Mohammad Ali Abam

Streaming Algorithms for Line Simplification

We study the following variant of the well-known line-simplification problem: we are getting a (possibly infinite) sequence of points p0,p1,p2,… in the plane defining a polygonal path, and as we receive the points, we wish to maintain a simplification of the path seen so far. We study this problem in a streaming setting, where we only have a limited amount of storage, so that we cannot store all the points. We analyze the competitive ratio of our algorithms, allowing resource augmentation: we let our algorithm maintain a simplification with 2k (internal) points and compare the error of our simplification to the error of the optimal simplification with k points. We obtain the algorithms with O(1) competitive ratio for three cases: convex paths, where the error is measured using the Hausdorff distance (or Fréchet distance), xy-monotone paths, where the error is measured using the Hausdorff distance (or Fréchet distance), and general paths, where the error is measured using the Fréchet distance. In the first case the algorithm needs O(k) additional storage, and in the latter two cases the algorithm needs O(k2) additional storage.

Streaming algorithms for line simplification

We study the following variant of the well-known line-simpli-ficationproblem: we are getting a possibly infinite sequence of points p0,p1,p2,... in the plane defining a polygonal path, and as wereceive the points we wish to maintain a simplification of the pathseen so far. We study this problem in a streaming setting, where weonly have a limited amount of storage so that we cannot store all thepoints. We analyze the competitive ratio of our algorithms, allowingresource augmentation: we let our algorithm maintain a simplificationwith 2k (internal) points, and compare the error of oursimplification to the error of the optimal simplification with k points. We obtain the algorithms with O(1) competitive ratio forthree cases: convex paths where the error is measured using theHausdorff distance (or Frechet distance), xy-monotone paths where the error is measured using theHausdorff distance (or Frechet distance), and general paths where the error is measured using theFrechet distance. In the first case the algorithm needs O(k) additionalstorage, and in the latter two cases the algorithm needs O(k2) additional storage.

A simple and efficient kinetic spanner

We present a new and simple (1+ε)-spanner of size <i>O</i>(<i>n</i>ε²) for a set of <i>n</i> points in the plane, which can be maintained efficiently as the points move. Assuming the trajectories of the points can be described by polynomials whose degrees are at most <i>s</i>, the number of topological changes to the spanner is <i>O</i>((<i>n</i>/ε²).λ_s+2(<i>n</i>)), and at each event the spanner can be updated in <i>O</i>(1) time.

Kinetic collision detection for convex fat objects

We design compact and responsive kinetic data structures for detecting collisions between n convex fat objects in 3-dimensional space that can have arbitrary sizes. Our main results are: (i) If the objects are 3-dimensional balls that roll on a plane, then we can detect collisions with a KDS of size O(nlogn) that can handle events in O (logn) time. This structure processes O(n2) events in the worst case, assuming that the objects follow constant-degree algebraic trajectories. (ii) If the objects are convex fat 3-dimensional objects of constant complexity that are free-flying in R3, then we can detect collisions with a KDS of O(nlog6n) size that can handle events in O(log6n) time. This structure processes O(n2) events in the worst case, assuming that the objects follow constant-degree algebraic trajectories. If the objects have similar sizes then the size of the KDS becomes O(n) and events can be handled in O(1) time.

Computing the Smallest Color-Spanning Axis-Parallel Square

For a given set of n colored points with k colors in the plane, we study the problem of computing the smallest color-spanning axis-parallel square. First, for a dynamic set of colored points on the real line, we propose a dynamic structure with O(log2 n) update time per insertion and deletion for maintaining the smallest color-spanning interval. Next, we use this result to compute the smallest color-spanning square. Although we show there could be Ω(kn) minimal color-spanning squares, our algorithm runs in O(nlog2 n) time and O(n) space.

Kinetic kd-Trees and Longest-Side kd-Trees

We propose a simple variant of kd-trees, called rank-based kd-trees, for sets of

A simple, faster method for kinetic proximity problems

n

On the Power of the Semi-Separated Pair Decomposition

points in

Kinetic spanners in R d

\mathbb{R}^d

Out-of-order event processing in kinetic data structures

. We show that a rank-based kd-tree, like an ordinary kd-tree, supports orthogonal range queries in