This paper initiates the study of active learning for exact recovery of partitions exclusively through access to a same-cluster oracle in the presence of bounded adversarial error. We first highlight a novel connection between learning partitions and correlation clustering. Then we use this connection to build a Rényi-Ulam style analytical framework for this problem, and prove upper and lower bounds on its worst-case query complexity. Further, we bound the expected performance of a relevant randomized algorithm. Finally, we study the relationship between adaptivity and query complexity for this problem and related variants.
We construct data structures for extremal and pairwise distances in directed graphs in the presence of transient edge failures. Henzinger et al. [ITCS 2017] initiated the study of fault-tolerant (sensitivity) oracles for the diameter and vertex eccentricities. We extend this with a special focus on space efficiency. We present several new data structures, among them the first fault-tolerant eccentricity oracle for dual failures in subcubic space. We further prove lower bounds that show limits to approximation vs. space and diameter vs. space trade-offs for fault-tolerant oracles. They highligh…
University of Sheffield（谢菲尔德大学）；University of Southern California（南加州大学）
Motivated by applications in crowdsourced entity resolution in database, signed edge prediction in social networks and correlation clustering, Mazumdar and Saha [NIPS 2017] proposed an elegant theoretical model for studying clustering with a faulty oracle. In this model, given a set of $n$ items which belong to $k$ unknown groups (or clusters), our goal is to recover the clusters by asking pairwise queries to an oracle. This oracle can answer the query that ``do items $u$ and $v$ belong to the same cluster?''. However, the answer to each pairwise query errs with probability $\varepsilon$, for …
In a graph $G$ with a source $s$, we design a distance oracle that can answer the following query: Query$(s,t,e)$ -- find the length of shortest path from a fixed source $s$ to any destination vertex $t$ while avoiding any edge $e$. We design a deterministic algorithm that builds such an oracle in $\tilde{O}(m\sqrt n)$ time. Our oracle uses $\tilde{O}(n\sqrt n)$ space and can answer queries in $\tilde{O}(1)$ time. Our oracle is an improvement of the work of Bilò et al. (ESA 2021) in the preprocessing time, which constructs the first deterministic oracle for this problem in $\tilde{O}(m\sqrt n+…
An approximate distance oracle is a succinct data structure that provides fast answers to distance queries between any two nodes. In this paper we consider approximate distance oracles for general undirected graphs with non-negative edge weights with constant query time. We present a distance oracle of size O(k n^{1+1/k}), with 2k-1 stretch and O(1) query time. This improves the O(log{k}) query time of Wulff-Nilsen's distance oracle [SODA '13], which in turn improved the O(k) query time of Thorup and Zwick's distance oracle [J. ACM '05].
We give a $(1+ε)$-approximate distance oracle with $O(1)$ query time for an undirected planar graph $G$ with $n$ vertices and non-negative edge lengths. For $ε>0$ and any two vertices $u$ and $v$ in $G$, our oracle gives a distance $\tilde{d}(u,v)$ with stretch $(1+ε)$ in $O(1)$ time. The oracle has size $O(n\log n ((\log n)/ε+f(ε)))$ and pre-processing time $O(n\log n((\log^3 n)/ε^2+f(ε)))$, where $f(ε)=2^{O(1/ε)}$. This is the first $(1+ε)$-approximate distance oracle with $O(1)$ query time independent of $ε$ and the size and pre-processing time nearly linear in $n$, and improves the query t…
We present new tradeoffs between space and query-time for exact distance oracles in directed weighted planar graphs. These tradeoffs are almost optimal in the sense that they are within polylogarithmic, sub-polynomial or arbitrarily small polynomial factors from the na\"ıve linear space, constant query-time lower bound. These tradeoffs include: (i) an oracle with space $\tilde{O}(n^{1+ε})$ and query-time $\tilde{O}(1)$ for any constant $ε>0$, (ii) an oracle with space $\tilde{O}(n)$ and query-time $\tilde{O}(n^ε)$ for any constant $ε>0$, and (iii) an oracle with space $n^{1+o(1)}$ and query-ti…
Indian Institute of Technology, Bombay（印度理工学院孟买分校）
We consider the problem of identifying the subset $\mathcal{S}^γ_{\mathcal{P}}$ of elements in the support of an underlying distribution $\mathcal{P}$ whose probability value is larger than a given threshold $γ$, by actively querying an oracle to gain information about a sequence $X_1, X_2, \ldots$ of $i.i.d.$ samples drawn from $\mathcal{P}$. We consider two query models: $(a)$ each query is an index $i$ and the oracle return the value $X_i$ and $(b)$ each query is a pair $(i,j)$ and the oracle gives a binary answer confirming if $X_i = X_j$ or not. For each of these query models, we design s…
Given an undirected, unweighted planar graph $G$ with $n$ vertices, we present a truly subquadratic size distance oracle for reporting exact shortest-path distances between any pair of vertices of $G$ in constant time. For any $\varepsilon > 0$, our distance oracle takes up $O(n^{5/3+\varepsilon})$ space and is capable of answering shortest-path distance queries exactly for any pair of vertices of $G$ in worst-case time $O(\log (1/\varepsilon))$. Previously no truly sub-quadratic size distance oracles with constant query time for answering exact all-pairs shortest paths distance queries existe…
We consider approximate distance oracles for edge-weighted n-vertex undirected planar graphs. Given fixed epsilon > 0, we present a (1+epsilon)-approximate distance oracle with O(n(loglog n)^2) space and O((loglog n)^3) query time. This improves the previous best product of query time and space of the oracles of Thorup (FOCS 2001, J. ACM 2004) and Klein (SODA 2002) from O(n log n) to O(n(loglog n)^5).
Tree path minimum query problem is a fundamental problem while processing trees, and is used widely in minimum spanning tree verification and randomized minimum spanning tree algorithms. In this paper, we study the possibility of building an oracle in advance, which is able to answer the queries efficiently. We present an algorithm based on Boruvka trees. Our algorithm is the first to achieve a near-optimal bound on query time, while matching the currently optimal trade-off between construction time and the number of comparisons required at query. Particularly, in order to answer each query wi…
A local property reconstructor for a graph property is an algorithm which, given oracle access to the adjacency list of a graph that is "close" to having the property, provides oracle access to the adjacency matrix of a "correction" of the graph, i.e. a graph which has the property and is close to the given graph. For this model, we achieve local property reconstructors for the properties of connectivity and $k$-connectivity in undirected graphs, and the property of strong connectivity in directed graphs. Along the way, we present a method of transforming a local reconstructor (which acts as a…
Indian Institute of Technology, Bombay（印度理工学院孟买分校）
We study a variant of the canonical k-center problem over a set of vertices in a metric space, where the underlying distances are apriori unknown. Instead, we can query an oracle which provides noisy/incomplete estimates of the distance between any pair of vertices. We consider two oracle models: Dimension Sampling where each query to the oracle returns the distance between a pair of points in one dimension; and Noisy Distance Sampling where the oracle returns the true distance corrupted by noise. We propose active algorithms, based on ideas such as UCB, Thompson Sampling and Track-and-Stop de…
University of L’Aquila（拉奎拉大学）；Indian Institute of Technology Delhi（德里印度理工学院）；Reichman University（雷希曼大学）；Hasso Plattner Institute, University of Potsdam（波茨坦大学哈索·普拉特纳研究所）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）
Given an undirected $n$-vertex planar graph $G=(V,E,ω)$ with non-negative edge weight function $ω:E\rightarrow \mathbb R$ and given an assigned label to each vertex, a vertex-labeled distance oracle is a data structure which for any query consisting of a vertex $u$ and a label $λ$ reports the shortest path distance from $u$ to the nearest vertex with label $λ$. We show that if there is a distance oracle for undirected $n$-vertex planar graphs with non-negative edge weights using $s(n)$ space and with query time $q(n)$, then there is a vertex-labeled distance oracle with $\tilde{O}(s(n))$ space…
In this paper, we consider reachability oracles and reachability preservers for directed graphs/networks prone to edge/node failures. Let $G = (V, E)$ be a directed graph on $n$-nodes, and $P\subseteq V\times V$ be a set of vertex pairs in $G$. We present the first non-trivial constructions of single and dual fault-tolerant pairwise reachability oracle with constant query time. Furthermore, we provide extremal bounds for sparse fault-tolerant reachability preservers, resilient to two or more failures. Prior to this work, such oracles and reachability preservers were widely studied for the spec…
Yale University（耶鲁大学）；University of California San Diego（加利福尼亚大学圣地亚哥分校）
There is increasing interest in learning algorithms that involve interaction between human and machine. Comparison-based queries are among the most natural ways to get feedback from humans. A challenge in designing comparison-based interactive learning algorithms is coping with noisy answers. The most common fix is to submit a query several times, but this is not applicable in many situations due to its prohibitive cost and due to the unrealistic assumption of independent noise in different repetitions of the same query. In this paper, we introduce a new weak oracle model, where a non-maliciou…
We describe a slightly sub-exponential time algorithm for learning parity functions in the presence of random classification noise. This results in a polynomial-time algorithm for the case of parity functions that depend on only the first O(log n log log n) bits of input. This is the first known instance of an efficient noise-tolerant algorithm for a concept class that is provably not learnable in the Statistical Query model of Kearns. Thus, we demonstrate that the set of problems learnable in the statistical query model is a strict subset of those problems learnable in the presence of noise i…
In this paper, we consider the problem of tolerant junta testing for boolean function. Compared with the prior work by Blais et al., we provide improved results in terms of both the parameter gap and query complexity. Compared with the prior work by De et al., our work did not close the parameter gap, but we removed the $2^k$ term of the query complexity.
A novel landmark-based oracle (CFLAT) is presented, which provides earliest-arrival-time route plans in time-dependent road networks. To our knowledge, this is the first oracle that preprocesses combinatorial structures (collections of time-stamped min-travel-time-path trees) rather than travel-time functions. The preprocessed data structure is exploited by a new query algorithm (CFCA) which computes (and pays for it), apart from earliest-arrival-time estimations, the actual connecting path that preserves the theoretical approximation guarantees. To make it practical and tackle the main burden…
A $(1+ε)$-approximate distance oracle of an edge-weighted graph is a data structure that returns an approximate shortest path distance between any two query vertices up to a $(1+ε)$ factor. Thorup (FOCS 2001, JACM 2004) and Klein (SODA 2002) independently constructed a $(1+ε)$-approximate distance oracle with $O(n\log n)$ space, measured in number of words, and $O(1)$ query time when $G$ is an undirected planar graph with $n$ vertices and $ε$ is a fixed constant. Many follow-up works gave $(1+ε)$-approximate distance oracles with various trade-offs between space and query time. However, improv…
We consider the basic problem of querying an expert oracle for labeling a dataset in machine learning. This is typically an expensive and time consuming process and therefore, we seek ways to do so efficiently. The conventional approach involves comparing each sample with (the representative of) each class to find a match. In a setting with $N$ equally likely classes, this involves $N/2$ pairwise comparisons (queries per sample) on average. We consider a $k$-ary query scheme with $k\ge 2$ samples in a query that identifies (dis)similar items in the set while effectively exploiting the associat…
We study the problem of reconstructing a hidden graph given access to a distance oracle. We design randomized algorithms for the following problems: reconstruction of a degree bounded graph with query complexity $\tilde{O}(n^{3/2})$; reconstruction of a degree bounded outerplanar graph with query complexity $\tilde{O}(n)$; and near-optimal approximate reconstruction of a general graph.
Given an undirected unweighted graph $G$ and a source set $S$ of $|S| = σ$ sources, we want to build a data structure which can process the following query {\sc Q}$(s,t,e):$ find the shortest distance from $s$ to $t$ avoiding an edge $e$, where $s \in S$ and $t \in V$. When $σ=n$, Demetrescu, Thorup, Chowdhury and Ramachandran (SIAM Journal of Computing, 2008) designed an algorithm with $\tilde O(n^2)$ space ($\tilde O(\cdot)$ hides poly $\log n$ factor.) and $O(1)$ query time. A natural open question is to generalize this result to any number of sources. Recently, Bil{ò} et. al. (STACS 2018) …
We study networks obeying \emph{time-dependent} min-cost path metrics, and present novel oracles for them which \emph{provably} achieve two unique features: % (i) \emph{subquadratic} preprocessing time and space, \emph{independent} of the metric's amount of disconcavity; % (ii) \emph{sublinear} query time, in either the network size or the actual Dijkstra-Rank of the query at hand.
We introduce a new model for testing graph properties which we call the \emph{rejection sampling model}. We show that testing bipartiteness of $n$-nodes graphs using rejection sampling queries requires complexity $\widetildeΩ(n^2)$. Via reductions from the rejection sampling model, we give three new lower bounds for tolerant testing of Boolean functions of the form $f\colon\{0,1\}^n\to \{0,1\}$: $\bullet$Tolerant $k$-junta testing with \emph{non-adaptive} queries requires $\widetildeΩ(k^2)$ queries. $\bullet$Tolerant unateness testing requires $\widetildeΩ(n)$ queries. $\bullet$Tolerant unaten…
We consider exact distance oracles for directed weighted planar graphs in the presence of failing vertices. Given a source vertex $u$, a target vertex $v$ and a set $X$ of $k$ failed vertices, such an oracle returns the length of a shortest $u$-to-$v$ path that avoids all vertices in $X$. We propose oracles that can handle any number $k$ of failures. We show several tradeoffs between space, query time, and preprocessing time. In particular, for a directed weighted planar graph with $n$ vertices and any constant $k$, we show an $\tilde{\mathcal{O}}(n)$-size, $\tilde{\mathcal{O}}(\sqrt{n})$-quer…
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…
Thorup [FOCS'01, JACM'04] and Klein [SODA'01] independently showed that there exists a $(1+ε)$-approximate distance oracle for planar graphs with $O(n (\log n)ε^{-1})$ space and $O(ε^{-1})$ query time. While the dependency on $n$ is nearly linear, the space-query product of their oracles depend quadratically on $1/ε$. Many follow-up results either improved the space \emph{or} the query time of the oracles while having the same, sometimes worst, dependency on $1/ε$. Kawarabayashi, Sommer, and Thorup [SODA'13] were the first to improve the dependency on $1/ε$ from quadratic to nearly linear (at …
In a recent breakthrough, Charalampopoulos, Gawrychowski, Mozes, and Weimann (STOC 2019) showed that exact distance queries on planar graphs could be answered in $n^{o(1)}$ time by a data structure occupying $n^{1+o(1)}$ space, i.e., up to $o(1)$ terms, optimal exponents in time (0) and space (1) can be achieved simultaneously. Their distance query algorithm is recursive: it makes successive calls to a point-location algorithm for planar Voronoi diagrams, which involves many recursive distance queries. The depth of this recursion is non-constant and the branching factor logarithmic, leading to…
In 2001 Thorup and Zwick devised a distance oracle, which given an $n$-vertex undirected graph and a parameter $k$, has size $O(k n^{1+1/k})$. Upon a query $(u,v)$ their oracle constructs a $(2k-1)$-approximate path $Π$ between $u$ and $v$. The query time of the Thorup-Zwick's oracle is $O(k)$, and it was subsequently improved to $O(1)$ by Chechik. A major drawback of the oracle of Thorup and Zwick is that its space is $Ω(n \cdot \log n)$. Mendel and Naor devised an oracle with space $O(n^{1+1/k})$ and stretch $O(k)$, but their oracle can only report distance estimates and not actual paths. In…
Given a vertex-labeled graph, each vertex $v$ is attached with a label from a set of labels. The vertex-label query desires the length of the shortest path from the given vertex to the set of vertices with the given label. We show how to construct an oracle if the given graph is planar, such that $O(\frac{1}εn\log n)$ storing space is needed, and any vertex-label query could be answered in $O(\frac{1}ε\log n\log ρ)$ time with stretch $1+ε$. $ρ$ is the radius of the given graph, which is half of the diameter. For the case that $ρ= O(\log n)$, we construct an oracle that achieves $O(\log n)$ que…
This paper addresses the problem of finding the nearest neighbor (or one of the R-nearest neighbors) of a query object q in a database of n objects. In contrast with most existing approaches, we can only access the ``hidden'' space in which the objects live through a similarity oracle. The oracle, given two reference objects and a query object, returns the reference object closest to the query object. The oracle attempts to model the behavior of human users, capable of making statements about similarity, but not of assigning meaningful numerical values to distances between objects.
We define and study a new type of quantum oracle, the quantum conditional oracle, which provides oracle access to the conditional probabilities associated with an underlying distribution. Amongst other properties, we (a) obtain speed-ups over the best known quantum algorithms for identity testing, equivalence testing and uniformity testing of probability distributions; (b) study the power of these oracles for testing properties of boolean functions, and obtain an algorithm for checking whether an $n$-input $m$-output boolean function is balanced or $ε$-far from balanced; and (c) give a sub-lin…
University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；Blueqat Research（Blueqat研究院）
Given a partition of a graph into connected components, the membership oracle asserts whether any two vertices of the graph lie in the same component or not. We prove that for $n\ge k\ge 2$, learning the components of an $n$-vertex hidden graph with $k$ components requires at least $(k-1)n-\binom k2$ membership queries. Our result improves on the best known information-theoretic bound of $Ω(n\log k)$ queries, and exactly matches the query complexity of the algorithm introduced by [Reyzin and Srivastava, 2007] for this problem. Additionally, we introduce an oracle, with access to which one can …
Consider the family of bounded degree graphs in any minor-closed family (such as planar graphs). Let d be the degree bound and n be the number of vertices of such a graph. Graphs in these classes have hyperfinite decompositions, where, for a sufficiently small \e > 0, one removes \edn edges to get connected components of size independent of n. An important tool for sublinear algorithms and property testing for such classes is the partition oracle, introduced by the seminal work of Hassidim-Kelner-Nguyen-Onak (FOCS 2009). A partition oracle is a local procedure that gives consistent access to a…
For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an $O(n^{5/3})$-space distance oracle which answers exact distance queries in $O(\log n)$ time for $n$-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space i.e., space $O(n^{2 - ε})$ for some constant $ε> 0$) either required query time polynomial in $n$ or could only answer approximate distance queries. Furthermore, we show how to trade-off time and space: for any $S \ge n^{3/2}$, we show how to obtain an…
Accelerating Matroid Optimization through Fast Imprecise Oracles
Institute of Mathematics, Technical University of Berlin（柏林工业大学数学研究所）；Faculty of Mathematics and Computer Science, University of Bremen（不来梅大学数学与计算机科学学院）；Zhejiang University（浙江大学）；Centrum Wiskunde & Informatica（荷兰国家数学和计算机科学研究院）
We consider the problem of preprocessing two strings $S$ and $T$, of lengths $m$ and $n$, respectively, in order to be able to efficiently answer the following queries: Given positions $i,j$ in $S$ and positions $a,b$ in $T$, return the optimal alignment of $S[i \mathinner{.\,.} j]$ and $T[a \mathinner{.\,.} b]$. Let $N=mn$. We present an oracle with preprocessing time $N^{1+o(1)}$ and space $N^{1+o(1)}$ that answers queries in $\log^{2+o(1)}N$ time. In other words, we show that we can query the alignment of every two substrings in almost the same time it takes to compute just the alignment of…
We present new and improved data structures that answer exact node-to-node distance queries in planar graphs. Such data structures are also known as distance oracles. For any directed planar graph on n nodes with non-negative lengths we obtain the following: * Given a desired space allocation $S\in[n\lg\lg n,n^2]$, we show how to construct in $\tilde O(S)$ time a data structure of size $O(S)$ that answers distance queries in $\tilde O(n/\sqrt S)$ time per query. As a consequence, we obtain an improvement over the fastest algorithm for k-many distances in planar graphs whenever $k\in[\sqrt n,n)…
In this paper we provide sub-linear algorithms for several fundamental problems in the setting in which the input graph excludes a fixed minor, i.e., is a minor-free graph. In particular, we provide the following algorithms for minor-free unbounded degree graphs. (1) A tester for Hamiltonicity with two-sided error with $poly(1/ε)$-query complexity, where $ε$ is the proximity parameter. (2) A local algorithm, as defined by Rubinfeld et al. (ICS 2011), for constructing a spanning subgraph with almost minimum weight, specifically, at most a factor $(1+ε)$ of the optimum, with $poly(1/ε)$-query co…
In the sensitive distance oracle problem, there are three phases. We first preprocess a given directed graph $G$ with $n$ nodes and integer weights from $[-W,W]$. Second, given a single batch of $f$ edge insertions and deletions, we update the data structure. Third, given a query pair of nodes $(u,v)$, return the distance from $u$ to $v$. In the easier problem called sensitive reachability oracle problem, we only ask if there exists a directed path from $u$ to $v$. Our first result is a sensitive distance oracle with $\tilde{O}(Wn^{ω+(3-ω)μ})$ preprocessing time, $\tilde{O}(Wn^{2-μ}f^{2}+Wnf^ω…
Let $G$ be an $n$-node and $m$-edge positively real-weighted undirected graph. For any given integer $f \ge 1$, we study the problem of designing a sparse \emph{f-edge-fault-tolerant} ($f$-EFT) $σ${\em -approximate single-source shortest-path tree} ($σ$-ASPT), namely a subgraph of $G$ having as few edges as possible and which, following the failure of a set $F$ of at most $f$ edges in $G$, contains paths from a fixed source that are stretched at most by a factor of $σ$. To this respect, we provide an algorithm that efficiently computes an $f$-EFT $(2|F|+1)$-ASPT of size $O(f n)$. Our structure…
We consider the problem of building Distance Sensitivity Oracles (DSOs). Given a directed graph $G=(V, E)$ with edge weights in $\{1, 2, \dots, M\}$, we need to preprocess it into a data structure, and answer the following queries: given vertices $u,v\in V$ and a failed vertex or edge $f\in (V\cup E)$, output the length of the shortest path from $u$ to $v$ that does not go through $f$. Our main result is a simple DSO with $\tilde{O}(n^{2.7233}M)$ preprocessing time and $O(1)$ query time. Moreover, if the input graph is undirected, the preprocessing time can be improved to $\tilde{O}(n^{2.6865}…
University of Pennsylvania（宾夕法尼亚大学）；MIT（麻省理工学院）；UT Austin（德克萨斯大学奥斯汀分校）
A natural problem in high-dimensional inference is to decide if a classifier $f:\mathbb{R}^n \rightarrow \{-1,1\}$ depends on a small number of linear directions of its input data. Call a function $g: \mathbb{R}^n \rightarrow \{-1,1\}$, a linear $k$-junta if it is completely determined by some $k$-dimensional subspace of the input space. A recent work of the authors showed that linear $k$-juntas are testable. Thus there exists an algorithm to distinguish between: 1. $f: \mathbb{R}^n \rightarrow \{-1,1\}$ which is a linear $k$-junta with surface area $s$, 2. $f$ is $ε$-far from any linear $k$-j…
