Journal of Computational Geometry 

About: Journal of Computational Geometry is an academic journal published by Carleton University. The journal publishes majorly in the area(s): Time complexity & Planar graph. It has an ISSN identifier of 1920-180X. Over the lifetime, 175 publications have been published receiving 1550 citations. The journal is also known as: JoCG.

Induced matchings and the algebraic stability of persistence barcodes

Abstract: $\DeclareMathOperator{\ker}{ker}\DeclareMathOperator{\coker}{coker}$We define a simple, explicit map sending a morphism $f\colon M\to N$ of pointwise finite dimensional persistence modules to a matching between the barcodes of $M$ and $N$. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of $\ker f$ and $\coker f$. As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes, a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a $\delta$-interleaving morphism between two persistence modules induces a $\delta$-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules, and yields a novel “single-morphism” characterization of the interleaving relation on persistence modules.

Stochastic convergence of persistence landscapes and silhouettes

Abstract: Persistent homology is a widely used tool in Topological Data Analysis that encodes multi-scale topological information as a multiset of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly to a random sample of diagrams. Instead, we summarize persistent homology with a persistence landscape, introduced by Bubenik, which converts a diagram into a well-behaved real-valued function. We investigate the statistical properties of landscapes, such as weak convergence of the average landscapes and convergence of the bootstrap. In addition, we introduce an alternate functional summary of persistent homology, which we call the silhouette, and derive an analogous statistical theory.

Approximability of the discrete Fréchet distance

Abstract: The Frechet distance is a popular and widespread distance measure for point sequences and for curves. About two years ago, Agarwal et al. [SIAM J. Comput. 2014] presented a new (mildly) subquadratic algorithm for the discrete version of the problem. This spawned a flurry of activity that has led to several new algorithms and lower bounds. In this paper, we study the approximability of the discrete Frechet distance. Building on a recent result by Bringmann [FOCS 2014], we present a new conditional lower bound showing that strongly subquadratic algorithms for the discrete Frechet distance are unlikely to exist, even in the one-dimensional case and even if the solution may be approximated up to a factor of 1.399. This raises the question of how well we can approximate the Frechet distance (of two given $d$-dimensional point sequences of length $n$) in strongly subquadratic time. Previously, no general results were known. We present the first such algorithm by analysing the approximation ratio of a simple, linear-time greedy algorithm to be $2^{\Theta(n)}$. Moreover, we design an $\alpha$-approximation algorithm that runs in time $O(n\log n + n^2/\alpha)$, for any $\alpha\in [1, n]$. Hence, an $n^\varepsilon$-approximation of the Frechet distance can be computed in strongly subquadratic time, for any $\varepsilon > 0$.

Computing multidimensional persistence

Abstract: The theory of multidimensional persistence captures the topology of a multifiltration - a multiparameter family of increasing spaces. Multifiltrations arise naturally in the topological analysis of scientific data. In this paper, we give a polynomial time algorithm for computing multidimensional persistence. We recast this computation as a problem within computational commutative algebra and utilize algorithms from this area to solve it. While the resulting problem is EXPSPACE-complete and the standard algorithms take doubly-exponential time, we exploit the structure inherent withing multifiltrations to yield practical algorithms. We implement all algorithms in the paper and provide statistical experiments to demonstrate their feasibility.

Constant-work-space algorithms for geometric problems

Abstract: Constant-work-space algorithms may use only constantly many cells of storage in addition to their input, which is provided as a read-only array. We show how to construct several geometric structures efficiently in the constant-work-space model. Traditional algorithms process the input into a suitable data structure (like a doubly-connected edge list) that allows efficient traversal of the structure at hand. In the constant-work-space setting, however, we cannot afford to do this. Instead, we provide operations that compute the desired features on the fly by accessing the input with no extra space. The whole geometric structure can be obtained by using these operations to enumerate all the features. Of course, we must pay for the space savings by slower running times. While the standard data structure allows us to implement traversal operations in constant time, our schemes typically take linear time to read the input data in each step. We begin with two simple problems: triangulating a planar point set and finding the trapezoidal decomposition of a simple polygon. In both cases adjacent features can be enumerated in linear time per step, resulting in total quadratic running time to output the whole structure. Actually, we show that the former result carries over to the Delaunay triangulation, and hence the Voronoi diagram. This also means that we can compute the largest empty circle of a planar point set in quadratic time and constant work-space. As another application, we demonstrate how to enumerate the features of an Euclidean minimum spanning tree (EMST) in quadratic time per step, so that the whole EMST can be found in cubic time using constant work-space. Finally, we describe how to compute a shortest geodesic path between two points in a simple polygon. Although the shortest path problem in general graphs is NL-complete (Jakoby and Tantau 2003), this constrained problem can be solved in quadratic time using only constant work-space.