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1. Two-Sided Kirszbraun Theorem

   https://doi.org/10.4230/LIPIcs.SoCG.2021.13  

   Backurs, Arturs ; Mahabadi, Sepideh ; Makarychev, Konstantin ; Makarychev, Yury ( June 2021 , Leibniz international proceedings in informatics)

   Buchin, Kevin ; Colin de Verdiere, Eric (Ed.)

   In this paper, we prove a two-sided variant of the Kirszbraun theorem. Consider an arbitrary subset X of Euclidean space and its superset Y. Let f be a 1-Lipschitz map from X to ℝ^m. The Kirszbraun theorem states that the map f can be extended to a 1-Lipschitz map ̃ f from Y to ℝ^m. While the extension ̃ f does not increase distances between points, there is no guarantee that it does not decrease distances significantly. In fact, ̃ f may even map distinct points to the same point (that is, it can infinitely decrease some distances). However, we prove that there exists a (1 + ε)-Lipschitz outer extension f̃:Y → ℝ^{m'} that does not decrease distances more than "necessary". Namely, ‖f̃(x) - f̃(y)‖ ≥ c √{ε} min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) for some absolutely constant c > 0. This bound is asymptotically optimal, since no L-Lipschitz extension g can have ‖g(x) - g(y)‖ > L min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) even for a single pair of points x and y. In some applications, one is interested in the distances ‖f̃(x) - f̃(y)‖ between images of points x,y ∈ Y rather than in the map f̃ itself. The standard Kirszbraun theorem does not provide any method of computing these distances without computing the entire map ̃ f first. In contrast, our theorem provides a simple approximate formula for distances ‖f̃(x) - f̃(y)‖. 

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2. Computing β-Stretch Paths in Drawings of Graphs

   Arkin, E ; Darabi, F ; Efrat, A ; Frank, F ; Fulek, R ; Kobourov, S ; Mitchell, J ( October 2020 , 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT))

   Let f be a drawing in the Euclidean plane of a graph G, which is understood to be a 1-dimensional simplicial complex. We assume that every edge of G is drawn by f as a curve of constant algebraic complexity, and the ratio of the length of the longest simple path to the the length of the shortest edge is poly(n). In the drawing f, a path P of G, or its image in the drawing π = f(P), is β-stretch if π is a simple (non-self-intersecting) curve, and for every pair of distinct points p ∈ P and q ∈ P , the length of the sub-curve of π connecting f(p) with f(q) is at most β∥f(p) − f(q)∥, where ∥.∥ denotes the Euclidean distance. We introduce and study the β-stretch Path Problem (βSP for short), in which we are given a pair of vertices s and t of G, and we are to decide whether in the given drawing of G there exists a β-stretch path P connecting s and t. We also output P if it exists. The βSP quantifies a notion of “near straightness” for paths in a graph G, motivated by gerrymandering regions in a map, where edges of G represent natural geographical/political boundaries that may be chosen to bound election districts. The notion of a β-stretch path naturally extends to cycles, and the extension gives a measure of how gerrymandered a district is. Furthermore, we show that the extension is closely related to several studied measures of local fatness of geometric shapes. We prove that βSP is strongly NP-complete. We complement this result by giving a quasi-polynomial time algorithm, that for a given ε > 0, β ∈ O(poly(log |V (G)|)), and s, t ∈ V (G), outputs a β-stretch path between s and t, if a (1 − ε)β-stretch path between s and t exists in the drawing. 

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3. Communication Complexity of Inner Product in Symmetric Normed Spaces

   Andoni, Alexandr ; Blasiok, Jaroslaw ; Filtser, Arnold ( January 2023 , Leibniz international proceedings in informatics)

   Tauman Kalai, Yael (Ed.)

   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'). 

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4. Approximate Nearest Neighbors in Limited Space

   https://doi.org/10.1137/1.9781611975031  

   Indyk, Piotr ; Wagner, Tal ( January 2018 , Conference on Learning Theory)

   We consider the (1+ϵ)-approximate nearest neighbor search problem: given a set X of n points in a d-dimensional space, build a data structure that, given any query point y, finds a point x∈X whose distance to y is at most (1+ϵ)minx∈X ‖x−y‖ for an accuracy parameter ϵ∈(0,1). Our main result is a data structure that occupies only O(ϵ^−2 n log(n)log(1/ϵ)) bits of space, assuming all point coordinates are integers in the range {−n^O(1)…n^O(1)}, i.e., the coordinates have O(logn) bits of precision. This improves over the best previously known space bound of O(ϵ^−2 n log(n)^2), obtained via the randomized dimensionality reduction method of Johnson and Lindenstrauss (1984). We also consider the more general problem of estimating all distances from a collection of query points to all data points X, and provide almost tight upper and lower bounds for the space complexity of this problem. 

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5. Triforce and corners

   https://doi.org/10.1017/s0305004119000173  

   FOX, JACOB ; SAH, ASHWIN ; SAWHNEY, MEHTAAB ; STONER, DAVID ; ZHAO, YUFEI ( July 2020 , Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract May the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ 4– o (1) but not O ( δ 4 ). Let M ( δ ) be the maximum number such that the following holds: for every ∊ > 0 and $G = {\mathbb{F}}_2^n$ with n sufficiently large, if A ⊆ G × G with A ≥ δ | G | 2 , then there exists a nonzero “popular difference” d ∈ G such that the number of “corners” ( x , y ), ( x + d , y ), ( x , y + d ) ∈ A is at least ( M ( δ )–∊)| G | 2 . As a corollary via a recent result of Mandache, we conclude that M ( δ ) = δ 4– o (1) and M ( δ ) = ω ( δ 4 ). On the other hand, for 0 < δ < 1/2 and sufficiently large N , there exists A ⊆ [ N ] 3 with | A | ≥ δN 3 such that for every d ≠ 0, the number of corners ( x , y , z ), ( x + d , y , z ), ( x , y + d , z ), ( x , y , z + d ) ∈ A is at most δ c log(1/ δ ) N 3 . A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3. 

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