A Framework for Algorithm Stability

We say that an algorithm is stable if small changes in the input result in small changes in the output. This kind of algorithm stability is particularly relevant when analyzing and visualizing time-varying data. Stability in general plays an important role in a wide variety of areas, such as numerical analysis, machine learning, and topology, but is poorly understood in the context of (combinatorial) algorithms. In this paper we present a framework for analyzing the stability of algorithms. We focus in particular on the tradeoff between the stability of an algorithm and the quality of the solution it computes. Our framework allows for three types of stability analysis with increasing degrees of complexity: event stability, topological stability, and Lipschitz stability. We demonstrate the use of our stability framework by applying it to kinetic Euclidean minimum spanning trees

Topological Stability of Kinetic kk-Centers

We study the $k$-center problem in a kinetic setting: given a set of continuously moving points $P$ in the plane, determine a set of $k$ (moving) disks that cover $P$ at every time step, such that the disks are as small as possible at any point in time. Whereas the optimal solution over time may exhibit discontinuous changes, many practical applications require the solution to be stable: the disks must move smoothly over time. Existing results on this problem require the disks to move with a bounded speed, but this model is very hard to work with. Hence, the results are limited and offer little theoretical insight. Instead, we study the topological stability of $k$-centers. Topological stability was recently introduced and simply requires the solution to change continuously, but may do so arbitrarily fast. We prove upper and lower bounds on the ratio between the radii of an optimal but unstable solution and the radii of a topologically stable solution---the topological stability ratio---considering various metrics and various optimization criteria. For $k = 2$ we provide tight bounds, and for small $k > 2$ we can obtain nontrivial lower and upper bounds. Finally, we provide an algorithm to compute the topological stability ratio in polynomial time for constant $k$

Stability analysis of kinetic orientation-based shape descriptors

We study three orientation-based shape descriptors on a set of continuously moving points: the first principal component, the smallest oriented bounding box and the thinnest strip. Each of these shape descriptors essentially defines a cost capturing the quality of the descriptor and uses the orientation that minimizes the cost. This optimal orientation may be very unstable as the points are moving, which is undesirable in many practical scenarios. If we bound the speed with which the orientation of the descriptor may change, this may lower the quality of the resulting shape descriptor. In this paper we study the trade-off between stability and quality of these shape descriptors. We first show that there is no stateless algorithm, an algorithm that keeps no state over time, that both approximates the minimum cost of a shape descriptor and achieves continuous motion for the shape descriptor. On the other hand, if we can use the previous state of the shape descriptor to compute the new state, we can define "chasing" algorithms that attempt to follow the optimal orientation with bounded speed. We show that, under mild conditions, chasing algorithms with sufficient bounded speed approximate the optimal cost at all times for oriented bounding boxes and strips. The analysis of such chasing algorithms is challenging and has received little attention in literature, hence we believe that our methods used in this analysis are of independent interest

Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time

A fundamental question in computational geometry is for a dynamic collection of geometric objects in Euclidean space, whether it is possible to maintain a maximum independent set in polylogarithmic update time. Already, for a set of intervals, it is known that no dynamic algorithm can maintain an exact maximum independent set with sublinear update time. Therefore, the typical objective is to explore the trade-off between update time and solution size. Substantial efforts have been made in recent years to understand this question for various families of geometric objects, such as intervals, hypercubes, hyperrectangles, and fat objects. We present the first fully dynamic approximation algorithm for disks of arbitrary radii in the plane that maintains a constant-factor approximate maximum independent set in polylogarithmic update time. First, we show that for a fully dynamic set of $n$ unit disks in the plane, a $12$-approximate maximum independent set can be maintained with worst-case update time $O(\log^2 n)$, and optimal output-sensitive reporting. Moreover, this result generalizes to fat objects of comparable sizes in any fixed dimension $d$, where the approximation ratio depends on the dimension and the fatness parameter. Our main result is that for a fully dynamic set of disks of arbitrary radii in the plane, an $O(1)$-approximate maximum independent set can be maintained in polylogarithmic expected amortized update time.Comment: Abstract is shortened to meet Arxiv's requirement on the number of character

Lower Bounds for Protrusion Replacement by Counting Equivalence Classes

Garnero et al. [SIAM J. Discrete Math. 2015, 29(4):1864-1894] recently introduced a framework based on dynamic programming to make applications of the protrusion replacement technique constructive and to obtain explicit upper bounds on the involved constants. They show that for several graph problems, for every boundary size t one can find an explicit set R_t of representatives. Any subgraph H with a boundary of size t can be replaced with a representative H\u27 in R_t such that the effect of this replacement on the optimum can be deduced from H and H\u27 alone. Their upper bounds on the size of the graphs in R_t grow triple-exponentially with t. In this paper we complement their results by lower bounds on the sizes of representatives, in terms of the boundary size t. For example, we show that each set of planar representatives R_t for the Independent Set problem contains a graph with Omega(2^t / sqrt{4t}) vertices. This lower bound even holds for sets that only represent the planar subgraphs of bounded pathwidth. To obtain our results we provide a lower bound on the number of equivalence classes of the canonical equivalence relation for Independent Set on t-boundaried graphs. We also find an elegant characterization of the number of equivalence classes in general graphs, in terms of the number of monotone functions of a certain kind. Our results show that the number of equivalence classes is at most 2^{2^t}, improving on earlier bounds of the form (t+1)^{2^t}

Access of energetic particles to Titan's exobase: a study of Cassini's T9 flyby

We study how the local electromagnetic disturbances introduced by Titan affect the ionization rates of the atmosphere. For this, we model the precipitation of energetic particles, specifically hydrogen and oxygen ions with energies between 1 keV and 1 MeV, into Titan's exobase for the specific magnetospheric configuration of the T9 flyby. For the study, a particle tracing software package is used which consists of an integration of the single particle Lorentz force equation using a 4th order Runge-Kutta numerical method. For the electromagnetic disturbances, the output of the A.I.K.E.F. hybrid code (kinetic ions, fluid electrons) is used, allowing the possibility of analyzing the disturbances and asymmetries in the access of energetic particles originated by their large gyroradii. By combining these methods, 2D maps showing the access of each set of particles were produced. We show that the access of different particles is largely dominated by their gyroradii, with the complexity of the maps increasing with decreasing gyroradius, due to the larger effect that local disturbances introduced by the presence of the moon have in the trajectory of the particles with lower energies. We also show that for particles with gyroradii much larger than the moon's radius, simpler descriptions of the electromagnetic environment can reproduce similar results to those obtained when using the full hybrid simulation description, with simple north-south fields being sufficient to reproduce the hybrid code results for O+ ions with energies larger than 10 keV but not enough to reproduce those for H+H+ ions at any of the energies covered in the present study. Finally, by combining the maps created with upstream plasma flow measurements by the MIMI/CHEMS instrument, we are able to estimate normalized fluxes arriving at different selected positions of the moon's exobase. We then use these fluxes to calculate energy deposition and non-dissociative N2 ionization rates for precipitating O+O+ and H+H+ ions and find differences in the ion production rates of up to almost 80% at the selected positions. All these results combined show that the electromagnetic field disturbances present in the vicinity of Titan significantly affect the contribution of energetic ions to local ionization profiles