Smallest k point enclosing rectangle of arbitrary orientation

Given a set of points in 2D, the problem of identifying the smallest rectangle of arbitrary orientation, and containing exactly points is studied in this paper. The worst case time and space complexities of the proposed algorithm are ! " # $ % and & respectively. The algorithm is then used to identify the smallest square of arbitrary orientation containing points in # ' ( ) * + ' time. , .0/21%354&6 7'/2893 . Given a set : of ; points in 2D, and an integer <>=@?A;CB , we consider the problem of identifying the smallest rectangle on the plane which encloses exactly < points of : . A restricted variation of this problem has already been investigated, where the desired rectangle is axis-parallel. The first result on this problem appeared in [1] with time and space complexities D(=E< F#;!GIH JK;CB and D(=L<0;CB respectively. Both time and space complexity results are finally improved to =L< F2;NMO;!GPHKJ ;CB and D(=Q;CB respectively in [2]. In [6], it is mentioned that all the aforesaid algorithms are efficient when < is small. It also proposes an efficient algorithm when < is large (very close to ; ). The time and space complexities of this algorithm are D(=Q;)MOST< BUFVB and D(=Q;CB respectively. In W ( XZY B dimensions, the algorithm proposed in [6] runs in D(=QW ;[MTW0<!=R;)S\< BUF^]P_#`!a@bcB time using D(=QW ;CB space. In all these variations, the points are assumed to be in general position, i.e., no two points lie on the same horizontal or vertical line, and the desired rectangle is isothetic and closed (i.e., enclosed points may lie on the boundary of the rectangle). A similar problem is studied in [5], where ; points are distributed on the plane, and the proposed algorithm identifies the smallest circle containing exactly < points in D(=Q;!GIH J ;dMe=Q;fSg< Bih^;Cj@B time for some klXnm . The motivation of studying all these problems come from pattern recognition and facility location, where essential features are represented as a point set, and the objective is to identify a precise cluster (region) containing desired number of features. We consider the generalized version of this problem, where the desired rectangle may be of arbitrary orientation. We assume that the lines joining pairs of points have distinct slopes. Our proposed algorithm runs in D(=Q;oF#GIH J ;pM+<q;r=R;sS < B^=R;sSt<uM[GIH J$< BiB time and D(=R;CB space. The proposed technique can also identify the smallest < -enclosing square of arbitrary orientation in D(=Q;rF^GPHKJ ;tMl<q;r=R;[Sv< BcF^GIH JK;CB time w Indian Statistical Institute, Kolkata 700 108, India x Calcutta University, Kolkata 700 029, India y Indian Statistical Institute, Kolkata 700 108, India and D(=R;CFVB space. The time complexity result of identifying smallest < -enclosing square is comparable with that of identifying smallest < -enclosing circle proposed in [5]. z {0301@|}6 ~9$/2893 .