Finding the largest rectangle in several classes of polygons The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Daniels, Karen, Victor J. Milenkovic, and Dan Roth. 1995. Finding the largest rectangle in several classes of polygons. Harvard Computer Science Group Technical Report TR-22-95. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:27030936 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Finding the Largest Rectangle in Several Classes of Polygons Karen Daniels Victor J Milenkovic Dan Roth TR September Center for Research in Computing Technology Harvard University Cambridge Massachusetts Finding the Largest Rectangle in Several Classes of Polygons y z x Karen Daniels Victor Milenkovic Dan Roth September Abstract This pap er considers the geometric optimization problem of nding the Largest area axisparallel Rectangle LR in an nvertex general p olygon We characterize the LR for general p olygons by considering dierent cases based on the types of con tacts b etween the rectangle and the p olygon A general framework is presented for solving a key subproblem of the LR problem which dominates the running time for a variety of p olygon types This framework p ermits us to transform an algorithm for orthogonal p olygons into an algorithm for nonorthogonal p olygons Using this framework we obtain the following LR time results n for xy monotone p olygons Onn for orthogonally convex p olygons where n is the slowly growing inverse of Ackermanns function Onn log n for horizontally vertically convex p olygons On log n for a sp ecial type of horizontally convex p olygon whose b oundary consists of two y monotone chains on opp osite sides of a vertical line and On log n for general p olygons allowing holes For all these types of nonorthogonal p olygons we match the running time of the b est known algorithms for their orthogonal counterparts A lower b ound of time in n log n is established for nding the LR in b oth self intersecting p olygons and general p olygons with holes The latter result gives us b oth a lower b ound of n log n and an upp er b ound of On log n for general p olygons Keywords rectangles geometric optimization fast matrix searching An earlier version of this pap er app eared in the Proceedings of the Fifth Annual Canadian Conference on Computational Geometry The general p olygon result will app ear in a sp ecial issue of Computational Geometry Theory and Applications y This research was funded by the TextileClothing Technology Corp oration from funds awarded to them by the Alfred P Sloan Foundation z This research was funded by the TextileClothing Technology Corp oration from funds awarded to them by the Alfred P Sloan Foundation and by NSF grants CCR and CCR x Supp orted by NSF grants CCR and CCR and by DARPA AFOSRFJ Introduction The problem of nding the Largest area axisparallel Rectangle LR inside a general p olygon of n vertices is a geometric optimization problem in the class of p olygon inclusion problems Dene I ncP Q Given P P nd the largest Q Q inside P where P and Q are families of p olygons and is a real function on p olygons such that 0 0 0 Q Q Q Q Q Q Q Our problem is an inclusion problem where Q is the set of axisparallel rectangles P is the set of general p olygons and gives the area of a rectangle This rectangle problem arises naturally in applications where a quick internal approxi mation to a p olygon is useful It is needed for example in the industrial problem of laying out apparel pattern pieces on clothing markers with minimal cloth waste see Section Related Work Despite its practical imp ortance work on nding the LR has b een restricted to orthogonal p olygons and recently convex p olygons see Figure Amenta has shown that the LR in a convex p olygon can b e found in linear time by phrasing it as a convex programming problem For a constrained type of orthogonal p olygon Aggarwal and Wein give a n time algorithm for nding the LR using the monotonicity of an area matrix asso ciated with the p olygon McKenna et al use a divideandconquer approach to nd the LR in an orthogonal p olygon in On log n time For the merge step at the rst level of divideandconquer they obtain an orthogonal vertically separated horizontally convex p olygon At the second level their merge step pro duces an orthogonal orthogonally convex p olygon for which they solve the LR problem in On log n time They also establish a lower b ound of time A general p olygon is a p olygonal region in the plane with an arbitrary number of comp onents and holes A rectangle is inside if it is a subset The rectangle can share part of its b oundary with the p olygons We use ORourkes denition An orthogonal p olygon is one whose edges are all aligned with a pair of orthogonal co ordinate axes which we take to b e horizontal and vertical without loss of generality In the context of this pap er this might b e called an axisparallel p olygon The b oundary of a vertically separated p olygon consists of two chains which extend from the highest p oint of the p olygon to the lowest p oint and which are on opp osite sides of some vertical line A horizontal ly convex p olygon contains every horizontal line segment whose endp oints lie inside the p olygon For a vertically separated horizontal ly convex p olygon the two chains are y monotone An orthogonally convex p olygon is b oth horizontally and vertically convex This class contains the class of convex p olygons Constrained Staircase Convex Orthogonal, Orthogonally Convex Orthogonal Largest Empty Rectangle Problem Θ 3 5 3 Θ (n) (n) O (n log n) O (n log n) O (n log n) Amenta 93 Aggarwal and Wein 88 McKenna, O'Rourke, Suri 85 McKenna, O'Rourke, Suri 85 Chazelle, Drysdale, Lee 86 (convex programming) (fast matrix searching) 2 2 O (n α (n)) O (n log n) O (n log n) (using Klawe and Kleitman 90) (using Aggarwal and Suri 87) Aggarwal and Suri 87 (fast matrix searching) (fast matrix searching) (fast matrix searching) Figure Related Work in n log n for nding the LR in orthogonal p olygons with degenerate holes which implies the same lower b ound for general p olygons with degenerate holes McKenna et al note without giving details that the LR in an orthogonal p olygon can also b e found using the more complicated On log n time divideandconquer algorithm of Chazelle et al for the largest empty rectangle LER problem The LER problem is stated as follows given a rectangle containing a set S of n p oints nd the largest area rectangular subset with sides parallel to those of the original rectangle whose interior contains no p oints from S Chazelle et al observe that the running time of the merge step of their algorithm is dominated by the largest empty corner rectangle LECR problem given two subsets S and S of S nd the largest rectangle containing no p oint of S which has left right lowerleft corner in S and upp erright corner in S The fastest solution to LECR is left right Aggarwal and Suris On log n time algorithm which they present as part of an On log n time solution to the LER problem Their LECR algorithm relies on fast searching of area matrices We observe that a sp eedup in the LECR algorithm automatically improves the running time for nding the LR in an orthogonal p olygon This sp eedup o ccurs b ecause a fast LECR algorithm implies a fast algorithm for the Largest Corner Rectangle LCR in an orthogonal vertically separated horizontally convex p olygon Computing the LCR in turn dominates the running time of the LR problem for orthogonal vertically separated horizontally convex p olygons Finally as we have previously stated this sp ecial case is required for the merge step of a divideandconquer algorithm for general orthogonal p olygons Thus the On log n time algorithm for LECR yields an On log n time algorithm for nding the LR in an orthogonal p olygon The On log n time algorithm of for orthogonal orthogonally convex p olygons can The LCR of an orthogonal p olygon is the largest area rectangle with diagonally opp osite corners on the b oundary of the p olygon Our denition of LCR for nonorthogonal p olygons is somewhat more sp ecic see Section also b e improved by applying recent results in fast matrix searching Aggarwal and Suri note that for the LECR problem which can b e asso ciated with the vertices of this type of p olygon there is a corresp onding area matrix whose maximum can b e found in On log n time by decomp osing it into a set of simpler area matrices They note in that Klawe and Kleitmans results for this simpler type of matrix imply Onn search time for the more complex matrix where n is the slowly growing inverse of Ackermanns function It is easy to see that this yields Onn time for nding the LR in an orthogonal orthogonally convex p olygon Melissaratos and Souvaine use the visibility techniques of to solve several ge ometric optimization problems In particular they nd the largest triangle contained in a p olygon in On time by considering the types of contacts b etween the p olygon and the triangle A similar approach can b e applied to the LR problem by using the concept of rectangular visibility but this leads to an On algorithm which is much slower than the On log n one we prop ose in this pap er Another p ossible approach