arXiv subjects

Songhua He

Publications and source records attributed to Songhua He.

3 recordsLinked to original sources

arXiv2026

Systematic Data Structure Lower Bounds via the Query-with-Sketch Model

We study data structure lower bounds for the Approximate Matrix Powering (AMP) problem. Given a substochastic, symmetric matrix $\mathbf{M}\in\mathbb{R}^{n\times n}$ and parameters $k$ and $α$, the goal is to preprocess $\mathbf{M}$ so as to answer entry queries $(u,v)\mapsto \mathbf{M}^{k}[u,v]$ up to additive error $1/n^α$. We focus on AMP in the succinct and systematic regime, in which the data structure stores $\mathbf{M}$ verbatim, uses an additional $r$ bits of redundancy, and must answer queries by probing only a small number of entries of $\mathbf{M}$. Our main conceptual contribution is a general framework for proving probe--redundancy trade-offs for systematic data structures. We introduce the query-with-sketch model and develop a min-entropy-based approach that lifts conditional min-entropy bounds in the absence of redundancy to probe lower bounds in the presence of redundancy. We then establish these min-entropy bounds using problem-specific analytic and algebraic tools, for the downstream applications to AMP and its variants. As a consequence, our results provide new unconditional evidence toward a conjecture of Patrascu and Roditty (2010) on the space required for constant-time set-disjointness queries.

cs.DS↗

arXiv2026

Should Tables Be Sorted? Revisited with a Large Language Model

We revisit the implicit membership problem in Yao's full-table model [Yao, 1981] and obtain, to our knowledge, the first quantitative improvements to his 45-year-old Ramsey bounds, most notably reducing the two-probe bound from tower-type to polynomial. In this model, an $n$-set $S\subseteq\{1,\ldots,m\}$ is stored as a permutation in an $n$-cell table, and queries decide whether $x\in S$. Let $G_q(n)$ be the largest universe size admitting a $q$-probe membership scheme for all $n$-sets. Yao determined the one-probe case exactly, proving $G_1(n)=2n-2$ for $n>2$, but the behavior for $q\ge2$ remained wide open. Fiat and Naor [1993] constructed schemes for universes of size $\exp(n^c)$ for some constant $c>0$ and sufficiently large constant $q$. For the first adaptive case, $q=2$, we prove $G_2(n)=O(n^2(\log n)^2)$. For every fixed integer $q\ge3$, we show that $G_q(n)$ is at most a tower of height $q-1$ with top $n^{1+o(1)}$; in particular, $G_3(n)\le\exp(n^{1+o(1)})$. The two-probe proof avoids Ramsey theory altogether; for larger fixed $q$, we use Ramsey theory only to make the first $q-1$ probes follow a fixed pattern, and then handle the last probe by the same non-Ramsey argument. Somewhat surprisingly, for each fixed $q$, we also show that implicit membership is as hard as implicit search up to a polynomial loss in universe size. Implicit search must return the cell containing $x$ when present and reject otherwise. For the analogous search threshold $H_q(n)$, we prove $H_q(n)\le G_q(n)\le n^q(H_q(n)+1)^{q+1}$ for every $q,n$. Thus, for every fixed $q$, one threshold is at most $\exp(n^{O(1)})$ if and only if the other is. The proofs were first generated by ChatGPT 5.5 Pro without mathematical hints; the membership-search equivalence emerged while pursuing an improved four-probe bound. The authors have validated and edited the proofs and assume responsibility for all content.

cs.DS↗

arXiv2026

Query Lower Bounds for Correlation Clustering under Memory Constraints

This work initiates the study of memory-query tradeoffs for graph problems, with a focus on correlation clustering. Correlation clustering asks for a partition of the vertices that minimizes disagreements: non-edges inside clusters plus edges across clusters. Our first result is a tight query lower bound: to output a partition whose cost approximates the optimum up to an additive error of $\varepsilon n^2$, any algorithm requires $Ω(n/\varepsilon^2)$ adjacency-matrix queries. Under memory constraints, we show that even for the seemingly easier task of approximating the optimal clustering cost (without producing a partition), any algorithm in the random query model must make $\gg n/\varepsilon^2$ adjacency-matrix queries. Finally, we prove the first general graph model query lower bound for correlation clustering, where algorithms are allowed adjacency-matrix, neighbor, and degree queries. The latter two bounds are not yet tight, leaving room for sharper results.

cs.CC↗