An Efficient Multi-Dimensional Searching Technique and itsApplications

This paper describes an improved algorithm for the multi-dimensional searching problem introduced by Megiddo. As a result, we obtain a $d^{O(d)} n$ time deterministic algorithms for linear programming in $\reals^d$ with $n$ constraints, for computing the Euclidean $1$-center of a set of $n$ points in $\Re^d$, for computing the minimum enclosing ellipsoid of a set of $n$ points in $\Re^d$, etc. Our techniques also improve the running time of known algorithms for a number of parametric graph searching problems, including that of finding zero cycles in dynamic graphs.