Lower bounds for k-distance approximation

Consider a set P of N random points on the unit sphere of dimension $d-1$, and the symmetrized set S = P union (-P). The halving polyhedron of S is defined as the convex hull of the set of centroids of N distinct points in S. We prove that after appropriate rescaling this halving polyhedron is Hausdorff close to the unit ball with high probability, as soon as the number of points grows like $Omega(d log(d))$. From this result, we deduce probabilistic lower bounds on the complexity of approximations of the distance to the empirical measure on the point set by distance-like functions

Tight Bounds for Approximate Near Neighbor Searching for Time Series under the {F}r\'{e}chet Distance

We study the $c$-approximate near neighbor problem under the continuous Fr\'echet distance: Given a set of $n$ polygonal curves with $m$ vertices, a radius $\delta > 0$, and a parameter $k \leq m$, we want to preprocess the curves into a data structure that, given a query curve $q$ with $k$ vertices, either returns an input curve with Fr\'echet distance at most $c\cdot \delta$ to $q$, or returns that there exists no input curve with Fr\'echet distance at most $\delta$ to $q$. We focus on the case where the input and the queries are one-dimensional polygonal curves -- also called time series -- and we give a comprehensive analysis for this case. We obtain new upper bounds that provide different tradeoffs between approximation factor, preprocessing time, and query time. Our data structures improve upon the state of the art in several ways. We show that for any $0 < \varepsilon \leq 1$ an approximation factor of $(1+\varepsilon)$ can be achieved within the same asymptotic time bounds as the previously best result for $(2+\varepsilon)$. Moreover, we show that an approximation factor of $(2+\varepsilon)$ can be obtained by using preprocessing time and space $O(nm)$, which is linear in the input size, and query time in $O(\frac{1}{\varepsilon})^{k+2}$, where the previously best result used preprocessing time in $n \cdot O(\frac{m}{\varepsilon k})^k$ and query time in $O(1)^k$. We complement our upper bounds with matching conditional lower bounds based on the Orthogonal Vectors Hypothesis. Interestingly, some of our lower bounds already hold for any super-constant value of $k$. This is achieved by proving hardness of a one-sided sparse version of the Orthogonal Vectors problem as an intermediate problem, which we believe to be of independent interest

Approximation Algorithms and Hardness for nn-Pairs Shortest Paths and All-Nodes Shortest Cycles

We study the approximability of two related problems on graphs with $n$ nodes and $m$ edges: $n$-Pairs Shortest Paths ($n$-PSP), where the goal is to find a shortest path between $O(n)$ prespecified pairs, and All Node Shortest Cycles (ANSC), where the goal is to find the shortest cycle passing through each node. Approximate $n$-PSP has been previously studied, mostly in the context of distance oracles. We ask the question of whether approximate $n$-PSP can be solved faster than by using distance oracles or All Pair Shortest Paths (APSP). ANSC has also been studied previously, but only in terms of exact algorithms, rather than approximation. We provide a thorough study of the approximability of $n$-PSP and ANSC, providing a wide array of algorithms and conditional lower bounds that trade off between running time and approximation ratio. A highlight of our conditional lower bounds results is that for any integer $k\ge 1$, under the combinatorial $4k$-clique hypothesis, there is no combinatorial algorithm for unweighted undirected $n$-PSP with approximation ratio better than $1+1/k$ that runs in $O(m^{2-2/(k+1)}n^{1/(k+1)-\epsilon})$ time. This nearly matches an upper bound implied by the result of Agarwal (2014). A highlight of our algorithmic results is that one can solve both $n$-PSP and ANSC in $\tilde O(m+ n^{3/2+\epsilon})$ time with approximation factor $2+\epsilon$ (and additive error that is function of $\epsilon$), for any constant $\epsilon>0$. For $n$-PSP, our conditional lower bounds imply that this approximation ratio is nearly optimal for any subquadratic-time combinatorial algorithm. We further extend these algorithms for $n$-PSP and ANSC to obtain a time/accuracy trade-off that includes near-linear time algorithms.Comment: Abstract truncated to meet arXiv requirement. To appear in FOCS 202

Dynamic Time Warping in Strongly Subquadratic Time: Algorithms for the Low-Distance Regime and Approximate Evaluation

Dynamic time warping distance (DTW) is a widely used distance measure between time series. The best known algorithms for computing DTW run in near quadratic time, and conditional lower bounds prohibit the existence of significantly faster algorithms. The lower bounds do not prevent a faster algorithm for the special case in which the DTW is small, however. For an arbitrary metric space $\Sigma$ with distances normalized so that the smallest non-zero distance is one, we present an algorithm which computes $\operatorname{dtw}(x, y)$ for two strings $x$ and $y$ over $\Sigma$ in time $O(n \cdot \operatorname{dtw}(x, y))$. We also present an approximation algorithm which computes $\operatorname{dtw}(x, y)$ within a factor of $O(n^\epsilon)$ in time $\tilde{O}(n^{2 - \epsilon})$ for $0 < \epsilon < 1$. The algorithm allows for the strings $x$ and $y$ to be taken over an arbitrary well-separated tree metric with logarithmic depth and at most exponential aspect ratio. Extending our techniques further, we also obtain the first approximation algorithm for edit distance to work with characters taken from an arbitrary metric space, providing an $n^\epsilon$-approximation in time $\tilde{O}(n^{2 - \epsilon})$, with high probability. Additionally, we present a simple reduction from computing edit distance to computing DTW. Applying our reduction to a conditional lower bound of Bringmann and K\"unnemann pertaining to edit distance over $\{0, 1\}$, we obtain a conditional lower bound for computing DTW over a three letter alphabet (with distances of zero and one). This improves on a previous result of Abboud, Backurs, and Williams. With a similar approach, we prove a reduction from computing edit distance to computing longest LCS length. This means that one can recover conditional lower bounds for LCS directly from those for edit distance, which was not previously thought to be the case

Metrical Service Systems with Multiple Servers

We study the problem of metrical service systems with multiple servers (MSSMS), which generalizes two well-known problems -- the $k$-server problem, and metrical service systems. The MSSMS problem is to service requests, each of which is an $l$-point subset of a metric space, using $k$ servers, with the objective of minimizing the total distance traveled by the servers. Feuerstein initiated a study of this problem by proving upper and lower bounds on the deterministic competitive ratio for uniform metric spaces. We improve Feuerstein's analysis of the upper bound and prove that his algorithm achieves a competitive ratio of $k({{k+l}\choose{l}}-1)$. In the randomized online setting, for uniform metric spaces, we give an algorithm which achieves a competitive ratio $\mathcal{O}(k^3\log l)$, beating the deterministic lower bound of ${{k+l}\choose{l}}-1$. We prove that any randomized algorithm for MSSMS on uniform metric spaces must be $\Omega(\log kl)$-competitive. We then prove an improved lower bound of ${{k+2l-1}\choose{k}}-{{k+l-1}\choose{k}}$ on the competitive ratio of any deterministic algorithm for $(k,l)$-MSSMS, on general metric spaces. In the offline setting, we give a pseudo-approximation algorithm for $(k,l)$-MSSMS on general metric spaces, which achieves an approximation ratio of $l$ using $kl$ servers. We also prove a matching hardness result, that a pseudo-approximation with less than $kl$ servers is unlikely, even for uniform metric spaces. For general metric spaces, we highlight the limitations of a few popular techniques, that have been used in algorithm design for the $k$-server problem and metrical service systems.Comment: 18 pages; accepted for publication at COCOON 201