In this paper, we consider the problem of reconstructing a directed graph using path queries. In this query model of learning, a graph is hidden from the learner, and the learner can access information about it with path queries. For a source and destination node, a path query returns whether there is a directed path from the source to the destination node in the hidden graph. In this paper we first give bounds for learning graphs on $n$ vertices and $k$ strongly connected components. We then study the case of bounded degree directed trees and give new algorithms for learning "almost-trees" -- directed trees to which extra edges have been added. We also give some lower bound constructions justifying our approach.
We present a randomized algorithm for reconstructing directed rooted trees of $n$ nodes and node degree at most $d$, by asking at most $O(dn\log^2 n)$ path queries. Each path query takes as input an origin node and a target node, and answers whether there is a directed path from the origin to the target. Regarding lower bounds, we show that any randomized algorithm requires at least $Ω(n\log n)$ queries, while any deterministic algorithm requires at least $Ω(dn)$ queries. Additionally, we present a $O(dn\log^3 n)$ randomized algorithm for noisy queries, and a $O(dn\log^2 n)$ randomized algorit…
Model extraction attacks have renewed interest in the classic problem of learning neural networks from queries. In this work we give the first polynomial-time algorithm for learning arbitrary one hidden layer neural networks activations provided black-box access to the network. Formally, we show that if $F$ is an arbitrary one hidden layer neural network with ReLU activations, there is an algorithm with query complexity and running time that is polynomial in all parameters that outputs a network $F'$ achieving low square loss relative to $F$ with respect to the Gaussian measure. While a number…
Given a directed graph, G=(V,E), a path query, path(u,v), returns whether there is a directed path from u to v in G, for u,v vertices in V. Given only V, exactly learning all the edges in G using path queries is often impossible, since path queries cannot detect transitive edges. In this paper, we study the query complexity of exact learning for cases when learning G is possible using path queries. In particular, we provide efficient learning algorithms, as well as lower bounds, for multitrees and almost-trees, including butterfly networks.
The Quantitative Group Testing (QGT) is about learning a (hidden) subset $K$ of some large domain $N$ using a sequence of queries, where a result of a query provides information about the size of the intersection of the query with the unknown subset $K$. Almost all previous work focused on randomized algorithms minimizing the number of queries; however, in case of large domains $N$, randomization may result in a significant deviation from the expected precision. Others assumed unlimited computational power (existential results) or adaptiveness of queries. In this work we propose efficient non-…
We study a basic problem of approximating the size of an unknown set $S$ in a known universe $U$. We consider two versions of the problem. In both versions the algorithm can specify subsets $T\subseteq U$. In the first version, which we refer to as the group query or subset query version, the algorithm is told whether $T\cap S$ is non-empty. In the second version, which we refer to as the subset sampling version, if $T\cap S$ is non-empty, then the algorithm receives a uniformly selected element from $T\cap S$. We study the difference between these two versions under different conditions on th…
We develop the data structure PReaCH (for Pruned Reachability Contraction Hierarchies) which supports reachability queries in a directed graph, i.e., it supports queries that ask whether two nodes in the graph are connected by a directed path. PReaCH adapts the contraction hierarchy speedup techniques for shortest path queries to the reachability setting. The resulting approach is surprisingly simple and guarantees linear space and near linear preprocessing time. Orthogonally to that, we improve existing pruning techniques for the search by gathering more information from a single DFS-traversa…
We introduce a new dynamic data structure for maintaining the strongly connected components (SCCs) of a directed graph (digraph) under edge deletions, so as to answer a rich repertoire of connectivity queries. Our main technical contribution is a decremental data structure that supports sensitivity queries of the form "are $ u $ and $ v $ strongly connected in the graph $ G \setminus w $?", for any triple of vertices $ u, v, w $, while $ G $ undergoes deletions of edges. Our data structure processes a sequence of edge deletions in a digraph with $n$ vertices in $O(m n \log{n})$ total time and …
We consider the problem of finding an edge in a hidden undirected graph $G = (V, E)$ with $n$ vertices, in a model where we only allowed queries that ask whether or not a subset of vertices contains an edge. We study the non-adaptive model and show that while in the deterministic model the optimal algorithm requires $\binom{n}{2}$ queries (i.e., querying for any possible edge separately), in the randomized model $\tildeΘ(n)$ queries are sufficient (and needed) in order to find an edge. In addition, we study the query complexity for specific families of graphs, including Stars, Cliques, and Mat…
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 an unweighted, directed graph $G$ with the diameter $D$. In this paper, we introduce the framework for counting cycles and walks of given length in matrix multiplication time $\widetilde{O}(n^ω)$. The framework is based on the fast decomposition into Frobenius normal form and the Hankel matrix-vector multiplication. It allows us to solve the All-Nodes Shortest Cycles, All-Pairs All Walks problems efficiently and also give some improvement upon distance queries in unweighted graphs.
This paper investigates two types of graph queries: {\em single source distance (SSD)} queries and {\em single source shortest path (SSSP)} queries. Given a node $v$ in a graph $G$, an SSD query from $v$ asks for the distance from $v$ to any other node in $G$, while an SSSP query retrieves the shortest path from $v$ to any other node. These two types of queries are fundamental building blocks of numerous graph algorithms, and they find important applications in graph analysis, especially in the computation of graph measures. Most of the existing solutions for SSD and SSSP queries, however, req…
We study reachability and shortest paths problems in dynamic directed graphs. Whereas algebraic dynamic data structures supporting edge updates and reachability/distance queries have been known for quite a long time, they do not, in general, allow reporting the underlying paths within the same time bounds, especially against an adaptive adversary. In this paper we develop the first known fully dynamic reachability data structures working against an adaptive adversary and supporting edge updates and path queries for two natural variants: (1) point-to-point path reporting, and (2) single-source …
Let $G = (V,E,w)$ be a weighted, digraph subject to a sequence of adversarial edge deletions. In the decremental single-source reachability problem (SSR), we are given a fixed source $s$ and the goal is to maintain a data structure that can answer path-queries $s \rightarrowtail v$ for any $v \in V$. In the more general single-source shortest paths (SSSP) problem the goal is to return an approximate shortest path to $v$, and in the SCC problem the goal is to maintain strongly connected components of $G$ and to answer path queries within each component. All of these problems have been very acti…
Massive network exploration is an important research direction with many applications. In such a setting, the network is, usually, modeled as a graph $G$, whereas any structural information of interest is extracted by inspecting the way nodes are connected together. In the case where the adjacency matrix or the adjacency list of $G$ is available, one can directly apply graph mining algorithms to extract useful knowledge. However, there are cases where this is not possible because the graph is \textit{hidden} or \textit{implicit}, meaning that the edges are not recorded explicitly in the form o…
In this paper, we initiate the study of the dynamic maintenance of $2$-edge-connectivity relationships in directed graphs. We present an algorithm that can update the $2$-edge-connected blocks of a directed graph with $n$ vertices through a sequence of $m$ edge insertions in a total of $O(mn)$ time. After each insertion, we can answer the following queries in asymptotically optimal time: (i) Test in constant time if two query vertices $v$ and $w$ are $2$-edge-connected. Moreover, if $v$ and $w$ are not $2$-edge-connected, we can produce in constant time a "witness" of this property, by exhibit…
We consider distance queries in vertex-labeled planar graphs. For any fixed $0 < ε\leq 1/2$ we show how to preprocess a directed planar graph with vertex labels and arc lengths into a data structure that answers queries of the following form. Given a vertex $u$ and a label $λ$ return a $(1+ε)$-approximation of the distance from $u$ to its closest vertex with label $λ$. For a directed planar graph with $n$ vertices, such that the ratio of the largest to smallest arc length is bounded by $N$, the preprocessing time is $O(ε^{-2}n\lg^{3}{n}\lg(nN))$, the data structure size is $O(ε^{-1}n\lg{n}\lg(…
We consider deterministic algorithms for the well-known hidden subgroup problem ($\mathsf{HSP}$): for a finite group $G$ and a finite set $X$, given a function $f:G \to X$ and the promise that for any $g_1, g_2 \in G, f(g_1) = f(g_2)$ iff $g_1H=g_2H$ for a subgroup $H \le G$, the goal of the decision version is to determine whether $H$ is trivial or not, and the goal of the identification version is to identify $H$. An algorithm for the problem should query $f(g)$ for $g\in G$ at least as possible. Nayak asked whether there exist deterministic algorithms with $O(\sqrt{\frac{|G|}{|H|}})$ query …
We study the computational problem of checking whether a quantified conjunctive query (a first-order sentence built using only conjunction as Boolean connective) is true in a finite poset (a reflexive, antisymmetric, and transitive directed graph). We prove that the problem is already NP-hard on a certain fixed poset, and investigate structural properties of posets yielding fixed-parameter tractability when the problem is parameterized by the query. Our main algorithmic result is that model checking quantified conjunctive queries on posets of bounded width is fixed-parameter tractable (the wid…
We consider the problem of recovering (that is, interpolating) and identity testing of a "hidden" monic polynomial $f$, given an oracle access to $f(x)^e$ for $x\in{\mathbb F_q}$ (extension fields access is not permitted). The naive interpolation algorithm needs $O(e\, \mathrm{deg}\, f)$ queries and thus requires $e\, \mathrm{deg}\, f<q$. We design algorithms that are asymptotically better in certain cases; requiring only $e^{o(1)}$ queries to the oracle. In the randomized (and quantum) setting, we give a substantially better interpolation algorithm, that requires only $O(\mathrm{deg}\, f \log…
Given a directed graph G = (V,E) and an integer k>=1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V, E_H) that has (1) the same transitive-closure as G and (2) diameter at most k. These spanners were implicitly studied in access control, data structures, and property testing, and properties of these spanners have been rediscovered over the span of 20 years. The main goal in each of these applications is to obtain the sparsest k-TC-spanners. We bring these diverse areas under the unifying framework of TC-spanners. We initiate the study of approximability of the s…
We study property testing in directed graphs in the bounded degree model, where we assume that an algorithm may only query the outgoing edges of a vertex, a model proposed by Bender and Ron in 2002. As our first main result, we we present a property testing algorithm for strong connectivity in this model, having a query complexity of $\mathcal{O}(n^{1-ε/(3+α)})$ for arbitrary $α>0$; it is based on a reduction to estimating the vertex indegree distribution. For subgraph-freeness we give a property testing algorithm with a query complexity of $\mathcal{O}(n^{1-1/k})$, where $k$ is the number of …
Hub labeling schemes are popular methods for computing distances on road networks and other large complex networks, often answering to a query within a few microseconds for graphs with millions of edges. In this work, we study their algorithmic applications beyond distance queries. We focus on eccentricity queries and distance-sum queries, for several versions of these problems on directed weighted graphs, that is in part motivated by their importance in facility location problems. On the negative side, we show conditional lower bounds for these above problems on unweighted undirected sparse g…
We present an $O(n^{1.5})$-space distance oracle for directed planar graphs that answers distance queries in $O(\log n)$ time. Our oracle both significantly simplifies and significantly improves the recent oracle of Cohen-Addad, Dahlgaard and Wulff-Nilsen [FOCS 2017], which uses $O(n^{5/3})$-space and answers queries in $O(\log n)$ time. We achieve this by designing an elegant and efficient point location data structure for Voronoi diagrams on planar graphs. We further show a smooth tradeoff between space and query-time. For any $S\in [n,n^2]$, we show an oracle of size $S$ that answers querie…
We present deterministic algorithms for the Hidden Subgroup Problem. The first algorithm, for abelian groups, achieves the same asymptotic worst-case query complexity as the optimal randomized algorithm, namely O($\sqrt{ n}\,$), where $n$ is the order of the group. The analogous algorithm for non-abelian groups comes within a $\sqrt{ \log n}$ factor of the optimal randomized query complexity. The best known randomized algorithm for the Hidden Subgroup Problem has expected query complexity that is sensitive to the input, namely O($\sqrt{ n/m}\,$), where $m$ is the order of the hidden subgroup. …
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…
Consider the following "local" cut-detection problem in a directed graph: We are given a seed vertex $x$ and need to remove at most $k$ edges so that at most $ν$ edges can be reached from $x$ (a "local" cut) or output $\bot$ to indicate that no such cut exists. If we are given query access to the input graph, then this problem can in principle be solved without reading the whole graph and with query complexity depending on $k$ and $ν$. In this paper we consider a slack variant of this problem where, when such a cut exists, we can output a cut with up to $O(kν)$ edges reachable from $x$. We pre…
Consider a variant of the Mastermind game in which queries are $\ell_p$ distances, rather than the usual Hamming distance. That is, a codemaker chooses a hidden vector $\mathbf{y}\in\{-k,-k+1,\dots,k-1,k\}^n$ and answers to queries of the form $\Vert\mathbf{y}-\mathbf{x}\Vert_p$ where $\mathbf{x}\in\{-k,-k+1,\dots,k-1,k\}^n$. The goal is to minimize the number of queries made in order to correctly guess $\mathbf{y}$. Motivated by this question, in this work, we develop a nonadaptive polynomial time algorithm that works for a natural class of separable distance measures, i.e.\ coordinate-wise s…
