Convex Partitions with 2-Edge Connected Dual Graphs

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1 Noname manuscipt No. (will be inseted by the edito) Conex Patitions with 2-Edge Connected Dual Gaphs Mawan Al-Jubeh Michael Hoffmann Mashhood Ishaque Diane L. Souaine Csaba D. Tóth Receied: date / Accepted: date Abstact It is shown that fo eey finite set of disjoint conex polygonal obstacles in the plane, with a total of n etices, the fee space aound the obstacles can be patitioned into open conex cells whose dual gaph (defined below) is 2-edge connected. Intuitiely, eey edge of the dual gaph coesponds to a pai of adjacent cells that ae both incident to the same etex. Aichholze et al. ecently conjectued that gien an een numbe of line-segment obstacles, one can constuct a conex patition by successiely extending the segments along thei suppoting lines such that the dual gaph is the union of two edge-disjoint spanning tees. Hee we pesent counteexamples to this conjectue, with n disjoint line segments fo any n 15, such that the dual gaph of any conex patition constucted by this method has a bidge edge, and thus the dual gaph cannot be patitioned into two spanning tees. Keywods Conex Patitions Dual Gaphs Geometic Matchings 1 Intoduction Fo a finite set S of disjoint conex polygonal obstacles in the plane R 2, a conex patition of the fee space R 2 \ ( S) is a set C of open conex egions (called cells) lying in the fee space such that the cells ae paiwise disjoint and thei closues coe the entie fee space. Since eey etex of an obstacle is a eflex etex of the fee M. Al-Jubeh, M. Ishaque, D. L. Souaine Depatment of Compute Science, Tufts Uniesity, Mefod, MA, USA {maljub01, mishaque, dls}@cs.tufts.edu This mateial is based upon wok suppoted by the National Science Foundation unde Gant No. CCF and CCF , and Tufts Poost s Summe Scholas Pogam. M. Hoffmann Institute of Theoetical Compute Science, ETH Züich, Switzeland hoffmann@inf.ethz.ch C. D. Tóth Depatment of Mathematics, Uniesity of Calgay, Calgay, AB, Canada cdtoth@ucalgay.ca Suppoted by NSERC gant RGPIN Reseach done at Tufts Uniesity.

2 2 space, it must be incident to at least two cells. Let σ be an assignment of eey etex to two adjacent conex cells in C. A conex patition C and an assignment σ define a dual gaph D(C, σ): the cells in C coespond to the nodes of the dual gaph, and each etex of an obstacle coesponds to an edge between the two cells assigned to (see Fig. 1). Double edges ae possible, coesponding to two endpoints of a line-segment obstacle on the common bounday of two cells. (a) (b) (c) (d) Fig. 1 (a) Fie obstacles with a total of 12 etices. (b) A conex patition. (c) An assignment σ. (d) The esulting dual gaph (dashed lines). It is staightfowad to constuct an abitay conex patition fo a set of conex polygons as follows. Let V denote the set of etices of the obstacles; and let π be a pemutation on V. Pocess the etices in the ode π. Fo a etex V, daw a diected line segment (called extension) that stats fom the etex along the angle bisecto (fo a line-segment obstacle, the extension is collinea with the obstacle), and ends whee it hits anothe obstacle, a peious extension, o infinity. Fo k conex obstacles with a total of n etices, this naïe algoithm poduces a conex patition with n k 1 cells, if no two extensions ae collinea. Fo example, fo n disjoint line segments (with 2n endpoints) in geneal position, we obtain n 1 cells. If the obstacles ae in geneal position, then each etex is incident to exactly two cells, lying on opposite sides of the extension emanating fom. Hence the assignment σ is unique, and the choice of pemutation π completely detemines the dual gaph, which we denote by D(π). We call this a Staight-Fowad conex patition, and a Staight-Fowad dual gaph, which depends only on the pemutation π of the etices. Results. We show instances whee no pemutation π poduces a Staight-Fowad dual gaph D(π) that is 2-edge connected (Section 2). This is a counteexample to a conjectue by Aichholze et al. [1]. We show that fo eey finite set of disjoint conex polygons in the plane, with no thee collinea etices, thee is a conex patition C (not necessaily Staight- Fowad) and an assignment σ such that D(C, σ) is 2-edge connected (Section 3). We define a wide class of conex patitions, which includes all Staight-Fowad conex patitions. In this class, a bidge in the dual gaph is chaacteized by a cetain simple polygon in the conex patition (a fobidden patten). To build a conex patition with a 2-edge connected dual gaph, we stat with an abitay Staight- Fowad conex patition and apply a sequence of local modifications until all fobidden pattens ae eliminated. A local modification continuously defoms a simple

3 3 polygon, which coesponds to a fobidden patten. Simila continuous motion aguments hae peiously been used fo poing combinatoial esults in [2, 3, 16, 19, 21]. Motiation. A plane matching is a set of n disjoint line segments in the plane, which is a pefect matching on the 2n endpoints. Two plane matchings on the same etex set ae compatible if thee ae no two edges that coss, and ae disjoint if thee is no shaed edge. Aichholze et al. [1] conjectued that fo eey plane matching on 4n etices, thee is a disjoint compatible plane matching (compatible geometic matchings conjectue). They poed that thei conjectue holds if the 2n segments in the matching admit a conex patition whose dual gaph is the union of two edge-disjoint spanning tees, and the two endpoints of each segment coesponds to distinct spanning tees. Aichholze et al. futhe conjectued fo the 4n endpoints of 2n line segments in the plane, thee is a pemutation π such that the Staight-Fowad dual gaph D(π) is the union of two edge-disjoint spanning tees (two spanning tees conjectue). The conjectue would immediately imply that such a dual gaph is 2-edge connected. Benbenou et al. [4] claimed that thee is always a pemutation π such that D(π) is 2-edge connected but thee was a flaw in thei agument [5]. Ou fist esult shows that such pemutation π does not always exist, and it also efutes the two spanning tees conjectue of Aichholze et al. [1]. Related Wok. Gien a set of conex polygonal obstacles and a bounding box, we may think of the bounding box as a simple polygon and the obstacles as polygonal holes. Then the poblem of ceating a conex patition becomes that of decomposing the polygon with holes into conex pats. Conex polygonal decomposition has eceied consideable attention in the field of computational geomety. The focus has been to poduce a decomposition with as few conex pats as possible. Lingas [18] showed that finding the minimal conex decomposition (decomposing the polygon into the fewest numbe of conex pats) is NP-had fo polygons with holes. Thee ae appoximation algoithm [15] and it is fixed paamete tactable [10]. Howee, fo polygons without holes, minimal conex decompositions can be computed in polynomial time [9, 14] see [13] fo a suey on polygonal decomposition. While minimal conex decomposition is desiable, the numbe of conex pats is not the only measue of the quality of a conex patition (decomposition). In Lien s and Amato s wok on appoximate conex decomposition [17] with applications in skeleton extaction, the goal is to poduce an appoximate conex patition (whee not all cells ae conex) that highlights salient featues. In the equitable conex patitioning poblem, all conex cells ae equied to hae the same alue of some measue e.g. the same numbe of ed and blues points [12], o the same aea [8]. Anothe citeion fo the quality of a conex patition might be some popety of its dual gaph (the definition of dual gaph aies fom application to application). Tan et al. [20] show that a Staight-Fowad conex patition fo senso netwoks can be computed in a distibuted manne, and demonstate how it impoes the pefomance of the geogaphic outing algoithms. A leade senso is chosen fo each cell. Communication between sensos belonging to diffeent conex cells is outed though these leade sensos. Two leade sensos can communicate with each othe if and only if they shae a bounday etex. The netwok of leade sensos can be modeled by the dual gaph of the conex patition (although Tan et al. [20] did not explicitly efe to a dual gaph). We beliee that the communication in thei model can be made fault-toleant using a conex patition poduced by ou algoithm.

4 4 2 A Counteexample fo the Two Spanning Tees Conjectue Theoem 1 Fo eey n 15, thee ae n disjoint line segments in the plane such that the Staight-Fowad dual gaph D(π) has a bidge edge fo eey pemutation π on the 2n segment endpoints. C 2 C 1 B 4 B 5 C 3 B 3 C 5 O B 2 C 4 B 1 A 1 A2 A 3 Fig. 2 A counteexample with n = 15 segments. Eey pemutation poduces a Staight- Fowad dual gaph with a bidge edge. A 5 A 4 Poof. We show that fo the 15 line segments in Fig. 2, eey pemutation π poduces a Staight-Fowad dual gaph D(π) with a bidge edge (that is, emoing this edge disconnects the dual gaph). We can geneate lage constuctions by adding segments whose suppoting lines aoid the conex hull of this configuation. Ou configuation in Fig. 2 is otationally symmetic about the oigin O. It consists of thee otationally symmetic configuations {A 1, A 2,..., A 5 }, {B 1, B 2, B 3, B 4, B 5 }, and {C 1, C 2, C 3, C 4, C 5 }, which we call sta stuctues. In addition, the sta stuctue A i, (esp., B i and C i ) has a eflection symmety in the othogonal bisecto of segment A 3 (esp., B 3 and C 3 ). Label the endpoints of each segment X by X and X such that the tiangle OX X has counteclockwise oientation (whee O is the oigin). In Fig. 2 the dotted lines epesent the aangement of all possible extensions of the gien line segments. Note that both endpoints of A 3, B 3, and C 3 lie in the inteio of the conex hull of the set of all segments, and the emaining segments each hae one endpoint on the conex hull. The extension fom an endpoint on the conex hull goes

5 5 to infinity o teminates at a peious extension. The extension fom an endpoint in the inteio of the conex hull teminates at anothe segment o a peious extension. It is enough to show that any Staight-Fowad conex patition has a cell incident to exactly one segment endpoint. Such a cell necessaily coesponds to a leaf node in the dual gaph. We distinguish two cases. Case 1. Fo at least one of A 3, B 3, and C 3, the extension fom each endpoint teminates at some line segment. Assume w.l.o.g. that the extensions fom endpoints A 3 and A 3 teminate at segments A 1 and A 4, espectiely (Fig. 2). Then the extensions fom A 2 and A 5 teminate at o befoe eaching the extensions fom A 3 and A 3, espectiely (Fig. 2). It can be easily eified that in this case eey pemutation of the fou endpoints {A 1, A 2, A 4, A 5 } poduces a leaf in the dual gaph. A 1 A 3 A A 4 1 A 3 A 4 A1 A 3 A 4 A 2 A 5 A 2 A 5 A 2 A 5 A 1 A 3 A 4 Case 1: Extensions fom A 3 and A 3 hit the segments A 1 and A 4, espectiely. A 2 A 5 A 1 e A 4 A 1 e A 4 A 1 e A 4 A 3 A 2 A 5 A 3 A 2 A 5 A 3 A 2 A 5 Case 2: The segment A 3 is hit by an extension e. Fig. 3 Possible pemutations in one sta-stuctue of ou constuction. Case 2. Fo A 3, B 3, and C 3, the extension fom at least one endpoint does not teminate at a line segment. If the extension fom endpoint A 3 (esp. A 3, B 3, B 3, C 3, C 3 ) does not teminate at a line segment, then it is blocked by the extension emanating fom A 2 (esp., A 5, B 2, B 5, C 2, C 5 ). In this case, we say that the extension fom endpoint A 2 (esp., A 5, B 2, B 5, C 2, C 5 ) escapes (intuitiely, it escapes fom its sta stuctue). We may assume by symmety that the fist extension in pemutation π that escapes is the one emanating fom C 2. Then the extension e fom C 2 hits segment A 3. By ou assumption, the extension fom A 2 o A 5 also escapes, and hits e. The extension e, an extension fom A 2 o A 5, and fom A 3 o A 3, espectiely, ceates a leaf in the dual gaph (Fig. 2). It is not essential in ou constuction that all obstacles ae line segments. We can epeat the constuction using conex polygons with abitaily many etices. Theoem 2 Fo eey n 15, and integes k i 2, i = 1, 2,..., n, thee ae n disjoint obstacles in the plane such that obstacle i is a conex polygon with k i etices and the Staight-Fowad dual gaph D(π) has a bidge edge fo eey pemutation π.

6 6 Poof. Fo n = 15, stat with the constuction in Fig. 2. Label the 15 segments abitaily by integes If k i 3, we eplace segment i by a conex polygon with k i etices as follows. If segment I has an endpoint incident to the conex hull, then eplace it by a long and skinny conex k i -gon with k i 1 etices in the small neighbohood of the segment endpoint on the conex hull, and one etex at the othe segment endpoint. If segment i lies in the inteio of the conex hull, then eplace it by a long and skinny conex k i -gon with one etex at each segment endpoint and k i 2 etices in the small neighbohood of the midpoint facing the oigin. The poof of Theoem 1 shows that fo any pemutation π of the etices, the Staight-Fowad dual gaph has a leaf. If n > 15, we can add conex polygon such that all angle bisectos aoid the conex hull of this configuation. 3 Constucting a Conex Patition We showed in Section 2 that in some instances, no Staight-Fowad dual gaph is 2-edge connected. In this section we pesent an algoithm that poduces a conex patition with a 2-edge connected dual gaph. We will stat fom an abitay Staight-Fowad conex patition, and apply a sequence of local modifications, if necessay, until the dual gaph becomes 2-edge connected. Ou local modifications will not change the numbe of cells. We define a class of conex patitions (Diected- Foest) that includes all Staight-Fowad conex patitions and is closed unde the local modifications we popose. The basis fo local modifications is a simple idea. In a Staight-Fowad conex patition, extensions ae ceated sequentially (each etex emits a diected ay) and whenee two diected extensions meet at a Steine etex (defined below), the ealie extension continues in its oiginal diection, and the late one teminate (Fig. 3(a)). Hee, howee, we allow the two diected extensions to mege and continue as one edge in any diection that maintains the conexity of all the angles incident to (Fig. 3(b)). Meged extensions poide consideable flexibility. q q (a) Fig. 4 (a) If two incoming extensions meet at q, the ealie extension continues in its oiginal diection, and the late one teminates. (b) If two incoming extensions meet at q, the meged extension may continue in any diection within the opposing wedge. (b)

7 7 Definition 1 Fo a gien set S of disjoint conex polygonal obstacles, the class of Diected-Foest conex patitions is defined as follows (efe to Fig. 3): The fee space R 2 \ ( S) is decomposed into conex cells by diected edges (including diected ays going to infinity). Denote by V the set of etices of the polygonal obstacles in S. Each endpoint of a diected edge is eithe a etex in V o a Steine etex (lying in the inteio of the fee space, o on the bounday of an obstacle). We equie that eey etex in V emits exactly one outgoing edge; eey Steine point in the inteio of the fee space is incident to exactly one outgoing edge; no Steine point on a conex obstacle is incident to any outgoing edge; and the diected edges do not fom a cycle. It is easy to see that a Staight-Fowad conex patition belongs to the class of Diected-Foest conex patitions. The dual gaph of a Diected-Foest conex patitions has no isolated nodes. Poposition 1 Thee is an obstacle etex on the bounday of eey cell. Poof. Conside a diected edge on the bounday of a cell. Follow diected edges in eese oientation along the bounday. Since diected edges cannot fom a cycle, and the out-degee of eey Steine etex is at most one, thee must be at least one obstacle etex on the bounday of the cell. In a Diected-Foest, we can also follow diected edges (in fowad diection) fom any etex in V to an obstacle o to infinity, since the out-degee of each etex is always exactly one, unless the etex lies on the bounday of an obstacle o at infinity. Fo connected components of extensions (diected edges), we use the concept of extension tees intoduced by Bose et al. [7]. γ (a) (b) (c) Fig. 5 (a) A conex patition fomed by diected line segments. The extended path γ oiginates at and teminates at, two points on the same obstacle. The edge at is a bidge in the dual gaph, and γ is called fobidden. (b) A single extended-path emitted by. (c) A single extension tee ooted at. Definition 2 The extended-path of a etex V is a diected path along diected edges stating fom and ending on an obstacle o at infinity. Its (elatie) inteio is disjoint fom all obstacles.

8 8 Definition 3 An extension tee is the union of all extended-paths that end at the same point, which is called the oot of the extension tee. The size of an extension tee is the numbe of extended-paths included in the tee. A etex V may be incident to moe than two cells. It is incident to l 2 cells if it is incident to l incoming edges. In ou constuction, we let σ assign a etex of an obstacle to the two cells adjacent to the unique outgoing edge incident to. With this conention, a bidge edge in the dual gaphs D(C, σ) can be chaacteized by a fobidden patten (see Fig. 3(b)). Definition 4 An extended-path stating at V is called fobidden if it ends at the obstacle incident to. A fobidden extended-path, togethe with the bounday of the incident obstacle, foms a simple closed cue, which encloses a bounded egion. Lemma 1 A dual gaph D(C, σ) of a Diected-Foest conex patition is 2-edge connected if and only if no etex V emits a fobidden extended-path. Poof. Fist we show that a fobidden extended-path implies a bidge in the dual gaph. Let γ be a fobidden extended-path, stating fom etex of an obstacle, and ending at point on the bounday of the same obstacle (see Figues 3(b),3.2, 8). Extended-path γ togethe with the obstacle bounday between and foms a simple closed cue and patitions the fee space into two egions R 1 and R 2, each of which is the union of some conex cells. Let V 1 and V 2 be the set of nodes in the dual gaph coesponding to the conex cells in these egions, espectiely. Point is the only obstacle etex along γ. If an edge e of the dual gaph connects some node in V 1 to a node in V 2, then e coesponds to a etex of an obstacle whose unique outgoing edge is pat of γ. But is the only such etex. This implies that thee is a bidge in the dual gaph, whose emoal disconnects V 1 fom V 2. Next we show that a bidge in the dual gaph implies a fobidden extended-path. Assume that V 1 and V 2 fom a patition of V in D(C, σ) such that V 1 and V 2 ae connected by a bidge edge e. The two node sets coespond to two egions, R 1 and R 2, in the fee space. Let β be bounday sepaating the two egions. We fist show that one of these egions is bounded. Suppose fo contadiction that both egions R 1 and R 2 ae unbounded. Note that β must contain at least two diected edges of the conex patition that go to infinity. Since eey Steine etex in the inteio of the fee space has an outgoing edge, β must contain at least two extended-paths. Hence β contains at least two etices of some obstacles, and the adjacent outgoing edges. Thus thee ae at least two edges in the dual gaph between the node sets V 1 and V 2, theefoe, e is not a bidge edge. Now assume without loss of geneality that the egion R 1 is bounded, and thus the sepaating bounday β is a closed cue. If we pick an abitay diected extension along β and follow β in eese diection, then we aie to a segment endpoint. Assume that coesponds to the bidge edge e. Then we aie to the same segment endpoint stating fom any diected extension along β. This means that all diected edges along β ae in the extended-path of. Since β is a closed cue, the extended-path of must end on the bounday of the obstacle incident to, and thus it a fobidden extended-path.

9 9 3.1 Conex Patitioning Algoithm We constuct a conex patition as follows. We fist ceate a Staight-Fowad conex patition, which is in the class of Diected-Foest conex patitions. Let T denote the set of extension tees. Each extension tee may contain one o moe fobidden extended-paths. If an extension tee t T contains a fobidden extendedpath γ stating fom a etex, then we continuously defom t with a sequence of local modifications until a etex of an obstacle collides with the elatie inteio of t (suboutine FlexTee(t)). At that time, t splits into two extension tees t 1 and t 2 (whee t 1 contains all the extended-paths of t leading to, and t 2 contains all the emaining extended-paths of t still leading to ). Each of these two tees (t 1 and t 2 ) is stictly smalle in size than t. An extension tee of size one is a staight-line extension, and cannot contain a fobidden extended-path. Since the numbe of extended-paths is fixed (equal to the numbe of etices in V ), eentually no extension tee contains any fobidden extended-path, and we obtain a conex patition whose dual gaph has no bidges by Lemma 1. Fo a finite set S of disjoint conex polygonal obstacles in the plane, the main loop of ou patition algoithm is CeateConexPatition(S). It calls suboutine FlexTee(t) fo eey extension tee that contains a fobidden extended-path, which in tun calls suboutine Expand(t, γ) fo a fobidden extended path γ, as descibed in Section 3.2. Algoithm 1 CeateConexPatition(S) Gien: A set S of disjoint conex polygons haing n etices in total. Ceate a Staight-Fowad conex patition. Let T be set of extension tees in the patition. while thee is an extension tee t T containing a fobidden extended-path do FlexTee(t) end while Algoithm 2 FlexTee(t) Let γ be a fobidden extended-path contained in t. while γ is still a fobidden extended-path do (t, γ) = Expand(t, γ) end while Let V be a etex of an obstacle whee the extended-path γ now teminates. Split tee t into two extension tees t 1 and t 2. Subtee t 1 consists of the extended-paths that now teminate at. Subtee t 2 consists of the extended-paths that teminate at the oiginal endpoint of γ. 3.2 Local Modifications: Expand(t, γ) Conside a fobidden extended-path γ contained in an extension tee t T. Path γ stats fom a etex V, and ends at a oot lying on the bounday of the obstacle

10 10 γ t t 1 t 2 γ t 1 t 2 t 3 (a) (b) (c) Fig. 6 (a) An extension tee t with a fobidden extended-path. (b) Afte defoming and splitting t into two tees, t 2 contains a fobidden extended-path. (c) Defoming and splitting t 2 eliminates all fobidden extended-paths. s S incident to. Let P be the simple polygon fomed by path γ and a potion of the bounday of s between and such that s is disjoint fom the inteio of P. We continuously defom the bounday of P, togethe with the extension tee t, until it collides with a new etex V that is not incident to s. We defom P in a sequence of local defomations, o steps. What is a local defomation step? Each step inoles eithe one edge o two consecutie edges of the polygon P. The etices of P ae, and the Steine points whee P has an inteio angle diffeent fom 180. Steine etices whee P has an inteio angle of 180 ae consideed inteio points of edges of P. Duing the defomation of an edge of P, one endpoint of the edge moes along a staight line tajectoy and the othe endpoint is fixed. Each step of the defomation will (i) incease the inteio of the polygon P, (ii) keep a etex of P, and (iii) maintain a alid Diected-Foest conex patition. The thid condition implies, in paticula, that eey cell has to emain conex. Since the inteio of P is inceasing, some cells in the exteio of P (and adjacent to P ) will shink we maintain that eey cell adjacent to P has a nonempty inteio. Note that an edge of P may contain Steine etices of the conex patition. At each such Steine point, an extended path stating fom eithe the inteio of o the exteio of P joins the fobidden extended-path along the bounday of P (Fig. 3.2(a)). In the aea swept by the defoming edges, the adjacent edges of the conex subdiision ae eithe tuncated (in the exteio of P ) o extended (in the inteio of P ), while maintaining a alid Diected-Foest conex patition at all times (Fig. 3.2(b-c)). In paticula, the Steine points along the defoming edges also moe continuously The continuous defomation of the bounday of P can be discetized based on the sweep-line paadigm of Bentley and Ottmann [6]. Since the defoming edges follow algebaic tajectoies (one endpoint is fixed, and the othe moes along a staight line), we can maintain a queue of combinatoial changes, which ae: (1) two Steine points along a defoming edge mege; (2) A Steine point along a defoming edge splits into seeal Steine points; (3) A defoming edge of P becomes collinea with an adjacent diected edge of the conex patition.

11 11 y y y γ γ γ x w xw x w (a) (b) (c) Fig. 7 (a) An extension tee t with a fobidden extended-path. (b) A local defomation step stetches edges xy and x of P such that x continuously modes along the edge xw towads w. (c) We update the extended paths in the aea swept by the defoming edges. Whee to pefom a local defomation step? The polygon P is modified eithe at a conex etex x on the conex hull of P o at a cetain eflex etex x. The etex x o x is calculated at the stat of each local defomation step. x z y x P s l Fig. 8 Polygon P coesponding to a fobidden extended-path,..., ; conex etex x; inflexible edges xy and xz; eflex etex x. Conside the edge of the obstacle s that is incident to the point, and is pat of the bounday of the polygon P. Let l be the suppoting line though this edge. The obstacle s lies completely in one of the closed halfplanes bounded by l (since s is

12 12 conex). Let x be a etex of P futhest away fom the suppoting line l in the othe halfplane. Clealy, x is a conex etex of P (inteio angle less than 180 ), othewise it will not be the futhest. The goal is to expand the polygon P by modifying the edges xy and xz incident to x. Imagine gabbing the etex x and pulling it away fom the polygon P stetching the edges xy and xz. But this expansion can only occu if both the edge xy and xz ae flexible. An edge of P is inflexible if thee is a conex cell in the inteio of P such that pat of the edge lies on the bounday of the cell and the cell has an angle of 180 at one endpoints of the edge. Since x is a conex etex, the edge xy o xz can be inflexible if and only if some conex cell has an angle of 180 at y o z, espectiely (Fig. 8). In the case that at least one of the edges incident to x is inflexible, local modification of P takes place at a eflex etex x. Assume w.l.o.g. that xy is inflexible. Then y must be a eflex etex of P (eey inflexible edge of P is incident to a eflex etex). Stating with the eflex etex y, moe along the bounday of P in the diection away fom x. Let x be the fist eflex etex encounteed such that one of the edges incident to x is flexible. By Poposition 2 (below), thee is always one such etex x. Poposition 2 If etex x is incident to an inflexible edge, then thee is a eflex polygonal chain along P that includes this inflexible edge and teminates at a eflex etex x of P that has exactly one flexible edge. Poof. Let xy be an inflexible edge incident to x. Conside the maximal eflex polygonal chain λ along P stating fom x and containing xy. Each intenal etex of λ is eflex, and it is incident on at most one cell in the inteio of P which has an angle of 180 at that etex. Since each cell is conex, its bounday can oelap with at most one edge of λ. That is, a cell at each intenal etex of λ is esponsible fo making at most one edge of λ inflexible. Since a polygonal path has one fewe intenal etices than edges, λ has at least one flexible edge. Let x be the fist intenal etex of λ incident to a flexible edge along λ. How to pefom a local defomation step? Local defomation of P takes place eithe at a conex etex x (Cases 1 and 2), o at a eflex etex x (Case 3). Since the numbe of cells in the conex patition must emain the same, it is necessay to check fo the collapse of a cell in the exteio of P (Case 4). Case 1. Both edges xy and xz of P incident to x ae flexible, and thee is an edge wx in the opposing wedge of yxz. Fig. 3.2(a). Then continuously moe x along xw towads w while stetching the edges xy and xz. Case 2. Both edges xy and xz of P incident to x, ae flexible, and thee is no edge in the opposing wedge of yxz. Fig. 3.2(b). Let l x be the line paallel to l passing though x, and let w be a neighbo of x on the opposite side of l x. Assume that z and w ae on the same side of the angle bisecto of yxz. Then split x into two etices x 1 and x 2. Now x 1 emains fixed at x and x 2 moes continuously along xw towads w stetching the edge x 2 z. Case 3. At least one edge incident to x is inflexible; then thee is a eflex etex x such that edge x z is inflexible, and x y is flexible. Fig. 3.2(c). Continuously moe x along x z towads z while stetching the edge x y. Case 4. The next combinatoial change duing stetching some edge ab to position ab, whee etex b continuously moes along segment bb, would collapse a cell c in the exteio of P. Fig. 3.2(d). Then the inteio of tiangle abb is disjoint fom obstacles,

13 13 z w l P P P x w z x x l y x y z y l b b P s s a w x x 2 w z x y x 1 P P P z y z y l l (a) (b) (c) Fig. 9 Thee local opeations: (a) Conex etex x, incoming edge w in the wedge. (b) Conex etex x, no incoming edge in the wedge. (c) Reflex etex x. (d) The case of a collapsing cell. b b s P (d) s a and a side of some obstacle lies in segment ab and is adjacent to cell c (cf. Poposition 3 below). Let ab be the etex of this obstacle that lies close to a. Stetch edge ab of P into the path (a,, b). When to stop a local defomation step? Continuously defom one o two edges of P, at eithe a conex etex x o a eflex etex x, until one of the following conditions occus: an angle of a conex cell inteio to P o an angle of P becomes 180 ; two etices of the polygon P collide; one of the edges of P collides eithe in its inteio o at its endpoint with a etex of an obstacle; the next combinatoial change in the defomation would collapse a cell in the exteio of P. Since a local defomation step does not always teminate in a collision with an obstacle etex, the suboutine FlexTee(t) decides at the end of each step whethe moe local modifications ae needed. 3.3 Coectness of the Algoithm We poe that we can eliminate all fobidden extended-paths and obtain a Diected- Foest conex patition with a 2-edge connected dual gaph. Let t be a extension tee, containing a fobidden extended-path γ stating fom V and ending at oot. Fist we show that in Expand(t, γ), the fou cases coe all possibilities.

14 14 Poposition 3 If the next combinatoial change while defoming some edge ab to position ab, whee b continuously moes along segment bb, would collapse a cell c in the exteio of P, then the inteio of tiangle abb is disjoint fom obstacles, and and a side of some obstacle lies in segment ab and is adjacent to cell c. Poof. A continuous defomation of ab to ab, whee b moes along segment bb, sweeps tiangle abb. Hence the inteio of abb is disjoint fom obstacles, and if any extended-path intesects the inteio of abb, then it is pat of the extension tee t. Assume that cell c C would collapse if ab eaches position ab. By Poposition 1, thee is a etex V on the bounday of cell c, and so must lie on the line segment ab. Since eey extended-path in inteio of tiangle abb is pat of t, no extension in abb teminates at. Hence the two edges along the bounday of c incident to ae the extension emitted by and a side of the obstacle s containing. It follows that segment ab contains a side of obstacle s. Poposition 4 Eey step Expand(t, γ) pesees a Diected-Foest conex patition and it also pesees the numbe of cells. Poof. It is clea that the continuous defomations in Cases 1-3 maintain a alid Diected-Foest with the same numbe of cells. Conside Case 4. Denoting by ab the etex of this side that lies close to a, we stetch edge ab of P into the path (a,, b). By the choice of, edge a does not contains any side of obstacles. Edge b taeses tiangle abb, so it does not any side of obstacles, eithe. By Poposition 3, a side of some obstacle s lies in segment ab and is adjacent to cell c, and so it cannot collapse in this step of Expand(t, γ). A alid Diected-Foest conex patition is peseed with the same numbe of cells. Lemma 2 The suboutine FlexTee(t) modifies an extension tee t T, with a fobidden extended-path γ, in a finite numbe of Expand(t, γ) steps until an obstacle etex V appeas along γ. Poof. FlexTee(t) epeatedly calls Expand(t, γ) fo a fobidden extended-path γ. We associate an intege count(t, γ) to t and γ and show that Expand(t, γ) eithe defoms t to collide with an obstacle s s o count(t, γ) stictly deceases. This implies that FlexTee(t) teminates in a finite numbe of steps. Let k denote the size of t (i.e., the numbe of extended-paths in t). Then t has at most k 1 Steine etices in the fee space, since each coesponds to the meging of two o moe extended-paths. Let k ex be the numbe of Steine etices of t in the exteio of P, let P be the numbe of etices of P, let f P be the numbe of flexible edges of P, and let m p be the numbe of extended-paths in t that ente etex x of P fom the exteio of P. Then let count(t, γ) = 2k k ex P f P 2m P. Recall that a Steine etex whee P has an intenal angle of 180 is not a etex of P. The etices of P ae, and Steine etices in the inteio of the fee space whee P has a non-staight intenal angle, hence p, f P, m P < k. Conside a sequence of Expand(t, γ) steps whee t does not collide with an obstacle. Since in Case 4, a etex V appeas in the elatie inteio of t, we may assume that only Case 1 3 ae applied. Case 1 3 expand the inteio of polygon P, and the diected edges in the exteio of P ae not defomed. Hence k ex nee inceases, and it deceases if P expands and eaches a Steine point in the exteio of P.

15 15 Now conside a sequence of Expand(t, γ) steps whee k ex emains fixed and Case 4 does not apply. Then m P can only decease in Case 1 3. Case 2 initially intoduces a new edge of P (inceasing P and f P by one each) but it also deceases m P by at least one. Case 1 and 3 nee incease P o f P. In Case 1 3, the defomation step teminates when an inteio angle of a conex cell within P becomes 180 (and an edge becomes inflexible, deceasing f P ) o an inteio angle of P becomes 180 (and P loses a etex, deceasing P ). In both eents, P f P deceases by at least one. Theefoe, count(t, γ) = 2k k ex P f P 2m P stictly deceases in eey step Expand(t, γ), until the elatie inteio of t collides with an obstacle. Theoem 3 Fo eey finite set of disjoint conex polygonal obstacles in the plane, thee is a conex patition and an assignment σ such that the dual gaph D(C, σ) is 2-edge connected. Fo k conex polygonal obstacles with a total of n etices, the conex patition consists of n k 1 conex cells. Poof. The conex patitioning algoithm fist ceates a Staight-Fowad conex patition fo the gien set of disjoint polygonal obstacles. Fo k disjoint obstacles with a total of n etices, it consists of n k 1 conex cells. The extensions in the conex patition can be epesented as a set of extension tees T. We showed in Lemma 1 that thee is a bidge in the dual gaph iff some extension tee contains a fobidden extendedpath. Suboutine FlexTee(t) splits eey extension tee t containing a fobidden extended-path into two smalle tees. (The extended-paths in t ae distibuted between the two esulting tees.) An extension tee that consists of a single extended-path is a staight-line extension, and cannot be fobidden (a staight-line extension emitted fom a etex of an obstacle cannot hit the same obstacle, since each obstacle is conex.) Theefoe, afte at most V /2 calls to FlexTee(t), no extended-path is fobidden, and so the dual gaph of the conex patition is 2-edge connected. 4 Conclusion We hae shown that fo eey finite set of conex polygonal obstacles, thee is a conex patition with a 2-edge connected dual gaph. We hae pesented a polynomial time algoithm fo constucting the dual gaph. It is staightfowad to implemented ou algoithm in O(n 4 log n) time fo obstacles with a total of n etices. Each Expand(t, γ) opeation inoles stetching two edges of a simple polygon, and can be implemented in O(n log n) time. By Lemma 2, each FlexTee(t) equies O(n 2 ) calls to Expand(t, γ). Finally, each FlexTee(t) inceases splits the extension tee t into two, whee the numbe of extension tees is at most n the numbe of etices. Fo compaison, the Staight-Fowad conex patition fo any pemutation π can be computed in O(n log 2 n) time [11]. Fo a pemutation π whee the etices with ightwad pointing angle bisectos come befoe the etices with leftwad pointing angle bisectos, a Staight-Fowad conex patition can be computed in O(n log n) time by two line sweeps. Tan et al. [20] computes Staight-Fowad conex patition in a distibuted manne, which makes the algoithm suitable fo senso netwoks. Howee, it emains to be seen whethe the algoithm to poduce conex patitions with 2-edge connected dual gaphs pesented in this pape could be modified to wok in a distibuted manne. A elated question is how to suppot efficient insetion and deletion of conex polygonal obstacles. In othe wods: how local the local modifications eally ae?

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