The Hamiltonian Structures of the super KP hierarchy Associated with an Even Parity SuperLax Operator

We consider the even parity superLax operator for the supersymmetric KP hierarchy of the form $L~=~D^2 + \sum_{i=0}^\infty u_{i-2} D^{-i+1}$ and obtain the two Hamiltonian structures following the standard method of Gelfand and Dikii. We observe that the first Hamiltonian structure is local and linear whereas the second Hamiltonian structure is non-local and nonlinear among the superfields appearing in the Lax operator. We discuss briefly on their connections with the super $w_{\infty}$ algebra.Comment: 14 pages, Plain tex, IC/93/17

Finding Axis-Parallel Rectangles of Fixed Perimeter or Area Containing the Largest Number of Points

Let P be a set of n points in the plane in general position, and consider the problem of finding an axis-parallel rectangle with a given perimeter, or area, or diagonal, that encloses the maximum number of points of P. We present an exact algorithm that finds such a rectangle in O(n^{5/2} log n) time, and, for the case of a fixed perimeter or diagonal, we also obtain (i) an improved exact algorithm that runs in O(nk^{3/2} log k) time, and (ii) an approximation algorithm that finds, in O(n+(n/(k epsilon^5))*log^{5/2}(n/k)log((1/epsilon) log(n/k))) time, a rectangle of the given perimeter or diagonal that contains at least (1-epsilon)k points of P, where k is the optimum value. We then show how to turn this algorithm into one that finds, for a given k, an axis-parallel rectangle of smallest perimeter (or area, or diagonal) that contains k points of P. We obtain the first subcubic algorithms for these problems, significantly improving the current state of the art

Querying for the Largest Empty Geometric Object in a Desired Location

We study new types of geometric query problems defined as follows: given a geometric set $P$, preprocess it such that given a query point $q$, the location of the largest circle that does not contain any member of $P$, but contains $q$ can be reported efficiently. The geometric sets we consider for $P$ are boundaries of convex and simple polygons, and point sets. While we primarily focus on circles as the desired shape, we also briefly discuss empty rectangles in the context of point sets.Comment: This version is a significant update of our earlier submission arXiv:1004.0558v1. Apart from new variants studied in Sections 3 and 4, the results have been improved in Section 5.Please note that the change in title and abstract indicate that we have expanded the scope of the problems we stud

Minimum Enclosing Circle with Few Extra Variables

Asano et al. [JoCG 2011] proposed an open problem of computing the minimum enclosing circle of a set of n points in R^2 given in a read-only array in sub-quadratic time. We show that Megiddo\u27s prune and search algorithm for computing the minimum radius circle enclosing the given points can be tailored to work in a read-only environment in O(n^{1+epsilon}) time using O(log n) extra space, where epsilon is a positive constant less than 1. As a warm-up, we first solve the same problem in an in-place setup in linear time with O(1) extra space