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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Chan, Har-Peled, and Jones [SICOMP 2020] developed locality-sensitive orderings (LSO) for Euclidean space. A (τ,ρ)-LSO is a collection Σ of orderings such that for every x,y ∈ ℝ^d there is an ordering σ ∈ Σ, where all the points between x and y w.r.t. σ are in the ρ-neighborhood of either x or y. In essence, LSO allow one to reduce problems to the 1-dimensional line. Later, Filtser and Le [STOC 2022] developed LSO’s for doubling metrics, general metric spaces, and minor free graphs.
For Euclidean and doubling spaces, the number of orderings in the LSO is exponential in the dimension, which made them mainly useful for the low dimensional regime. In this paper, we develop new LSO’s for Euclidean, 𝓁_p, and doubling spaces that allow us to trade larger stretch for a much smaller number of orderings. We then use our new LSO’s (as well as the previous ones) to construct path reporting low hop spanners, fault tolerant spanners, reliable spanners, and light spanners for different metric spaces.
While many nearest neighbor search (NNS) data structures were constructed for metric spaces with implicit distance representations (where the distance between two metric points can be computed using their names, e.g. Euclidean space), for other spaces almost nothing is known. In this paper we initiate the study of the labeled NNS problem, where one is allowed to artificially assign labels (short names) to metric points. We use LSO’s to construct efficient labeled NNS data structures in this model.

Arnold Filtser. Labeled Nearest Neighbor Search and Metric Spanners via Locality Sensitive Orderings. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 33:1-33:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{filtser:LIPIcs.SoCG.2023.33, author = {Filtser, Arnold}, title = {{Labeled Nearest Neighbor Search and Metric Spanners via Locality Sensitive Orderings}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {33:1--33:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.33}, URN = {urn:nbn:de:0030-drops-178839}, doi = {10.4230/LIPIcs.SoCG.2023.33}, annote = {Keywords: Locality sensitive ordering, nearest neighbor search, high dimensional Euclidean space, doubling dimension, planar and minor free graphs, path reporting low hop spanner, fault tolerant spanner, reliable spanner, light spanner} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

We introduce and study the communication complexity of computing the inner product of two vectors, where the input is restricted w.r.t. a norm N on the space ℝⁿ. Here, Alice and Bob hold two vectors v,u such that ‖v‖_N ≤ 1 and ‖u‖_{N^*} ≤ 1, where N^* is the dual norm. The goal is to compute their inner product ⟨v,u⟩ up to an ε additive term. The problem is denoted by IP_N, and generalizes important previously studied problems, such as: (1) Computing the expectation 𝔼_{x∼𝒟}[f(x)] when Alice holds 𝒟 and Bob holds f is equivalent to IP_{𝓁₁}. (2) Computing v^TAv where Alice has a symmetric matrix with bounded operator norm (denoted S_∞) and Bob has a vector v where ‖v‖₂ = 1. This problem is complete for quantum communication complexity and is equivalent to IP_{S_∞}.
We systematically study IP_N, showing the following results, near tight in most cases:
1) For any symmetric norm N, given ‖v‖_N ≤ 1 and ‖u‖_{N^*} ≤ 1 there is a randomized protocol using 𝒪̃(ε^{-6} log n) bits of communication that returns a value in ⟨u,v⟩±ε with probability 2/3 - we will denote this by ℛ_{ε,1/3}(IP_N) ≤ 𝒪̃(ε^{-6} log n). In a special case where N = 𝓁_p and N^* = 𝓁_q for p^{-1} + q^{-1} = 1, we obtain an improved bound ℛ_{ε,1/3}(IP_{𝓁_p}) ≤ 𝒪(ε^{-2} log n), nearly matching the lower bound ℛ_{ε, 1/3}(IP_{𝓁_p}) ≥ Ω(min(n, ε^{-2})).
2) One way communication complexity ℛ^{→}_{ε,δ}(IP_{𝓁_p}) ≤ 𝒪(ε^{-max(2,p)}⋅ log n/ε), and a nearly matching lower bound ℛ^{→}_{ε, 1/3}(IP_{𝓁_p}) ≥ Ω(ε^{-max(2,p)}) for ε^{-max(2,p)} ≪ n.
3) One way communication complexity ℛ^{→}_{ε,δ}(N) for a symmetric norm N is governed by the distortion of the embedding 𝓁_∞^k into N. Specifically, while a small distortion embedding easily implies a lower bound Ω(k), we show that, conversely, non-existence of such an embedding implies protocol with communication k^𝒪(log log k) log² n.
4) For arbitrary origin symmetric convex polytope P, we show ℛ_{ε,1/3}(IP_{N}) ≤ 𝒪(ε^{-2} log xc(P)), where N is the unique norm for which P is a unit ball, and xc(P) is the extension complexity of P (i.e. the smallest number of inequalities describing some polytope P' s.t. P is projection of P').

Alexandr Andoni, Jarosław Błasiok, and Arnold Filtser. Communication Complexity of Inner Product in Symmetric Normed Spaces. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 4:1-4:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{andoni_et_al:LIPIcs.ITCS.2023.4, author = {Andoni, Alexandr and B{\l}asiok, Jaros{\l}aw and Filtser, Arnold}, title = {{Communication Complexity of Inner Product in Symmetric Normed Spaces}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {4:1--4:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.4}, URN = {urn:nbn:de:0030-drops-175077}, doi = {10.4230/LIPIcs.ITCS.2023.4}, annote = {Keywords: communication complexity, symmetric norms} }

Document

**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

In this paper we initiate the study of expander decompositions of a graph G = (V, E) in the streaming model of computation. The goal is to find a partitioning 𝒞 of vertices V such that the subgraphs of G induced by the clusters C ∈ 𝒞 are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model.
Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of V) to within a (δ, ε)-multiplicative/additive error with high probability. The power cut sparsifier uses Õ(n/εδ) space and edges, which we show is asymptotically tight up to polylogarithmic factors in n for constant δ.

Arnold Filtser, Michael Kapralov, and Mikhail Makarov. Expander Decomposition in Dynamic Streams. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 50:1-50:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{filtser_et_al:LIPIcs.ITCS.2023.50, author = {Filtser, Arnold and Kapralov, Michael and Makarov, Mikhail}, title = {{Expander Decomposition in Dynamic Streams}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {50:1--50:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.50}, URN = {urn:nbn:de:0030-drops-175534}, doi = {10.4230/LIPIcs.ITCS.2023.50}, annote = {Keywords: Streaming, expander decomposition, graph sparsifiers} }

Document

**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric t-spanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a t-spanner G_i for S_i for all i, while minimizing the number of edges, and their total weight.
Under the L₂-norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)-spanner algorithm with competitive ratio O_d(ε^{-d} log n), improving the previous bound of O_d(ε^{-(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1-d}log ε^{-1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1-d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{-3/2}logε^{-1}log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{-d}) lower bound for the competitive ratio for online (1+ε)-spanner algorithms in ℝ^d under the L₁-norm.
Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k-1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{-1}logε^{-1})⋅ n^{1+1/k} edges and O(ε^{-1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)-spanner for ultrametrics with O(ε^{-1}logε^{-1})⋅ n edges and O(ε^{-2}) lightness.

Sujoy Bhore, Arnold Filtser, Hadi Khodabandeh, and Csaba D. Tóth. Online Spanners in Metric Spaces. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bhore_et_al:LIPIcs.ESA.2022.18, author = {Bhore, Sujoy and Filtser, Arnold and Khodabandeh, Hadi and T\'{o}th, Csaba D.}, title = {{Online Spanners in Metric Spaces}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {18:1--18:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.18}, URN = {urn:nbn:de:0030-drops-169564}, doi = {10.4230/LIPIcs.ESA.2022.18}, annote = {Keywords: spanner, online algorithm, lightness, sparsity, minimum weight} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

A partition 𝒫 of a weighted graph G is (σ,τ,Δ)-sparse if every cluster has diameter at most Δ, and every ball of radius Δ/σ intersects at most τ clusters. Similarly, 𝒫 is (σ,τ,Δ)-scattering if instead for balls we require that every shortest path of length at most Δ/σ intersects at most τ clusters. Given a graph G that admits a (σ,τ,Δ)-sparse partition for all Δ > 0, Jia et al. [STOC05] constructed a solution for the Universal Steiner Tree problem (and also Universal TSP) with stretch O(τσ²log_τ n). Given a graph G that admits a (σ,τ,Δ)-scattering partition for all Δ > 0, we construct a solution for the Steiner Point Removal problem with stretch O(τ³σ³). We then construct sparse and scattering partitions for various different graph families, receiving many new results for the Universal Steiner Tree and Steiner Point Removal problems.

Arnold Filtser. Scattering and Sparse Partitions, and Their Applications. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 47:1-47:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{filtser:LIPIcs.ICALP.2020.47, author = {Filtser, Arnold}, title = {{Scattering and Sparse Partitions, and Their Applications}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {47:1--47:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.47}, URN = {urn:nbn:de:0030-drops-124547}, doi = {10.4230/LIPIcs.ICALP.2020.47}, annote = {Keywords: Scattering partitions, sparse partitions, sparse covers, Steiner point removal, Universal Steiner tree, Universal TSP} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

In the (1+ε,r)-approximate near-neighbor problem for curves (ANNC) under some similarity measure δ, the goal is to construct a data structure for a given set 𝒞 of curves that supports approximate near-neighbor queries: Given a query curve Q, if there exists a curve C ∈ 𝒞 such that δ(Q,C)≤ r, then return a curve C' ∈ 𝒞 with δ(Q,C') ≤ (1+ε)r. There exists an efficient reduction from the (1+ε)-approximate nearest-neighbor problem to ANNC, where in the former problem the answer to a query is a curve C ∈ 𝒞 with δ(Q,C) ≤ (1+ε)⋅δ(Q,C^*), where C^* is the curve of 𝒞 most similar to Q.
Given a set 𝒞 of n curves, each consisting of m points in d dimensions, we construct a data structure for ANNC that uses n⋅ O(1/ε)^{md} storage space and has O(md) query time (for a query curve of length m), where the similarity measure between two curves is their discrete Fréchet or dynamic time warping distance. Our method is simple to implement, deterministic, and results in an exponential improvement in both query time and storage space compared to all previous bounds.
Further, we also consider the asymmetric version of ANNC, where the length of the query curves is k ≪ m, and obtain essentially the same storage and query bounds as above, except that m is replaced by k. Finally, we apply our method to a version of approximate range counting for curves and achieve similar bounds.

Arnold Filtser, Omrit Filtser, and Matthew J. Katz. Approximate Nearest Neighbor for Curves - Simple, Efficient, and Deterministic. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 48:1-48:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{filtser_et_al:LIPIcs.ICALP.2020.48, author = {Filtser, Arnold and Filtser, Omrit and Katz, Matthew J.}, title = {{Approximate Nearest Neighbor for Curves - Simple, Efficient, and Deterministic}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {48:1--48:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.48}, URN = {urn:nbn:de:0030-drops-124555}, doi = {10.4230/LIPIcs.ICALP.2020.48}, annote = {Keywords: polygonal curves, Fr\'{e}chet distance, dynamic time warping, approximation algorithms, (asymmetric) approximate nearest neighbor, range counting} }

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APPROX

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Given a weighted graph G=(V,E,w), a partition of V is Delta-bounded if the diameter of each cluster is bounded by Delta. A distribution over Delta-bounded partitions is a beta-padded decomposition if every ball of radius gamma Delta is contained in a single cluster with probability at least e^{-beta * gamma}. The weak diameter of a cluster C is measured w.r.t. distances in G, while the strong diameter is measured w.r.t. distances in the induced graph G[C]. The decomposition is weak/strong according to the diameter guarantee.
Formerly, it was proven that K_r free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r^2) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric d_G has doubling dimension ddim, a weak O(ddim)-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known.
We construct strong O(r)-padded decompositions for K_r free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension ddim we construct a strong O(ddim)-padded decomposition, which is also tight. We use this decomposition to construct (O(ddim),O~(ddim))-sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles.

Arnold Filtser. On Strong Diameter Padded Decompositions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{filtser:LIPIcs.APPROX-RANDOM.2019.6, author = {Filtser, Arnold}, title = {{On Strong Diameter Padded Decompositions}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {6:1--6:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.6}, URN = {urn:nbn:de:0030-drops-112217}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.6}, annote = {Keywords: Padded decomposition, Strong Diameter, Sparse Cover, Doubling Dimension, Minor free graphs, Unique Games, Spanners, Distance Oracles} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, Chechik and Wulff-Nilsen [Shiri Chechik and Christian Wulff-Nilsen, 2018] improved the state-of-the-art for light spanners by constructing a (2k-1)(1+epsilon)-spanner with O(n^(1+1/k)) edges and O_epsilon(n^(1/k)) lightness. Soon after, Filtser and Solomon [Arnold Filtser and Shay Solomon, 2016] showed that the classic greedy spanner construction achieves the same bounds. The major drawback of the greedy spanner is its running time of O(mn^(1+1/k)) (which is faster than [Shiri Chechik and Christian Wulff-Nilsen, 2018]). This makes the construction impractical even for graphs of moderate size. Much faster spanner constructions do exist but they only achieve lightness Omega_epsilon(kn^(1/k)), even when randomization is used.
The contribution of this paper is deterministic spanner constructions that are fast, and achieve similar bounds as the state-of-the-art slower constructions. Our first result is an O_epsilon(n^(2+1/k+epsilon')) time spanner construction which achieves the state-of-the-art bounds. Our second result is an O_epsilon(m + n log n) time construction of a spanner with (2k-1)(1+epsilon) stretch, O(log k * n^(1+1/k) edges and O_epsilon(log k * n^(1/k)) lightness. This is an exponential improvement in the dependence on k compared to the previous result with such running time. Finally, for the important special case where k=log n, for every constant epsilon>0, we provide an O(m+n^(1+epsilon)) time construction that produces an O(log n)-spanner with O(n) edges and O(1) lightness which is asymptotically optimal. This is the first known sub-quadratic construction of such a spanner for any k = omega(1).
To achieve our constructions, we show a novel deterministic incremental approximate distance oracle. Our new oracle is crucial in our construction, as known randomized dynamic oracles require the assumption of a non-adaptive adversary. This is a strong assumption, which has seen recent attention in prolific venues. Our new oracle allows the order of the edge insertions to not be fixed in advance, which is critical as our spanner algorithm chooses which edges to insert based on the answers to distance queries. We believe our new oracle is of independent interest.

Stephen Alstrup, Søren Dahlgaard, Arnold Filtser, Morten Stöckel, and Christian Wulff-Nilsen. Constructing Light Spanners Deterministically in Near-Linear Time. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{alstrup_et_al:LIPIcs.ESA.2019.4, author = {Alstrup, Stephen and Dahlgaard, S{\o}ren and Filtser, Arnold and St\"{o}ckel, Morten and Wulff-Nilsen, Christian}, title = {{Constructing Light Spanners Deterministically in Near-Linear Time}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {4:1--4:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.4}, URN = {urn:nbn:de:0030-drops-111255}, doi = {10.4230/LIPIcs.ESA.2019.4}, annote = {Keywords: Spanners, Light Spanners, Efficient Construction, Deterministic Dynamic Distance Oracle} }

Document

**Published in:** OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

We reprove three known algorithmic bounds for terminal-clustering problems, using a single framework that leads to simpler proofs. In this genre of problems, the input is a metric space (X,d) (possibly arising from a graph) and a subset of terminals K subset X, and the goal is to partition the points X such that each part, called a cluster, contains exactly one terminal (possibly with connectivity requirements) so as to minimize some objective. The three bounds we reprove are for Steiner Point Removal on trees [Gupta, SODA 2001], for Metric 0-Extension in bounded doubling dimension [Lee and Naor, unpublished 2003], and for Connected Metric 0-Extension [Englert et al., SICOMP 2014].
A natural approach is to cluster each point with its closest terminal, which would partition X into so-called Voronoi cells, but this approach can fail miserably due to its stringent cluster boundaries. A now-standard fix, which we call the Relaxed-Voronoi framework, is to use enlarged Voronoi cells, but to obtain disjoint clusters, the cells are computed greedily according to some order. This method, first proposed by Calinescu, Karloff and Rabani [SICOMP 2004], was employed successfully to provide state-of-the-art results for terminal-clustering problems on general metrics. However, for restricted families of metrics, e.g., trees and doubling metrics, only more complicated, ad-hoc algorithms are known. Our main contribution is to demonstrate that the Relaxed-Voronoi algorithm is applicable to restricted metrics, and actually leads to relatively simple algorithms and analyses.

Arnold Filtser, Robert Krauthgamer, and Ohad Trabelsi. Relaxed Voronoi: A Simple Framework for Terminal-Clustering Problems. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{filtser_et_al:OASIcs.SOSA.2019.10, author = {Filtser, Arnold and Krauthgamer, Robert and Trabelsi, Ohad}, title = {{Relaxed Voronoi: A Simple Framework for Terminal-Clustering Problems}}, booktitle = {2nd Symposium on Simplicity in Algorithms (SOSA 2019)}, pages = {10:1--10:14}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-099-6}, ISSN = {2190-6807}, year = {2019}, volume = {69}, editor = {Fineman, Jeremy T. and Mitzenmacher, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.10}, URN = {urn:nbn:de:0030-drops-100369}, doi = {10.4230/OASIcs.SOSA.2019.10}, annote = {Keywords: Clustering, Steiner point removal, Zero extension, Doubling dimension, Relaxed voronoi} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har-Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any n-point Euclidean metric has an O(t)-spanner with O~(n^{1+1/t^2}) edges, little is known.
In this paper we study several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of l_p with 1<p <=2. Second, our construction yields a spanner which is both sparse and also light, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any n-point subset of l_p for 1<p <=2 has an O(t)-spanner with n^{1+O~(1/t^p)} edges and lightness n^{O~(1/t^p)}.
In fact, our results are more general, and they apply to any metric space admitting a certain low diameter stochastic decomposition. It is known that arbitrary metric spaces have an O(t)-spanner with lightness O(n^{1/t}). We exhibit the following tradeoff: metrics with decomposability parameter nu=nu(t) admit an O(t)-spanner with lightness O~(nu^{1/t}). For example, n-point Euclidean metrics have nu <=n^{1/t}, metrics with doubling constant lambda have nu <=lambda, and graphs of genus g have nu <=g. While these families do admit a (1+epsilon)-spanner, its lightness depend exponentially on the dimension (resp. log g). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch.

Arnold Filtser and Ofer Neiman. Light Spanners for High Dimensional Norms via Stochastic Decompositions. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{filtser_et_al:LIPIcs.ESA.2018.29, author = {Filtser, Arnold and Neiman, Ofer}, title = {{Light Spanners for High Dimensional Norms via Stochastic Decompositions}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {29:1--29:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.29}, URN = {urn:nbn:de:0030-drops-94922}, doi = {10.4230/LIPIcs.ESA.2018.29}, annote = {Keywords: Spanners, Stochastic Decompositions, High Dimensional Euclidean Space, Doubling Dimension, Genus Graphs} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

In this paper we study terminal embeddings, in which one is given a finite metric (X,d_X) (or a graph G=(V,E)) and a subset K of X of its points are designated as terminals. The objective is to embed the metric into a normed space, while approximately preserving all distances among pairs that contain a terminal. We devise such embeddings in various settings, and conclude that even though we have to preserve approx |K| * |X| pairs, the distortion depends only on |K|, rather than on |X|.
We also strengthen this notion, and consider embeddings that approximately preserve the distances between all pairs, but provide improved distortion for pairs containing a terminal. Surprisingly, we show that such embeddings exist in many settings, and have optimal distortion bounds both with respect to X \times X and with respect to K * X.
Moreover, our embeddings have implications to the areas of Approximation and Online Algorithms. In particular, Arora et. al. devised an ~O(sqrt(log(r))-approximation algorithm for sparsest-cut instances with r demands. Building on their framework, we provide an ~O(sqrt(log |K|)-approximation for sparsest-cut instances in which each demand is incident on one of the vertices of K (aka, terminals). Since |K| <= r, our bound generalizes that of Arora et al.

Michael Elkin, Arnold Filtser, and Ofer Neiman. Terminal Embeddings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 242-264, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{elkin_et_al:LIPIcs.APPROX-RANDOM.2015.242, author = {Elkin, Michael and Filtser, Arnold and Neiman, Ofer}, title = {{Terminal Embeddings}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {242--264}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.242}, URN = {urn:nbn:de:0030-drops-53064}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.242}, annote = {Keywords: embedding, distortion, terminals} }

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