We study edit distance computation with preprocessing: the preprocessing algorithm acts on each string separately, and then the query algorithm takes as input the two preprocessed strings. This model is inspired by scenarios where we would like to compute edit distance between many pairs in the same pool of strings.
Our results include:
Permutation-LCS: If the LCS between two permutations has length $n-k$, we can compute it \textit{ exactly} with $O(n \log(n))$ preprocessing and $O(k \log(n))$ query time.
Small edit distance: For general strings, if their edit distance is at most $k$, we can compute it \textit{ exactly} with $O(n\log(n))$ preprocessing and $O(k^2 \log(n))$ query time.
Approximate edit distance: For the most general input, we can approximate the edit distance to within factor $(7+o(1))$ with preprocessing time $\tilde{O}(n^2)$ and query time $\tilde{O}(n^{1.5+o(1)})$.
All of these results significantly improve over the state of the art in edit distance computation without preprocessing. Interestingly, by combining ideas from our algorithms with preprocessing, we provide new improved results for approximating edit distance without preprocessing in subquadratic time.
A New Number Theoretic Transform(NTT), which is a form of FFT, is introduced, that is faster than FFTs. Also, a multiplication algorithm is introduced that uses this to perform integer multiplication faster than O(n log n). It uses preprocessing to achieve an upper bounds of (n log n/(log log n/ log log log n). Also, we explore the possibility of O(n) time multiplication via NTTs that require only O(n) operations, using preprocessing.
Adobe Research（Adobe研究院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Deep neural networks have achieved impressive performance in many areas. Designing a fast and provable method for training neural networks is a fundamental question in machine learning. The classical training method requires paying $Ω(mnd)$ cost for both forward computation and backward computation, where $m$ is the width of the neural network, and we are given $n$ training points in $d$-dimensional space. In this paper, we propose two novel preprocessing ideas to bypass this $Ω(mnd)$ barrier: $\bullet$ First, by preprocessing the initial weights of the neural networks, we can train the neural…
Columbia University（哥伦比亚大学）；University of California, Berkeley（加州大学伯克利分校）；Adobe Research（奥多比研究院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Duke University（杜克大学）
Over the last decade, deep neural networks have transformed our society, and they are already widely applied in various machine learning applications. State-of-art deep neural networks are becoming larger in size every year to deliver increasing model accuracy, and as a result, model training consumes substantial computing resources and will only consume more in the future. Using current training methods, in each iteration, to process a data point $x \in \mathbb{R}^d$ in a layer, we need to spend $Θ(md)$ time to evaluate all the $m$ neurons in the layer. This means processing the entire layer …
Under the word RAM model, we design three data structures that can be constructed in $O(n\sqrt{\lg n})$ time over $n$ points in an $n \times n$ grid. The first data structure is an $O(n\lg^ε n)$-word structure supporting orthogonal range reporting in $O(\lg\lg n+k)$ time, where $k$ denotes output size and $ε$ is an arbitrarily small constant. The second is an $O(n\lg\lg n)$-word structure supporting orthogonal range successor in $O(\lg\lg n)$ time, while the third is an $O(n\lg^ε n)$-word structure supporting sorted range reporting in $O(\lg\lg n+k)$ time. The query times of these data structu…
This paper introduces a novel algorithm combination designed for fast one-to-many multicriteria shortest path search. A preprocessing algorithm excludes irrelevant vertices by building a smaller cover graph. A modified version of multicriteria label-setting algorithm operates on the cover graph and employs a dimensionality reduction technique for swifter domination checks. While the method itself maintains solution optimality, it is able to additionally incorporate existing heuristics for further speedups. The proposed algorithm has been tested on multiple criteria combinations of varying corr…
In this paper, we study the problem of speeding up a type of optimization algorithms called Frank-Wolfe, a conditional gradient method. We develop and employ two novel inner product search data structures, improving the prior fastest algorithm in [Shrivastava, Song and Xu, NeurIPS 2021]. * The first data structure uses low-dimensional random projection to reduce the problem to a lower dimension, then uses efficient inner product data structure. It has preprocessing time $\tilde O(nd^{ω-1}+dn^{1+o(1)})$ and per iteration cost $\tilde O(d+n^ρ)$ for small constant $ρ$. * The second data structure…
We design $f$-edge fault-tolerant diameter oracles ($f$-FDOs). We preprocess a given graph $G$ on $n$ vertices and $m$ edges, and a positive integer $f$, to construct a data structure that, when queried with a set $F$ of $|F| \leq f$ edges, returns the diameter of $G-F$. For a single failure ($f=1$) in an unweighted directed graph of diameter $D$, there exists an approximate FDO by Henzinger et al. [ITCS 2017] with stretch $(1+\varepsilon)$, constant query time, space $O(m)$, and a combinatorial preprocessing time of $\widetilde{O}(mn + n^{1.5} \sqrt{Dm/\varepsilon})$.We present an FDO for dir…
We study the problem of quickly computing point-to-point shortest paths in massive road networks with traffic predictions. Incorporating traffic predictions into routing allows, for example, to avoid commuter traffic congestions. Existing techniques follow a two-phase approach: In a preprocessing step, an index is built. The index depends on the road network and the traffic patterns but not on the path start and end. The latter are the input of the query phase, in which shortest paths are computed. All existing techniques have large index size, slow query running times or may compute suboptima…
In parameterized complexity, it is well-known that a parameterized problem is fixed-parameter tractable if and only if it has a kernel - an instance equivalent to the input instance, whose size is just a function of the parameter. The size of the kernel can be exponential or worse, resulting in a quest for fixed-parameter tractable problems with a polynomial-sized kernel. The developments in machinery to show lower bounds for the sizes of the kernel gave rise to the question of the asymptotically optimum size for the kernel of fixed-parameter tractable problems. In this article, we survey a to…
We study the problem of approximating the edit distance of two strings in sublinear time, in a setting where one or both string(s) are preprocessed, as initiated by Goldenberg, Rubinstein, Saha (STOC '20). Specifically, in the $(k, K)$-gap edit distance problem, the goal is to distinguish whether the edit distance of two strings is at most $k$ or at least $K$. We obtain the following results: * After preprocessing one string in time $n^{1+o(1)}$, we can solve $(k, k \cdot n^{o(1)})$-gap edit distance in time $(n/k + k) \cdot n^{o(1)}$. * After preprocessing both strings separately in time $n^{…
This paper introduces a new algorithm for the fundamental problem of generating a random integer from a discrete probability distribution using a source of independent and unbiased random coin flips. We prove that this algorithm, which we call the Fast Loaded Dice Roller (FLDR), is highly efficient in both space and time: (i) the size of the sampler is guaranteed to be linear in the number of bits needed to encode the input distribution; and (ii) the expected number of bits of entropy it consumes per sample is at most 6 bits more than the information-theoretically optimal rate. We present fast…
University of Trento（特伦托大学）；Politecnico di Milano（米兰理工大学）；University of Verona（维罗纳大学）
In this paper, we present a novel algorithm for the maximum a posteriori decoding (MAPD) of time-homogeneous Hidden Markov Models (HMM), improving the worst-case running time of the classical Viterbi algorithm by a logarithmic factor. In our approach, we interpret the Viterbi algorithm as a repeated computation of matrix-vector $(\max, +)$-multiplications. On time-homogeneous HMMs, this computation is online: a matrix, known in advance, has to be multiplied with several vectors revealed one at a time. Our main contribution is an algorithm solving this version of matrix-vector $(\max,+)$-multip…
We consider online algorithms for the $k$-server problem on trees. There is a $k$-competitive algorithm for this problem, and it is the best competitive ratio. M. Chrobak and L. Larmore provided it. At the same time, the existing implementation has $O(n)$ time complexity for processing a query and $O(n)$ for prepossessing, where $n$ is the number of nodes in a tree. Another implementation of the algorithm has $O(k^2+k\log n)$ time complexity for processing a query and $O(n\log n)$ for prepossessing. We provide a new time-efficient implementation of the algorithm. It has $O(n)$ time complexity …
We study efficient and exact shortest path algorithms for routing on road networks with realistic traffic data. For navigation applications, both current (i.e., live) traffic events and predictions of future traffic flows play an important role in routing. While preprocessing-based speedup techniques have been employed successfully to both settings individually, a combined model poses significant challenges. Supporting predicted traffic typically requires expensive preprocessing while live traffic requires fast updates for regular adjustments. We propose an A*-based solution to this problem. B…
We consider online algorithms for the $k$-server problem on trees. Chrobak and Larmore proposed a $k$-competitive algorithm for this problem that has the optimal competitive ratio. However, a naive implementation of their algorithm has $O(n)$ time complexity for processing each query, where $n$ is the number of nodes in the tree. We propose a new time-efficient implementation of this algorithm that has $O(n\log n)$ time complexity for preprocessing and $O\left(k^2 + k\cdot \log n\right)$ time for processing a query. We also propose a quantum algorithm for the case where the nodes of the tree a…
We continue the study of distance sensitivity oracles (DSOs). Given a directed graph $G$ with $n$ vertices and edge weights in $\{1, 2, \dots, M\}$, we want to build a data structure such that given any source vertex $u$, any target vertex $v$, and any failure $f$ (which is either a vertex or an edge), it outputs the length of the shortest path from $u$ to $v$ not going through $f$. Our main result is a DSO with preprocessing time $O(n^{2.5794}M)$ and constant query time. Previously, the best preprocessing time of DSOs for directed graphs is $O(n^{2.7233}M)$, and even in the easier case of und…
