nd then let s i be the line dul to s i, defined by y = x x i y i. Let = x, y be point in the plne, not equl to ny of the sites, nd similrly define its dul line by y = x x y. These points nd lines re shown in Figure 3. The ntge point ws chosen to be the origin for conenience in mking nd understnding the figure; its imge in dul spce is the x xis. The sorted sequence of the segments (, s i ) round corresponds to n ordered sequence of intersections between the lines nd s i, s line connecting two points in priml spce corresponds to the intersection of two lines in dul spce. Let L be the erticl line pssing through. L diides the plne into two hlf-plnes. In Figure 3(), the right hlf-plne contins,, nd, nd the left hlf-plne contins, s 2, nd. In Figure 3(b), the crossing points for s 3, s 5, nd s 6 (shown in white) pper consecutiely right to left. Similrly, the crossing points for s 1, s 2, nd s 4 (shown in blck) pper consecutiely left to right. We cn conctente the two lists to obtin rdil permuttion π of the sites round :, s 2,,,,. The dul spce nturlly suggests nother sequence of the sites, tht found by exmining ll the crossings (not blck nd white seprtely) long the line. From right to left tht sequence is s 1, s 2, s 3, s 5, s 4, s 6. In the priml spce it corresponds to rotting line, not ry, pssing through, strting t erticl nd then 180 counterclockwise until it becomes erticl gin, nd enumerting the sites in the order the line encounters them. This rition of rdil permuttions corresponds to the undirected strs described by Streinu [7]. If we dd sign to ech element in the sequence describing whether it ws hit by the hed or the til of the line during the rdil sweep, the result is directed str (or simply str) s described by Streinu; in Figure 3, using Streinu s nottion, the str would be , where x nd x denote tht element x ws met by the hed or til end, respectiely, of the rotting line. Gien directed str, it is strightforwrd to construct the corresponding rdil permuttion in liner time.

4 23 d Cndin Conference on Computtionl Geometry, 2011 s 2 s 4 s 6 s 5 s 3 s 2 s 1 () (b) Figure 3: A rdil permuttion in () priml nd (b) dul spces. 6 Results s π(3) s π(1) The bisectors for sweep-line permuttions suggest simple liner-time lgorithm for relizing permuttions; in fct, becuse the cells re simply interls round the circle, we not only compute single unit ector to relize the permuttion, but lso completely describe the set of ll such ectors in the sme symptotic time. Theorem 1 There exists liner-time lgorithm tht gien sites, s 2,..., s n nd permuttion π on the indices, finds the set of ll directions u for which the sweep-line permuttion is π. Proof. By trnsitiity, it suffices to enforce the n 1 constrints tht the sweep line reches s π(1) before s π(2), s π(2) before s π(3), nd so on. Ech of those constrints corresponds to n interl of llowed lues for u in S 1. Ech interl is open nd hs length 180 ; therefore the intersection of ny two of them is single, possibly smller, interl; nd by ssocitiity we cn compute the intersection of ll of them in O(n) time. In the cse of distnce permuttions, liner time does not llow us to exmine ll of the qudrtic number of bisectors; but becuse the bisectors correspond to the trnsitie less thn reltion on distnces, we cn obtin ll the necessry informtion by exmining linersized subset of them. Theorem 2 There exists liner-time lgorithm tht gien sites, s 2,..., s n nd permuttion π on the indices, finds ntge point for which the distnce permuttion is π, if such exists. Proof. By trnsitiity, it suffices to enforce the n 1 constrints tht is closer to s π(1) thn to s π(2), closer to s π(2) thn to s π(3), nd so on. Ech of those corresponds to hlf-plne (liner) constrint. By the linertime two-dimensionl liner progrmming lgorithm of Megiddo [5], we cn find point stisfying ll the constrints in O(n) time. Rdil permuttions present greter chllenge, primrily becuse we re seeking not single permuttion s π(2) θ 2 θ 3 θ 1 Figure 4: Angles mesured round. but n equilence clss of permuttions. We begin by proing connection between the reliztion problem nd liner progrmming. Lemm 3 There exists liner-time lgorithm tht gien sites, s 2,..., s n nd permuttion π on the indices, constructs set of liner constrints such tht ny ntge point for which the rdil permuttion is π up to rottion stisfies ll, or ll but one, of the constrints. Proof. Where is the ntge point, for ech integer 1 i n, let θ i denote the ngle mesured counterclockwise round from the ry pointing t s π(i) to the ry pointing t s π(j), where j = (i mod n)+1, s shown in Figure 4. Let Θ = n i=1 θ i, tht is, the sum of ll the θ i. For ech pir of successie sites s π(i) nd s π(j) where θ i < 180, it must be tht, s π(i), nd s π(j) form tringle in counterclockwise order, like, s π(1), nd s π(2) in Figure 4. Tht is equilent to the sttement tht is on the left side of the directed line from s π(i) to s π(j), nd we cn express tht sttement s hlf-plne constrint. We crete such constrint for ech pir of successie sites. One wy might relize π would be if it were in the kernel of str-shped polygon formed by the sites in the order described by π; then eery θ i < 180 nd ll the liner constrints would be stisfied. This sitution is illustrted in Figure 5().

5 CCCG 2011, Toronto ON, August 10 12, 2011 s 2 s 2 () (b) Figure 5: Relizing permuttion while iolting () zero or (b) one of the liner constrints (π is the identity). It is lso possible for the ntge point to lie outside the kernel of the str-shped polygon, s shown in Figure 5(b). Howeer, if relizes π, then summing ll the θ i corresponds to mking one full sweep round ; Θ = 360. Thus, t most one of the θ i cn be greter thn 180, corresponding to iolted constrint; ll the others must be stisfied. Therefore, t most one of the constrints cn be iolted. To ctully sole the reliztion problem we must not only perform liner progrmming but lso determine which constrint to iolte, if ny. Here we exploit the specil properties of Megiddo s liner progrmming lgorithm [5], which either finds solution to the reliztion problem immeditely, or gies us clue to where the permuttion must strt. Theorem 4 There exists liner-time lgorithm tht gien sites, s 2,..., s n nd permuttion π on the indices, finds ntge point for which the rdil permuttion is π up to rottion, if such exists. Proof. We inoke the liner-time two-dimensionl liner progrmming lgorithm of Megiddo [5] to find point stisfying ll the constrints of Lemm 3, if possible. We cn then distinguish two cses: (1) such point exists; or (2) no such point exists. Cse 1. It is possible tht point could stisfy ll the constrints but not relize the permuttion π, if the sequence of sites described by π winds more thn once round. An exmple demonstrting this sitution is shown in Figure 6. When the liner progrm returns solution, it is esy to test in liner time whether relizes the permuttion π. If it does, the lgorithm returns it immeditely. Suppose does not relize π. The cumultie ngle Θ must be n integer multiple of 360 ; nd when is solution to the liner progrm but does not relize the desired permuttion, it must be t lest 720. Remoing one constrint (chnging the polygon to pth, which might still be self-intersecting) reduces the sum for the remining pirs of sites by strictly less thn 360, leing it strictly greter thn 360. Now suppose we strt our ry sweep with ry pointing from, solution to the liner progrm with one constrint relxed, to the site t the strt of the pth. If we sweep to ech successie site on the pth in turn, we will complete full ngle (360 ) nd see the strt of the pth gin, before we complete the sweep t the end of the pth. Tht mens we must lredy he seen the end of the pth, before its proper plce t the end of the sweep. Therefore cnnot relize the permuttion π. In intuitie terms, if the polygon wrps more thn once round some solution, it must wrp t lest twice, nd then the pth formed by deleting one edge from the polygon (which subtrcts less thn 360 ) must still wrp more thn once round eery solution. We he tht if there exists ntge point tht is solution to the liner progrm but does not relize the permuttion π, then no point which is solution to ny liner progrm formed by relxing one of the originl constrints, cn relize the permuttion π. Since eery point relizing π must be solution to our originl liner progrm with t most one constrint relxed, then there cn be no point relizing π t ll. Thus, in Cse 1, where the liner progrm is fesible, it suffices to test whether the solution relizes π, return it if it does relize π, nd return filure if it does not. Cse 2. If the first liner progrm is infesible, then ny relizing π must be solution to the liner progrm with exctly one constrint relxed. Megiddo s lgorithm [5] works by exmining constnt-sized subsets of the input constrints nd, t ech one, ttempting to proe tht t lest one of the constrints is unnecessry for the optiml solution. His nlysis shows tht in ech of the subsets t lest one constrint cn lwys be remoed if the input is fesible, llowing the lgorithm to stop fter liner number of steps with either solution or proof of infesibility. Tht pproch hs the impor-

6 23 d Cndin Conference on Computtionl Geometry, s 7 0 s 8 s2 s 9 ge three liner-time lgorithms for this kind of problem, corresponding to three schemes of determining permuttions: sweeping line in direction u, mesuring distnce from ntge point, nd sweeping ry counterclockwise round. One obious direction for future work is to consider other wys of determining permuttion; for exmple, rotting line through insted of ry strting t. We might lso consider more generl kinds of constrint stisfction inoling site permuttions; for instnce, finding point tht relizes ny permuttion contining gien contiguous subsequence. Figure 6: The sites wind more thn once round (π is the identity). Acknowledgments We wish to thnk Pk Ching Li nd Json Morrison for insightful discussions on these nd relted problems. References Figure 7: An infesibility certificte. tnt consequence tht in cse of n infesible input, the lgorithm ctully finds constnt-sized certificte of infesibility, nmely the lst subset of input constrints it exmined before hlting. The lgorithm cn esily be modified to produce the certificte s output, in the form of t most three constrints tht cnnot ll be stisfied. In generl those constrints will be rrnged s shown in Figure 7; with input not in generl position certificte consisting of two non-intersecting prllel hlf-plnes would lso be possible. In order to be ntge point relizing the desired rdil permuttion, would he to stisfy ll except t most one of the constrints in the originl liner progrmming problem. If cn stisfy ll except one, but not ll of the constrints, then eery infesible subset of the constrints must include tht one constrint, so it must be mong the t most three returned when Megiddo s lgorithm filed. By inoking Megiddo s lgorithm t most three more times, with ech constrint from the certificte remoed in turn, we cn find lue for tht relizes the permuttion π, if ny exists. [1] H. Bieri nd P.-M. Schmidt. On the permuttions generted by rottionl sweeps of plnr point sets. In Proceedings of the 8th Cndin Conference on Computtionl Geometry, Ottw, Cnd (CCCG 1996), pges , August [2] A. Björner, M. L. Vergns, B. Sturmfels, N. White, nd G. M. Ziegler. Oriented Mtroids. Cmbridge Uniersity Press, 2nd edition, [3] E. Cháez, K. Figuero, nd G. Nrro. Proximity serching in high dimensionl spces with proximity presering order. In Proceedings of the 4th Mexicn Interntionl Conference on Artificil Intelligence (MICAI 2005), Monterrey, Mexico, olume 3789 of Lecture Notes in Computer Science, pge Springer, Noember [4] J. E. Goodmn. On the combintoril clssifiction of nondegenerte configurtions in the plne. Journl of Combintoril Theory, Series A, 29(2): , September [5] N. Megiddo. Liner-time lgorithms for liner progrmming in R 3 nd relted problems. SIAM Journl on Computing, 12(4): , Noember [6] M. Skl. Counting distnce permuttions. Journl of Discrete Algorithms, 7(1):49 61, Mrch [7] I. Streinu. Clusters of strs. In Proceedings of the 13th Annul Symposium on Computtionl Geometry (SCG 1997), Nice, Frnce, pge39 441, June [8] B. Tor, L. Fred, nd S. M. LVlle. Lerning combintoril mp informtion from permuttions of lndmrks. Interntionl Journl of Robotics Reserch, October Conclusion In this pper, we considered the problem of relizing permuttion π on set of n sites in the plne. We