On intersecting a set of parallel line segments with a convex polygon of minimum area

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1 On intesecting a set of aae ine segments with a convex oygon of minimum aea Asish Mukhoadhyay Schoo of Comute Science Univesity of Windso, Canada Eugene Geene Schoo of Comute Science Univesity of Windso, Canada Chancha Kuma Indian Administative Sevice, India Binay Bhattachaya Schoo of Comuting Science Simon Fase Univesity, Canada Keywods: Agoithms; Comutationa Geomety; Geometic Otimization; Geometic Intesection 1 Intoduction et S = { 1, 2, 3,..., n } be a set of n vetica ine segments in the ane. Though not essentia, to simify oofs we assume that no two i s ae on the same vetica ine. A convex oygon weaky intesects S if it contains a oint of each ine segment on its bounday o inteio. In this ae, we oose an O(n og n) agoithm fo the obem of finding a minimum aea convex oygon that weaky intesects S. The incia motivation behind this ae is the oen obem oosed by Tami [Tam88] at the fouth Comutationa Geomety day at NYU to decide if thee exists a convex oygon whose bounday intesects a set of abitaiy oiented ine segments. 2 Pio Wok Edesbunne et a [EMP + 82] oosed an O(n og n) time agoithm based on duaity that comutes a descition of a the ine tansvesas of a set of ine segments in the ane. Goodich and Snoeyink [GS90] extended this esut to oose an otima O(n og n) time agoithm to decide if thee exists a convex oygon whose bounday intesects each i in S by educing the obem to biatite stabbing; this atte obem is soved by duaization as in Edesbunne et a [EMP + 82]. If thee exists one, they show how to find one of minimum eimete o aea in O(n 2 ) time. This is a atia soution to Tami s oen obem [Tam88]. Key et a [M] soved a sime vesion of Tami s obem (as we do, too) by oosing an O(n og n) agoithm that comutes a minimum eimete convex oygon which weaky intesects a set of n othogona ine segments. Thei techniue is inteesting as it educes the obem to shotest-ath comutation, but uite imactica as it is based on Chazee s inea-time agoithm fo oygon tianguation. aaot [a95] extended this esut to find a minimum eimete convex oygon which weaky intesects a set of n ine segments in a fixed numbe of oientations in O(n og n) time. Suisingy, not much seems to be known on comuting a minimum aea convex oygon that weaky intesects a ana set of ine segments. This is a cometey evised vesion of a ae that oiginay aeaed in the oceedings of CCCG

2 3 Notations A ine segment in S with end-oints and wi be denoted by. The functions to(.) and bot(.) etun its ue and owe end-oints. The ue chain of the convex hu of the owe end-oints of the ine segments in S has the oety that bot( i ) fo each i ies on o beow it. If we atiay ode convex chains ove a given ange of x-vaues by defining a chain to be ess than o eua to anothe if at evey oint of the ange the coesonding y-vaue of the fome is ess than o eua to the coesonding y-vaue of the atte, then the ue hu of the owe end-oints is the smaest one in the given atia ode to have the above oety. To efect this we denote this owest uwad-convex chain by uc(s). Simiay, the owe chain of the convex hu of the ue end-oints is the agest among a convex chains which have to( i ) fo each i on o above it. We denote this highest downwad-convex chain by hdc(s). We assume, without oss of geneaity, that thee is a uniue eftmost ine segment and a uniue ightmost ine segment. Ceay, the end-oints of uc(s) ae and and those of hdc(s), and. et < u 1, u 2,..., u > be the odeed set of vetices on uc(s) fom to and < v 1, v 2,..., v > those on hdc(s) fom to. et be the convex egion that consists of the oints ying on o beow uc(s) and on o above hdc(s) (Fig. 1). Figue 1: The coe aea defined by uc(s) and hdc(s) 4 Chaacteization In this section we discuss ou chaacteization of a minimum aea convex oygon, P, that weaky intesects S. Fact 1: A convex oygon that weaky intesects a the segments must contain the coe aea. Poof: A convex oygon, P, that weaky intesects a the segments can be obtained by constucting the convex hu of a set of oints { 1, 2, 3,..., n }, whee each i is a oint of i. Conside the convex egion u fomed by the ue hu of P and the ays to - fom the exteme vetices of P on and. Fo two adjacent vetices u i and u i+1 on uc(s), the edge joining them must be contained in the convex egion unde consideation as u i and u i+1 being bottom end oints of thei esective segments must be on o beow the ue convex chain of P and hence in u. Thus uc(s) must be contained in u. We can simiay ague that hdc(s) must be contained in the convex egion fomed by the owe hu of P 2

3 and the ays to fom the exteme vetices of P on and. Thus must be contained in P = u. It foows fom Fact 1 that the minimum aea convex oygon, P, that weaky intesects S aso contains. We make a note hee egading the uniueness of the segments and. Say thee is a set S of eft-most segments, a of them on the same vetica ine. Conside the segment defined by the to-most bottom endoint, a, and the bottom-most to endoint, b, of S. If a is above b then the detemination of a minimum oygon is made easie. In this case, uc(s) and hdc(s) wi not coss on the eft side. Instead of having a eft-most segment on which we have to choose a oint, we just have a segment ab that we have to incude in the minimum aea stabbe P by joining it to uc(s) and hdc(s). See Figue 2 fo an exame. When a is beow b, uc(s) and hdc(s) coss, and we can eace the entie set S by a new segment ab. This modified set S = (S S ) {ab} is teated exacty ike S in what foows, and ab wi be an exteme segment. uc(s) hdc(s) Figue 2: The minimum aea convex oygon wi consist of uc(s) joined to hdc(s) by adding edges on the eft and ight Fact 2: The ightmost vetex and the eftmost vetex of a minimum aea convex oygon, P, must be invisibe with esect to uc(s) and hdc(s). Poof: Note that the eftmost and ightmost vetices cannot be visibe with esect to both uc(s) and hdc(s), excet in the degeneate case that the ine segments have a common tansvesa. Suose then that the eftmost vetex is visibe to the ightmost vetex with esect to hdc(s). It is cea fom the skew of the tiange o in Fig. 3, that a sight uwad etubation of the vetex can educe the aea of P. Hence it cannot be of minimum aea. et e be an edge on uc(s) o hdc(s), with suoting ine that intesects o (see Fig. 4). We wi atition and into intevas using a such oints of intesection. Fom a oint on o, one tangent can be dawn to uc(s) and one tangent to hdc(s). So any oint on o can be associated with one oint of tangency on each of these chains. A oints in a given atition inteva wi shae thei oints of tangency. Fact 3: If the exteme vetex ies in the inteio of a atition inteva, then the aea of P does not incease if we move to one of the end oints of this inteva. Poof: et the eftmost vetex be sticty inside a atition inteva. et t u and t be oints of tangency to uc(s) and hdc(s) fom as shown in Fig. 4. Fist, assume that is not joined to by a staight ine. If t is to the eft of the vetica ine defined by t u, then a sight downwad etubation of the vetex deceases the aea of P. If t is to the ight of the vetica ine defined by t u (as in Fig. 4), then a sight uwad etubation of deceases the aea of P. 3

4 o Figue 3: and ae visibe with esect to hdc(s) Patition oint tu A atition inteva t Figue 4: and ae intena oints of a atition inteva Now assume that is joined to by a staight ine (so that eithe, t u, and ae coinea, o, t, and ae). et us say that, t u, and ae coinea (see Fig. 5). Then t u wi be the oint of tangency fom to uc(s). et s be the oint of tangency fom to hdc(s). s has to be on o to the ight of t. Thee ae two ossibiities fo the osition of : (a) is in the inteio of a atition inteva. If t is to the eft of the vetica ine defined by t u, then a sight downwad etubation of the vetex deceases the aea of P (see Fig. 5). If t is to the ight of the vetica ine defined by t u, then s must aso be to the ight of it. Given the skews of t u s and t u t, a sight downwad etubation of, foowed by a sight uwad etubation of, wi decease the aea of P. (b) is a atition oint (see Fig. 6). The edge geneating woud have to be on hdc(s) (if it was on uc(s), then woud be a atition oint as we). et us say that s is the ightmost oint on this edge. If t is to the eft of the vetica ine defined by t u, then a sight downwad etubation of the vetex deceases the aea of P. Assume t (and hence s ) is to the ight of the vetica ine defined by t u. cannot be the owest atition oint on (if it was, then woud be a atition oint, geneated by the ightmost edge on uc(s)). et be the atition oint just beow. The tangent fom to uc(s) wi ass though t u. et s be the eftmost oint on the edge that geneated. The tangent fom to hdc(s) wi ass though 4

5 tu t s Figue 5: Case when,, and t u ae on a staight ine s. s can not be to the eft of t, and so it must be to the ight of the vetica ine defined by t u. The skew of t u s means that the choice woud esut in a smae oygon P than woud the choice of. Given that is now the ightmost vetex of P, and given the skew of t u t, a sight uwad etubation of wi decease the aea of P. tu t s s Figue 6: Case when, and t u ae coinea and is a atition oint If t u t is vetica with esect to then an uwad o downwad etubation of does not change the aea of P, and in this case we can aways assume that is a atition oint. Thus the exteme vetices must be atition oints that ae invisibe with esect to the above hus. Howeve, this is not sufficient. In Fig. 7 beow, we have an exame that shows a ai of atition oints that ae invisibe with esect to both hus. The oygon P cannot be of minimum aea as we can educe its aea by a sma etubation of eithe the eftmost vetex o the ightmost vetex. This suggests that we must ook fo atition oints which minimize the aea ocay. The foowing fact ovides an inteesting chaacteization of the atition oints whose choice fo and minimizes the aea ocay on the eft as we as the ight. Fact 4: If is a eft atition oint which is due to en edge on the ue hu, a vetica ine though the 5

6 Figue 7: and ae atition oints, but P is not of minimum aea oint of tangency to the owe hu intesects this edge on the ue hu (see Fig. 8). Poof: Fom the skew of t u t with esect to, it is cea that a sma downwad etubation of wi incease the aea of P, whie fom the skew of t ut with esect to, it is cea that a sma uwad etubation wi incease the aea of P. Thus this must be a minimum aea configuation. t u tu t Figue 8: and ae atition oints, and P has minimum aea We can simiay chaacteize the ightmost vetex. Fact 5: A ai of atition oints, one on and the othe on, that minimizes the aea ocay on the eft and the ight is uniue. Poof: Assume thee is a atition oint, geneated by edge e on uc(s), and with tangent oint t, on hdc(s), in the vetica sti defined by e. et e be an edge on uc(s) to the ight of e. The associated atition oint wi have to be above. So the tangent oint t wi be on o to the eft of t, and hence to the eft of the vetica sti defined by e. Simiay, the tangent oint associated with an edge e to the eft of e wi be to the ight of the vetica sti defined by e. Hence thee can be no othe edge on uc(s) with the same oety as e. The same can 6

7 be said fo edges e, e, and e on hdc(s). Now et e be an edge on hdc(s), associated with a tangent oint t u on uc(s) that is in the vetica sti defined by e. et e 1 and e 2 be the eft and ight endoints of e. et e 1 and e 2 be the eft and ight endoints of e. e can not be whoy contained by the vetica sti defined by e, since thee woud be no t u above e. Assume e 1 (es. e 2 ) is to the eft of e 1 (es. e 2 ) (see Fig. 9). Since t u is diecty above e, it woud have to be on o to the eft of e 1. Simiay, t woud have to be on o to the ight of e 2. It is imossibe fo both of these to be tue at the same time, and hence thee can be no such e. A othe cases ae symmetic. e 1 e e 2 e 1 e e 2 Figue 9: The endoints of e ae to the eft of the coesonding endoints of e If we aow two segments to be on the same vetica ine, then thee wi be at most 2 atition oints on the eft segment and two atition oints on the ight segment that satisfy the condition in Fact 4. A ai of oints satisfying the condition in Fact 2 can be chosen in constant time. Fact 6: et be the vetex found on, using the chaacteization imied by Fact 4. et be the vetex found on. Then is invisibe to with esect to uc(s) and hdc(s). Poof: et u 1 be the eft vetex on the edge extended to ceate, and et u 2 be the ight vetex. et u 3 be the oint of tangency fom to the othe chain. et v 1 be the ight vetex on the edge extended to ceate, and et v 2 be the eft vetex. et v 3 be the oint of tangency fom to the othe chain. So, u 3 is between the vetica ines defined by u 1 and u 2. Aso, v 3 is between the vetica ines defined by v 1 and v 2 (see Fig. 10). When consideing the eative ositions of u 1, u 2, v 1, and v 2, thee ae 11 cases (6 cases in which the extended edges ae on diffeent chains, and 5 cases in which the extended edges ae on the same chain). A of these cases can fit into one of the foowing: Case 1: u 2 is sticty to the eft of v 2. (a) If u 2 and v 2 occu on the same chain: is invisibe to with esect to that chain, because of the section of chain between u 1 and v 1 ; and is invisibe to with esect to the othe chain, because of the section of chain between u 3 and v 3. (b) If u 2 and v 2 occu on diffeent chains: is invisibe to with esect to the chain containing u 1 and u 2, because of the section of chain between u 1 and v 3 ; and is invisibe to with esect to the chain containing v 1 and v 2, because of the section of chain between v 1 and u 3. 7

8 u 1 u 2 v 3 u 3 v 1 v 2 Figue 10: Patition oints that esut in oca minimum aeas must be invisibe to each othe Case 2: The extended edges ae on the same chain, and u 2 and v 2 ae the same oint. Assume the extended edges ae on uc(s). Then wi be invisibe to with esect to uc(s), because of the section of chain between u 1 and v 1. Since no two segments ae on the same vetica ine, u 3 has to be sticty to the eft of v 3. wi be invisibe to with esect to hdc(s), because of the section of chain between u 3 and v 3. Case 3: The same edge is extended to ceate and (u 2 and v 1 ae the same oint, and u 1 and v 2 ae the same oint). Assume uc(s) contains the extended edge. Then is invisibe to with esect to uc(s), because of the section of chain between u 1 and u 2. Aso, u 3 and v 3 wi have to be on hdc(s), beow the ine connecting to. Point wi be invisibe to with esect to hdc(s), because of u 3 and v 3 (and any section of chain otentiay between u 3 and v 3 ). Case 4: The extended edges, occuing on diffeent chains, ovea so that: u 2 is to the ight of the vetica ine defined by v 2 and to the eft of the vetica ine defined by v 1, and v 2 is to the ight of the vetica ine defined by u 1 (see Fig. 11). Assume uc(s) contains u 1 and u 2. Since u 3 is between the vetica ines defined by u 1 and u 2, then u 3 has to be v 2 o a oint to the eft of v 2. So, is above the ine defined by v 1 and v 2, and is invisibe to with esect to hdc(s). Since v 3 is between the vetica ines defined by v 1 and v 2, then v 3 has to be u 2 o a oint to the ight of u 2. So, is beow the ine defined by u 1 and u 2, and is invisibe to with esect to uc(s). Case 5: (a) u 1 and u 2 ae sticty between the vetica ines defined by v 1 and v 2, o (b) vice vesa. u 1 and u 2 wi have to be on one chain, and v 1 and v 2 wi have to be on the othe. Assume uc(s) contains u 1 and u 2. This case wi neve occu. In (a), thee ae no oints between v 1 and v 2 on hdc(s), and so no u 3, on hdc(s), between the vetica ines defined by u 1 and u 2. In (b), thee ae no oints between u 1 and u 2 on uc(s), and so no v 3, on uc(s), between the vetica ines defined by v 1 and v 2. Case 6: The extended edges ae on diffeent chains, v 2 is to the eft of the vetica ine defined by u 1, and v 1 is to the eft of the vetica ine defined by u 2 (see Fig. 12). 8

9 u 2 v 3 u 1 u 3 v 1 v 2 Figue 11: One case of oveaing edges u 1 u 2 v 1 v 2 Figue 12: A case that wi not aise Assume u 1 and u 2 ae on uc(s). Since u 3 is to the ight of u 1, then u 3 is on hdc(s), to the ight of v 2. This means that is beow the ine defined by v 1 and v 2. Since v 3 is to the eft of v 1, then v 3 is on uc(s), to the eft of u 2. Then must be above the ine defined by u 1 and u 2. This is ony ossibe if uc(s) is beow hdc(s). This means that thee is a ine stabbing the segments, and we ae not consideing this case. Case 7: The extended edges ae on the same chain, and u 1 is on o to the ight of v 1. Assume the extended edges ae on uc(s). Since no two segments ae on the same vetica ine, u 3 is sticty to the ight of v 3. woud have to be beow the ine defined by u 3 and v 3. Aso, woud have to be beow the ine defined by u 3 and v 3. To maintain the convexity of uc(s), this is ony ossibe if uc(s) is beow hdc(s). Again, this case esuts in a ine stabbe. Putting Facts 4, 5, and 6 togethe we concude that the choice of a ai of oints which minimizes the aea ocay on the eft and the ight gives the gobay minimum aea convex oygon P that we ae ooking fo. 9

10 5 The Agoithm The agoithm oceeds by comuting uc(s) and hdc(s). We extend the edges of these two hus to detemine thei intesections with and that atition these segments. We associate a tiet of abes with each such atition oint. If it is due to, say uc(s), we stoe the end-oints of the edge that geneated this atition oint as the fist two abes, whie the thid is the abe of the vetex on hdc(s) to which we can daw a tangent fom this atition oint (see Fig. 13). v 5 1 : (v 4, v 5, u 3) 1 : (v 2, v 1, u 4) v 1 2 : (v 3, v 2, u 3) v 2 u 3 u 4 2 : (v 3, v 4, u 4) v 4 u 2 v 3 u 1 3 : (v 2, v 3, u 4) u 5 Figue 13: abeing atition oints geneated by edges on hdc(s) Once we have detemined the atition oints on and, in inea time we can find the ai that satisfies the citeion ovided by Fact 4. The comexity of the agoithm is in O(n og n) as finding uc(s) and hdc(s) dominates the unning time. 6 Concusions In this ae we have descibed an O(n og n) agoithm fo finding a minimum aea convex oygon that weaky intesects a set of n vetica segments. We have imemented the agoithm and it is avaiabe at htt://cs.uwindso.ca/~asishm (cick on the ink softwae). An obvious oen obem is to extend this to an abitay coection of ine segments. 7 Acknowedgements We acknowedge the contibution made by M Samidh Chattejee in imementing an initia vesion of the agoithm. efeences [EMP + 82] H. Edesbunne, H. A. Maue, F. P. Peaata, A.. osenbeg, E. Wez, and D. Wood. Stabbing ine segments. BIT, 22: , [GS90] M. Goodich and J. Snoeyink. Stabbing aae segments with a convex oygon. Comute vision, Gahics and Image Pocessing, 49: ,

11 [M] [a95] [Tam88] Key A. yons, Henk Meije, and David aaot. Minimum oygon stabbes of isothetic ine segments. Deatment of Comuting and Infomation Science, Queen s Univesity, Ontaio, Canada. David aaot. Minimum oygon tansvesas of ine segments. Int. J. Comut. Geomety A., 5(3): , Aie Tami. Imoved comexity bounds fo cente ocation obems on netwoks by using dynamic data stuctues. SIAM J. Discete Math., 1(3): ,