arXiv subjects

Karl Bringmann

Publications and source records attributed to Karl Bringmann.

At least 73 records · Page 4Linked to original sources

arXiv2019

Approximating APSP without Scaling: Equivalence of Approximate Min-Plus and Exact Min-Max

Zwick's $(1+\varepsilon)$-approximation algorithm for the All Pairs Shortest Path (APSP) problem runs in time $\widetilde{O}(\frac{n^ω}{\varepsilon} \log{W})$, where $ω\le 2.373$ is the exponent of matrix multiplication and $W$ denotes the largest weight. This can be used to approximate several graph characteristics including the diameter, radius, median, minimum-weight triangle, and minimum-weight cycle in the same time bound. Since Zwick's algorithm uses the scaling technique, it has a factor $\log W$ in the running time. In this paper, we study whether APSP and related problems admit approximation schemes avoiding the scaling technique. That is, the number of arithmetic operations should be independent of $W$; this is called strongly polynomial. Our main results are as follows. - We design approximation schemes in strongly polynomial time $O(\frac{n^ω}{\varepsilon} \text{polylog}(\frac{n}{\varepsilon}))$ for APSP on undirected graphs as well as for the graph characteristics diameter, radius, median, minimum-weight triangle, and minimum-weight cycle on directed or undirected graphs. - For APSP on directed graphs we design an approximation scheme in strongly polynomial time $O(n^{\frac{ω+ 3}{2}} \varepsilon^{-1} \text{polylog}(\frac{n}{\varepsilon}))$. This is significantly faster than the best exact algorithm. - We explain why our approximation scheme for APSP on directed graphs has a worse exponent than $ω$: Any improvement over our exponent $\frac{ω+ 3}{2}$ would improve the best known algorithm for Min-Max Product In fact, we prove that approximating directed APSP and exactly computing the Min-Max Product are equivalent.

cs.DS↗

arXiv2019

Walking the Dog Fast in Practice: Algorithm Engineering of the Fréchet Distance

The Fréchet distance provides a natural and intuitive measure for the popular task of computing the similarity of two (polygonal) curves. While a simple algorithm computes it in near-quadratic time, a strongly subquadratic algorithm cannot exist unless the Strong Exponential Time Hypothesis fails. Still, fast practical implementations of the Fréchet distance, in particular for realistic input curves, are highly desirable. This has even lead to a designated competition, the ACM SIGSPATIAL GIS Cup 2017: Here, the challenge was to implement a near-neighbor data structure under the Fréchet distance. The bottleneck of the top three implementations turned out to be precisely the decision procedure for the Fréchet distance. In this work, we present a fast, certifying implementation for deciding the Fréchet distance, in order to (1) complement its pessimistic worst-case hardness by an empirical analysis on realistic input data and to (2) improve the state of the art for the GIS Cup challenge. We experimentally evaluate our implementation on a large benchmark consisting of several data sets (including handwritten characters and GPS trajectories). Compared to the winning implementation of the GIS Cup, we obtain running time improvements of up to more than two orders of magnitude for the decision procedure and of up to a factor of 30 for queries to the near-neighbor data structure.

cs.CG↗

arXiv2018

Average Distance in a General Class of Scale-Free Networks with Underlying Geometry

In Chung-Lu random graphs, a classic model for real-world networks, each vertex is equipped with a weight drawn from a power-law distribution, and two vertices form an edge independently with probability proportional to the product of their weights. Chung-Lu graphs have average distance O(log log n) and thus reproduce the small-world phenomenon, a key property of real-world networks. Modern, more realistic variants of this model also equip each vertex with a random position in a specific underlying geometry. The edge probability of two vertices then depends, say, inversely polynomial on their distance. In this paper we study a generic augmented version of Chung-Lu random graphs. We analyze a model where the edge probability of two vertices can depend arbitrarily on their positions, as long as the marginal probability of forming an edge (for two vertices with fixed weights, one fixed position, and one random position) is as in Chung-Lu random graphs. The resulting class contains Chung-Lu random graphs, hyperbolic random graphs, and geometric inhomogeneous random graphs as special cases. Our main result is that every random graph model in this general class has the same average distance as Chung-Lu random graphs, up to a factor 1+o(1). This shows in particular that specific choices, such as the underlying geometry being Euclidean or the dependence on the distance being inversely polynomial, do not significantly influence the average distance. The proof also yields that our model has a giant component and polylogarithmic diameter with high probability.

cs.DM↗

arXiv2018

Sketching, Streaming, and Fine-Grained Complexity of (Weighted) LCS

We study sketching and streaming algorithms for the Longest Common Subsequence problem (LCS) on strings of small alphabet size $|Σ|$. For the problem of deciding whether the LCS of strings $x,y$ has length at least $L$, we obtain a sketch size and streaming space usage of $\mathcal{O}(L^{|Σ| - 1} \log L)$. We also prove matching unconditional lower bounds. As an application, we study a variant of LCS where each alphabet symbol is equipped with a weight that is given as input, and the task is to compute a common subsequence of maximum total weight. Using our sketching algorithm, we obtain an $\mathcal{O}(\textrm{min}\{nm, n + m^{\lvert Σ\rvert}\})$-time algorithm for this problem, on strings $x,y$ of length $n,m$, with $n \ge m$. We prove optimality of this running time up to lower order factors, assuming the Strong Exponential Time Hypothesis.

cs.DS↗

arXiv2018

Approximation Algorithms for $\ell_0$-Low Rank Approximation

We study the $\ell_0$-Low Rank Approximation Problem, where the goal is, given an $m \times n$ matrix $A$, to output a rank-$k$ matrix $A'$ for which $\|A'-A\|_0$ is minimized. Here, for a matrix $B$, $\|B\|_0$ denotes the number of its non-zero entries. This NP-hard variant of low rank approximation is natural for problems with no underlying metric, and its goal is to minimize the number of disagreeing data positions. We provide approximation algorithms which significantly improve the running time and approximation factor of previous work. For $k > 1$, we show how to find, in poly$(mn)$ time for every $k$, a rank $O(k \log(n/k))$ matrix $A'$ for which $\|A'-A\|_0 \leq O(k^2 \log(n/k)) \mathrm{OPT}$. To the best of our knowledge, this is the first algorithm with provable guarantees for the $\ell_0$-Low Rank Approximation Problem for $k > 1$, even for bicriteria algorithms. For the well-studied case when $k = 1$, we give a $(2+ε)$-approximation in {\it sublinear time}, which is impossible for other variants of low rank approximation such as for the Frobenius norm. We strengthen this for the well-studied case of binary matrices to obtain a $(1+O(ψ))$-approximation in sublinear time, where $ψ= \mathrm{OPT}/\lVert A\rVert_0$. For small $ψ$, our approximation factor is $1+o(1)$.

cs.DS↗

arXiv2018

More Consequences of Falsifying SETH and the Orthogonal Vectors Conjecture

The Strong Exponential Time Hypothesis and the OV-conjecture are two popular hardness assumptions used to prove a plethora of lower bounds, especially in the realm of polynomial-time algorithms. The OV-conjecture in moderate dimension states there is no $ε>0$ for which an $O(N^{2-ε})\mathrm{poly}(D)$ time algorithm can decide whether there is a pair of orthogonal vectors in a given set of size $N$ that contains $D$-dimensional binary vectors. We strengthen the evidence for these hardness assumptions. In particular, we show that if the OV-conjecture fails, then two problems for which we are far from obtaining even tiny improvements over exhaustive search would have surprisingly fast algorithms. If the OV conjecture is false, then there is a fixed $ε>0$ such that: (1) For all $d$ and all large enough $k$, there is a randomized algorithm that takes $O(n^{(1-ε)k})$ time to solve the Zero-Weight-$k$-Clique and Min-Weight-$k$-Clique problems on $d$-hypergraphs with $n$ vertices. As a consequence, the OV-conjecture is implied by the Weighted Clique conjecture. (2) For all $c$, the satisfiability of sparse TC1 circuits on $n$ inputs (that is, circuits with $cn$ wires, depth $c\log n$, and negation, AND, OR, and threshold gates) can be computed in time ${O((2-ε)^n)}$.

cs.CC↗

arXiv2018

Multivariate Analysis of Orthogonal Range Searching and Graph Distances Parameterized by Treewidth

We show that the eccentricities, diameter, radius, and Wiener index of an undirected $n$-vertex graph with nonnegative edge lengths can be computed in time $O(n\cdot \binom{k+\lceil\log n\rceil}{k} \cdot 2^k k^2 \log n)$, where $k$ is the treewidth of the graph. For every $ε>0$, this bound is $n^{1+ε}\exp O(k)$, which matches a hardness result of Abboud, Vassilevska Williams, and Wang (SODA 2015) and closes an open problem in the multivariate analysis of polynomial-time computation. To this end, we show that the analysis of an algorithm of Cabello and Knauer (Comp. Geom., 2009) in the regime of non-constant treewidth can be improved by revisiting the analysis of orthogonal range searching, improving bounds of the form $\log^d n$ to $\binom{d+\lceil\log n\rceil}{d}$, as originally observed by Monier (J. Alg. 1980). We also investigate the parameterization by vertex cover number.

cs.DS↗

arXiv2018

Tighter Connections Between Formula-SAT and Shaving Logs

A noticeable fraction of Algorithms papers in the last few decades improve the running time of well-known algorithms for fundamental problems by logarithmic factors. For example, the $O(n^2)$ dynamic programming solution to the Longest Common Subsequence problem (LCS) was improved to $O(n^2/\log^2 n)$ in several ways and using a variety of ingenious tricks. This line of research, also known as "the art of shaving log factors", lacks a tool for proving negative results. Specifically, how can we show that it is unlikely that LCS can be solved in time $O(n^2/\log^3 n)$? Perhaps the only approach for such results was suggested in a recent paper of Abboud, Hansen, Vassilevska W. and Williams (STOC'16). The authors blame the hardness of shaving logs on the hardness of solving satisfiability on Boolean formulas (Formula-SAT) faster than exhaustive search. They show that an $O(n^2/\log^{1000} n)$ algorithm for LCS would imply a major advance in circuit lower bounds. Whether this approach can lead to tighter barriers was unclear. In this paper, we push this approach to its limit and, in particular, prove that a well-known barrier from complexity theory stands in the way for shaving five additional log factors for fundamental combinatorial problems. For LCS, regular expression pattern matching, as well as the Fréchet distance problem from Computational Geometry, we show that an $O(n^2/\log^{7+\varepsilon} n)$ runtime would imply new Formula-SAT algorithms. Our main result is a reduction from SAT on formulas of size $s$ over $n$ variables to LCS on sequences of length $N=2^{n/2} \cdot s^{1+o(1)}$. Our reduction is essentially as efficient as possible, and it greatly improves the previously known reduction for LCS with $N=2^{n/2} \cdot s^c$, for some $c \geq 100$.

cs.CC↗

arXiv2018

Fast Fencing

We consider very natural "fence enclosure" problems studied by Capoyleas, Rote, and Woeginger and Arkin, Khuller, and Mitchell in the early 90s. Given a set $S$ of $n$ points in the plane, we aim at finding a set of closed curves such that (1) each point is enclosed by a curve and (2) the total length of the curves is minimized. We consider two main variants. In the first variant, we pay a unit cost per curve in addition to the total length of the curves. An equivalent formulation of this version is that we have to enclose $n$ unit disks, paying only the total length of the enclosing curves. In the other variant, we are allowed to use at most $k$ closed curves and pay no cost per curve. For the variant with at most $k$ closed curves, we present an algorithm that is polynomial in both $n$ and $k$. For the variant with unit cost per curve, or unit disks, we present a near-linear time algorithm. Capoyleas, Rote, and Woeginger solved the problem with at most $k$ curves in $n^{O(k)}$ time. Arkin, Khuller, and Mitchell used this to solve the unit cost per curve version in exponential time. At the time, they conjectured that the problem with $k$ curves is NP-hard for general $k$. Our polynomial time algorithm refutes this unless P equals NP.

cs.CG↗

arXiv2018

Fine-Grained Complexity of Analyzing Compressed Data: Quantifying Improvements over Decompress-And-Solve

Can we analyze data without decompressing it? As our data keeps growing, understanding the time complexity of problems on compressed inputs, rather than in convenient uncompressed forms, becomes more and more relevant. Suppose we are given a compression of size $n$ of data that originally has size $N$, and we want to solve a problem with time complexity $T(\cdot)$. The naive strategy of "decompress-and-solve" gives time $T(N)$, whereas "the gold standard" is time $T(n)$: to analyze the compression as efficiently as if the original data was small. We restrict our attention to data in the form of a string (text, files, genomes, etc.) and study the most ubiquitous tasks. While the challenge might seem to depend heavily on the specific compression scheme, most methods of practical relevance (Lempel-Ziv-family, dictionary methods, and others) can be unified under the elegant notion of Grammar Compressions. A vast literature, across many disciplines, established this as an influential notion for Algorithm design. We introduce a framework for proving (conditional) lower bounds in this field, allowing us to assess whether decompress-and-solve can be improved, and by how much. Our main results are: - The $O(nN\sqrt{\log{N/n}})$ bound for LCS and the $O(\min\{N \log N, nM\})$ bound for Pattern Matching with Wildcards are optimal up to $N^{o(1)}$ factors, under the Strong Exponential Time Hypothesis. (Here, $M$ denotes the uncompressed length of the compressed pattern.) - Decompress-and-solve is essentially optimal for Context-Free Grammar Parsing and RNA Folding, under the $k$-Clique conjecture. - We give an algorithm showing that decompress-and-solve is not optimal for Disjointness.

cs.CC↗

arXiv2018

Multivariate Fine-Grained Complexity of Longest Common Subsequence

We revisit the classic combinatorial pattern matching problem of finding a longest common subsequence (LCS). For strings $x$ and $y$ of length $n$, a textbook algorithm solves LCS in time $O(n^2)$, but although much effort has been spent, no $O(n^{2-\varepsilon})$-time algorithm is known. Recent work indeed shows that such an algorithm would refute the Strong Exponential Time Hypothesis (SETH) [Abboud, Backurs, Vassilevska Williams + Bringmann, Künnemann FOCS'15]. Despite the quadratic-time barrier, for over 40 years an enduring scientific interest continued to produce fast algorithms for LCS and its variations. Particular attention was put into identifying and exploiting input parameters that yield strongly subquadratic time algorithms for special cases of interest, e.g., differential file comparison. This line of research was successfully pursued until 1990, at which time significant improvements came to a halt. In this paper, using the lens of fine-grained complexity, our goal is to (1) justify the lack of further improvements and (2) determine whether some special cases of LCS admit faster algorithms than currently known. To this end, we provide a systematic study of the multivariate complexity of LCS, taking into account all parameters previously discussed in the literature: the input size $n:=\max\{|x|,|y|\}$, the length of the shorter string $m:=\min\{|x|,|y|\}$, the length $L$ of an LCS of $x$ and $y$, the numbers of deletions $δ:= m-L$ and $Δ:= n-L$, the alphabet size, as well as the numbers of matching pairs $M$ and dominant pairs $d$. For any class of instances defined by fixing each parameter individually to a polynomial in terms of the input size, we prove a SETH-based lower bound matching one of three known algorithms. Specifically, we determine the optimal running time for LCS under SETH as $(n+\min\{d, δΔ, δm\})^{1\pm o(1)}$. [...]

cs.CC↗

arXiv2018

A fast implementation of near neighbors queries for Fréchet distance (GIS Cup)

This paper describes an implementation of fast near-neighbours queries (also known as range searching) with respect to the Fréchet distance. The algorithm is designed to be efficient on practical data such as GPS trajectories. Our approach is to use a quadtree data structure to enumerate all curves in the database that have similar start and endpoints as the query curve. On these curves we run positive and negative filters to narrow the set of potential results. Only for those trajectories where these heuristics fail, we compute the Fréchet distance exactly, by running a novel recursive variant of the classic free-space diagram algorithm. Our implementation won the ACM SIGSPATIAL GIS Cup 2017.

cs.CG↗

arXiv2018

Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

Tree-adjoining grammars are a generalization of context-free grammars that are well suited to model human languages and are thus popular in computational linguistics. In the tree-adjoining grammar recognition problem, given a grammar $Γ$ and a string $s$ of length $n$, the task is to decide whether $s$ can be obtained from $Γ$. Rajasekaran and Yooseph's parser (JCSS'98) solves this problem in time $O(n^{2ω})$, where $ω< 2.373$ is the matrix multiplication exponent. The best algorithms avoiding fast matrix multiplication take time $O(n^6)$. The first evidence for hardness was given by Satta (J. Comp. Linguist.'94): For a more general parsing problem, any algorithm that avoids fast matrix multiplication and is significantly faster than $O(|Γ| n^6)$ in the case of $|Γ| = Θ(n^{12})$ would imply a breakthrough for Boolean matrix multiplication. Following an approach by Abboud et al. (FOCS'15) for context-free grammar recognition, in this paper we resolve many of the disadvantages of the previous lower bound. We show that, even on constant-size grammars, any improvement on Rajasekaran and Yooseph's parser would imply a breakthrough for the $k$-Clique problem. This establishes tree-adjoining grammar parsing as a practically relevant problem with the unusual running time of $n^{2ω}$, up to lower order factors.

cs.CC↗

arXiv2018

Maximum Volume Subset Selection for Anchored Boxes

Let $B$ be a set of $n$ axis-parallel boxes in $\mathbb{R}^d$ such that each box has a corner at the origin and the other corner in the positive quadrant of $\mathbb{R}^d$, and let $k$ be a positive integer. We study the problem of selecting $k$ boxes in $B$ that maximize the volume of the union of the selected boxes. This research is motivated by applications in skyline queries for databases and in multicriteria optimization, where the problem is known as the hypervolume subset selection problem. It is known that the problem can be solved in polynomial time in the plane, while the best known running time in any dimension $d \ge 3$ is $Ω\big(\binom{n}{k}\big)$. We show that: - The problem is NP-hard already in 3 dimensions. - In 3 dimensions, we break the bound $Ω\big(\binom{n}{k}\big)$, by providing an $n^{O(\sqrt{k})}$ algorithm. - For any constant dimension $d$, we present an efficient polynomial-time approximation scheme.

cs.CG↗

arXiv2018

A Note on Hardness of Diameter Approximation

We revisit the hardness of approximating the diameter of a network. In the CONGEST model of distributed computing, $ \tilde Ω(n) $ rounds are necessary to compute the diameter [Frischknecht et al. SODA'12], where $ \tilde Ω(\cdot) $ hides polylogarithmic factors. Abboud et al. [DISC 2016] extended this result to sparse graphs and, at a more fine-grained level, showed that, for any integer $ 1 \leq \ell \leq \operatorname{polylog} (n) $, distinguishing between networks of diameter $ 4 \ell + 2 $ and $ 6 \ell + 1 $ requires $ \tilde Ω(n) $ rounds. We slightly tighten this result by showing that even distinguishing between diameter $ 2 \ell + 1 $ and $ 3 \ell + 1 $ requires $ \tilde Ω(n) $ rounds. The reduction of Abboud et al. is inspired by recent conditional lower bounds in the RAM model, where the orthogonal vectors problem plays a pivotal role. In our new lower bound, we make the connection to orthogonal vectors explicit, leading to a conceptually more streamlined exposition.

cs.DS↗

arXiv2017

Truly Sub-cubic Algorithms for Language Edit Distance and RNA Folding via Fast Bounded-Difference Min-Plus Product

It is a major open problem whether the $(\min,+)$-product of two $n\times n$ matrices has a truly sub-cubic (i.e. $O(n^{3-ε})$ for $ε>0$) time algorithm, in particular since it is equivalent to the famous All-Pairs-Shortest-Paths problem (APSP) in $n$-vertex graphs. Some restrictions of the $(\min,+)$-product to special types of matrices are known to admit truly sub-cubic algorithms, each giving rise to a special case of APSP that can be solved faster. In this paper we consider a new, different and powerful restriction in which all matrix entries are integers and one matrix can be arbitrary, as long as the other matrix has "bounded differences" in either its columns or rows, i.e. any two consecutive entries differ by only a small amount. We obtain the first truly sub-cubic algorithm for this bounded-difference $(\min,+)$-product (answering an open problem of Chan and Lewenstein). Our new algorithm, combined with a strengthening of an approach of L.~Valiant for solving context-free grammar parsing with matrix multiplication, yields the first truly sub-cubic algorithms for the following problems: Language Edit Distance (a major problem in the parsing community), RNA-folding (a major problem in bioinformatics) and Optimum Stack Generation (answering an open problem of Tarjan).

cs.DS↗

arXiv2017

On algebraic branching programs of small width

In 1979 Valiant showed that the complexity class VP_e of families with polynomially bounded formula size is contained in the class VP_s of families that have algebraic branching programs (ABPs) of polynomially bounded size. Motivated by the problem of separating these classes we study the topological closure VP_e-bar, i.e. the class of polynomials that can be approximated arbitrarily closely by polynomials in VP_e. We describe VP_e-bar with a strikingly simple complete polynomial (in characteristic different from 2) whose recursive definition is similar to the Fibonacci numbers. Further understanding this polynomial seems to be a promising route to new formula lower bounds. Our methods are rooted in the study of ABPs of small constant width. In 1992 Ben-Or and Cleve showed that formula size is polynomially equivalent to width-3 ABP size. We extend their result (in characteristic different from 2) by showing that approximate formula size is polynomially equivalent to approximate width-2 ABP size. This is surprising because in 2011 Allender and Wang gave explicit polynomials that cannot be computed by width-2 ABPs at all! The details of our construction lead to the aforementioned characterization of VP_e-bar. As a natural continuation of this work we prove that the class VNP can be described as the class of families that admit a hypercube summation of polynomially bounded dimension over a product of polynomially many affine linear forms. This gives the first separations of algebraic complexity classes from their nondeterministic analogs.

cs.CC↗

arXiv2017

Sampling Geometric Inhomogeneous Random Graphs in Linear Time

Real-world networks, like social networks or the internet infrastructure, have structural properties such as large clustering coefficients that can best be described in terms of an underlying geometry. This is why the focus of the literature on theoretical models for real-world networks shifted from classic models without geometry, such as Chung-Lu random graphs, to modern geometry-based models, such as hyperbolic random graphs. With this paper we contribute to the theoretical analysis of these modern, more realistic random graph models. Instead of studying directly hyperbolic random graphs, we use a generalization that we call geometric inhomogeneous random graphs (GIRGs). Since we ignore constant factors in the edge probabilities, GIRGs are technically simpler (specifically, we avoid hyperbolic cosines), while preserving the qualitative behaviour of hyperbolic random graphs, and we suggest to replace hyperbolic random graphs by this new model in future theoretical studies. We prove the following fundamental structural and algorithmic results on GIRGs. (1) As our main contribution we provide a sampling algorithm that generates a random graph from our model in expected linear time, improving the best-known sampling algorithm for hyperbolic random graphs by a substantial factor O(n^0.5). (2) We establish that GIRGs have clustering coefficients in Ω(1), (3) we prove that GIRGs have small separators, i.e., it suffices to delete a sublinear number of edges to break the giant component into two large pieces, and (4) we show how to compress GIRGs using an expected linear number of bits.

cs.SI↗