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1. Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time

   https://doi.org/10.4230/LIPIcs.SoCG.2024.19  

   Bhore, Sujoy; Nöllenburg, Martin; Tóth, Csaba D; Wulms, Jules (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   A fundamental question is whether one can maintain a maximum independent set (MIS) in polylogarithmic update time for a dynamic collection of geometric objects in Euclidean space. For a set of intervals, it is known that no dynamic algorithm can maintain an exact MIS in 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 MIS in polylogarithmic expected amortized update time. Moreover, for a fully dynamic set of n unit disks in the plane, we show that a 12-approximate MIS can be maintained with worst-case update time O(log n), and optimal output-sensitive reporting. 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. Further, we note that, even for a dynamic set of disks of unit radius in the plane, it is impossible to maintain O(1+ε)-approximate MIS in truly sublinear update time, under standard complexity assumptions. Our results build on two recent technical tools: (i) The MIX algorithm by Cardinal et al. (ESA 2021) that can smoothly transition from one independent set to another; hence it suffices to maintain a family of independent sets where the largest one is an O(1)-approximate MIS. (ii) A dynamic nearest/farthest neighbor data structure for disks by Kaplan et al. (DCG 2020) and Liu (SICOMP 2022), which generalizes the dynamic convex hull data structure by Chan (JACM 2010), and quickly yields a "replacement" disk (if any) when a disk in one of our independent sets is deleted. 

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2. Fast Dynamic Cuts, Distances and Effective Resistances via Vertex Sparsifiers

   https://doi.org/10.1109/FOCS46700.2020.00109  

   Chen, Li; Goranci, Gramoz; Henzinger, Monika; Peng, Richard; Saranurak, Thatchaphol (November 2020, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020)

   null (Ed.)

   We present a general framework of designing efficient dynamic approximate algorithms for optimization on undirected graphs. In particular, we develop a technique that, given any problem that admits a certain notion of vertex sparsifiers, gives data structures that maintain approximate solutions in sub-linear update and query time. We illustrate the applicability of our paradigm to the following problems. (1) A fully-dynamic algorithm that approximates all-pair maximum-flows/minimum-cuts up to a nearly logarithmic factor in $$\tilde{O}(n^{2/3})$$ amortized time against an oblivious adversary, and $$\tilde{O}(m^{3/4})$$ time against an adaptive adversary. (2) An incremental data structure that maintains $O(1)$-approximate shortest path in $$n^{o(1)}$$ time per operation, as well as fully dynamic approximate all-pair shortest path and transshipment in $$\tilde{O}(n^{2/3+o(1)})$$ amortized time per operation. (3) A fully-dynamic algorithm that approximates all-pair effective resistance up to an $$(1+\eps)$$ factor in $$\tilde{O}(n^{2/3+o(1)} \epsilon^{-O(1)})$$ amortized update time per operation. The key tool behind result (1) is the dynamic maintenance of an algorithmic construction due to Madry [FOCS' 10], which partitions a graph into a collection of simpler graph structures (known as j-trees) and approximately captures the cut-flow and metric structure of the graph. The $O(1)$-approximation guarantee of (2) is by adapting the distance oracles by [Thorup-Zwick JACM `05]. Result (3) is obtained by invoking the random-walk based spectral vertex sparsifier by [Durfee et al. STOC `19] in a hierarchical manner, while carefully keeping track of the recourse among levels in the hierarchy. 

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3. A New Deterministic Algorithm for Fully Dynamic All-Pairs Shortest Paths

   https://doi.org/10.1145/3564246.3585196  

   Chuzhoy, Julia; Zhang, Ruimin (June 2023, Proceedings of the 55th Annual {ACM} Symposium on Theory of Computing, {STOC} June 2023)

   We study the fully dynamic All-Pairs Shortest Paths (APSP) problem in undirected edge-weighted graphs. Given an n-vertex graph G with non-negative edge lengths, that undergoes an online sequence of edge insertions and deletions, the goal is to support approximate distance queries and shortest-path queries. We provide a deterministic algorithm for this problem, that, for a given precision parameter є, achieves approximation factor (loglogn)2O(1/є3), and has amortized update time O(nєlogL) per operation, where L is the ratio of longest to shortest edge length. Query time for distance-query is O(2O(1/є)· logn· loglogL), and query time for shortest-path query is O(|E(P)|+2O(1/є)· logn· loglogL), where P is the path that the algorithm returns. To the best of our knowledge, even allowing any o(n)-approximation factor, no adaptive-update algorithms with better than Θ(m) amortized update time and better than Θ(n) query time were known prior to this work. We also note that our guarantees are stronger than the best current guarantees for APSP in decremental graphs in the adaptive-adversary setting. In order to obtain these results, we consider an intermediate problem, called Recursive Dynamic Neighborhood Cover (RecDynNC), that was formally introduced in [Chuzhoy, STOC ’21]. At a high level, given an undirected edge-weighted graph G undergoing an online sequence of edge deletions, together with a distance parameter D, the goal is to maintain a sparse D-neighborhood cover of G, with some additional technical requirements. Our main technical contribution is twofolds. First, we provide a black-box reduction from APSP in fully dynamic graphs to the RecDynNC problem. Second, we provide a new deterministic algorithm for the RecDynNC problem, that, for a given precision parameter є, achieves approximation factor (loglogm)2O(1/є2), with total update time O(m1+є), where m is the total number of edges ever present in G. This improves the previous algorithm of [Chuzhoy, STOC ’21], that achieved approximation factor (logm)2O(1/є) with similar total update time. Combining these two results immediately leads to the deterministic algorithm for fully-dynamic APSP with the guarantees stated above. 

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4. On Efficient Range-Summability of IID Random Variables in Two or Higher Dimensions

   Jingfan Meng, Huayi Wang (January 2023, 26th International Conference on Database Theory (ICDT 2023))

   d-dimensional (for d > 1) efficient range-summability (dD-ERS) of random variables (RVs) is a fundamental algorithmic problem that has applications to two important families of database problems, namely, fast approximate wavelet tracking (FAWT) on data streams and approximately answering range-sum queries over a data cube. Whether there are efficient solutions to the dD-ERS problem, or to the latter database problem, have been two long-standing open problems. Both are solved in this work. Specifically, we propose a novel solution framework to dD-ERS on RVs that have Gaussian or Poisson distribution. Our dD-ERS solutions are the first ones that have polylogarithmic time complexities. Furthermore, we develop a novel k-wise independence theory that allows our dD-ERS solutions to have both high computational efficiencies and strong provable independence guarantees. Finally, we show that under a sufficient and likely necessary condition, certain existing solutions for 1D-ERS can be generalized to higher dimensions. 

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5. Graph Sketching against Adaptive Adversaries Applied to the Minimum Degree Algorithm

   https://doi.org/􏰇􏰈􏰫􏰇􏰇􏰈􏰑􏰩􏰠􏰮􏰣􏰙􏰫􏰌􏰈􏰇􏰍􏰫􏰈􏰈􏰈􏰇􏰑􏰇􏰈􏰫􏰇􏰇􏰈􏰑􏰩􏰠􏰮􏰣􏰙􏰫􏰌􏰈􏰇􏰍􏰫􏰈􏰈􏰈􏰇􏰑􏰇􏰈􏰫􏰇􏰇􏰈􏰑􏰩􏰠􏰮􏰣􏰙􏰫􏰌􏰈􏰇􏰍􏰫􏰈􏰈􏰈􏰇􏰑􏰇􏰈􏰫􏰇􏰇􏰈􏰑􏰩􏰠􏰮􏰣􏰙􏰫􏰌􏰈􏰇􏰍􏰫􏰈􏰈􏰈􏰇􏰑10.1109/FOCS.2018.00019  

   Fahrbach, Matthew; Miller, Gary L.; Peng, Richard; Sawlani, Saurabh; Wang, Junxing; Xu, Shen Chen (October 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science)

   Abstract—Motivated by the study of matrix elimination orderings in combinatorial scientific computing, we utilize graph sketching and local sampling to give a data structure that provides access to approximate fill degrees of a matrix undergoing elimination in polylogarithmic time per elimination and query. We then study the problem of using this data structure in the minimum degree algorithm, which is a widely- used heuristic for producing elimination orderings for sparse matrices by repeatedly eliminating the vertex with (approx- imate) minimum fill degree. This leads to a nearly-linear time algorithm for generating approximate greedy minimum degree orderings. Despite extensive studies of algorithms for elimination orderings in combinatorial scientific computing, our result is the first rigorous incorporation of randomized tools in this setting, as well as the first nearly-linear time algorithm for producing elimination orderings with provable approximation guarantees. While our sketching data structure readily works in the oblivious adversary model, by repeatedly querying and greed- ily updating itself, it enters the adaptive adversarial model where the underlying sketches become prone to failure due to dependency issues with their internal randomness. We show how to use an additional sampling procedure to circumvent this problem and to create an independent access sequence. Our technique for decorrelating interleaved queries and updates to this randomized data structure may be of independent interest. 

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