A Framework for Algorithm Stability and Its Application to Kinetic Euclidean MSTs

Abstract

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Multicriteria Optimization for Dynamic Demers Cartograms

“…Yet they are used in such manner, e.g., as a sequence of contiguous cartograms showing the evolution of the Internet [29]. There has however been recent attention on stability in algorithms and visualization for both spatial [30], [31], [32] and nonspatial data [33], [34] that can be used as a basis.…”

Stability analysis of kinetic orientation-based shape descriptors

“…This requires a different analysis approach. Here, we use the framework introduced by Meulemans et al [22], restricting their definitions to our setting as described below.…”

“…The K-Lipschitz stability ratio ρ LS (Π, K) is defined almost exactly the same: the only difference is that we now take the infimum over all algorithms A for which A(P (t)) is K-Lipschitz, that is, the output moves continuously with speed bounded by K. We measure the output in radians: the orientation or angle can change with a speed of at most K radians per time unit. Lipschitz stability requires bounded input speed and scale invariance [22]. Here, we assume that points move with at most unit speed and that their diameter is at least 1 at all times.…”

“…Letscher et al [21] show that the medial axis for a union of disks changes continuously under certain conditions or if it is pruned appropriately . Meulemans et al [22] introduced a framework for analyzing stability and apply it to Euclidean minimum spanning trees on a set of moving points. They show bounded topological stability ratios for various topology definitions on the space of spanning trees, and that the Lipschitz stability ratio is at most linear, but also at least linear if the allowed speed for the changes in the tree is too low.…”

“…To derive meaningful bounds on the Lipschitz stability ratio, we assume the points move with at most unit speed. Furthermore, the relation between distances and speeds in input and output space should be scale invariant [22]. This is currently not the case: if we scale the coordinates of the points, then the distances in the input space change accordingly, but the distances in the output space (between orientations) do not.…”