递归组合函数的蒙特卡洛到拉斯维加斯转换

Probabilistic Computability and Choice

We study the computational power of randomized computations on infinite objects, such as real numbers. In particular, we introduce the concept of a Las Vegas computable multi-valued function, which is a function that can be computed on a probabilistic Turing machine that receives a random binary sequence as auxiliary input. The machine can take advantage of this random sequence, but it always has to produce a correct result or to stop the computation after finite time if the random advice is not successful. With positive probability the random advice has to be successful. We characterize the c…

Las Vegas算法的黑盒加速与算法化逆向Jensen不等式

Black-box Acceleration of Las Vegas Algorithms and Algorithmic Reverse Jensen's Inequalities

本文针对Las Vegas算法提出一种新的通用重启策略，用更小的量替代经典结果中的对数项以降低期望运行时间，并建立算法化逆向Jensen不等式的存在性条件。

Partial randomness and dimension of recursively enumerable reals

A real αis called recursively enumerable ("r.e." for short) if there exists a computable, increasing sequence of rationals which converges to α. It is known that the randomness of an r.e. real αcan be characterized in various ways using each of the notions; program-size complexity, Martin-Löf test, Chaitin Ωnumber, the domination and Ω-likeness of α, the universality of a computable, increasing sequence of rationals which converges to α, and universal probability. In this paper, we generalize these characterizations of randomness over the notion of partial randomness by parameterizing each of …

The Adversary Bound Revisited: From Optimal Query Algorithms to Optimal Control

This note complements the paper "One-Way Ticket to Las Vegas and the Quantum Adversary" (arxiv:2301.02003). I develop the ideas behind the adversary bound - universal algorithm duality therein in a different form, using the same perspective as Barnum-Saks-Szegedy in which query algorithms are defined as sequences of feasible reduced density matrices rather than sequences of unitaries. This form may be faster to understand for a general quantum information audience: It avoids defining the "unidirectional relative $γ_{2}$-bound" and relating it to query algorithms explicitly. This proof is also …

Randomized Polynomial-Time Root Counting in Prime Power Rings

Suppose $k,p\!\in\!\mathbb{N}$ with $p$ prime and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial with degree $d$ and all coefficients having absolute value less than $p^k$. We give a Las Vegas randomized algorithm that computes the number of roots of $f$ in $\mathbb{Z}/\!\left(p^k\right)$ within time $d^3(k\log p)^{2+o(1)}$. (We in fact prove a more intricate complexity bound that is slightly better.) The best previous general algorithm had (deterministic) complexity exponential in $k$. We also present some experimental data evincing the potential practicality of our algorithm.

更快速的模多项式复合算法

Faster Modular Composition

该研究提出一种新型Las Vegas算法，通过计算小规模代数关系矩阵并结合随机化方法，实现任意域上多项式模复合的$O(n^{1.43})$域运算复杂度，首次突破3/2指数瓶颈，大幅优于1978年以来的经典算法。