arXiv subjects

Andrew Krapivin

Publications and source records attributed to Andrew Krapivin.

5 recordsLinked to original sources

arXiv2026

Linear Probing with Non-Greedy Insertions

Linear probing hash tables classically use a \emph{greedy} insertion strategy, placing a key $u$ in the first available position out of $h(u), h(u) + 1, h(u) + 2, \ldots$. If the hash table is filled to $1 - 1/x$ full, this results in $\Theta(x^{2})$ worst-case expected insertion time. In this note, we show that there is a simple \emph{non-greedy} insertion strategy that does better, and without requiring elements to be reordered within the table over time. Given $x$ in advance, the strategy is able to bring the worst-case expected insertion time down to $O(x \log x)$.

cs.DS↗

arXiv2026

Toward Satisfiability Modulo Realizability

Problems complete for the existential theory of the reals ($\exists \mathbb{R}$) arise throughout discrete geometry. We introduce satisfiability modulo realizability, a SAT-based approach for solving satisfiable instances of $\exists \mathbb{R}$ whose solutions correspond to realizable geometric configurations. Our method encodes an underapproximation of a geometric problem as a SAT instance over abstract order types. Since almost all abstract order types are unrealizable, naive search is infeasible. We guide the search toward realizable order types using diversity-driven sampling, partial realizability feedback, and a novel flippability heuristic that passes only limited information between components. We apply our method to discrete geometry problems and resolve an open problem by showing that the largest set of points avoiding empty convex hexagons and convex heptagons is of size 23.

cs.CG↗

arXiv2026

Near-Optimal Encodings of Cardinality Constraints

We present several novel encodings for cardinality constraints, which use fewer clauses than previous encodings and, more importantly, introduce new generally applicable techniques for constructing compact encodings. First, we present a CNF encoding for the $\text{AtMostOne}(x_1,\dots,x_n)$ constraint using $2n + 2 \sqrt{2n} + O(\sqrt[3]{n})$ clauses, thus refuting the conjectured optimality of Chen's product encoding. Our construction also yields a smaller monotone circuit for the threshold-2 function, improving on a 50-year-old construction of Adleman and incidentally solving a long-standing open problem in circuit complexity. On the other hand, we show that any encoding for this constraint requires at least $2n + \sqrt{n+1} - 2$ clauses, which is the first nontrivial unconditional lower bound for this constraint and answers a question of Kučera, Savický, and Vorel. We then turn our attention to encodings of $\text{AtMost}_k(x_1,\dots,x_n)$, where we introduce "grid compression", a technique inspired by hash tables, to give encodings using $2n + o(n)$ clauses as long as $k = o(\sqrt[3]{n})$ and $4n + o(n)$ clauses as long as $k = o(n)$. Previously, the smallest known encodings were of size $(k+1)n + o(n)$ for $k \le 5$ and $7n - o(n)$ for $k \ge 6$.

cs.CC↗

arXiv2025

Optimal and Efficient Partite Decompositions of Hypergraphs

We study the problem of partitioning the edges of a $d$-uniform hypergraph $H$ into a family $F$ of complete $d$-partite hypergraphs ($d$-cliques). We show that there is a partition $F$ in which every vertex $v \in V(H)$ belongs to at most $(\frac{1}{d!} + o_d(1))n^{d-1}/\lg n$ members of $F$. This settles the central question of a line of research initiated by Erdős and Pyber (1997) for graphs, and more recently by Csirmaz, Ligeti, and Tardos (2014) for hypergraphs. The $d=2$ case of this theorem answers a 40-year-old question of Chung, Erdős, and Spencer (1983). An immediate corollary of our result is an improved upper bound for the maximum share size for binary secret sharing schemes on uniform hypergraphs. Building on results of Nechiporuk (1969), we prove that every graph with fixed edge density $γ\in (0,1)$ has a biclique partition of total weight at most $(\tfrac{1}{2}+o(1))\cdot h_2(γ) \frac{n^2}{\lg n}$, where $h_2$ is the binary entropy function. Our construction implies that such biclique partitions can be constructed in time $O(m)$, which answers a question of Feder and Motwani (1995) and also improves upon results of Mubayi and Turán (2010) as well as Chavan, Rabinia, Grosu, and Brocanelli (2025). Using similar techniques, we also give an $n^{1+o(1)}$ algorithm for finding a subgraph $K_{t,t}$ with $t = (1-o(1)) \fracγ{h_2(γ)} \lg n$. Our results show that biclique partitions are information-theoretically optimal representations for graphs at every fixed density. We show that with this succinct representation one can answer independent set queries and cut queries in time $O(n^2/ \lg n)$, and if we increase the space usage by a constant factor, we can compute a $2α$-approximation for the densest subgraph problem in time $O(n^2/\lg α)$ for any $α> 1$.

math.CO↗

arXiv2025

Optimal Bounds for Open Addressing Without Reordering

In this paper, we revisit one of the simplest problems in data structures: the task of inserting elements into an open-addressed hash table so that elements can later be retrieved with as few probes as possible. We show that, even without reordering elements over time, it is possible to construct a hash table that achieves far better expected search complexities (both amortized and worst-case) than were previously thought possible. Along the way, we disprove the central conjecture left by Yao in his seminal paper ``Uniform Hashing is Optimal''. All of our results come with matching lower bounds.

cs.DS↗