Given a k-CNF formula and an integer s ≥ 2, we study algorithms that obtain s solutions to the formula that are as dispersed as possible. For s = 2, this problem of computing the diameter of a k-CNF formula was initiated by Creszenzi and Rossi, who showed strong hardness results even for k = 2. The current best upper bound [Angelsmark and Thapper ’04] goes to 4 n as k → ∞. As our first result, we show that this quadratic blow up is not necessary by utilizing the Fast-Fourier transform (FFT) to give a O ∗ (2n ) time exact algorithm for computing the diameter of any k-CNF formula. For s > 2, the problem was raised in the SAT community (Nadel ’11) and several heuristics have been proposed for it, but no algorithms with theoretical guarantees are known. We give exact algorithms using FFT and clique-finding that run in O ∗ (2(s−1)n ) and O ∗ (s 2 |ΩF| ω⌈s/3⌉ ) respectively, where |ΩF| is the size of the solutions space of the formula F and ω is the matrix multiplication exponent. However, current SAT algorithms for finding one solution run in time O ∗ (2εkn ) for εk ≈ 1−Θ(1/k), which is much faster than all above run times. As our main result, we analyze two popular SAT algorithms - PPZ (Paturi, Pudlák, Zane ’97) and Schöning’s (’02) algorithms, and show that in time poly(s)O ∗ (2εkn ), they can be used to approximate diameter as well as the dispersion (s > 2) problem. While we need to modify Schöning’s original algorithm for technical reasons, we show that the PPZ algorithm, without any modification, samples solutions in a geometric sense. We believe this geometric sampling property of PPZ may be of independent interest. Finally, we focus on diverse solutions to NP-complete optimization problems, and give biapproximations running in time poly(s)O ∗ (2εn) with ε < 1 for several problems such as Maximum Independent Set, Minimum Vertex Cover, Minimum Hitting Set, Feedback Vertex Set, Multicut on Trees and Interval Vertex Deletion. For all of these problems, all existing exact methods for finding optimal diverse solutions have a runtime with at least an exponential dependence on the number of solutions s. Our methods show that by relaxing to bi-approximations, this dependence on s can be made polynomial.

Range filters are compact probabilistic data structures that answer approximate range emptiness queries. They are used in many domains, e.g., in key-value stores, to quickly rule out the existence of keys in a given query range and avoid having to search for them in storage. However, all existing range filters exhibit at least one of the following shortcomings: (1) they do not provide robust false positive rate and performance guarantees, (2) they do not support variable-length keys and query ranges, and (3) they do not allow dynamic operations such as insertions, deletions, or expansions. We introduce Diva, the first range filter to address all the above challenges simultaneously. Diva learns the dataset’s distribution by sampling keys and storing them in a cache-efficient trie. It compresses the keys in-between samples by removing their longest common prefix and truncating their suffixes while leaving enough bits in the middle (i.e., an infix) to allow differentiating between the keys in the sorted order. It stores infixes in constant time dynamic data blocks, which it splits to handle insertions and expansions. It processes a range query by traversing the trie and checking for the inclusion of infixes in the target query range. We show, theoretically and empirically, that Diva achieves a false positive rate on par with the state of the art on real-world datasets while supporting dynamicity and variable-length queries and keys.

Filter data structures are widely used in various areas of computer science to answer approximate set-membership queries. In many applications, the data grows dynamically, requiring their filters to expand along with the data. However, existing methods for expanding filters cannot maintain stable performance, memory footprint, and false positive rate (FPR) simultaneously. We address this prob lem with Aleph Filter, which makes the following contributions. (1) It supports all operations (insertions, queries, deletes, etc.) in constant time, no matter how much the data grows. (2) Given an estimate of how much the data will ultimately grow, Aleph Filter provides a memory vs. FPR trade-offs on par with static filters.

A filter is a widely used data structure for storing an approximation of a given set S of elements from some universe U (a countable set). It represents a superset S′ ⊇ S that is “close to S” in the sense that for x ∉ S, the probability that x ∈ S′ is bounded by some ε > 0. The advantage of using a Bloom filter, when some false positives are acceptable, is that the space usage becomes smaller than what is required to store S exactly. Though filters are well-understood from a worst-case perspective, it is clear that state-of-the-art constructions may not be close to optimal for particular distributions of data and queries. Suppose, for instance, that some elements are in S with probability close to 1. Then it would make sense to always include them in S′, saving space by not having to represent these elements in the filter. Questions like this have been raised in the context of Weighted Bloom filters (Bruck, Gao and Jiang, ISIT 2006) and Bloom filter implementations that make use of access to learned components (Vaidya, Knorr, Mitzenmacher, and Krask, ICLR 2021). In this paper, we present a lower bound for the expected space that such a filter requires. We also show that the lower bound is asymptotically tight by exhibiting a filter construction that executes queries and insertions in worst-case constant time, and has a false positive rate at most ε with high probability over input sets drawn from a product distribution. We also present a Bloom filter alternative, which we call the Daisy Bloom filter, that executes operations faster and uses significantly less space than the standard Bloom filter.

In the online sorting problem, n items are revealed one by one and have to be placed (immediately and irrevocably) into empty cells of a size-n array. The goal is to minimize the sum of absolute differences between items in consecutive cells. This natural problem was recently introduced by Aamand, Abrahamsen, Beretta, and Kleist (SODA 2023) as a tool in their study of online geometric packing problems. They showed that when the items are reals from the interval [0, 1] a competitive ratio of O(√n) is achievable, and no deterministic algorithm can improve this ratio asymptotically. In this paper, we extend and generalize the study of online sorting in three directions: randomized: we settle the open question of Aamand et al. by showing that the O(√n) competitive ratio for the online sorting of reals cannot be improved even with the use of randomness; stochastic: we consider inputs consisting of n samples drawn uniformly at random from an interval, and give an algorithm with an improved competitive ratio of Oe(n1/4). The result reveals connections between online sorting and the design of efficient hash tables; high-dimensional: we show that Oe(√n)-competitive online sorting is possible even for items from Rd, for arbitrary fixed d, in an adversarial model. This can be viewed as an online variant of the classical TSP problem where tasks (cities to visit) are revealed one by one and the salesperson assigns each task (immediately and irrevocably) to its timeslot. Along the way, we also show a tight O(log n)-competitiveness result for uniform metrics, i.e., where items are of different types and the goal is to order them so as to minimize the number of switches between consecutive items of different types.