Constant-Work-Space Algorithms for Shortest Paths in Trees and Simple Polygons

Transcription

1 Journal of Graph Algorihms and Applicaions hp://jgaa.info/ vol. 15, no. 5, pp (2011) Consan-Work-Space Algorihms for Shores Pahs in Trees and Simple Polygons Tesuo Asano 1 Wolfgang Mulzer 2 Yajun Wang 3 1 School of Informaion Science, JAIST, Japan 2 Insiu für Informaik, Freie Universiä Berlin, Germany 3 Microsof Research Asia, Beijing, China Absrac Consan-work-space algorihms model compuaion when space is a a premium: he inpu is given as a read-only array ha allows random access o is enries, and he algorihm may use consanly many addiional regisers of O(log n) bis each, where n denoes he inpu size. We presen such algorihms for wo resriced varians of he shores pah problem. Firs, we describe how o repor a simple pah beween wo arbirary nodes in a given ree. Using a echnique called compuing insead of soring, we obain a naive algorihm ha needs quadraic ime and a consan number of addiional regisers. We hen show how o improve he running ime o linear by applying a echnique named simulaed parallelizaion. Second, we show how o compue a shores geodesic pah beween wo poins in a simple polygon in quadraic ime and wih consan work-space. Submied: April 2010 Aricle ype: Regular Paper Reviewed: November 2010 Final: May 2011 Revised: December 2010 Published: Ocober 2011 Acceped: May 2011 Communicaed by: M. S. Rahman and S. Fujia A preliminary version of his paper appeared as T. Asano, W. Mulzer, and Y. Wang, A Consan-Work-Space Algorihm for Shores Pahs in Simple Polygons. in Proc. 4h WAL- COM, pp. 9 20, T. Asano was parially suppored by he Minisry of Educaion, Science, Spors and Culure, Gran-in-Aid for Scienific Research on Prioriy Areas and Scienific Research (B). W. Mulzer was suppored by a Wallace Memorial Fellowship in Engineering, and in par by NSF gran CCF and NSF CCF addresses: -asano@jais.ac.jp (Tesuo Asano) mulzer@inf.fu-berlin.de (Wolfgang Mulzer) yajunw@microsof.com (Yajun Wang)

2 570 T. Asano e al. Shores Pahs wih Consan Work-Space 1 Inroducion We consider wo resriced varians of he shores pah problem in a compuaional model ha we call consan-work-space compuaion. In his model, which is also known as log-space [1], he inpu is given as a read-only array. Each cell of he array sores O(log n) bis and can be accessed in consan ime. Addiionally, he algorihm may use consanly many regisers, each of which sores O(log n) bis and can be read and wrien in consan ime. These regisers are referred o as he work-space of he algorihm, and he number of regisers is he size of he work-space. This model has been invesigaed before. One of he mos imporan resuls in his area is he selecion algorihm by Munro and Raman [22] which runs in O(n 1+ε ) ime using work-space O(1/ε), for any small consan ε > 0. More recenly, a consan-work-space algorihm by Reingold [23] for deermining wheher here exiss a pah beween wo arbirary verices in an undireced graph solved a long-sanding open problem in complexiy heory. Asano [2 5] describes applicaions o image processing. Consan-work-space algorihms for some geomeric problems are also known: Several auhors describe algorihms for enumeraing he verices and faces of geomeric arrangemens wihou addiional space [7,9,10,16,17,24]. Bose and Morin [11] describe how o find a pah beween wo given verices in a planar riangulaion in an online seing ha allows only consan space. Asano e al [6] give efficien consan-work-space algorihms for drawing he Delaunay riangulaion and he Voronoi diagram of a planar poin se, and hey also show how he Euclidean minimum spanning ree for a planar poin se can be consruced quickly in his model. They also show a differen algorihm for geodesic shores pahs in polygons. Our seing is similar o he sric daa-sreaming model, where he algorihm also has only a resriced amoun of work-space a is disposal. However, in he sreaming model he inpu can be read only once in a sequenial manner. Chan and Chen [12] give algorihms in differen compuaional models varying from a mulipass daa-sreaming model o he random access consan-work-space model considered here. In his paper, we focus on he design of fas algorihms in he consanwork-space model. Using wo resriced versions of he shores pah problem, we showcase some echniques for designing such algorihms. The firs echnique, named compuing insead of soring, is applied o he problem of finding a simple pah beween wo nodes in a given ree. A simple soluion using linear work-space goes as follows: compue an Eulerian pah beween he wo nodes and coun how ofen each edge appears on he pah. Then remove hose edges ha appear wice. This gives he desired simple pah. We can implemen his idea using only consanly many regisers: insead of soring a coun in each edge, we compue i direcly whenever we need o decide wheher o include an edge in he pah or no. This akes linear ime per edge, so we obain a quadraic-ime consan-work-space algorihm for he problem. Anoher imporan echnique is called simulaed parallelizaion. I may be considered as a generalizaion of he baby-sep, gian-sep mehod [19,

3 JGAA, 15(5) (2011) ], which uses wo poiners wih differen speeds o deec a loop in a given linked lis. In simulaed parallelizaion, we proceed as follows: given a verex u on he simple pah, we wan o deermine which edge o follow oward he arge verex. For his, we use wo poiners for scanning he subrees of u for. We alernaely advance hese poiners unil one of hem exhauss is subree or encouners. If a poiner exhauss is subree, we move i o he nex remaining subree of u. If no subree is lef, we can conclude ha he subree searched by he oher poiner mus conain. Simulaed parallelizaion is quie powerful. In fac, jus by incorporaing his echnique ino he naive quadraic-ime algorihm we can improve is running ime o linear. The above algorihms can be exended o an algorihm for finding shores geodesic pahs in a simple polygon. Given a simple polygon P wih n verices and wo poins s and in is inerior, we would like o find he shores pah beween s and ha says wihin P. A naive approach leads o a cubic-ime algorihm, bu a more careful implemenaion yields a quadraic-ime algorihm. See also Asano e al [6] for a differen approach ha also yields a quadraic ime algorihm. Wha abou general weighed graphs? Of course, in he linear-work-space model here is he classic and popular shores pah algorihm by Dijksra [15, Chaper 24], which can be implemened in O(n 2 ) ime using very simple daa srucures. Is i sill possible o find an analogous algorihm in he consanwork-space model? Unforunaely, no such algorihm is known, and i is unlikely o exis, since he shores pah problem for general weighed graphs is NLcomplee [18]. 2 Simple Pahs in Trees Using Eulerian Tours As our firs problem, we consider he following quesion: le T be a ree wih n nodes. Given wo disinc nodes s and in T, find a simple pah from s o wih no node visied more han once. Here is a simple algorihm using O(n) ime and space: i is well known ha every ree has an Eulerian our, i.e., a closed walk ha visis every edge exacly wice. Compue such a our and ake a subour E ha goes from s o. Remove from E all edges ha occur wice. This yields a simple pah beween s and. Unforunaely, in our consan-work-space model no exra array can be used for soring he our and finding he duplicae edges. To obain a consan-space algorihm, we apply he echnique compuing insead of soring, where we recompue a subour of he Eulerian our whenever we need i. We assume ha he ree is given by adjacency liss sored in a read-only array. Le Adj(u) be he adjacency lis of a node u T. The following wo funcions suffice o generae an Eulerian our. FirsNeighbor(u): given a node u, reurn he firs node in he adjacency lis Adj(u).

4 572 T. Asano e al. Shores Pahs wih Consan Work-Space NexNeighbor(u, v): Given a node u and an adjacen node v, reurn he successor of v in he adjacency lis Adj(u). If v is he las node, reurn he firs node in he lis. The funcion FirsNeighbor can easily be performed in consan ime, bu he ime required for NexNeighbor depends on which daa srucure we assume. More precisely, when generaing he Eulerian our, afer following an edge from a verex v o a verex u, we need o find he enry for v in Adj(u) so ha we can execue NexNeighbor(u, v). If he ree is given by a doublyconneced edge lis (DCEL) [8, Chaper 2], his can be done in consan ime. On he oher hand, if we represen Adj(u) as a simple lis, we may need o search he whole lis for he successor of v, which akes ime O( ), where is he maximum degree of a node in he ree. Given a ree T, a saring node s, and a arge node, an algorihm ha finds he shores pah from s o is shown in Algorihm 1. The algorihm repeaedly calls he funcion FindFeasibleSubree o obain he nex edge on he shores pah by deermining he subree of he curren node ha conains. In Find- FeasibleSubree, we use a funcion SubreeSearch o deermine wheher a given subree conains : SubreeSearch sars from an edge (u, v) inciden o u and follows he Eulerian pah by applying he funcion NexNeighbor. If we encouner he win edge (v, u) before, hen is no conained in he corresponding subree, i.e., (u, v) and all edges in he subree appear wice in he our and can be omied (see Figure 1). v k T k u v i T i s v 1 v 2 T 2 T 1 Figure 1: Which subree of u conains he arge node? Theorem 1 Given a ree T wih n nodes, a saring verex s and a arge verex, le l be he lengh of he shores pah from s o. There is an algorihm ha repors he pah from s o in O(lnd) ime and wih consan work-space. Here, d depends on which daa srucure is used: if T is given by a DCEL, hen d = O(1). If T is given by mere adjacency liss, hen d = O( ), where is he maximum node degree in T. The running ime is always O(n 2 d).

5 JGAA, 15(5) (2011) 573 Algorihm 1: Finding a simple pah from s o. Inpu: A ree T, wo nodes s and in T. Oupu: A simple pah from s o. begin currennode = s; sarneighbor = FirsNeighbor(s); repea repor currennode; u = currennode; currennode = FindFeasibleSubree(u, sarneighbor, ); sarneighbor = NexNeighbor(currenNode, u) : unil currennode == // reurns a child of u whose subree conains funcion FindFeasibleSubree(u, v, ) begin for each node w in Adj(u), in order saring from v do if SubreeSearch(u, w, ) hen reurn w; // checks wheher he subree of u rooed a v conains funcion SubreeSearch(u, v, ) begin currennode = u; neighbor = v; repea nexnode = NexNeighbor(neighbor, currennode); currennode = neighbor; neighbor = nexnode; unil (currennode == or (currennode == v and neighbor == u)) reurn (currennode == ); Proof: Refer o Algorihm 1. The algorihm sars from s and checks for each neighbor of s wheher he corresponding subree conains. Then i proceeds o he appropriae neighbor and coninues. Thus, i suffices o show correcness of he funcion SubreeSearch(u, v, ) which checks wheher he subree of u rooed a v conains. This is proved by inducion on he heigh of he subree. This algorihm runs in O(lnd) ime, as FindFeasibleSubree is called a mos l imes, and each such call akes O(nd) ime. Since l n, i follows ha his is always O(n 2 d). Figure 2 illusraes how he search proceeds. A Linear Time Algorihm. We now improve he running ime of Algorihm 1 o O(n), using simulaed parallelizaion. Consider Algorihm 1. The wors case happens when each call o Find- FeasibleSubree(u, v, ) akes almos linear ime. In oher words, he subrees conaining are large. Consider he edge (u, v). If is in he subree rooed a

6 574 T. Asano e al. Shores Pahs wih Consan Work-Space B C A E G D H Sar F J I N M P L K Goal O S Q R Adjacency liss: A (D) B (D) C (E) D (A, B, E, H) E (C, F, D) F (E) G (H) H (D, G, I, M, P) Figure 2: Canonical raversal of a ree given by adjacency liss. v and if his subree is large, SubreeSearch(u, v, ) could ake a long ime. However, if insead we call SubreeSearch(u, w, ), for all oher children of u, we can quickly conclude ha is in he subree for child v, wih running ime proporional o he size of hose oher subrees. If we could guaranee ha all subrees no conaining are cu off from he search space a mos once, we could safely charge he running ime o he nodes in hose subrees, resuling in a linear-ime algorihm. The only apparen difficuly is ha we do no know which subree conains. Le us sar wih a rivial soluion ha runs SearchSubree(u, v, ) in parallel, one process for each subree of u excep he one conaining s, such ha he seps of he parallel processes are synchronized. Le N(u, ) be he oal number of nodes in he subrees of u which do no conain s or. We claim ha we can find he subree wih in O(N(u, )) seps, where we sum he number of seps over all parallel processes. For his, we sop all parallel processes as soon as one of he following condiions is saisfied: (a) one process reurns rue; (b) all processes bu one reurn false. Clearly, he oal number of seps in all processes is a mos 2N(u, ), as claimed. Since we are working wih a sequenial machine, we mus simulae he parallel execuion by ieraively advancing each process in urn. However, if he maximum degree is linear, we canno afford o simulae O(n) processes since we have o sore he inernal saes for all of hem. Insead, we only mainain wo copies of SubreeSearch(u, v, ) a a ime. We sar anoher copy if one process finishes and if here are more subrees o explore. Remember ha we can sop if here is only one process lef. In his way, we can find he simple pah from s o in linear ime. For compleeness, we presen pseudocode for FindFeasibleSubree(u, v, ) in Algorihm 2. Theorem 2 Given wo nodes s and in a ree T represened by a DCEL, here is an algorihm ha finds he simple pah from s o in T in O(n) ime using O(1) addiional space. Proof: In Algorihm 2, FindFeasibleSubree(u, v, ) akes ime linear in he number of nodes in he subrees of v which do no conain. Noice ha every subree ha does no conain is explored a mos once, due o our seing

7 JGAA, 15(5) (2011) 575 of numofneighbors in he algorihm. Therefore, Algorihm 1 akes linear ime wih he new implemenaion of FindFeasibleSubree(u, v, ) in Algorihm 2, as claimed. 3 Geodesic Shores Pahs in Polygons We now urn o he problem of finding a shores pah beween wo arbirary poins s and wihin a given polygon P. If linear work-space is allowed, here is a classic linear-ime algorihm due o Lee and Preparaa [20]. Their algorihm works as follows: We firs pariion he inerior of P ino riangles using Chazelle s mehod [13]. Then, we compue he dual graph G of he riangulaion: he verices of G correspond o he riangles, and wo verices are adjacen precisely if heir corresponding riangles share an edge. We locae he poins s and in he riangulaion. Since G is a ree, any wo verices in G are conneced by a unique simple pah. Consider he pah beween he riangle conaining s and he riangle conaining. I defines a sequence (e 0, e 1,..., e m ) of diagonals hi by he pah. The algorihm walks along his sequence while mainaining a funnel. The funnel consiss of a cusp p, iniially se o s, and of wo concave chains from p o he wo endpoins of he curren diagonal e i. In each sep, here are wo cases: (i) if he nex diagonal remains visible from he cusp, we jus updae he appropriae concave chain, similar o Graham s scan; (ii) if he nex diagonal is no visible from he cusp, we proceed along he appropriae chain unil we find he cusp for he nex funnel, and we oupu he verices encounered along he way as par of he shores pah. Implemened properly, all his can be done in linear ime (see Lee and Preparaa [20] for deails). 3.1 A Shores-Pah Algorihm Using he Dual Graph We adap he algorihm by Lee and Preparaa [20] o use consan work-space. For his, we need o solve wo problems: (i) we need o develop an algorihm for riangulaing a given simple polygon and hen finding a simple pah in he dual graph; and (ii) we mus mainain he funnel during he raversal. The difficuly here is, of course, ha we canno sore any inermediae resuls. Compuing he riangulaion. In order o mainain he riangulaion of our polygon P efficienly, we need a canonical riangulaion of P. This means ha for every riple of verices in P, here should be an easily checkable condiion o deermine wheher he riple defines a riangle or no. This allows us o recompue he individual riangles encounered on he pah as needed. Specifically, our canonical riangulaion will be he consrained Delaunay riangulaion [14]. For a poin se S, hree poins of S deermine a Delaunay riangle if and only if he circle defined by he hree poins conains no poin of S in is proper inerior. Such a circle is called an empy circle. The Delaunay

8 576 T. Asano e al. Shores Pahs wih Consan Work-Space riangles pariion he convex hull of he se S, and he resuling srucure is called he Delaunay riangulaion of S [8, Chaper 9]. Consrained Delaunay riangulaions offer a way o exend his noion o a simple polygon P [14]. 1 The verices of P define a poin se V and he edges define a se E of line segmens. Consrained Delaunay edges are defined using he noion of a diagonal. A diagonal is an open line segmen beween wo polygon verices ha does no inersec he boundary of P. A pair (p, q) of verices defines a consrained Delaunay edge if and only if here is a hird poin r in V such ha (i) (p, q) is a diagonal; (ii) (p, r) and (q, r) are diagonals polygon edges; and (iii) he circle hrough p, q, r does no conain any oher poin s V ha is visible from r, i.e., for no oher poin s in he circle does (r, s) define a diagonal. p r q (a) (b) (c) Figure 3: An example of a consrained Delaunay riangle. (a) The Delaunay riangulaion of a planar poin se; (b) he overlay of a polygon wih he Delaunay riangulaion of is verex se; and (c) a Delaunay riangle (p, q, r) and is associaed empy circle. Noe ha he verex in he circle is no visible from r. I is known ha here is a unique consrained Delaunay riangulaion DT(P ) for any simple polygon whose verices are in general posiion. We denoe he dual graph by DT(P ). Recall ha since a simple polygon is simply conneced, DT(P ) is a ree. We would like o use Theorem 2 o find he shores pah from s o in DT(P ). For his, we need o implemen he funcion NexNeighbor(). By he definiion of he dual graph and he fac ha each DT(P ) has maximum degree a mos hree, he nex neighbor is deermined by he clockwise or counerclockwise nex Delaunay edge, as shown in Figure 5. Hence, we need o find he hird verex of a Delaunay riangle for a given edge. More precisely, given a Delaunay edge (u, v), we wan o find a verex w such ha (i) w is visible from he edge (u, v); and (ii) he circle defined by u, v, and w is empy, ha is, i does no conain any oher verex visible from he edge (u, v). Thus, i akes O(n 2 ) ime o find a verex which complees a Delaunay edge o a Delaunay riangle, and his is also he ime for an invocaion of NexNeighbor(). We emphasize 1 More generally, consrained Delaunay riangulaions allow us o define a riangulaion ha conains a prespecified se of edges and is as close o he usual Delaunay riangulaion as possible.

9 JGAA, 15(5) (2011) 577 s (a) (b) (c) Figure 4: The unique pah on he dual graph and is corresponding sequence of Delaunay edges. (a) The consrained Delaunay riangulaion of a given simple polygon; (b) he dual graph of he riangulaion (a ree); and (c) he unique pah beween s and. ha we never sore DT(P ) explicily, bu we only access i on he fly by direc compuaion. This is anoher applicaion of he compuing insead of soring principle. (a) (b) Figure 5: Finding a shores pah beween wo poins wihin a simple polygon. (a) Walking along a pah in he dual graph while finding he clockwise or counerclockwise nex riangular edge; (b) he evoluion of he visibiliy angles. Mainaining he funnel. During he walk in DT(P ) from s o, we need o mainain he curren funnel. Unforunaely, we do no have space o sore he wo concave chains. Therefore, we proceed as follows: while walking, we only mainain he cusp p of he funnel and he wo verices a and b ha deermine he visibiliy angle from p (a, b are he firs verices on he wo concave chains). We iniialize p o he saring poin s and a, b o he endpoins of he firs diagonal inerseced by he shores pah, e 0. In order o process a new diagonal e i, we ake he inersecion of he curren visibiliy angle wih ha defined by e i. If he new visibiliy angle is nonempy, we do nohing. See Figure 6. Oherwise, we mus updae he cusp of he curren funnel. For his, we sar from a or b, depending on wheher e i lies above or below he old visibiliy angle. Then, we perform a Jarvis march (i.e., a sequence of gif-wrapping seps [15, Chaper 33.3]) along he boundary of he polygon, oupuing he verices of he concave chain, unil we find a verex from which e i is visible again (his

10 578 T. Asano e al. Shores Pahs wih Consan Work-Space can be checked in O(n) ime per verex). This verex becomes our new cusp p, and a, b are se o he highes and lowes poin on e i ha is visible from p; see Figure 7 for an example. e i θ i θ i+1 e i+1 Figure 6: Angle changes while walking along he edge sequence. Since every sep in Jarvis s march needs O(n) ime, and since here are O(n) such seps overall, he oal running ime for mainaining he funnel is O(n 2 ). We have hus shown: Theorem 3 There is a consan-work-space algorihm for finding a shores pah beween any wo poins inside a simple n-gon P in ime O(n 3 ). Proof: By Theorem 2, he shores pah in DT(P ) can be found by O(n) applicaions of he funcion NexNeighbor(). Since each such call akes O(n 2 ) ime, and since he oal running ime for mainaining he funnel as described above is O(n 2 ), we obain he bound in he heorem. 3.2 A Shores-Pah Algorihm Using Poin Locaion The algorihm from Theorem 3 comes from a direc adapaion of he algorihm by Lee and Preparaa [20]. The dual graph gives us he correc direcion oward a given arge poin. In his secion we show ha here is a more direc way o find his direcion, see also Asano e al [6] for anoher approach. We assume wihou loss of generaliy ha polygon verices are numbered sequenially 0 hrough n 1 around he boundary (since he verices of he polygon are given sequenially in our inpu array, hese numbers could correspond o he addresses of he cells soring he verices). Suppose we are in some riangle A in DT(P ). Removing A divides P ino a mos hree pars, and we need o find he par ha conains. To do his, we use he indices associaed wih he polygon edge e jus above. The par of he polygon ha conains is he par whose boundary conains he edge e, unless he upward verical line segmen from o e crosses A before reaching e. The process of finding he par of he polygon ha conains is called poin locaion. Since we jus need o compare indices and check for inersecion wih A, poin locaion akes jus O(1) ime once we have precompued e. Now we know which way o go from any given riangle. Unforunaely, finding adjacen riangles in a canonical riangulaion can be slow, and i urns

11 JGAA, 15(5) (2011) 579 e 0 e 1 e 2 s s p e 5 e 8 e 12 s p Figure 7: Mainaining he funnel. Iniially, he cusp of he funnel is se o s, and a, b are he endpoins of he diagonal e 0. The visibiliy angle needs o be updaed on encounering e 1 (op). A e 5, he visibiliy angle vanishes, and we mus advance he cusp. Now, he visibiliy angle is updaed a e 8. A e 12, he visibiliy angle becomes empy again, and we need o perform a Jarvis march on he lower boundary of he polygon unil we find he nex cusp.

12 580 T. Asano e al. Shores Pahs wih Consan Work-Space ou o be more efficien o replace he consrained Delaunay riangulaion by he rapezoidal decomposiion [8, Chaper 6]. Tha is, we pariion he inerior of P by drawing a verical chord a each verex oward he inerior of he polygon. This decomposiion is canonical and easy o compue. Moreover, i offers he same properies ha he riangulaion used o have for shores pahs. s s s (a) (b) (c) Figure 8: Trapezoidal decomposiion of a simple polygon for finding a shores pah in a simple polygon. (a) A simple polygon and wo inernal poins s and o be inerconneced wihin he polygon. (b) Trapezoidal decomposiion of P. (c) A sequence of rapezoids beween wo conaining s and. The rapezoidal decomposiion defined above is uniquely deermined for any simple polygon. However, degeneracies can cause one rapezoid o be adjacen o arbirarily many oher rapezoids, as shown in Figure 9. We perform a symbolic perurbaion o avoid his issue: he verices of P and he wo poins s and all have inegral coordinaes wih O(log n) bis. Each inegral poin (x, y) is reaed as a poin (x + yε, y), i.e., i is shifed o he righ by yε for a small parameer ε such ha y ε < 1 for he larges y-coordinae y appearing in he inpu. Afer his perurbaion, no wo verices share he same x-coordinae, as shown o he righ in Figure 9. Figure 9: Removing degeneracies by shifing verices o he righ. An original polygon is given o he lef. The conversion resuls in he righ polygon in which no wo verices share he same x-coordinae. From now on, we assume ha every rapezoid is adjacen o a mos four oher rapezoids. Our firs goal is o find he sequence of rapezoids beween hose conaining s and. For his, i suffices o find he correc neighbor a each rapezoid along he pah. A characerizaion of a rapezoid is given in Figure 10. Given an arbirary poin q in he inerior of P, we can deermine he rapezoid conaining q as follows: firs find he polygon edges which are hi by a verical ray emanaing

13 JGAA, 15(5) (2011) 581 from q upward. The one closes o q is he op edge e a (T ) of he rapezoid T conaining q. In a similar fashion we can find he polygon edge e b (T ) jus below q, which is he boom edge of T. Then, we compue he lef and righ verical sides of he rapezoid T, denoed by v l (T ) and v r (T ), respecively. We sar wih four endpoins of e a (T ) and e b (T ). v l (T ) is iniially deermined by he righmos of he wo lef endpoins of e a (T ) and e b (T ). The iniial value of v r (T ) is similarly deermined. Then, we scan each polygon verex. If i lies inside he curren rapezoid and is inciden polygon edge eners he rapezoid from is lef, hen we updae he value v l (T ) o be he x-coordinae of he verex. If i lies in T and is inciden edge eners T from he righ, we updae v r (T ). In his way we can obain he rapezoid in O(n) ime. A a rapezoid T we have o find is neighbors and deermine how o proceed oward he arge poin. The firs quesion can be solved as follows: Suppose we wan o find a rapezoid T r which shares a righ boundary wih T. To do his, ake a poin q which is locaed o he righ of he side a a small enough disance. Using he poin q, he rapezoid T r is compued in he same manner as described above. Thus, we can find all adjacen rapezoids in O(n) ime. To deermine which rapezoid o follow, simply raverse he boundaries of he regions divided by he adjacen rapezoids o find which region conains he edge e jus above. This is done using indices of endpoins of hose subpolygons associaed wih he curren rapezoid. e e a (T ) e s v l (T ) P T v r (T ) e b (T ) s (a) (b) Figure 10: Characerizaion of a rapezoid T by wo polygon edges bounded from above and below and wo verical sides. (a) A rapezoid adjacen o hree rapezoids. (b) A polygon edge e s jus above he poin s and a polygon edge e jus above. Hence, we can find he correc neighbor in O(n) ime wih consan workspace. Since we need o raverse O(n) rapezoids, he oal ime is O(n 2 ). We sill need o describe how o find he shores pah from s o, bu his works jus as in he previous algorihm: we know how o walk on he sequence using O(n) ime per sep, and o find a shores pah we mainain he funnel from he curren saring poin, as before. Theorem 4 Given a simple polygon P wih n verices and wo arbirary poins

14 582 T. Asano e al. Shores Pahs wih Consan Work-Space s and in P, we can find a geodesic shores pah beween s and in O(n2 ) ime wih consan work-space. We have implemened our algorihm using LEDA [21] o check he deails. An experimenal resul is shown in Figure 11. The inpu configuraion is shown in (a). Afer finding he iniial segmen on he shores pah in (b), we repeaedly generae nex rapezoids unil he visibiliy angle vanishes, i.e., he las verical edge is no visible from he curren cusp of he funnel. This happens in (c), so he shores pah is exended and he cusp is moved as shown in (c). The final resul is appears in (d). The shores pah is given by bold lines in he figure. (a) (b) (c) (d) Figure 11: Experimenal resuls. The cusp of he curren funnel is shown as he las verex of he green pah.

15 JGAA, 15(5) (2011) Concluding Remarks We have presened consan-work-space algorihms for finding shores pahs in rees and polygons. One obvious fuure direcion is o exend he laer algorihm o he general Euclidean shores pah problem in he presence of polygonal obsacles. If he number of obsacles is bounded by some small consan k, hen here are O(n k ) differen ways o conver he problem ino one on a simple polygon by connecing he obsacles by chords, so we can obain a polynomialime algorihm. However, finding such an algorihm for an unbounded number of obsacles, or proving ha no polynomial-ime consan-work-space algorihm exiss, seems more challenging. A number of geomeric problems are open in he consan work-space model. For example, does here exis an efficien consan-work-space algorihm for compuing he visibiliy polygon from a poin in a simple polygon. Anoher ineresing direcion is o invesigae ime-space rade-offs: how much workspace is necessary o find a shores pah in a simple polygon in linear ime?

16 584 T. Asano e al. Shores Pahs wih Consan Work-Space Algorihm 2: New implemenaion of FindFeasibleSubree(u, v, ). // reurn a child of u whose subree conains. funcion FindFeasibleSubree(u, v, ) begin fneighbor = v lasneighbor = sneighbor = NexNeighbor(u, v) numofneighbors = deg(u) 1 // in case of s, we mus explore all subrees. if u == s hen numofneighbors++ if numofneighbors == 1 hen reurn fneighbor one = u; onenex = fneighbor; wo = u; wonex = sneighbor; while rue do // advance wo copies. (sigf1, sigc1, one, onenex) = AdvSearch(u, fneighbor, one, onenex) (sigf2, sigc2, wo, wonex) = AdvSearch(u, sneighbor, wo, wonex) if sigf1 hen reurn fneighbor if sigf2 hen reurn sneighbor if no sigc1 hen // copy 1 finishes wihou finding. onenex = fneighbor = NexNeighbor(u, lasneighbor) lasneighbor = fneighbor; one = u numofneighbors-- if numofneighbors == 1 hen reurn sneighbor if no sigc2 hen // copy 2 finishes wihou finding. wonex = sneighbor = NexNeighbor(u, lasneighbor) lasneighbor = sneighbor; wo = u numofneighbors-- if numofneighbors == 1 hen reurn fneighbor // advance he Eulerian our in he subree rooed a v. funcion AdvSearch(u, v, u, v ) begin v = NexNeighbor(v, u ); u = v ; if u == v and v == u hen reurn (found = false, coninue = false, u,v ) if v == hen reurn (found = rue, coninue = false, u,v ) reurn (found = false, coninue = rue, u,v )

17 JGAA, 15(5) (2011) 585 References [1] S. Arora and B. Barak. Compuaional complexiy. A modern approach. Cambridge Universiy Press, Cambridge, UK, [2] T. Asano. Consan-work-space algorihms: how fas can we solve problems wihou using any exra array? In Proc. 19h Annu. Inerna. Sympos. Algorihms Compu. (ISAAC), invied alk, volume 5369 of Lecure Noes in Compuer Science, page 1. Springer-Verlag, [3] T. Asano. Consan-working space algorihm for image processing. In Proc. of he Firs AAAC Annual meeing, page 3, Hong Kong, April [4] T. Asano. Consan-work-space algorihms for image processing. In F. Nielsen, edior, Emerging Trends in Visual Compuing (ETVC 2008), volume 5416 of Lecure Noes in Compuer Science, pages Springer-Verlag, [5] T. Asano. Consan-working-space image scan wih a given angle. In Proc. 24h European Workshop Compu. Geom. (EWCG), pages , Nancy, France, March [6] T. Asano, W. Mulzer, G. Roe, and Y. Wang. Consan-working-space algorihms for geomeric problems. Journal on Compuaional Geomery, page o appear, See also CCCG [7] D. Avis and K. Fukuda. A pivoing algorihm for convex hulls and verex enumeraion of arrangemens and polyhedra. Discree Compu. Geom., 8(3): , [8] M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars. Compuaional Geomery: Algorihms and Applicaions. Springer-Verlag, Berlin, hird ediion, [9] M. de Berg, M. J. van Kreveld, R. van Oosrum, and M. H. Overmars. Simple raversal of a subdivision wihou exra sorage. Inernaional Journal of Geographical Informaion Science, 11(4): , [10] P. Bose and P. Morin. An improved algorihm for subdivision raversal wihou exra sorage. Inerna. J. Compu. Geom. Appl., 12(4): , [11] P. Bose and P. Morin. Online rouing in riangulaions. SIAM J. Compu., 33(4): , [12] T. M. Chan and E. Y. Chen. Muli-pass geomeric algorihms. Discree Compu. Geom., 37(1):79 102, [13] B. Chazelle. Triangulaing a simple polygon in linear ime. Discree Compu. Geom., 6(5): , 1991.

18 586 T. Asano e al. Shores Pahs wih Consan Work-Space [14] L. P. Chew. Consrained Delaunay riangulaions. Algorihmica, 4(1):97 108, [15] T. H. Cormen, C. E. Leiserson, R. L. Rives, and C. Sein. Inroducion o algorihms. MIT Press, Cambridge, MA, hird ediion, [16] H. Edelsbrunner, L. J. Guibas, and J. Solfi. Opimal poin locaion in a monoone subdivision. SIAM J. Compu., 15(2): , [17] C. M. Gold and S. Cormack. Spaially ordered neworks and opographic reconsrucions. In. J. Geographical Informaion Sysems, 1(2): , [18] A. Jakoby and T. Tanau. Logspace algorihms for compuing shores and longes pahs in series-parallel graphs. Technical Repor TR03-077, ECCC Repors, [19] D. Knuh. The Ar of Compuer Programming. Addison-Wesley, Reading, Massachuses, [20] D. T. Lee and F. P. Preparaa. Euclidean shores pahs in he presence of recilinear barriers. Neworks, 14(3): , [21] K. Mehlhorn and S. Näher. LEDA. A plaform for combinaorial and geomeric compuing. Cambridge Universiy Press, Cambridge, [22] J. I. Munro and V. Raman. Selecion from read-only memory and soring wih minimum daa movemen. Theore. Compu. Sci., 165(2): , [23] O. Reingold. Undireced conneciviy in log-space. J. ACM, 55(4):Ar. #17, 24 pp., [24] G. Roe. Degenerae convex hulls in high dimensions wihou exra sorage. In Proc. 8h Annu. ACM Sympos. Compu. Geom. (SoCG), pages 26 32, [25] D. Shanks. The infrasrucure of a real quadraic field and is applicaions. In Proc. Number Theory Conference, pages , 1972.