Covering many points with a small-area box

Let $P$ be a set of $n$ points in the plane. We show how to find, for a given integer $k>0$, the smallest-area axis-parallel rectangle that covers $k$ points of $P$ in $O(nk^2 \log n+ n\log^2 n)$ time. We also consider the problem of, given a value $\alpha>0$, covering as many points of $P$ as possible with an axis-parallel rectangle of area at most $\alpha$. For this problem we give a probabilistic $(1-\varepsilon)$-approximation that works in near-linear time: In $O((n/\varepsilon^4)\log^3 n \log (1/\varepsilon))$ time we find an axis-parallel rectangle of area at most $\alpha$ that, with high probability, covers at least $(1-\varepsilon)\mathrm{\kappa^*}$ points, where $\mathrm{\kappa^*}$ is the maximum possible number of points that could be covered

Covering Points by Disjoint Boxes with Outliers

For a set of n points in the plane, we consider the axis--aligned (p,k)-Box Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together contain n-k points. In this paper, we consider the boxes to be either squares or rectangles, and we want to minimize the area of the largest box. For general p we show that the problem is NP-hard for both squares and rectangles. For a small, fixed number p, we give algorithms that find the solution in the following running times: For squares we have O(n+k log k) time for p=1, and O(n log n+k^p log^p k time for p = 2,3. For rectangles we get O(n + k^3) for p = 1 and O(n log n+k^{2+p} log^{p-1} k) time for p = 2,3. In all cases, our algorithms use O(n) space.Comment: updated version: - changed problem from 'cover exactly n-k points' to 'cover at least n-k points' to avoid having non-feasible solutions. Results are unchanged. - added Proof to Lemma 11, clarified some sections - corrected typos and small errors - updated affiliations of two author

Finding k points with a smallest enclosing square

Let $S$ be a set of $n$ points in $d$-space, let $R$ be an axes-parallel hyper-rectangle and let $1 \leq k \leq n$ be an integer. An algorithm is given that decides if $R$ can be translated such that it contains at least $k$ points of $S$. After a presorting step, this algorithm runs in $O(n)$ time, with a constant factor that is doubly-exponential in~$d$. Two applications are given. First, a translate of $R$ containing the maximal number of points can be computed in $O(n \log n)$ time. Second, a $k$-point subset of $S$ with minimal $L_{\infty}$-diameter can be computed, also in $O(n \log n)$ time. Using known dynamization techniques, the latter result gives improved dynamic data structures for maintaining such a $k$-point subset

Computing the smallest k-enclosing circle and related problems

AbstractWe present an efficient algorithm for solving the “smallest k-enclosing circle” (kSC) problem: Given a set of n points in the plane and an integer k ⩽ n, find the smallest disk containing k of the points. We present two solutions. When using O(nk) storage, the problem can be solved in time O(nk log2 n). When only O(n log n) storage is allowed, the running time is O(nk log2 n log n/k). We also extend our technique to obtain efficient solutions of several related problems (with similar time and storage bounds). These related problems include: finding the smallest homothetic copy of a given convex polygon P which contains k points from a given planar set, and finding the smallest disk intersecting k segments from a given planar set of non-intersecting segments