Speeding up Simplification of Polygonal Curves using Nested Approximations

We develop a multiresolution approach to the problem of polygonal curve approximation. We show theoretically and experimentally that, if the simplification algorithm A used between any two successive levels of resolution satisfies some conditions, the multiresolution algorithm MR will have a complexity lower than the complexity of A. In particular, we show that if A has a O(N2/K) complexity (the complexity of a reduced search dynamic solution approach), where N and K are respectively the initial and the final number of segments, the complexity of MR is in O(N).We experimentally compare the outcomes of MR with those of the optimal "full search" dynamic programming solution and of classical merge and split approaches. The experimental evaluations confirm the theoretical derivations and show that the proposed approach evaluated on 2D coastal maps either shows a lower complexity or provides polygonal approximations closer to the initial curves.Comment: 12 pages + figure

Progressive Simplification of Polygonal Curves

Simplifying polygonal curves at different levels of detail is an important problem with many applications. Existing geometric optimization algorithms are only capable of minimizing the complexity of a simplified curve for a single level of detail. We present an $O(n^3m)$-time algorithm that takes a polygonal curve of n vertices and produces a set of consistent simplifications for m scales while minimizing the cumulative simplification complexity. This algorithm is compatible with distance measures such as the Hausdorff, the Fr\'echet and area-based distances, and enables simplification for continuous scaling in $O(n^5)$ time. To speed up this algorithm in practice, we present new techniques for constructing and representing so-called shortcut graphs. Experimental evaluation of these techniques on trajectory data reveals a significant improvement of using shortcut graphs for progressive and non-progressive curve simplification, both in terms of running time and memory usage.Comment: 20 pages, 20 figure

Approximating Dynamic Time Warping and Edit Distance for a Pair of Point Sequences

We give the first subquadratic-time approximation schemes for dynamic time warping (DTW) and edit distance (ED) of several natural families of point sequences in $\mathbb{R}^d$, for any fixed $d \ge 1$. In particular, our algorithms compute $(1+\varepsilon)$-approximations of DTW and ED in time near-linear for point sequences drawn from k-packed or k-bounded curves, and subquadratic for backbone sequences. Roughly speaking, a curve is $\kappa$-packed if the length of its intersection with any ball of radius $r$ is at most $\kappa \cdot r$, and a curve is $\kappa$-bounded if the sub-curve between two curve points does not go too far from the two points compared to the distance between the two points. In backbone sequences, consecutive points are spaced at approximately equal distances apart, and no two points lie very close together. Recent results suggest that a subquadratic algorithm for DTW or ED is unlikely for an arbitrary pair of point sequences even for $d=1$. Our algorithms work by constructing a small set of rectangular regions that cover the entries of the dynamic programming table commonly used for these distance measures. The weights of entries inside each rectangle are roughly the same, so we are able to use efficient procedures to approximately compute the cheapest paths through these rectangles