This paper explores the notion of approximate data structures, which return approximately correct answers to queries, but run faster than their exact counterparts. The paper describes approximate variants of the van Emde Boas data structure, which support the same dynamic operations as the standard van Emde Boas data structure (min, max, successor, predecessor, and existence queries, as well as insertion and deletion), except that answers to queries are approximate. The variants support all operations in constant time provided the performance guarantee is 1+1/polylog(n), and in O(loglogn) time provided the performance guarantee is 1+1/polynomial(n), for n elements in the data structure.
Applications described include Prim's minimum-spanning-tree algorithm, Dijkstra's single-source shortest paths algorithm, and an on-line variant of Graham's convex hull algorithm. To obtain output which approximates the desired output with the performance guarantee tending to 1, Prim's algorithm requires only linear time, Dijkstra's algorithm requires O(mloglogn) time, and the on-line variant of Graham's algorithm requires constant amortized time per operation.
This paper introduces "Multi-Level Spherical LSH": parameter-free, a multi-level, data-dependant Locality Sensitive Hashing data structure for solving the Approximate Near Neighbors Problem (ANN). This data structure uses a modified version of a multi-probe adaptive querying algorithm, with the potential of achieving a $O(n^p + t)$ query run time, for all inputs n where $t <= n$.
We show that approximate near neighbor search in high dimensions can be solved in a Las Vegas fashion (i.e., without false negatives) for $\ell_p$ ($1\le p\le 2$) while matching the performance of optimal locality-sensitive hashing. Specifically, we construct a data-independent Las Vegas data structure with query time $O(dn^ρ)$ and space usage $O(dn^{1+ρ})$ for $(r, c r)$-approximate near neighbors in $\mathbb{R}^{d}$ under the $\ell_p$ norm, where $ρ= 1/c^p + o(1)$. Furthermore, we give a Las Vegas locality-sensitive filter construction for the unit sphere that can be used with the data-depen…
Given an $n$-bit array $A$, the succinct rank data structure problem asks to construct a data structure using space $n+r$ bits for $r\ll n$, supporting rank queries of form $\mathtt{rank}(x)=\sum_{i=0}^{x-1} A[i]$. In this paper, we design a new succinct rank data structure with $r=n/(\log n)^{Ω(t)}+n^{1-c}$ and query time $O(t)$ for some constant $c>0$, improving the previous best-known by Patrascu [Pat08], which has $r=n/(\frac{\log n}{t})^{Ω(t)}+\tilde{O}(n^{3/4})$ bits of redundancy. For $r>n^{1-c}$, our space-time tradeoff matches the cell-probe lower bound by Patrascu and Viola [PV10], w…
Much recent work has been devoted to approximate nearest neighbor queries. Motivated by applications in recommender systems, we consider approximate furthest neighbor (AFN) queries and present a simple, fast, and highly practical data structure for answering AFN queries in high- dimensional Euclidean space. The method builds on the technique of In- dyk (SODA 2003), storing random projections to provide sublinear query time for AFN. However, we introduce a different query algorithm, improving on Indyk's approximation factor and reducing the running time by a logarithmic factor. We also present …
Anonymous social networks present a number of new and challenging problems for existing Social Network Analysis techniques. Traditionally, existing methods for analysing graph structure, such as community detection, required global knowledge of the graph structure. That implies that a centralised entity must be given access to the edge list of each node in the graph. This is impossible for anonymous social networks and other settings where privacy is valued by its participants. In addition, using their graph structure inputs for learning tasks defeats the purpose of anonymity. In this work, we…
We consider the problem of maintaining a $(1-ε)$-approximation to the densest subgraph (DSG) in an undirected multigraph as it undergoes edge insertions and deletions (the fully dynamic setting). Sawlani and Wang [SW20] developed a data structure that, for any given $ε> 0$, maintains a $(1-ε)$-approximation with $O(\log^4 n/ε^6)$ worst-case update time for edge operations, and $O(1)$ query time for reporting the density value. Their data structure was the first to achieve near-optimal approximation, and improved previous work that maintained a $(1/4 - ε)$ approximation in amortized polylogarit…
A retrieval data structure for a static function $f:S\rightarrow \{0,1\}^r$ supports queries that return $f(x)$ for any $x \in S$. Retrieval data structures can be used to implement a static approximate membership query data structure (AMQ), i.e., a Bloom filter alternative, with false positive rate $2^{-r}$. The information-theoretic lower bound for both tasks is $r|S|$ bits. While succinct theoretical constructions using $(1+o(1))r|S|$ bits were known, these could not achieve very small overheads in practice because they have an unfavorable space--time tradeoff hidden in the asymptotic costs…
For any $ε\in (0,1)$, a $(1+ε)$-approximate range mode query asks for the position of an element whose frequency in the query range is at most a factor $(1+ε)$ smaller than the true mode. For this problem, we design an $O(n/ε)$ bit data structure supporting queries in $O(\lg(1/ε))$ time. This is an encoding data structure which does not require access to the input sequence; we prove the space cost is asymptotically optimal for constant $ε$. Our solution improves the previous best result of Greve et al. (Cell Probe Lower Bounds and Approximations for Range Mode, ICALP'10) by reducing the space …
Modern big data applications integrate data from various sources. As a result, these datasets may not satisfy perfect constraints, leading to sparse schema information and non-optimal query performance. The existing approach of PatchIndexes enable the definition of approximate constraints and improve query performance by exploiting the materialized constraint information. As real world data warehouse workloads are often not limited to read-only queries, we enhance the PatchIndex structure towards an update-conscious design in this paper. Therefore, we present a sharded bitmap as the underlying…
Approximate Membership Query structures (AMQs) rely on randomisation for time- and space-efficiency, while introducing a possibility of false positive and false negative answers. Correctness proofs of such structures involve subtle reasoning about bounds on probabilities of getting certain outcomes. Because of these subtleties, a number of unsound arguments in such proofs have been made over the years. In this work, we address the challenge of building rigorous and reusable computer-assisted proofs about probabilistic specifications of AMQs. We describe the framework for systematic decompositi…
We consider the problem of maintaining an approximately maximum (fractional) matching and an approximately minimum vertex cover in a dynamic graph. Starting with the seminal paper by Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. There remains, however, a polynomial gap between the best known worst case update time and the best known amortised update time for this problem, even after allowing for randomisation. Specifically, Bernstein and Stein [ICALP 2015, SODA 2016] have the best known worst case update time. They present a deterministic data…
We present a general framework of designing efficient dynamic approximate algorithms for optimization on undirected graphs. In particular, we develop a technique that, given any problem that admits a certain notion of vertex sparsifiers, gives data structures that maintain approximate solutions in sub-linear update and query time. We illustrate the applicability of our paradigm to the following problems. (1) A fully-dynamic algorithm that approximates all-pair maximum-flows/minimum-cuts up to a nearly logarithmic factor in $\tilde{O}(n^{2/3})$ amortized time against an oblivious adversary, and…
We study the fundamental problem of approximate nearest neighbor search in $d$-dimensional Hamming space $\{0,1\}^d$. We study the complexity of the problem in the famous cell-probe model, a classic model for data structures. We consider algorithms in the cell-probe model with limited adaptivity, where the algorithm makes $k$ rounds of parallel accesses to the data structure for a given $k$. For any $k\ge 1$, we give a simple randomized algorithm solving the approximate nearest neighbor search using $k$ rounds of parallel memory accesses, with $O(k(\log d)^{1/k})$ accesses in total. We also gi…
Pattern matching is a fundamental process in almost every scientific domain. The problem involves finding the positions of a given pattern (usually of short length) in a reference stream of data (usually of large length). The matching can be an exact or as an approximate (inexact). Exact matching is to search for the pattern without allowing for mismatches (or insertions and deletions) of one or more characters in the pattern), while approximate matching is the opposite. For exact matching, several data structures that can be built in linear time and space are used and in practice nowadays. Fo…
Columbia University（哥伦比亚大学）；Northeastern University（东北大学）；University of Toronto（多伦多大学）；MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）
We show that every symmetric normed space admits an efficient nearest neighbor search data structure with doubly-logarithmic approximation. Specifically, for every $n$, $d = n^{o(1)}$, and every $d$-dimensional symmetric norm $\|\cdot\|$, there exists a data structure for $\mathrm{poly}(\log \log n)$-approximate nearest neighbor search over $\|\cdot\|$ for $n$-point datasets achieving $n^{o(1)}$ query time and $n^{1+o(1)}$ space. The main technical ingredient of the algorithm is a low-distortion embedding of a symmetric norm into a low-dimensional iterated product of top-$k$ norms. We also sho…
