arXiv subjects

Chengzhang Wan

Publications and source records attributed to Chengzhang Wan.

4 recordsLinked to original sources

arXiv2026

A Simple Analysis of Quadratic Probing and Other Open Addressing Schemes

In open addressed hashing, quadratic probing is attractive for striking a nice balance between having a high locality of reference and a low number of probes per search. However, these are empirical observations, not theoretical guarantees. Indeed, until recently, it was not known whether quadratic probing had constant expected insertion cost under any positive load factor $\alpha > 0$, even with uniformly random hash functions. In a recent breakthrough---albeit a numerically understated breakthrough---Kuszmaul and Xi (2024) proved that any fixed offset sequence (including quadratic probing) does, in fact, have constant expected insertion cost for load factors $\alpha \leq 8.9\%$. This is well below what we would like to prove, that quadratic probing has constant insertion cost for any load factor $\alpha < 1-\epsilon$ bounded away from 1. In this paper, we prove that open addressed hashing with any fixed offset sequence has constant expected insertion cost for load factors up to $35.74\%$, and that for quadratic probing in particular, we can increase the load factor to $37.61\%$. Our main innovation is a new type of witness forest for recording collisions among the probe sequences.

cs.DS↗

arXiv2026

The Greedy Binary Search Tree is Non-trivially Competitive

We prove that the $\textsf{Greedy}$ binary search tree is $2^{O(\sqrt{\log\log n})}$-competitive. It is widely conjectured that $\textsf{Greedy}$ is $O(1)$-competitive, but before this work it was not known to be $f$-competitive, for any non-trivial $f(n)=o(\log n)$. Our analysis differs from prior analyses of binary search trees. It takes what might be called a "scaling" approach, where the cost at a refined scale is related to the cost at a coarser scale, and Wilber's interleave lower bound.

cs.DS↗

arXiv2026

$k$-Clustering via Iterative Randomized Rounding

In this work we propose a single rounding algorithm for the fractional solutions of the standard LP relaxation for $k$-clustering. As a starting point, we obtain an iterative rounding $(\frac{3^p + 1}{2})$-Lagrangian Multiplier-Perserving (LMP) approximation for the $k$-clustering problem with the cost function being the $p$-th power of the distance. Such an algorithm outputs a random solution that opens $k$ facilities \emph{in expectation}, whose cost in expectation is at most $\frac{3^p + 1}{2}$ times the optimum cost. Thus, we recover the $2$-LMP approximation for $k$-median by Jain et al.~[JACM'03], which played a central role in deriving the current best $2$ approximation for $k$-median. Unlike the result of Jain et al., our algorithm is based on LP rounding, and it can be easily adapted to the $L_p^p$-cost setting. For the Euclidean $k$-means problem, the LMP factor we obtain is $\frac{11}{3}$, which is better than the $5$ approximation given by this framework for general metrics. Then, we show how to convert the LMP-approximation algorithms to a true-approximation, with only a $(1+\varepsilon)$ factor loss in the approximation ratio. We obtain a ($\frac{3^p + 1}{2}+\varepsilon$)-approximation algorithm for $k$-clustering with cost function being the $p$-th power of the distance, for $p \geq 1$. This reproduces the best known ($2+\varepsilon$)-approximation for $k$-median by Cohen-Addad et al. [STOC'25], and improves the approximation factor for metric $k$-means from 5.83 by Charikar at al. [FOCS'25] to $5+\varepsilon$ in our framework. Moreover, the same algorithm, but with a specialized analysis, attains ($4+\varepsilon$)-approximation for Euclidean $k$-means matching the recent result by Charikar et al. [STOC'26].

cs.DS↗

arXiv2024

Electrically functionalized body surface for deep-tissue bioelectrical recording

Directly probing deep tissue activities from body surfaces offers a noninvasive approach to monitoring essential physiological processes1-3. However, this method is technically challenged by rapid signal attenuation toward the body surface and confounding motion artifacts4-6 primarily due to excessive contact impedance and mechanical mismatch with conventional electrodes. Herein, by formulating and directly spray coating biocompatible two-dimensional nanosheet ink onto the human body under ambient conditions, we create microscopically conformal and adaptive van der Waals thin films (VDWTFs) that seamlessly merge with non-Euclidean, hairy, and dynamically evolving body surfaces. Unlike traditional deposition methods, which often struggle with conformality and adaptability while retaining high electronic performance, this gentle process enables the formation of high-performance VDWTFs directly on the body surface under bio-friendly conditions, making it ideal for biological applications. This results in low-impedance electrically functionalized body surfaces (EFBS), enabling highly robust monitoring of biopotential and bioimpedance modulations associated with deep-tissue activities, such as blood circulation, muscle movements, and brain activities. Compared to commercial solutions, our VDWTF-EFBS exhibits nearly two-orders of magnitude lower contact impedance and substantially reduces the extrinsic motion artifacts, enabling reliable extraction of bioelectrical signals from irregular surfaces, such as unshaved human scalps. This advancement defines a technology for continuous, noninvasive monitoring of deep-tissue activities during routine body movements.

physics.med-ph↗