arXiv subjects

Kristin Sheridan

Publications and source records attributed to Kristin Sheridan.

6 recordsLinked to original sources

arXiv2026

Bi-Lipschitz extensions and outlier embeddings into trees

We develop low distortion embeddings with outliers from arbitrary metrics into hierarchically separated trees (HSTs). In particular, we develop an efficient algorithm that for any $ε>0$, given an input metric $(X,d)$, and a probabilistic embedding of all but $k$ points from $X$ into HSTs with distortion $c$, samples from a probabilistic embedding of all but $O(\frac{k}ε\log k)$ points into HSTs that achieves distortion at most $(32+ε)c$. Our results are based on two key technical components. First, we extend an algorithm of Munagala et al. [2023] for minimizing the distortion of embeddings without outliers into HSTs to the setting with outliers. We combine this with new results on bi-Lipschitz extensions into trees and $\ell_1$ space. In particular, we show that any probabilistic embedding into HSTs can be extended to $k$ additional points with only a factor $O(\log k)$ of additional distortion. This bi-Lipschitz extension result utilizes a new probabilistic partitioning scheme that we call onion partitioning.

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arXiv2026

Brief announcement: A special case of maximum flow over time with network changes

We consider the problem of finding the value of a maximum flow over time in a network with uniform edge lengths where the edge capacities change at specific time instants. To solve this problem, we show how to construct a condensed version of a Time Expanded Network (cTEN) whose standard max flow value is the same as the max flow over time on the original network. In particular, for a graph with $n$ nodes, $m$ edges, and $μ$ {\em critical times} where some edge capacity changes, we obtain a cTEN with $O(n^2μ)$ nodes and $O(μmn)$ edges. This implies that the problem can be solved in $O(μ^2n^3m)$ time using the combinatorial max flow algorithm of Orlin [Orl13], or in $O(μ^{(1+o(1))}(nm)^{1+o(1)}\log (UT))$ time using the algorithm of Chen et al. [CKL+22], where $U$ is the maximum capacity of any edge and $T$ is the time horizon. We focus on graphs that experience many time changes across the period of interest, as in such graphs the $μ$ term dominates the runtime.

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arXiv2025

Faster feasibility for dynamic flows and transshipments on temporal networks

In this paper we study flow problems on temporal networks, where edge capacities and travel times change over time. We consider a network with $n$ nodes and $m$ edges where the capacity and length of each edge is a piecewise constant function, and use $μ=Ω(m)$ to denote the total number of pieces in all of the $2m$ functions. Our goal is to design exact algorithms for various flow problems that run in time polynomial in the parameter $μ$. Importantly, the algorithms we design are strongly polynomial, i.e. have no dependence on the capacities, flow value, or the time horizon of the flow process, all of which can be exponentially large relative to the other parameters; and return an integral flow when all input parameters are integral. Our main result is an algorithm for checking feasibility of a dynamic transshipment problem on temporal networks -- given multiple sources and sinks with supply and demand values, is it possible to satisfy the desired supplies and demands within a given time horizon? We develop a fast ($O(μ^3)$ time) algorithm for this feasibility problem when the input network has a certain canonical form, by exploiting the cut structure of the associated time expanded network. We then adapt an approach of \cite{hoppe2000} to show how other flow problems on temporal networks can be reduced to the canonical format. For computing dynamic transshipments on temporal networks, this results in a $O(μ^7)$ time algorithm, whereas the previous best integral exact algorithm runs in time $\tilde O(μ^{19})$. We achieve similar improvements for other flow problems on temporal networks.

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arXiv2023

Composition of nested embeddings with an application to outlier removal

We study the design of embeddings into Euclidean space with outliers. Given a metric space $(X,d)$ and an integer $k$, the goal is to embed all but $k$ points in $X$ (called the ``outliers") into $\ell_2$ with the smallest possible distortion $c$. Finding the optimal distortion $c$ for a given outlier set size $k$, or alternately the smallest $k$ for a given target distortion $c$ are both NP-hard problems. In fact, it is UGC-hard to approximate $k$ to within a factor smaller than $2$ even when the metric sans outliers is isometrically embeddable into $\ell_2$. We consider bi-criteria approximations. Our main result is a polynomial time algorithm that approximates the outlier set size to within an $O(\log^2 k)$ factor and the distortion to within a constant factor. The main technical component in our result is an approach for constructing Lipschitz extensions of embeddings into Banach spaces (such as $\ell_p$ spaces). We consider a stronger version of Lipschitz extension that we call a \textit{nested composition of embeddings}: given a low distortion embedding of a subset $S$ of the metric space $X$, our goal is to extend this embedding to all of $X$ such that the distortion over $S$ is preserved, whereas the distortion over the remaining pairs of points in $X$ is bounded by a function of the size of $X\setminus S$. Prior work on Lipschitz extension considers settings where the size of $X$ is potentially much larger than that of $S$ and the expansion bounds depend on $|S|$. In our setting, the set $S$ is nearly all of $X$ and the remaining set $X\setminus S$, a.k.a. the outliers, is small. We achieve an expansion bound that is logarithmic in $|X\setminus S|$.

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arXiv2021

Isometric Hamming embeddings of weighted graphs

A mapping $α: V(G) \to V(H)$ from the vertex set of one graph $G$ to another graph $H$ is an isometric embedding if the shortest path distance between any two vertices in $G$ equals the distance between their images in $H$. Here, we consider isometric embeddings of a weighted graph $G$ into unweighted Hamming graphs, called Hamming embeddings, when $G$ satisfies the property that every edge is a shortest path between its endpoints. Using a Cartesian product decomposition of $G$ called its pseudofactorization, we show that every Hamming embedding of $G$ may be partitioned into Hamming embeddings for each irreducible pseudofactor graph of $G$, which we call its canonical partition. This implies that $G$ permits a Hamming embedding if and only if each of its irreducible pseudofactors is Hamming embeddable. This result extends prior work on unweighted graphs that showed that an unweighted graph permits a Hamming embedding if and only if each irreducible pseudofactor is a complete graph. When a graph $G$ has nontrivial pseudofactors, determining whether $G$ has a Hamming embedding can be simplified to checking embeddability of two or more smaller graphs.

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arXiv2021

Factorization and pseudofactorization of weighted graphs

For unweighted graphs, finding isometric embeddings is closely related to decompositions of $G$ into Cartesian products of smaller graphs. When $G$ is isomorphic to a Cartesian graph product, we call the factors of this product a factorization of $G$. When $G$ is isomorphic to an isometric subgraph of a Cartesian graph product, we call those factors a pseudofactorization of $G$. Prior work has shown that an unweighted graph's pseudofactorization can be used to generate a canonical isometric embedding into a product of the smallest possible pseudofactors. However, for arbitrary weighted graphs, which represent a richer variety of metric spaces, methods for finding isometric embeddings or determining their existence remain elusive, and indeed pseudofactorization and factorization have not previously been extended to this context. In this work, we address the problem of finding the factorization and pseudofactorization of a weighted graph $G$, where $G$ satisfies the property that every edge constitutes a shortest path between its endpoints. We term such graphs minimal graphs, noting that every graph can be made minimal by removing edges not affecting its path metric. We generalize pseudofactorization and factorization to minimal graphs and develop new proof techniques that extend the previously proposed algorithms due to Graham and Winkler [Graham and Winkler, '85] and Feder [Feder, '92] for pseudofactorization and factorization of unweighted graphs. We show that any $m$-edge, $n$-vertex graph with positive integer edge weights can be factored in $O(m^2)$ time, plus the time to find all pairs shortest paths (APSP) distances in a weighted graph, resulting in an overall running time of $O(m^2+n^2\log\log n)$ time. We also show that a pseudofactorization for such a graph can be computed in $O(mn)$ time, plus the time to solve APSP, resulting in an $O(mn+n^2\log\log n)$ running time.

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