On the Structure of Unique Shortest Paths in Graphs

Bodwin, Greg

doi:10.1137/1.9781611975482.125

Cited by 12 publications

(11 citation statements)

References 21 publications

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“…We start by presenting our reduction from linear queries to APSD. We follow the terminology of [Bod19] for the notion of strongly metrizable path systems.…”

Section: Reducing Linear Queries To Shortest Pathmentioning

confidence: 99%

Differentially Private All-Pairs Shortest Path Distances: Improved Algorithms and Lower Bounds

Ghazi¹,

Kumar²,

Manurangsi³

et al. 2022

Preprint

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We study the problem of releasing the weights of all-pair shortest paths in a weighted undirected graph with differential privacy (DP). In this setting, the underlying graph is fixed and two graphs are neighbors if their edge weights differ by at most 1 in the ℓ 1 -distance. We give an ǫ-DP algorithm with additive error Õ(n 2/3 /ǫ) and an (ǫ, δ)-DP algorithm with additive error Õ( √ n/ǫ) where n denotes the number of vertices. This positively answers a question of Sealfon [Sea16, Sea20], who asked whether a o(n)-error algorithm exists. We also show that an additive error of Ω(n 1/6 ) is necessary for any sufficiently small ǫ, δ > 0.Finally, we consider a relaxed setting where a multiplicative approximation is allowed. We show that, with a multiplicative approximation factor k, the additive error can be reduced to Õ n 1/2+O(1/k) /ǫ in the ǫ-DP case and Õ(n 1/3+O(1/k) /ǫ) in the (ǫ, δ)-DP case, respectively.

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“…We start by presenting our reduction from linear queries to APSD. We follow the terminology of [Bod19] for the notion of strongly metrizable path systems.…”

Section: Reducing Linear Queries To Shortest Pathmentioning

confidence: 99%

Differentially Private All-Pairs Shortest Path Distances: Improved Algorithms and Lower Bounds

Ghazi¹,

Kumar²,

Manurangsi³

et al. 2022

Preprint

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“…Similar to our setting, a system of paths is said to be consistent when the intersection of any two paths in the system is also a connected path in the system. The characterization of such system has been studied, see [3,10] for more details.…”

Section: Previous Workmentioning

confidence: 99%

Distance Bounds for High Dimensional Consistent Digital Rays and 2-D Partially-Consistent Digital Rays

Chiu

Korman²,

Suderland

et al. 2022

Discrete Comput Geom

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We consider the problem of digitalizing Euclidean segments. Specifically, we look for a constructive method to connect any two points in $$\mathbb {Z}^d$$ Z d . The construction must be consistent (that is, satisfy the natural extension of the Euclidean axioms) while resembling them as much as possible. Previous work has shown asymptotically tight results in two dimensions with $$\varTheta (\log N)$$ Θ ( log N ) error, where resemblance between segments is measured with the Hausdorff distance, and N is the $$L_1$$ L 1 distance between the two points. This construction was considered tight because of a $$\varOmega (\log N)$$ Ω ( log N ) lower bound that applies to any consistent construction in $$\mathbb {Z}^2$$ Z 2 . In this paper we observe that the lower bound does not directly extend to higher dimensions. We give an alternative argument showing that any consistent construction in d dimensions must have $$\varOmega (\log ^{1/(d-1)}\!N)$$ Ω ( log 1 / ( d - 1 ) N ) error. We tie the error of a consistent construction in high dimensions to the error of similar weak constructions in two dimensions (constructions for which some points need not satisfy all the axioms). This not only opens the possibility for having constructions with $$o(\log N)$$ o ( log N ) error in high dimensions, but also opens up an interesting line of research in the tradeoff between the number of axiom violations and the error of the construction. A side result, that we find of independent interest, is the introduction of the bichromatic discrepancy: a natural extension of the concept of discrepancy of a set of points. In this paper, we define this concept and extend known results to the chromatic setting.

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“…A graph for which there exists a path between every pair of nodes is considered a connected graph (Kolaczyk, 2009), which can be seen clearly in Figure 4C. Often there are many possible paths between two nodes, in which case the shortest possible path, which is the minimum number of edges needing to be traversed to connect two nodes, is often an interesting metric to consider (Bodwin, 2019). This concept is highlighted in Figures 5A,B which both illustrate paths between the same two nodes within the graph, where the first is a random path and the second is the shortest possible path.…”

Section: Paths In Graphsmentioning

confidence: 99%

Neuronal Graphs: A Graph Theory Primer for Microscopic, Functional Networks of Neurons Recorded by Calcium Imaging

Nelson

Bonner

2021

Front. Neural Circuits

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Connected networks are a fundamental structure of neurobiology. Understanding these networks will help us elucidate the neural mechanisms of computation. Mathematically speaking these networks are “graphs”—structures containing objects that are connected. In neuroscience, the objects could be regions of the brain, e.g., fMRI data, or be individual neurons, e.g., calcium imaging with fluorescence microscopy. The formal study of graphs, graph theory, can provide neuroscientists with a large bank of algorithms for exploring networks. Graph theory has already been applied in a variety of ways to fMRI data but, more recently, has begun to be applied at the scales of neurons, e.g., from functional calcium imaging. In this primer we explain the basics of graph theory and relate them to features of microscopic functional networks of neurons from calcium imaging—neuronal graphs. We explore recent examples of graph theory applied to calcium imaging and we highlight some areas where researchers new to the field could go awry.

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On the Structure of Unique Shortest Paths in Graphs

Cited by 12 publications

References 21 publications

Differentially Private All-Pairs Shortest Path Distances: Improved Algorithms and Lower Bounds

Differentially Private All-Pairs Shortest Path Distances: Improved Algorithms and Lower Bounds

Distance Bounds for High Dimensional Consistent Digital Rays and 2-D Partially-Consistent Digital Rays

Neuronal Graphs: A Graph Theory Primer for Microscopic, Functional Networks of Neurons Recorded by Calcium Imaging

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