The representation of a dynamic ordered set of $n$ integer keys drawn from a universe of size $m$ is a fundamental data structuring problem. Many solutions to this problem achieve optimal time but take polynomial space, therefore preserving time optimality in the \emph{compressed} space regime is the problem we address in this work. For a polynomial universe $m = n^{Θ(1)}$, we give a solution that takes $\textsf{EF}(n,m) + o(n)$ bits, where $\textsf{EF}(n,m) \leq n\lceil \log_2(m/n)\rceil + 2n$ is the cost in bits of the \emph{Elias-Fano} representation of the set, and supports random access to the $i$-th smallest element in $O(\log n/ \log\log n)$ time, updates and predecessor search in $O(\log\log n)$ time. These time bounds are optimal.
Reducing the database space overhead is critical in big-data processing. In this paper, we revisit oblivious RAM (ORAM) using big-data standard for the database space overhead. ORAM is a cryptographic primitive that enables users to perform arbitrary database accesses without revealing the access pattern to the server. It is particularly important today since cloud services become increasingly common making it necessary to protect users' private information from database access pattern analyses. Previous ORAM studies focused mostly on reducing the access overhead. Consequently, the access over…
The recent breakthrough paper by Calude et al. has given the first algorithm for solving parity games in quasi-polynomial time, where previously the best algorithms were mildly subexponential. We devise an alternative quasi-polynomial time algorithm based on progress measures, which allows us to reduce the space required from quasi-polynomial to nearly linear. Our key technical tools are a novel concept of ordered tree coding, and a succinct tree coding result that we prove using bounded adaptive multi-counters, both of which are interesting in their own right.
NTT Communication Science Laboratories（NTT通信科学实验室）；Graduate School of Information Science and Technology, The University of Tokyo（东京大学信息理工学系研究科）
The Sentential Decision Diagram (SDD) is a tractable representation of Boolean functions that subsumes the famous Ordered Binary Decision Diagram (OBDD) as a strict subset. SDDs are attracting much attention because they are more succinct than OBDDs, as well as having canonical forms and supporting many useful queries and transformations such as model counting and Apply operation. In this paper, we propose a more succinct variant of SDD named Variable Shift SDD (VS-SDD). The key idea is to create a unique representation for Boolean functions that are equivalent under a specific variable substi…
Succinct data structures give space-efficient representations of large amounts of data without sacrificing performance. They rely one cleverly designed data representations and algorithms. We present here the formalization in Coq/SSReflect of two different tree-based succinct representations and their accompanying algorithms. One is the Level-Order Unary Degree Sequence, which encodes the structure of a tree in breadth-first order as a sequence of bits, where access operations can be defined in terms of Rank and Select, which work in constant time for static bit sequences. The other represents…
Stanford University（斯坦福大学）；Massachusetts Institute of Technology（麻省理工学院）
We consider the problem of explaining the predictions of an arbitrary blackbox model $f$: given query access to $f$ and an instance $x$, output a small set of $x$'s features that in conjunction essentially determines $f(x)$. We design an efficient algorithm with provable guarantees on the succinctness and precision of the explanations that it returns. Prior algorithms were either efficient but lacked such guarantees, or achieved such guarantees but were inefficient. We obtain our algorithm via a connection to the problem of {\sl implicitly} learning decision trees. The implicit nature of this …
Consider an algorithm performing a computation on a huge random object. Is it necessary to generate the entire object up front, or is it possible to provide query access to the object and sample it incrementally "on-the-fly"? Such an implementation should emulate the object by answering queries in a manner consistent with a random instance sampled from the true distribution. Our first set of results focus on undirected graphs with independent edge probabilities, under certain assumptions. Then, we use this to obtain the first efficient implementations for the Erdos-Renyi model and the Stochast…
Given a set $S$ of $n$ (distinct) keys from key space $[U]$, each associated with a value from $Σ$, the \emph{static dictionary} problem asks to preprocess these (key, value) pairs into a data structure, supporting value-retrieval queries: for any given $x\in [U]$, $\mathtt{valRet}(x)$ must return the value associated with $x$ if $x\in S$, or return $\bot$ if $x\notin S$. The special case where $|Σ|=1$ is called the \emph{membership} problem. The "textbook" solution is to use a hash table, which occupies linear space and answers each query in constant time. On the other hand, the minimum possi…
University of Sydney（悉尼大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加利福尼亚大学圣迭戈分校）
We investigate the problem of testing whether a discrete probability distribution over an ordered domain is a histogram on a specified number of bins. One of the most common tools for the succinct approximation of data, $k$-histograms over $[n]$, are probability distributions that are piecewise constant over a set of $k$ intervals. The histogram testing problem is the following: Given samples from an unknown distribution $\mathbf{p}$ on $[n]$, we want to distinguish between the cases that $\mathbf{p}$ is a $k$-histogram versus $\varepsilon$-far from any $k$-histogram, in total variation distan…
