Cache-Oblivious Persistence

Partial persistence is a general transformation that takes a data structure and allows queries to be executed on any past state of the structure. The cache-oblivious model is the leading model of a modern multi-level memory hierarchy.We present the first general transformation for making cache-oblivious model data structures partially persistent

Lower Bounds for Oblivious Data Structures

An oblivious data structure is a data structure where the memory access patterns reveals no information about the operations performed on it. Such data structures were introduced by Wang et al. [ACM SIGSAC'14] and are intended for situations where one wishes to store the data structure at an untrusted server. One way to obtain an oblivious data structure is simply to run a classic data structure on an oblivious RAM (ORAM). Until very recently, this resulted in an overhead of $\omega(\lg n)$ for the most natural setting of parameters. Moreover, a recent lower bound for ORAMs by Larsen and Nielsen [CRYPTO'18] show that they always incur an overhead of at least $\Omega(\lg n)$ if used in a black box manner. To circumvent the $\omega(\lg n)$ overhead, researchers have instead studied classic data structure problems more directly and have obtained efficient solutions for many such problems such as stacks, queues, deques, priority queues and search trees. However, none of these data structures process operations faster than $\Theta(\lg n)$, leaving open the question of whether even faster solutions exist. In this paper, we rule out this possibility by proving $\Omega(\lg n)$ lower bounds for oblivious stacks, queues, deques, priority queues and search trees.Comment: To appear at SODA'1

Lower Bounds for Oblivious Near-Neighbor Search

We prove an $\Omega(d \lg n/ (\lg\lg n)^2)$ lower bound on the dynamic cell-probe complexity of statistically $\mathit{oblivious}$ approximate-near-neighbor search ($\mathsf{ANN}$) over the $d$-dimensional Hamming cube. For the natural setting of $d = \Theta(\log n)$, our result implies an $\tilde{\Omega}(\lg^2 n)$ lower bound, which is a quadratic improvement over the highest (non-oblivious) cell-probe lower bound for $\mathsf{ANN}$. This is the first super-logarithmic $\mathit{unconditional}$ lower bound for $\mathsf{ANN}$ against general (non black-box) data structures. We also show that any oblivious $\mathit{static}$ data structure for decomposable search problems (like $\mathsf{ANN}$) can be obliviously dynamized with $O(\log n)$ overhead in update and query time, strengthening a classic result of Bentley and Saxe (Algorithmica, 1980).Comment: 28 page

Cache-Oblivious Data Structures and Algorithms for Undirected Breadth-First Search and Shortest Paths

We present improved cache-oblivious data structures and algorithms for breadth-first search (BFS) on undirected graphs and the single-source shortest path (SSSP) problem on undirected graphs with non-negative edge weights. For the SSSP problem, our result closes the performance gap between the currently best cache-aware algorithm and the cache-oblivious counterpart. Our cache-oblivious SSSP-algorithm takes nearly full advantage of block transfers for dense graphs. The algorithm relies on a new data structure, called bucket heap, which is the first cache-oblivious priority queue to efficiently support a weak DECREASEKEY operation. For the BFS problem, we reduce the number of I/Os for sparse graphs by a factor of nearly sqrt{B}, where B is the cache-block size, nearly closing the performance gap between the currently best cache-aware and cache-oblivious algorithms

Axiomatizations for downward XPath on Data Trees

We give sound and complete axiomatizations for XPath with data tests by "equality" or "inequality", and containing the single "child" axis. This data-aware logic predicts over data trees, which are tree-like structures whose every node contains a label from a finite alphabet and a data value from an infinite domain. The language allows us to compare data values of two nodes but cannot access the data values themselves (i.e. there is no comparison by constants). Our axioms are in the style of equational logic, extending the axiomatization of data-oblivious XPath, by B. ten Cate, T. Litak and M. Marx. We axiomatize the full logic with tests by "equality" and "inequality", and also a simpler fragment with "equality" tests only. Our axiomatizations apply both to node expressions and path expressions. The proof of completeness relies on a novel normal form theorem for XPath with data tests

Decremental SSSP in Weighted Digraphs: Faster and Against an Adaptive Adversary

Given a dynamic digraph $G = (V,E)$ undergoing edge deletions and given $s\in V$ and $\epsilon>0$, we consider the problem of maintaining $(1+\epsilon)$-approximate shortest path distances from $s$ to all vertices in $G$ over the sequence of deletions. Even and Shiloach (J.~ACM'$81$) give a deterministic data structure for the exact version of the problem with total update time $O(mn)$. Henzinger et al. (STOC'$14$, ICALP'$15$) give a Monte Carlo data structure for the approximate version with improved total update time $O(mn^{0.9 + o(1)}\log W)$ where $W$ is the ratio between the largest and smallest edge weight. A drawback of their data structure is that they only work against an oblivious adversary, meaning that the sequence of deletions needs to be fixed in advance. This limits its application as a black box inside algorithms. We present the following $(1+\epsilon)$-approximate data structures: (1) the first data structure is Las Vegas and works against an adaptive adversary; it has total expected update time $\tilde O(m^{2/3}n^{4/3})$ for unweighted graphs and $\tilde O(m^{3/4}n^{5/4}\log W)$ for weighted graphs, (2) the second data structure is Las Vegas and assumes an oblivious adversary; it has total expected update time $\tilde O(\sqrt m n^{3/2})$ for unweighted graphs and $\tilde O(m^{2/3}n^{4/3}\log W)$ for weighted graphs, (3) the third data structure is Monte Carlo and is correct w.h.p.~against an oblivious adversary; it has total expected update time $\tilde O((mn)^{7/8}\log W) = \tilde O(mn^{3/4}\log W)$. Each of our data structures can be queried at any stage of $G$ in constant worst-case time; if the adversary is oblivious, a query can be extended to also report such a path in time proportional to its length. Our update times are faster than those of Henzinger et al.~for all graph densities.Comment: Appeared at SODA'2