We develop algorithms to compute shortest path edge sequences, Voronoi diagrams, the Fréchet distance, and the diameter for a polyhedral surface

B. Aronov

D.M. Mount

E.D. Demaine

G. Albers

H. Alt

J. Chen

J. Hershberger

J.S.B. Mitchell

L. Aleksandrov

L.J. Guibas

O. Devillers

P.K. Agarwal

V. Chandru

Y. Schreiber

Y.-H. Hwang

Wenk, Carola

Cook, Atlas F.

Dagstuhl Research Online Publication Server

Shortest Path Problems on a Polyhedral Surface

Cook IV, A.F.

Wenk, C.

Pure OAI Repository

 Shortest path problems on a polyhedral surfaceCitation for published version (APA):Cook IV, A. F., & Wenk, C. (2009). Shortest path problems on a polyhedral surface. In Abstracts 25th EuropeanWorkshop on Computational Geometry (EuroCG'09, Brussels, Belgium, March 16-18, 2009) (pp. 179-182)Document status and date:Published: 01/01/2009Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 05. Oct. 2023Shortest Path Problems on a Polyhedral Surface∗Atlas F. Cook IV † Carola Wenk †AbstractWe develop algorithms to compute shortest path edgesequences, Voronoi diagrams, the Fréchet distance,and the diameter for a polyhedral surface.1 IntroductionTwo questions are invariably encountered when deal-ing with shortest path problems. The first questionis how to represent the combinatorial structure of ashortest path. In the plane with polygonal obstacles,all shortest path vertices occur at obstacle vertices,so a shortest path can be combinatorially describedas a sequence of obstacle vertices [8]. On a polyhe-dral surface, a shortest path need not turn at vertices[11], so a shortest path is often described combinatori-ally by an edge sequence that represents the sequenceof edges encountered by the path [1].The second commonly encountered shortest pathquestion is how to compute shortest paths. Let Mbe the complexity of a polyhedral surface. On aconvex polyhedral surface, Mount [12] shows thatΘ(M4) combinatorially distinct shortest paths exist,and Schevon and O’Rourke [13] show that only Θ(M3)of these shortest paths are maximal (such paths can-not be extended at either end without creating a sub-optimal path). Agarwal et al. [1] use these propertiesto compute Θ(M4) shortest path edge sequences inO(M62α(M) log M) time, where α(M) is the inverseAckermann function. They also compute the diam-eter of a convex polyhedral surface in O(M8 log M)time.We improve the diameter and edge sequence al-gorithms of Agarwal et al. [1] on a convex poly-hedral surface by a linear factor. Our improvementcombines star unfolding and kinetic Voronoi diagramtechniques. In addition, we achieve a linear factorimprovement over the Fréchet distance algorithm ofMaheshwari and Yi [10] for polygonal curves on aconvex polyhedral surface, and we present the firstalgorithm to compute the Fréchet distance betweenpolygonal curves on a non-convex polyhedral surface.Our motivation for studying the Fréchet distance on apolyhedral surface is that teaming up two people for∗This work has been supported by the National ScienceFoundation grant NSF CAREER CCF-0643597.†Department of Computer Science, University of Texas atSan Antonio, One UTSA Circle, San Antonio, TX 78249-0667,USA, acook@cs.utsa.edu, carola@cs.utsa.edusafety reasons is common practice in many real-lifesituations, ranging from scouts in summer camp, tofire fighters and police officers, and even to astronautsexploring the moon. In all of these applications, twoteam members need to coordinate their movement inorder to stay within “walking distance” so that fastassistance can be offered in case of an emergency. TheFréchet distance is an ideal model for this scenario.We summarize our results in Table 1. The followingnotation is used throughout this paper. Let M bethe total complexity of a polyhedral surface and twopolygonal curves A and B that lie on the surface. Letπ(s, t) denote a shortest path on a surface betweenpoints s and t, and let d(s, t) be the Euclidean lengthof π(s, t).2 Shortest Path Problems on a Polyhedral Surface2.1 Shortest Path Edge SequencesThis section contains superset and exact algorithmsfor computing the Θ(M4) shortest path edge se-quences on a convex polyhedral surface P. Both of ouralgorithms are linear factor improvements over relatedresults of Agarwal et al. [1]. Let v1, ..., vM be the cor-ner vertices of P, and let Π = {π(s, v1), ..., π(s, vM )}be an angularly ordered set of non-crossing shortestpaths from a source point s ∈ P to each corner vertexvj ∈ P. The star unfolding S is a simple polygon [4]defined by cutting P along each of the shortest pathsin Π and unfolding the resulting shape into the plane.Since the source point s touches all M cuts, s ∈ Pmaps to M image points s1, ..., sM on the (two di-mensional) boundary of the unfolded simple polygonS (see Figure 1).An equator [5] in the star unfolding is a closedpolygonal curve through the points v1, ..., vM , v1. Theregion inside the equator contains no source imageand is called the core [7]. The disconnected regionsoutside the core each contain a source image and arecollectively referred to as the anti-core [7]. A coreedge is an image of an edge of P that was never cutduring the unfolding process. Each of the O(M) coreedges has both of its endpoints at vertices of S and isentirely contained in the core (see Figure 1b). Eachof the Θ(M2) anti-core edges is an image of an edgeof P that was cut during the unfolding process.The star unfolding of s can be used to computeπ(s, t) for points s, t ∈ P as follows. The shortest1Shortest Path Problems on a Polyhedral Surface179Time SpaceStar Unfolding O(M4)* O(M4)maintained over all edgesKinetic Voronoi Diagram O(M52α(M) log M) O(M52α(M))maintained over all edgesEdge Sequences (superset) O(M5) O(M5)Edge Sequences (exact) O(M52α(M) log M) O(M42α(M))Diameter O(M7 log M) O(M4)Fréchet O(M6 log2 M) O(M2)Distance O(M7 log2 M)* O(M3)Table 1: Summary of our contributions. An asterisk indicates a result for a non-convex polyhedral surface.path π(s, t) always originates from one of the sourceimages s1, ..., sM . If the image of t lies in an anti-coreregion containing si, then the optimal shortest pathmust originate from si (i.e., the shortest paths fromt to the remaining source images are suboptimal). Ifthe image of t lies in the core, then the nearest sourceimage is determined with Voronoi diagram techniques.Agarwal et al. [1] partition the M edges of the con-vex polyhedral surface P into O(M3) line segmentedgelets such that all source points on an edgelet canbe associated with the same combinatorial star un-folding. For each edgelet, a star unfolding is com-puted, and O(M3) edge sequences are extracted fromit. This yields an O(M6) superset of the Θ(M4)shortest path edge sequences for P in O(M6) timeand space [1]. Agarwal et al. [1] also show how tocompute the exact set of Θ(M4) shortest path edgesequences in O(M62α(M) log M) time. We speed upboth of these algorithms by a linear factor.Theorem 1 A star unfolding can be maintained as asource point s varies continuously over all M edges ofa (possibly non-convex) polyhedral surface in O(M4)time and space.Proof Sketch. The set Π of shortest paths to eachcorner vertex defines a combinatorial star unfolding.These paths can only change at edgelet endpoints, soΠ can be maintained over a discrete set of events. Foreach change to Π, we update O(1) anti-core regionsand possibly all O(M) core edges in the star unfold-ing. Theorem 2 A superset of the Θ(M4) shortest pathedge sequences for a convex polyhedral surface P canbe constructed in O(M5) time and space.Proof. Each edgelet defines a star unfolding withsource images s1, ..., sM . For each si, construct anedge sequence from si to each of the O(M) anti-coreedges in the anti-core region for si and to each ofthe O(M) core edges. This yields O(M2) edge se-quences per edgelet, and O(M5) edge sequences overall edgelets. This is sufficient to determine the desiredsuperset because only core edges have shortest pathedge sequences to multiple sites, and this approachconsiders all possibilities. The exact set of Θ(M4) shortest path edge se-quences can be determined with kinetic Voronoi dia-gram techniques. Albers et al. [2] show that for pointsites moving along line segments at constant speeds,each pair of sites defines O(M2α(M)) events, and eachevent is handled in O(log M) time.Theorem 3 A kinetic Voronoi diagram of source im-ages can be maintained in O(M42α(M) log M) timeand O(M42α(M)) space as a source point varies con-tinuously over one edge e of a convex polyhedral sur-face P.Proof. A kinetic Voronoi diagram for the firstedgelet on e has O(M2 · M2α(M)) events [2] due tothe linear motion of O(M2) pairs of source imagesin the star unfolding. Each of the O(M2) subsequentedgelets on e can be handled by removing one site (i.e.,a source image) and adding a new site. All other sitescontinue to be parameterized along the same line seg-ments as in the previous edgelet. Thus, each of theO(M2) edgelets contributes M − 1 new pairs of sitesand adds O(M · M2α(M)) new events to the eventqueue. We construct the exact set of shortest path edge se-quences for an edgelet α as follows. Maintaining a ki-netic Voronoi diagram for α yields a two-dimensionalparameterized Voronoi cell ϕi for each source imagesi. The unique edge sequence in the star unfolding’sdual graph from si to a core edge e represents a short-est path if and only if e intersects ϕi for some s ∈ α.To decide in logarithmic time whether e intersectsϕi, Agarwal et al.[1] represent each parameterized2EuroCG’09 - Brussels, Belgium180Voronoi vertex as an algebraic curve. For a vertexvi ∈ S that touches the anti-core region for the sourceimage si, they triangulate the region of the core thatis directly visible to vi such that every triangle ∆ hasapex vi. Using polar coordinates centered at vi, theycompute an upper envelope µ of the algebraic curvesdefining ϕi and partition these curves such that eachcurve segment is fully contained in one of these tri-angles. Inside each triangle ∆, the dual graph of thecore has at most one degree three vertex because eachcore edge is a chord of ∆. Agarwal et al. [1] computea dual graph tree D∆ for each triangle in O(M) time.Lemma 4 D∆ can be constructed implicitly inO(log M) time.Proof Sketch. Determine the portion of the core’sdual graph D that lies inside ∆ by point locating thethree vertices of ∆ in D. For each constant complexity algebraic curve on theupper envelope µ ∩ ∆, Agarwal et al. [1] perform abinary search on the two paths in D∆. The deepestedge on each of these two paths that intersects somearc of µ ∩ ∆ defines a maximal shortest path edgesequence. The set of all prefixes of these maximal se-quences defines Θ(M4) shortest path edge sequences.Theorem 5 The Θ(M4) shortest path edge se-quences for a convex polyhedral surface P with Medges can be constructed in O(M52α(M) log M) timeand O(M42α(M)) space.Proof. Let ni be the total number of parameterizedVoronoi vertices over all edgelets, and let t∆ be thetime to process each triangle ∆. The technique ofAgarwal et al. [1] requires O(ni log M + M5t∆) time.Since they assume ni ∈ O(M62α(M)) and t∆ ∈ O(M),they obtain O(M62α(M) log M) time. By Theorem 3,ni ∈ O(M52α(M)) over all O(M) edges of P, and byLemma 4, t∆ ∈ O(log M) time. 2.2 DiameterThe diameter of a convex polyhedral surface is thelargest shortest path distance between any pair ofpoints on the surface. Agarwal et al. [1] computethe diameter in O(M8 log M) time by defining O(M4)ridge-free regions on the surface such that all sourcepoints in a ridge-free region have the same combina-torial star unfolding.Theorem 6 The diameter of a convex polyhedralsurface P with M edges can be constructed inO(M7 log M) time and O(M4) space.Proof Sketch. Maintain a kinetic Voronoi diagramover all ridge-free regions. Each of the O(M4) ridge(a) (b) (c)v1 v2v3v4v5ss1 s2 s3s4s5v2v3v4v5v1sl1l2l3l4l5v1v5v2v3v4s1s2s3s4s5l1l2l3l4l5εFigure 1: The star unfolding of a polyhedral surface.free regions defines O(M3) new events, for a total ofO(M7) parameterized Voronoi vertices. Each vertexcan be associated with a function that defines the dis-tance from this vertex to its defining source image.The maximum value defined by these functions is thediameter. 2.3 Fréchet DistanceThe Fréchet distance [3] is a similarity metric forcontinuous shapes that is defined for two polygonalcurves A,B : [0, 1] → Rd asδF (A,B) = infα,β:[0,1]→[0,1]supt∈[0,1]d(A(α(t)), B(β(t)))where α and β range over continuous non-decreasingreparameterizations, and d is a distance metric forpoints. For a given constant ε ≥ 0, free space is de-fined as {(s, t) | s ∈ A, t ∈ B, d(s, t) ≤ ε}. LetδC(A,B) (resp. δN (A,B)) denote the Fréchet dis-tance between polygonal curves A and B on a con-vex (resp. non-convex) polyhedral surface. Mahesh-wari and Yi [10] have previously shown how to com-pute δC(A,B) in O(M7 log M) time by enumeratingall edge sequences. However, their approach relies on[9] whose key claim “has yet to be convincingly estab-lished” [1]. We compute δC(A,B) without enumerat-ing edge sequences in O(M6 log2 M) time and O(M2)space.We first describe all free space for a given constantε ≥ 0. The star unfolding S maps a fixed source points ∈ A to a set of source image points s1, ..., sM andmaps B to a set of core and anti-core edges. Free spaceis defined by the union of a set of disks d1, ..., dM ,where each disk di has radius ε and is centered atsi. As the source point s varies continuously on anedgelet, the core is fixed, and each si is parameter-ized along a line segment li in the star unfolding (seeFigure 1b).Theorem 7 δC(A,B) can be computed inO(M6 log2 M) time and O(M2) space.3Shortest Path Problems on a Polyhedral Surface181Proof Sketch. A star unfolding is maintained overall edgelets by Theorem 1. The free space for an anti-core edge is defined by the intersection of this edgewith one nearest disk di. The free space for each coreedge is defined by the intersection of this edge withd1∪...∪dM . After computing all free space, reachabil-ity information [3] can be propagated through the freespace via plane sweep [6], and parametric search [3]can be applied to a set of algebraic free space vertexcurves. For a non-convex polyhedral surface, the star un-folding can overlap itself [5] but is still defined by aset of shortest paths from the source to every cor-ner vertex [5, 11]. Thus, a core can still be definedby a polygonal equator with O(M) complexity. Sinceshortest paths only turn at vertices in the star unfold-ing [11], an anti-core region containing si must havean hourglass shape [8, 6] because it is bounded by twoline segments and two shortest paths in the plane.Theorem 8 δN (A,B) can be computed inO(M7 log2 M) time and O(M3) space.Proof Sketch. Maintain a star unfolding over alledgelets. One hourglass defines the free space to ananti-core edge, and O(M) hourglasses define the freespace to a core edge. After computing all free space,a combination of plane sweep and parametric searchtechniques yields δN (A,B). 3 ConclusionWe develop algorithms to compute edge sequences,Voronoi diagrams, the Fréchet distance, and the diam-eter for a polyhedral surface. Our work speeds up theedge sequence and diameter approaches of Agarwal etal. [1] by a linear factor and introduces many algo-rithms that apply to convex and non-convex polyhe-dral surfaces. Future work could attempt to computethe Θ(M4) shortest path edge sequences in Θ(M4)time. Additional shortest path map and link distanceresults were omitted from this version due to spaceconcerns.References[1] P. K. Agarwal, B. Aronov, J. O’Rourke, andC. A. Schevon. Star unfolding of a polytopewith applications. SIAM Journal on Computing,26(6):1689–1713, 1997.[2] G. Albers, J. S. B. Mitchell, L. J. Guibas, andT. Roos. Voronoi diagrams of moving points.Journal of Computational Geometry and Appli-cations, 8:365–380, 1998.[3] H. Alt and M. Godau. Computing the Fréchetdistance between two polygonal curves. Journalof Computational Geometry and Applications,5:75–91, 1995.[4] B. Aronov and J. O’Rourke. Nonoverlap of thestar unfolding. 7th Symposium on ComputationalGeometry (SoCG), pages 105–114, 1991.[5] J. Chen and Y. Han. Shortest paths on a poly-hedron. Journal of Computational Geometry andApplications, 6(2):127–144, 1996.[6] A. F. Cook IV and C. Wenk. Geodesic Fréchetdistance inside a simple polygon. 25th Sympo-sium on Theoretical Aspects of Computer Science(STACS), 2008.[7] E. D. Demaine and J. O’Rourke. GeometricFolding Algorithms: Linkages, Origami, Polyhe-dra. Cambridge University Press, New York, NY,USA, 2007.[8] L. J. Guibas, J. Hershberger, D. Leven,M. Sharir, and R. E. Tarjan. Linear-time al-gorithms for visibility and shortest path prob-lems inside triangulated simple polygons. Algo-rithmica, 2:209–233, 1987.[9] Y.-H. Hwang, R.-C. Chang, and H.-Y. Tu. Find-ing all shortest path edge sequences on a convexpolyhedron. Workshop on Algorithms and DataStructures (WADS), 1989.[10] A. Maheshwari and J. Yi. On computing Fréchetdistance of two paths on a convex polyhedron.21st European Workshop on Computational Ge-ometry (EuroCG), 2005.[11] J. S. B. Mitchell, D. M. Mount, and C. H.Papadimitriou. The discrete geodesic problem.SIAM Journal on Computing, 16(4):647–668,1987.[12] D. M. Mount. The number of shortest paths onthe surface of a polyhedron. SIAM Journal onComputing, 19(4):593–611, 1990.[13] C. Schevon and J. O’Rourke. The number ofmaximal edge sequences on a convex polytope.26th Allerton Conference on Communication,Control, and Computing, pages 49–57, 1988.4EuroCG’09 - Brussels, Belgium182

English

Shortest path problems on a polyhedral surface

 Shortest path problems on a polyhedral surfaceCitation for published version (APA):Cook IV, A. F., & Wenk, C. (2009). Shortest path problems on a polyhedral surface. In Abstracts 25th EuropeanWorkshop on Computational Geometry (EuroCG'09, Brussels, Belgium, March 16-18, 2009) (pp. 179-182)Document status and date:Published: 01/01/2009Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 04. Oct. 2023Shortest Path Problems on a Polyhedral Surface∗Atlas F. Cook IV † Carola Wenk †AbstractWe develop algorithms to compute shortest path edgesequences, Voronoi diagrams, the Fréchet distance,and the diameter for a polyhedral surface.1 IntroductionTwo questions are invariably encountered when deal-ing with shortest path problems. The first questionis how to represent the combinatorial structure of ashortest path. In the plane with polygonal obstacles,all shortest path vertices occur at obstacle vertices,so a shortest path can be combinatorially describedas a sequence of obstacle vertices [8]. On a polyhe-dral surface, a shortest path need not turn at vertices[11], so a shortest path is often described combinatori-ally by an edge sequence that represents the sequenceof edges encountered by the path [1].The second commonly encountered shortest pathquestion is how to compute shortest paths. Let Mbe the complexity of a polyhedral surface. On aconvex polyhedral surface, Mount [12] shows thatΘ(M4) combinatorially distinct shortest paths exist,and Schevon and O’Rourke [13] show that only Θ(M3)of these shortest paths are maximal (such paths can-not be extended at either end without creating a sub-optimal path). Agarwal et al. [1] use these propertiesto compute Θ(M4) shortest path edge sequences inO(M62α(M) log M) time, where α(M) is the inverseAckermann function. They also compute the diam-eter of a convex polyhedral surface in O(M8 log M)time.We improve the diameter and edge sequence al-gorithms of Agarwal et al. [1] on a convex poly-hedral surface by a linear factor. Our improvementcombines star unfolding and kinetic Voronoi diagramtechniques. In addition, we achieve a linear factorimprovement over the Fréchet distance algorithm ofMaheshwari and Yi [10] for polygonal curves on aconvex polyhedral surface, and we present the firstalgorithm to compute the Fréchet distance betweenpolygonal curves on a non-convex polyhedral surface.Our motivation for studying the Fréchet distance on apolyhedral surface is that teaming up two people for∗This work has been supported by the National ScienceFoundation grant NSF CAREER CCF-0643597.†Department of Computer Science, University of Texas atSan Antonio, One UTSA Circle, San Antonio, TX 78249-0667,USA, acook@cs.utsa.edu, carola@cs.utsa.edusafety reasons is common practice in many real-lifesituations, ranging from scouts in summer camp, tofire fighters and police officers, and even to astronautsexploring the moon. In all of these applications, twoteam members need to coordinate their movement inorder to stay within “walking distance” so that fastassistance can be offered in case of an emergency. TheFréchet distance is an ideal model for this scenario.We summarize our results in Table 1. The followingnotation is used throughout this paper. Let M bethe total complexity of a polyhedral surface and twopolygonal curves A and B that lie on the surface. Letπ(s, t) denote a shortest path on a surface betweenpoints s and t, and let d(s, t) be the Euclidean lengthof π(s, t).2 Shortest Path Problems on a Polyhedral Surface2.1 Shortest Path Edge SequencesThis section contains superset and exact algorithmsfor computing the Θ(M4) shortest path edge se-quences on a convex polyhedral surface P. Both of ouralgorithms are linear factor improvements over relatedresults of Agarwal et al. [1]. Let v1, ..., vM be the cor-ner vertices of P, and let Π = {π(s, v1), ..., π(s, vM )}be an angularly ordered set of non-crossing shortestpaths from a source point s ∈ P to each corner vertexvj ∈ P. The star unfolding S is a simple polygon [4]defined by cutting P along each of the shortest pathsin Π and unfolding the resulting shape into the plane.Since the source point s touches all M cuts, s ∈ Pmaps to M image points s1, ..., sM on the (two di-mensional) boundary of the unfolded simple polygonS (see Figure 1).An equator [5] in the star unfolding is a closedpolygonal curve through the points v1, ..., vM , v1. Theregion inside the equator contains no source imageand is called the core [7]. The disconnected regionsoutside the core each contain a source image and arecollectively referred to as the anti-core [7]. A coreedge is an image of an edge of P that was never cutduring the unfolding process. Each of the O(M) coreedges has both of its endpoints at vertices of S and isentirely contained in the core (see Figure 1b). Eachof the Θ(M2) anti-core edges is an image of an edgeof P that was cut during the unfolding process.The star unfolding of s can be used to computeπ(s, t) for points s, t ∈ P as follows. The shortest1Shortest Path Problems on a Polyhedral Surface179Time SpaceStar Unfolding O(M4)* O(M4)maintained over all edgesKinetic Voronoi Diagram O(M52α(M) log M) O(M52α(M))maintained over all edgesEdge Sequences (superset) O(M5) O(M5)Edge Sequences (exact) O(M52α(M) log M) O(M42α(M))Diameter O(M7 log M) O(M4)Fréchet O(M6 log2 M) O(M2)Distance O(M7 log2 M)* O(M3)Table 1: Summary of our contributions. An asterisk indicates a result for a non-convex polyhedral surface.path π(s, t) always originates from one of the sourceimages s1, ..., sM . If the image of t lies in an anti-coreregion containing si, then the optimal shortest pathmust originate from si (i.e., the shortest paths fromt to the remaining source images are suboptimal). Ifthe image of t lies in the core, then the nearest sourceimage is determined with Voronoi diagram techniques.Agarwal et al. [1] partition the M edges of the con-vex polyhedral surface P into O(M3) line segmentedgelets such that all source points on an edgelet canbe associated with the same combinatorial star un-folding. For each edgelet, a star unfolding is com-puted, and O(M3) edge sequences are extracted fromit. This yields an O(M6) superset of the Θ(M4)shortest path edge sequences for P in O(M6) timeand space [1]. Agarwal et al. [1] also show how tocompute the exact set of Θ(M4) shortest path edgesequences in O(M62α(M) log M) time. We speed upboth of these algorithms by a linear factor.Theorem 1 A star unfolding can be maintained as asource point s varies continuously over all M edges ofa (possibly non-convex) polyhedral surface in O(M4)time and space.Proof Sketch. The set Π of shortest paths to eachcorner vertex defines a combinatorial star unfolding.These paths can only change at edgelet endpoints, soΠ can be maintained over a discrete set of events. Foreach change to Π, we update O(1) anti-core regionsand possibly all O(M) core edges in the star unfold-ing. Theorem 2 A superset of the Θ(M4) shortest pathedge sequences for a convex polyhedral surface P canbe constructed in O(M5) time and space.Proof. Each edgelet defines a star unfolding withsource images s1, ..., sM . For each si, construct anedge sequence from si to each of the O(M) anti-coreedges in the anti-core region for si and to each ofthe O(M) core edges. This yields O(M2) edge se-quences per edgelet, and O(M5) edge sequences overall edgelets. This is sufficient to determine the desiredsuperset because only core edges have shortest pathedge sequences to multiple sites, and this approachconsiders all possibilities. The exact set of Θ(M4) shortest path edge se-quences can be determined with kinetic Voronoi dia-gram techniques. Albers et al. [2] show that for pointsites moving along line segments at constant speeds,each pair of sites defines O(M2α(M)) events, and eachevent is handled in O(log M) time.Theorem 3 A kinetic Voronoi diagram of source im-ages can be maintained in O(M42α(M) log M) timeand O(M42α(M)) space as a source point varies con-tinuously over one edge e of a convex polyhedral sur-face P.Proof. A kinetic Voronoi diagram for the firstedgelet on e has O(M2 · M2α(M)) events [2] due tothe linear motion of O(M2) pairs of source imagesin the star unfolding. Each of the O(M2) subsequentedgelets on e can be handled by removing one site (i.e.,a source image) and adding a new site. All other sitescontinue to be parameterized along the same line seg-ments as in the previous edgelet. Thus, each of theO(M2) edgelets contributes M − 1 new pairs of sitesand adds O(M · M2α(M)) new events to the eventqueue. We construct the exact set of shortest path edge se-quences for an edgelet α as follows. Maintaining a ki-netic Voronoi diagram for α yields a two-dimensionalparameterized Voronoi cell ϕi for each source imagesi. The unique edge sequence in the star unfolding’sdual graph from si to a core edge e represents a short-est path if and only if e intersects ϕi for some s ∈ α.To decide in logarithmic time whether e intersectsϕi, Agarwal et al.[1] represent each parameterized2EuroCG’09 - Brussels, Belgium180Voronoi vertex as an algebraic curve. For a vertexvi ∈ S that touches the anti-core region for the sourceimage si, they triangulate the region of the core thatis directly visible to vi such that every triangle ∆ hasapex vi. Using polar coordinates centered at vi, theycompute an upper envelope µ of the algebraic curvesdefining ϕi and partition these curves such that eachcurve segment is fully contained in one of these tri-angles. Inside each triangle ∆, the dual graph of thecore has at most one degree three vertex because eachcore edge is a chord of ∆. Agarwal et al. [1] computea dual graph tree D∆ for each triangle in O(M) time.Lemma 4 D∆ can be constructed implicitly inO(log M) time.Proof Sketch. Determine the portion of the core’sdual graph D that lies inside ∆ by point locating thethree vertices of ∆ in D. For each constant complexity algebraic curve on theupper envelope µ ∩ ∆, Agarwal et al. [1] perform abinary search on the two paths in D∆. The deepestedge on each of these two paths that intersects somearc of µ ∩ ∆ defines a maximal shortest path edgesequence. The set of all prefixes of these maximal se-quences defines Θ(M4) shortest path edge sequences.Theorem 5 The Θ(M4) shortest path edge se-quences for a convex polyhedral surface P with Medges can be constructed in O(M52α(M) log M) timeand O(M42α(M)) space.Proof. Let ni be the total number of parameterizedVoronoi vertices over all edgelets, and let t∆ be thetime to process each triangle ∆. The technique ofAgarwal et al. [1] requires O(ni log M + M5t∆) time.Since they assume ni ∈ O(M62α(M)) and t∆ ∈ O(M),they obtain O(M62α(M) log M) time. By Theorem 3,ni ∈ O(M52α(M)) over all O(M) edges of P, and byLemma 4, t∆ ∈ O(log M) time. 2.2 DiameterThe diameter of a convex polyhedral surface is thelargest shortest path distance between any pair ofpoints on the surface. Agarwal et al. [1] computethe diameter in O(M8 log M) time by defining O(M4)ridge-free regions on the surface such that all sourcepoints in a ridge-free region have the same combina-torial star unfolding.Theorem 6 The diameter of a convex polyhedralsurface P with M edges can be constructed inO(M7 log M) time and O(M4) space.Proof Sketch. Maintain a kinetic Voronoi diagramover all ridge-free regions. Each of the O(M4) ridge(a) (b) (c)v1 v2v3v4v5ss1 s2 s3s4s5v2v3v4v5v1sl1l2l3l4l5v1v5v2v3v4s1s2s3s4s5l1l2l3l4l5εFigure 1: The star unfolding of a polyhedral surface.free regions defines O(M3) new events, for a total ofO(M7) parameterized Voronoi vertices. Each vertexcan be associated with a function that defines the dis-tance from this vertex to its defining source image.The maximum value defined by these functions is thediameter. 2.3 Fréchet DistanceThe Fréchet distance [3] is a similarity metric forcontinuous shapes that is defined for two polygonalcurves A,B : [0, 1] → Rd asδF (A,B) = infα,β:[0,1]→[0,1]supt∈[0,1]d(A(α(t)), B(β(t)))where α and β range over continuous non-decreasingreparameterizations, and d is a distance metric forpoints. For a given constant ε ≥ 0, free space is de-fined as {(s, t) | s ∈ A, t ∈ B, d(s, t) ≤ ε}. LetδC(A,B) (resp. δN (A,B)) denote the Fréchet dis-tance between polygonal curves A and B on a con-vex (resp. non-convex) polyhedral surface. Mahesh-wari and Yi [10] have previously shown how to com-pute δC(A,B) in O(M7 log M) time by enumeratingall edge sequences. However, their approach relies on[9] whose key claim “has yet to be convincingly estab-lished” [1]. We compute δC(A,B) without enumerat-ing edge sequences in O(M6 log2 M) time and O(M2)space.We first describe all free space for a given constantε ≥ 0. The star unfolding S maps a fixed source points ∈ A to a set of source image points s1, ..., sM andmaps B to a set of core and anti-core edges. Free spaceis defined by the union of a set of disks d1, ..., dM ,where each disk di has radius ε and is centered atsi. As the source point s varies continuously on anedgelet, the core is fixed, and each si is parameter-ized along a line segment li in the star unfolding (seeFigure 1b).Theorem 7 δC(A,B) can be computed inO(M6 log2 M) time and O(M2) space.3Shortest Path Problems on a Polyhedral Surface181Proof Sketch. A star unfolding is maintained overall edgelets by Theorem 1. The free space for an anti-core edge is defined by the intersection of this edgewith one nearest disk di. The free space for each coreedge is defined by the intersection of this edge withd1∪...∪dM . After computing all free space, reachabil-ity information [3] can be propagated through the freespace via plane sweep [6], and parametric search [3]can be applied to a set of algebraic free space vertexcurves. For a non-convex polyhedral surface, the star un-folding can overlap itself [5] but is still defined by aset of shortest paths from the source to every cor-ner vertex [5, 11]. Thus, a core can still be definedby a polygonal equator with O(M) complexity. Sinceshortest paths only turn at vertices in the star unfold-ing [11], an anti-core region containing si must havean hourglass shape [8, 6] because it is bounded by twoline segments and two shortest paths in the plane.Theorem 8 δN (A,B) can be computed inO(M7 log2 M) time and O(M3) space.Proof Sketch. Maintain a star unfolding over alledgelets. One hourglass defines the free space to ananti-core edge, and O(M) hourglasses define the freespace to a core edge. After computing all free space,a combination of plane sweep and parametric searchtechniques yields δN (A,B). 3 ConclusionWe develop algorithms to compute edge sequences,Voronoi diagrams, the Fréchet distance, and the diam-eter for a polyhedral surface. Our work speeds up theedge sequence and diameter approaches of Agarwal etal. [1] by a linear factor and introduces many algo-rithms that apply to convex and non-convex polyhe-dral surfaces. Future work could attempt to computethe Θ(M4) shortest path edge sequences in Θ(M4)time. Additional shortest path map and link distanceresults were omitted from this version due to spaceconcerns.References[1] P. K. Agarwal, B. Aronov, J. O’Rourke, andC. A. Schevon. Star unfolding of a polytopewith applications. SIAM Journal on Computing,26(6):1689–1713, 1997.[2] G. Albers, J. S. B. Mitchell, L. J. Guibas, andT. Roos. Voronoi diagrams of moving points.Journal of Computational Geometry and Appli-cations, 8:365–380, 1998.[3] H. Alt and M. Godau. Computing the Fréchetdistance between two polygonal curves. Journalof Computational Geometry and Applications,5:75–91, 1995.[4] B. Aronov and J. O’Rourke. Nonoverlap of thestar unfolding. 7th Symposium on ComputationalGeometry (SoCG), pages 105–114, 1991.[5] J. Chen and Y. Han. Shortest paths on a poly-hedron. Journal of Computational Geometry andApplications, 6(2):127–144, 1996.[6] A. F. Cook IV and C. Wenk. Geodesic Fréchetdistance inside a simple polygon. 25th Sympo-sium on Theoretical Aspects of Computer Science(STACS), 2008.[7] E. D. Demaine and J. O’Rourke. GeometricFolding Algorithms: Linkages, Origami, Polyhe-dra. Cambridge University Press, New York, NY,USA, 2007.[8] L. J. Guibas, J. Hershberger, D. Leven,M. Sharir, and R. E. Tarjan. Linear-time al-gorithms for visibility and shortest path prob-lems inside triangulated simple polygons. Algo-rithmica, 2:209–233, 1987.[9] Y.-H. Hwang, R.-C. Chang, and H.-Y. Tu. Find-ing all shortest path edge sequences on a convexpolyhedron. Workshop on Algorithms and DataStructures (WADS), 1989.[10] A. Maheshwari and J. Yi. On computing Fréchetdistance of two paths on a convex polyhedron.21st European Workshop on Computational Ge-ometry (EuroCG), 2005.[11] J. S. B. Mitchell, D. M. Mount, and C. H.Papadimitriou. The discrete geodesic problem.SIAM Journal on Computing, 16(4):647–668,1987.[12] D. M. Mount. The number of shortest paths onthe surface of a polyhedron. SIAM Journal onComputing, 19(4):593–611, 1990.[13] C. Schevon and J. O’Rourke. The number ofmaximal edge sequences on a convex polytope.26th Allerton Conference on Communication,Control, and Computing, pages 49–57, 1988.4EuroCG’09 - Brussels, Belgium182

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We develop algorithms to compute edge sequences, Voronoi diagrams, shortest path maps, the Fr´echet distance, and the diameter of a polyhedral surface. Distances on the surface are measured by the length of a Euclidean shortest path. Our main result is a linear factor speedup for the computation of all shortest path edge sequences and the diameter of a convex polyhedral surface. This speedup is achieved with kinetic Voronoi diagrams. We also use the star unfolding to compute a shortest path map and the Fr´echet distance of a non-convex polyhedral surface

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Shortest Path Problems on a Polyhedral Surface

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