Ioana O. Bercea

Ioana O. Bercea, Jakob Bæk Tejs Houen, Rasmus Pagh

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 $S'\supseteq S$ that is ''close to $S$'' in the sense that for $x\not\in S$, the probability that $x\in S'$ is bounded by some $\varepsilon > 0$. 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 $\varepsilon $ with high probability over input sets drawn from a product distribution. We also present a Bloom filter alternative, which we call the $\textit{Daisy Bloom filter}$, that executes operations faster and uses significantly less space than the standard Bloom filter.

Navid Eslami, Ioana O. Bercea, Rasmus Pagh et al.

Modern stream processing systems must often track the frequency of distinct keys in a data stream in real-time. Since monitoring the exact counts often entails a prohibitive memory footprint, many applications rely on compact, probabilistic data structures called frequency estimation sketches to approximate them. However, mainstream frequency estimation sketches fall short in two critical aspects: (1) They are memory-inefficient under data skew. This is because they use uniformly-sized counters to track the key counts and thus waste memory on storing the leading zeros of many small counter values. (2) Their estimation error deteriorates at least linearly with the stream's length, which may grow indefinitely over time. This is because they count the keys using a fixed number~of~counters. We present Sublime, a framework that generalizes frequency estimation sketches to address these problems by dynamically adapting to the stream's skew and length. To save memory under skew, Sublime uses short counters upfront and elongates them with extensions stored within the same cache line as they overflow. It leverages novel bit manipulation routines to quickly access a counter's extension. It also controls the scaling of its error rate by expanding its number of approximate counters as the stream grows. We apply Sublime to Count-Min Sketch and Count Sketch. We show, theoretically and empirically, that Sublime significantly improves accuracy and memory over the state of the art while maintaining competitive or superior performance.

Ioana O. Bercea, Volkan Isler, Samir Khuller

We study the problem of sensor placement in environments in which localization is a necessity, such as ad-hoc wireless sensor networks that allow the placement of a few anchors that know their location or sensor arrays that are tracking a target. In most of these situations, the quality of localization depends on the relative angle between the target and the pair of sensors observing it. In this paper, we consider placing a small number of sensors which ensure good angular $α$-coverage: given $α$ in $[0,π/2]$, for each target location $t$, there must be at least two sensors $s_1$ and $s_2$ such that the $\angle(s_1 t s_2)$ is in the interval $[α, π-α]$. One of the main difficulties encountered in such problems is that since the constraints depend on at least two sensors, building a solution must account for the inherent dependency between selected sensors, a feature that generic Set Cover techniques do not account for. We introduce a general framework that guarantees an angular coverage that is arbitrarily close to $α$ for any $α<= π/3$ and apply it to a variety of problems to get bi-criteria approximations. When the angular coverage is required to be at least a constant fraction of $α$, we obtain results that are strictly better than what standard geometric Set Cover methods give. When the angular coverage is required to be at least $(1-1/δ)\cdotα$, we obtain a $\mathcal{O}(\log δ)$- approximation for sensor placement with $α$-coverage on the plane. In the presence of additional distance or visibility constraints, the framework gives a $\mathcal{O}(\logδ\cdot\log k_{OPT})$-approximation, where $k_{OPT}$ is the size of the optimal solution. We also use our framework to give a $\mathcal{O}(\log δ)$-approximation that ensures $(1-1/δ)\cdot α$-coverage and covers every target within distance $3R$.