We study the problem of identifying the source of a stochastic diffusion process spreading on a graph based on the arrival times of the diffusion at a few queried nodes. In a graph $G=(V,E)$, an unknown source node $v^* \in V$ is drawn uniformly at random, and unknown edge weights $w(e)$ for $e\in E$, representing the propagation delays along the edges, are drawn independently from a Gaussian distribution of mean $1$ and variance $σ^2$. An algorithm then attempts to identify $v^*$ by querying nodes $q \in V$ and being told the length of the shortest path between $q$ and $v^*$ in graph $G$ weighted by $w$. We consider two settings: non-adaptive, in which all query nodes must be decided in advance, and adaptive, in which each query can depend on the results of the previous ones. Both settings are motivated by an application of the problem to epidemic processes (where the source is called patient zero), which we discuss in detail.
We characterize the query complexity when $G$ is an $n$-node path. In the non-adaptive setting, $Θ(nσ^2)$ queries are needed for $σ^2 \leq 1$, and $Θ(n)$ for $σ^2 \geq 1$. In the adaptive setting, somewhat surprisingly, only $Θ(\log\log_{1/σ}n)$ are needed when $σ^2 \leq 1/2$, and $Θ(\log \log n)+O_σ(1)$ when $σ^2 \geq 1/2$. This is the first mathematical study of source identification with time queries in a non-deterministic diffusion process.
In this paper, we study the tradeoff between the approximation guarantee and adaptivity for the problem of maximizing a monotone submodular function subject to a cardinality constraint. The adaptivity of an algorithm is the number of sequential rounds of queries it makes to the evaluation oracle of the function, where in every round the algorithm is allowed to make polynomially-many parallel queries. Adaptivity is an important consideration in settings where the objective function is estimated using samples and in applications where adaptivity is the main running time bottleneck. Previous algo…
Columbia University（哥伦比亚大学）；Weizmann Institute（魏茨曼科学研究所）
Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of \emph{adaptive} testing algorithms, wherein each query may be determined by the answers received to prior queries, and their \emph{non-adaptive} counterparts, in which all queries are independent of answers obtained from previous queries. In this work, we investigate the role of adaptivity in property testing at a finer level. We first quantify the degree of adaptivity of a testing algorithm by considering the number of "rounds of adap…
We study the problem of estimating the number of edges in an $n$-vertex graph, accessed via the Bipartite Independent Set query model introduced by Beame et al. (ITCS '18). In this model, each query returns a Boolean, indicating the existence of at least one edge between two specified sets of nodes. We present a non-adaptive algorithm that returns a $(1\pm ε)$ relative error approximation to the number of edges, with query complexity $\tilde O(ε^{-5}\log^{5} n )$, where $\tilde O(\cdot)$ hides $\textrm{poly}(\log \log n)$ dependencies. This is the first non-adaptive algorithm in this setting a…
Filters (such as Bloom Filters) are data structures that speed up network routing and measurement operations by storing a compressed representation of a set. Filters are space efficient, but can make bounded one-sided errors: with tunable probability epsilon, they may report that a query element is stored in the filter when it is not. This is called a false positive. Recent research has focused on designing methods for dynamically adapting filters to false positives, reducing the number of false positives when some elements are queried repeatedly. Ideally, an adaptive filter would incur a fals…
In this paper we study the limitations of parallelization in convex optimization. A convenient approach to study parallelization is through the prism of \emph{adaptivity} which is an information theoretic measure of the parallel runtime of an algorithm [BS18]. Informally, adaptivity is the number of sequential rounds an algorithm needs to make when it can execute polynomially-many queries in parallel at every round. For combinatorial optimization with black-box oracle access, the study of adaptivity has recently led to exponential accelerations in parallel runtime and the natural question is w…
In many applications of clustering (for example, ontologies or clusterings of animal or plant species), hierarchical clusterings are more descriptive than a flat clustering. A hierarchical clustering over $n$ elements is represented by a rooted binary tree with $n$ leaves, each corresponding to one element. The subtrees rooted at interior nodes capture the clusters. In this paper, we study active learning of a hierarchical clustering using only ordinal queries. An ordinal query consists of a set of three elements, and the response to a query reveals the two elements (among the three elements i…
We consider a problem introduced by Feige, Gamarnik, Neeman, Rácz and Tetali [2020], that of finding a large clique in a random graph $G\sim G(n,\frac{1}{2})$, where the graph $G$ is accessible by queries to entries of its adjacency matrix. The query model allows some limited adaptivity, with a constant number of rounds of queries, and $n^δ$ queries in each round. With high probability, the maximum clique in $G$ is of size roughly $2\log n$, and the goal is to find cliques of size $α\log n$, for $α$ as large as possible. We prove that no two-rounds algorithm is likely to find a clique larger t…
Adaptivity is an important feature of data analysis---typically the choice of questions asked about a dataset depends on previous interactions with the same dataset. However, generalization error is typically bounded in a non-adaptive model, where all questions are specified before the dataset is drawn. Recent work by Dwork et al. (STOC '15) and Hardt and Ullman (FOCS '14) initiated the formal study of this problem, and gave the first upper and lower bounds on the achievable generalization error for adaptive data analysis. Specifically, suppose there is an unknown distribution $\mathcal{P}$ an…
We study the problem of finding a spanning forest in an undirected, $n$-vertex multi-graph under two basic query models. One is the Linear query model which are linear measurements on the incidence vector induced by the edges; the other is the weaker OR query model which only reveals whether a given subset of plausible edges is empty or not. At the heart of our study lies a fundamental problem which we call the {\em single element recovery} problem: given a non-negative real vector $x$ in $N$ dimension, return a single element $x_j > 0$ from the support. Queries can be made in rounds, and our …
University of California San Diego（加利福尼亚大学圣迭戈分校）；Pennsylvania State University（宾夕法尼亚州立大学）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）；Harvard University（哈佛大学）；Northeastern University（东北大学）；Columbia University（哥伦比亚大学）
Adaptivity is an important feature of data analysis---the choice of questions to ask about a dataset often depends on previous interactions with the same dataset. However, statistical validity is typically studied in a nonadaptive model, where all questions are specified before the dataset is drawn. Recent work by Dwork et al. (STOC, 2015) and Hardt and Ullman (FOCS, 2014) initiated the formal study of this problem, and gave the first upper and lower bounds on the achievable generalization error for adaptive data analysis. Specifically, suppose there is an unknown distribution $\mathbf{P}$ and…
We show that every algorithm for testing $n$-variate Boolean functions for monotonicity must have query complexity $\tildeΩ(n^{1/4})$. All previous lower bounds for this problem were designed for non-adaptive algorithms and, as a result, the best previous lower bound for general (possibly adaptive) monotonicity testers was only $Ω(\log n)$. Combined with the query complexity of the non-adaptive monotonicity tester of Khot, Minzer, and Safra (FOCS 2015), our lower bound shows that adaptivity can result in at most a quadratic reduction in the query complexity for testing monotonicity. By contras…
