Open Access A New Algorithm for the Shortest Path of Touring Disjoint Convex Polygons

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1 Send Orders for Reprnts to 1364 The Open Automaton and Control Systems Journal, 2015, 7, Open Access A New Algorthm for the Shortest Path of Tourng Dsjont Convex Polygons Wang Ljuan 1,2, Jang Bo *,1, Deng Ansheng 1 and We Q 2 1 College of Informaton Scence and Technology, Dalan Martme Unversty, Dalan, , P.R. Chna 2 Dept of Informaton and Scence, Dalan Insttute of Scence and Technology, Dalan, , P.R. Chna Abstract: Gven a start pont s, a target pont t, and a sequence of k dsjont convex polygons n the plane, fndng the shortest path from s to t whch vsts each convex polygon n the gven order s our focus. In ths paper, we present an mproved method to compute the shortest path based on the last step path maps by Dror et al. Instead of usng of pont locaton n prevous algorthm, we propose an effcent method of locatng the ponts n the path wth lnear query and make the data structures much smpler. Our mproved algorthm gves the O(nk) runnng tme whch mproves upon the tme O(nklog(n/k)) by Dror etal., where n s the total number of vertces of all polygons. Furthermore, we have mplemented ths algorthm by programmng. The result shows that our algorthm s correct and effcent.. Keywords: Dsjont convex polygon, last step path maps, lnear query. 1. INTRODUCTION Path plannng s one of the central problem areas n computatonal geometry [1]. The shortest path problem s the most classcal example of path plannng. Fndng the shortest path between two ponts for the gven order objects or obstacles s the man goal [2]. In ths paper, we manly study on the method of computng the shortest path between two ponts s and t of tourng a sequence of dsjont convex polygons gven n the plane. The problem can be descrbed as follows. Gven a start pont s, a target pont t, and a sequence P = (P 1,..., P k ) of smple dsjont convex polygons n the plane. The goal s to fnd the shortest path that starts from s, vsts all convex polygons accordng to ther order and ends at t [3]. In Fg. (1), the path lnked by bold lnes s the shortest path from s to t whch vsts the dsjont convex polygons. Fg. (1). The shortest path of tourng dsjont convex polygons, for k=5. The problem studed n ths paper s a sub-problem of the tourng polygons problem (TPP), ntroduced by Dror, Efrat, Lubw and Mtchell n STOC 03[4]. Algorthms for solvng the tourng convex polygons problem have many mportant applcatons n many geometrc problems, such as the zookeeper [5], safar [6], and watchman route problems [7-9]. In the fxed-source safar and zoo-keeper problems, gven a start pont s n a smple polygon P, and a set of dsjont convex polygons (cages) nsde P, each of whch havng a common edge wth P. In the safar problem, we need to seek the shortest route of tourng each cages, whle n the zookeeper problem, the cages can t be allowed to enter, see Fg. (2). In the fxed-source watchman route problem, gven a smple polygon P and a start pont s n t. The goal s to fnd a shortest route from s such that we can see each pont n P from at least one pont of the route, see Fg. (3). What all these problems have n common s that the shortest vstng path n the gven order needed to be found [3]. If the vsted order s not specfed, t becomes the classcal Travelng Salesperson Problem wth neghborhoods, whch s NP-hard [10]. *Address correspondence to ths author at the College of Informaton Scence and Technology, Dalan Martme Unversty, Dalan, Laonng, , P.R. Chna; Tel: ; Fax: ; E-mal: newdlhot@163.com Fg. (2). Safar problem / Bentham Open

2 A New Algorthm for the Shortest Path of Tourng Dsjont Convex Polygons The Open Automaton and Control Systems Journal, 2015, Volume Fg. (3). Watchman route problem. The tourng polygons problem has been ntensvely studed. Applyng the method of last step path maps, Dror et al. gave an algorthm runnng n tme O(nklog(n/k)) f the gven polygons are dsjont and convex, and an O(nk 2 logn) tme f the convex polygons are arbtrarly ntersected and the sub-path between any two consecutve polygons s constraned to le wthn a smply connected regon, where n s the total number of vertces specfyng the polygons [4]. They also proved the TPP s NP-hard for the case that they are ntersected and non-convex polygons [4, 11]. Arash Ahad et al. also proved that TPP s NP-hard when the polygons are parwse dsjont [3]. Several approxmaton algorthms have been proposed for the case of dsjont convex polygons, for example, Faje L and Renhard Kettle n 2007 gave an approxmate algorthm n K( ε ) O(n) tme by applyng the rubber-band algorthm for sequences of convex polygons, where K( ε )=(L 0 -L)/ ε, L 0 s an ntal path, L that of the true length of the shortest path of convex polygons set P, and n s the total number of vertces of the gven polygons [12]. The experment by Wang and Huo n 2011 showed that the tme complexty of rubber-band algorthm s O(n 2 ) when n s larger [13, 14]. Dror et al. has been solved the tourng dsjont convex polygons problem usng the last step shortest path maps method to compute teratvely [4]. Computng the shortest tourng path to a vertex v of P ( 0 k 1 ) wll be performed at most -1 pont locaton queres for the polygons P -1,..., P 1. It s clearly that a query for pont t n the fnal last step shortest path map determnes a query pont for the prevous last step shortest path map. The last step shortest path map method adopts the pont locaton data structure to save a polygon [4]. Thus, a query for the shortest path from s to any pont can be computed n tme O(klog(n/k)). In ths paper, nstead of computng the pont locatons ndependently, we compute the pont locatons for all the vertces of P once by a lnear scan on the boundary of P, thus, the pont locaton data structure s not needed. Ths method s smple and can reduce the tme complexty of the algorthm by a factor of O(logn). Thus the O(nk) runnng tme s obtaned whch mproved upon the tme O(nklog(n/k)) n the algorthms by Dror etal. The data structure s much smpler and the method of locatng the ponts s more effcent. 2. LOCAL OPTIMAL TOURING PATH We denote by opt(l) a shortest tourng path for the gven convex polygons P 1,..., P k ( k ), -path a path that starts at s and vsts the sequences of P 1,..., P ( k ) polygons, and π (m) a path that starts at s and vsts the sequences of P 1,..., P ( k ) polygons to m. We assume that all gven convex polygons P 1,..., P k are smple, and opt(l) vsts n order P 1,..., P k, then the local optmalty of opt(l) wth respect to the P 1,..., P k s equvalent to global optmalty [4]. Denote by the contact pont b of opt(l) wth the edge e e P ( k), after opt(l) vsts P 1,..., P -1. There are three cases of OPT(L) contacted wth P, whch are as follows. c p d 1 ' d 2 ' Fg. (4). Types of contact of OPT(L) wth an edge e polygon P.. (a) (b) (c) Case 1 Edge-reflecton contacts v" For a bend pont b on the nteror of an edge e of P, OPT(L) makes a reflecton contact wth an edge e f the angle of OPT(L) comng nto the edge e wth e (ncomng angle) s equal to the angle of OPT(L) gong away from e wth e (outgong angle), OPT(L) makes a perfect reflecton on the edge e. (see Fg. 4a), c' s the reflecton of c wth respect to the lne through e. c' b p p b

3 1366 The Open Automaton and Control Systems Journal, 2015, Volume 7 Ljuan et al. Case2 Vertex-bendng contacts For a bend pont at a vertex v = e h e j (e h P,e j P j ), consder the rays d 1 ' and d 2 ' formed by reflectng the segment pv n the two edges, respectvely, then, only when the outgong path segment from v leaves v n the cone ω, whch bounded by d 1 ' and d 2 ', the local path s the shortest (see Fg. 4b). Case 3 Passng-through contacts OPT(L) passes the edge e through an nteror pont b of e, we consder b as the second ntersecton pont of the boundary of P wth OPL(L), and thus OPT(L) reaches the pont b from the nteror of P (see Fg. 4c). 3. THE LAST STEP SHORTEST PATH MAP Let G be the frst contact set of P,.e., the ponts where the shortest path frst reaches a pont of P after vstng P 1,..., P -1. G s a (connected) chan on the boundary of P [4]. In Fg. (5), G 1 s the bold edge v 1 v 2 and v 2 v 3 of P 1. We denote by M the last step shortest path map for P. Suppose that all gven polygons are dsjont, let us compute the frst map M 1 as below. For every vertex v of P 1, we frst compute the shortest path from s to v. If ths path arrves at v from the nsde of P 1, then v s not a vertex of G 1, otherwse t s, and G 1 s a (connected) chan on the boundary of P 1. We can see that OPT(L) may make a reflecton on the ponts and edges of G or OPT(L) may go across e. The vertces and edges on the G 1 dvde the whole plane nto three types whch are passng-through regons(c), vertex-bendng regons (B) and edge-regons(r). () The passng-through regon s bounded by the boundary of G 1 and the extensons of the shortest paths from s to two endponts of G 1. C regon n the Fg. (5). () The edge-reflecton regon s bounded by one edge e of G 1 and the two rays reflected by the shortest paths from s to the two vertces of e. R(e ) regon n the Fg. (5). () Let v be a vertex of e, and e' be the other edge ncdent to v. The vertex-bendng regon of v s bounded by the two rays whch are used n defnng the edge-reflecton regon of e and the edge-reflecton or passng-through regon of e', whch s the trangular regon. B(v ) regon n the Fg. (5). Thus, all these subdvson regons n the plane form the last step shortest path map M 1 of P THE ALGORITHM 4.1. The Algorthm Shortest Tourng Path for Dsjont Convex Polygons Suppose Gven M 1,..., M, we can easly compute a shortest -path to any query pont m as follows. Case 1 m s contaned n a passng-through regon of M. In ths case, π (m) = π 1 (m), then we recursvely compute the (-1)-path to m, see Fg. (6a). Case 2 m s contaned n an edge-reflecton regon of M. In ths case, we let m' be the reflecton of m wth respect to e, then recursvely locate m' n M -1 and compute the (-1)- path to m', the segment from e to m s the -1 path to m, see Fg. (6b). Case 3 m s contaned n vertex-bendng regon of a vertex v. In ths case, the last segment of π (m) s vm, then we compute the (-1)-path to v (locatng v n M -1, etc.) recursvely, see Fg. (6c). B v 2 s R e 1 B v 1 v 1 v 2 R e 2 (a) B v 1 B v 2 s B v 2 s R e 1 m R e 1 v 1 v 2 (b) B v 1 P 1 v 3 v 5 v 3 v 5 m P 1 R e 2 v1 v 1 P1 P 1 v2 v 2 v 3 R e 2 m" v 4 B v 3 C v 4 C B v 3 v 5 v 4 B v 3 m" Fg. (5). The last step shortest path maps M 1. Fg. (6). Three Cases for Computng the Shortest (-1) Path to m. (c)

A New Algorthm for the Shortest Path of Tourng Dsjont Convex Polygons The Open Automaton and Control Systems Journal, 2015, Volume 7 1367 We construct each the maps M 1,..., M k teratvely.

4 A New Algorthm for the Shortest Path of Tourng Dsjont Convex Polygons The Open Automaton and Control Systems Journal, 2015, Volume We construct each the maps M 1,..., M k teratvely. The algorthm s follows. Suppose we have obtaned the maps M 1,..., M, n order to construct the next map M +1, we frst compute the shortest tourng paths from s to each vertex v of P +1 as descrbed above. If ths path arrves at v from the nsde of P, then v s not a vertex of G. Otherwse t s, the last segment of π 1 (v) determnes the subdvson of M +1. Thus, the edge-reflecton, passng-through regons and vertex-bendng regons of M +1 an be defned analogously. An example of M 2 s shown n Fg. (7). s v 1 v 2 P1 v 3 v 5 Fg. (7). The Last Step Shortest Path Maps M 2. Lemma1 Let v' denotes the pont n the map M whch results from the vertex v of P +1, called mappng-pont of v. Then the edges consst of the ponts sequences v ' (=1,.., m, m= p +1 ) n the map M whch results from the vertces P +1 form at most three edge chans along the boundary of P. Proof. Let the edges n the passng-through regon as one edge chan because of the way of locatng ponts s same, the edges n the other zones s dvded nto two cases accordng to the trend of rsng or fallng. Thus, for each convex polygon, the edges on the boundary s formed at most three edge chans, as shown n Fg. (8). Let a and b be two vertces of P +1, and a' and b' be ther mappng-ponts n M. The last portons of two shortest tourng paths (.e., from a' to a and from b' to b) can t cross n M, except for the followng stuatons: () a' and b' are possble the same pont, due to the vertex-bendng contacts of the shortest tourng paths to the vertces of P +1 ; or () one path completely s contaned by the other. A shortest -path to any query pont m s teratvely computed by the above mentoned method. So, f the pont a moves along the boundary of P +1, the shortest path ponts on P wll form two or three edge chans, whch depends on the poston of startng pont n P. For example, the mappngponts n P 1 whch result from the vertces of P 2 only form one edge chan, the bold lne s the edge chan n Fg. (7). v 4 B v 7 v 7 R e 7 v 8 B v 8 P2 R e 6 v 6 C B v 6 v 9 Fg. (8). The three Edge Chans of Polygon P. Lemma2 Gven M 1,..., M, the path π (q) can be determned n tme Ο constructed n tme Ο p j + p +1 One can easly see that t needs Ο ( p j=1 j ), and the map M +1 can be ( j=1 ). ( p ) tme to locate the shortest path pont n the M for m, and all the maps M 1,..., M needed to be vsted. The tme to fnd π (q) s then ( ), denoted by Ο p j Ο p + p 1 + p 1 ( j=1 ). To construct M +1, for each vertex v of P +1, we compute π (v). For example, we frst compute π (v 1 ), and the tme to fnd π (v 1 ) s Ο ( p j=1 j ). Accordng to Lemma1, locatng all the other shortest path ponts n M, for the other vertces of P +1 can be done by a constant number of lnear scans n M, and t needs Ο p +1 ( ) tme. So the map M +1 can be ( j=1 ). constructed n tme Ο p j + p +1 Each map M can be constructed teratvely, so all maps k 1 M 1,..., M k+1 can be computed nο p j + p +1 snce Ο p +1 Ο ( =1 ( j=1 )), k 1 ( ( ) =1 ) = Ο k k 1 ( ( p +1 ) =1 ) = Ο kn k 1 ( ( p =1 j=1 j )) = Ο k 1 ( n =1 ) = Ο kn can be computed n Ο kn ( ) tme. ( ), and ( ). Thus, all maps Therorem1 The Tourng polygons problem for k dsjont convex polygons wth nput sze n, a data structure of sze O(n) can be bult n tme O(kn) that enables shortest -path queres to any query pont m to be answered n tme O(n), where n s the total number of vertces of all the polygons The Implementaton of Algorthm The algorthm presented n ths paper has been mplemented by program, the dvson of the whole plane s shown n Fg. (9), and the runnng result of the shortest path s shown n Fg. (10). To make the result clearly vsble, we only present the result of 5 dsjont convex polygons. Ths

CONFLICT OF INTEREST The authors confrm that ths artcle content has no conflct of nterest. Fg. (9). The Last Step Shortest Path Map M for P ( =1,, 5). Fg. (10).

5 1368 The Open Automaton and Control Systems Journal, 2015, Volume 7 Ljuan et al. example has contaned edge-reflecton contacts, vertexbendng reflecton contacts, and passng-through contacts three cases of OPT(L) contacted wth P mentoned above. CONFLICT OF INTEREST The authors confrm that ths artcle content has no conflct of nterest. Fg. (9). The Last Step Shortest Path Map M for P ( =1,, 5). Fg. (10). The Runnng Result of Dsjont Convex Polygons, for k=5. CONCLUSION In ths paper, we present an fast algorthm of locatng the path ponts n computng the shortest path of tourng a sequence of dsjont convex polygons and we gve an O(kn) tme soluton, where k s the number of polygons and n s the total number of vertces of the polygons. Our results mprove upon the prevous tme O(nklog(n/k)). Ths research has made prelmnary results. A more effcent tme soluton to the problem of tourng dsjont and convex polygons problem s an open problem. In addton, fndng the shortest path of tourng the convex polygons possbly ntersected s also our further study. ACKNOWLEDGEMENTS Ths work s supported by the Natonal Natural Scence Foundaton of Chna (No , ), General Project of Lao Nng Provnce Scence and Research (No. L ), and the General Project of Dalan Insttute of Scence and Technology (No. KYZ1416). REFERENCES [1] Hakan Jonsson, The Travelng Salesman Problem for lnes n the plane, Informaton Processng Letters, vol.82, pp , [2] Mark de Berg, Otfred Cheong, Marc van Kreveld, et al., Computatonal Geometry: Algorths and Appl- catons (Thrd Edton), Sprnger-Verlag Publshers: Berln, pp , [3] Arash Ahad, Amrhossen Mozafar, Alreza Zare, Tourng a sequence of dsjont polygons: Complexty and extenson, Theoretcal Computer Scence, vol.556, pp , [4] M. Dror, A. Efrat, A. Lubw and J.S.B. Mtchell, Tourng a sequence of polygons, n Proc. 35th Annu. ACM Sympos. Theory Comput., Calforna, USA, pp , [5] John Hershberger, Jack Snoeynk, An effcent soluton to the Zookeeper s problem, n: CCCG, pp , [6] Xuehou Tan, Tomo Hrata, Shortest safar routes n smple polygons, n: Algorthms and Computaton, Sprnger, Berln, Hedelberg, pp , [7] Adran Dumtrescu, Joseph S. B. Mtchelland Paweł Żylńsk, Watchman Route for Lne and segments, Algorthm Theory SWAT 2012 Lecture Notes n Computer Scence, vol.7357, no. 14, pp , [8] Xuehou Tan, Tomo Hrata, Constructng shortest watchman routes by dvde-and-conquer, n: Algorthms and Computaton, Sprnger, Berln, Hedelberg, pp , [9] Xuehou Tan, A lnear-tme 2-approxmaton algorthm for the watchman route problem for smple polygons, Theoret. Comput. Sc, vol.384, no.1, pp ,2007. [10] Adran Dumtrescu, The travelng saleman problem for lnes and rays n the plane, The proceedng of the 22 nd Canadan Conference on Computatonal Geometry(CCCG2010), Cannada, pp ,2012. [11] A. Ahad, A. Mozafar and A. Zare, Tourng dsjont polygons problen s NP-hard, n Proc. Of COCOA 2013, LNCS 8287, pp , [12] LI Fa-je and KLETTE Renhard, Shortest path algorthms for sequences of polygons, CAAI Transactons on Intellgent Systems, vol.3, no.1, pp.23-30, [13] Ljuan Wang, L Huo, and Dandan He, An Improved Algorthm for Eucldean Shortest Paths of Vstng Lne Segments n the Plane, Journal of Convergence Informaton Technology, vol.6, no.6, pp ,2011. [14] Ljuan Wang, Bo Jang, Ansheng Deng, Q We, Fast Computaton of Shortest Path for Vstng Segments n the Plane, The Open Cybernetcs & Systemcs Journal, vol.8, no.4, pp ,2014. Receved: May 26, 2015 Revsed: July 14, 2015 Accepted: August 10, 2015 Ljuan et al.; Lcensee Bentham Open. Ths s an open access artcle lcensed under the terms of the ( whch permts unrestrcted, non-commercal use, dstrbuton and reproducton n any medum, provded the work s properly cted.

The Shortest Path of Touring Lines given in the Plane

The Shortest Path of Touring Lines given in the Plane Send Orders for Reprnts to reprnts@benthamscence.ae 262 The Open Cybernetcs & Systemcs Journal, 2015, 9, 262-267 The Shortest Path of Tourng Lnes gven n the Plane Open Access Ljuan Wang 1,2, Dandan He