Recognition of Largest Empty Orthoconvex Polygon in a Point Set

20th Canadian Conference on Computational Geometry, 2008RtlFtrRtlp jFtrFblp jp i Rbr Fblp iRbr(a)Figure 1: Orthoconvex polygons(b)combinatorial explosion holds for MEOP also. We nowpresent a polynomial time algorithm for computing themaximum-area MEOP.3 AlgorithmWe shall consider all possible pairs of points p i , p j ∈ P,and identify the maximum area MEOP with p i ∈ F blas the closest point of the bottom boundary of R, andp j ∈ F tr as the closest point of the top boundary of R.The points p i and p j are said to be the bottom-pivot andtop-pivot respectively, and the corresponding MEOP isdenoted by MEOP(p i , p j ). We will use S to denote thevertical slab bounded by V i and V j . The projections ofa point p k ∈ S on V i , V j , H i and H j are denoted by q k ,qk ′ , r k and rk ′ respectively. We now separately considertwo cases: (i) x(p i ) < x(p j ), and (ii) x(p i ) > x(p j ).In Case (i), the lines V i and V j split the point set Pinto three parts, P 1 , P 2 and P 3 , where P 1 and P 3 arethe set of points to the left of V i and to the right of V jrespectively, and the points in P 2 lie inside the verticalslab S. If V i hits the top and bottom boundaries ofR at t 1 and b 1 respectively, and V j hits the top andbottom boundaries of R at t 2 and b 2 respectively. Now,the portion of the MEOP inside the vertical slab S,denoted by M 2 (b 1 , t 2 ), is an empty staircase polygonwith diagonally opposite corners b 1 and t 2 among thepoints in P 2 . The two stairs of M 2 (b 1 , t 2 ) are parts ofthe rising stairs R br and R tl respectively. If R tl hitsV i at q α , then the portion of the MEOP to the left ofV i , denoted by M 1 (q α ), is an empty edge-visible polygonwith base [p i , q α ] among the points in P 1 such that everypoint inside the polygon is visible from its base [p i , q α ].Similarly, if R br hits V j at q β ′ then M 3(q β ′ ) is an emptyedge-visible polygon with base [p j , q β ′ ] among the pointsin P 3 .In Case (ii), V j is to the left of V i . Here M 2 is an emptystaircase polygon from p i to p j , and these are the partsof F bl and F tr of the MEOP respectively. If F bl (resp.F tr ) hits V j (resp. V i ) at q ′ α (resp. q β), then M 1 (q ′ α )(portion to the left of V j ) is an edge-visible polygon withbase [q ′ α , p j], and M 3 (q β ) (portion to the right of V i ) isan edge-visible polygon with base [q β , p i ]. After fixing p iand p j as the bottom-pivot and top-pivot respectively,we need to choose M 1 , M 2 and M 3 such that the sumof areas of these three polygons is maximum among allsuch polygons. We shall describe our algorithm for Case(i) only. Case (ii) can easily be handled using a similarmethod. For Case (i), we explain the method of computingthe desired M 1 and M 2 . The computation of M 3is the same as that of M 1 .3.1 Computation of M 1Let us consider a point p i ∈ P. Let P 1 = {p k |x(p k ) x(p i ) and y(p k ) > y(p i )}.Q includes the top-right corner of R, and |Q| = m +1. Let q 0 , q 1 , q 2 , . . . , q m denote the projections of thepoints in Q on the vertical line V i in decreasing orderof their y-coordinates. We create an array EV L(p i )whose elements are the maximum area empty edgevisiblepolygon M 1 (q k ) with [p i , q k ] as the base for allk = 0, 1, 2, . . ., m.We use vertical line sweep among the points in P 1 startingfrom the position of V i to create a height-balancedbinary tree T . Its each node v is represented as a 5-tuple(I, x val, y val, ∆, δ). I is the base of the edge-visiblepolygons attached to node v. (x val, y val) is the pointwhere the node v is generated, and ∆ contains the areaof the largest edge visible polygon rooted at that node.The ∆ parameters are computed in two passes. In theforward pass during the sweep, the ∆ parameter of anode contains the area of the edge visible polygon thatis computed so far at that node. At the end of thesweep, a backward pass is executed from the leaf levelof T up to its root, and ∆ value of each node is properlyset. The δ value of all nodes are 0 at the time of creationof T ; it will be set and used during the computation ofM 1 (q k ) for different q k .Creation of TThe root r of T corresponds to the entire intervalI = [p i , q 0 ]; its x val and ∆ parameters are set tox(p i ) and 0 respectively. A vertical line sweep is performedfrom x = x(p i ) towards left. When a pointp = (x(p), y(p)) ∈ P 1 is faced by the sweep line, theleaf nodes in T are searched. If y(p) lies in the inter-