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Hanlin Ren
I am a first-year DPhil student at the University of Oxford, where I'm very fortunate to be advised by Prof. Rahul Santhanam. I am a member of Christ Church and also a Clarendon scholar. Previously, I was an undergraduate student at Institute for Interdisciplinary Information Sciences, Tsinghua University.
I have a broad interest in theoretical computer science. Currently, I am interested in computational complexity, which studies the limitations of efficient computation. Some topics of complexity theory that I find fascinating are circuit complexity, meta-complexity, and average-case complexity.
At Tsinghua, I was advised by Prof. Ran Duan and worked on graph algorithms. One emphasis of my research was to design shortest-path data structures for graphs in the presence of failures.
[Google Scholar], [DBLP], [Twitter], [ORCID]
Contact:
hanlin {DOT} ren <AT> cs {DOT} ox {DOT} ac {DOT} uk.
h4n1in {DOT} r3n <AT> gmail {DOT} com.
rhl16 <AT> tsinghua {DOT} org {DOT} cn.
Publications
(In theoretical computer science, the list of authors are usually sorted in alphabetical order.)
Maintaining Exact Distances under Multiple Edge Failures,
with Ran Duan
Summary: For every constant \(d\ge 1\) and undirected weighted graph \(G\), we present a data structure of size \(O(n^4)\) that supports the following query in constant time: Given a set of \(d\) edge failures and two vertices \(u, v\), what is the exact distance between \(u\) and \(v\) with the failed edges removed? This is the first non-trivial data structure for handling exact distances under multiple failures. For non-constant \(d\), our query time is \(d^{O(d)}\) and space complexity is \(O(dn^4)\).
Robustness of Average-Case Meta-Complexity via Pseudorandomness,
with Rahul Ilango and Rahul Santhanam
Summary: We give reductions to show that the error-prone average-case complexity of many meta-complexity problems are very robust w.r.t. samplable distributions. For example, it is strongly (\(1/2+{\rm negl}(n)\)) average-case hard to approximate the (worst-case uncomputable) Kolmogorov complexity within a multiplicative factor of \(n^{0.9}\) if and only if it is weakly (\(1-1/{\rm poly}(n)\)) average-case hard to approximate the Kolmogorov complexity within an additive factor of \(\omega(\log n)\), both under some samplable distribution. Moreover, they are equivalent to the existence of one-way functions.
A Relativization Perspective on Meta-Complexity,
with Rahul Santhanam
Summary: We investigate the role of relativization in the study of meta-complexity (e.g. minimum circuit size problem). Among other results, we find a relativized world where \({\rm search}\text{-}{\rm MCSP}\) is "much harder" than \({\rm MCSP}\), and another world where Levin's \({\rm Kt}\) complexity can be approximated in (deterministic) polynomial time.
Hardness of \({\rm KT}\) Characterizes Parallel Cryptography,
with Rahul Santhanam
Summary: A recent breakthrough [LP'20] showed that one-way functions exist if and only if \({\rm K}^t\) is bounded-error hard on average, where \(t(n) > 2n\) is any polynomial. Building on this result, we show that one-way functions can be computed in (uniform-)\({\sf NC}^0\) if and only if the \({\rm KT}\) complexity is hard on average. Our techniques also imply average-case reductions between the \({\rm KT}\) complexity and an \({\sf NC}^1\)-variant of the \({\rm K}^t\) complexity.
Constructing a Distance Sensitivity Oracle in \(O(n^{2.5794}M)\) Time,
with Yong Gu
[arXiv],
[Poster at the 10-th year anniversary of IIIS],
[Slides at IJTCS 2021],
[Slides at Yaoclass seminar],
[Summary]
Summary: We improve the construction time of distance sensitivity oracles for direct graphs to \(O(n^{2.5794}M)\), where \(n\) is the number of vertices, and the edge weights are integers in \([1, M]\). The new ingredients are a faster algorithm for inverting a polynomial matrix modulo \(x^r\), adapted from [ZLS'15], and an algorithm that computes consistent shortest path trees in current APSP time.
Approximate Distance Oracles Subject to Multiple Vertex Failures,
with Ran Duan and Yong Gu
Summary: We design data structures that maintain approximate distances for weighted undirected graphs, and tolerate multiple vertex failures. In particular, given any two vertices \(u,v\) and any set of \(d\) failed vertices \(D\), we can output an \((1+\epsilon)\)-approximation of the length of shortest path from \(u\) to \(v\) that avoids \(D\). Roughly speaking, for any constant number of failures, our structure uses \(O(n^{2.01})\) space and \(\operatorname{poly}(\log n,\epsilon^{-1})\) query time.
Improved Distance Sensitivity Oracles with Subcubic Preprocessing Time
Summary: We show that on a directed graph with integer edge weights in \([1,W]\), we can construct a distance sensitivity oracle with \(\tilde{O}(n^{2.7233}M)\) preprocessing time and constant query time. For undirected graphs, the preprocessing time can be improved to \(\tilde{O}(n^{(3+\omega)/2}M)\). We think the main advantage of our oracle is its simplicity.
Strong Average-Case Circuit Lower Bounds from Non-trivial Derandomization,
with Lijie Chen
Summary: We show that for any integer \(k\), \(\mathsf{NQP} = \mathsf{NTIME}[2^{\mathrm{polylog}(n)}]\) cannot be \((1/2+1/2^{-\log^k n})\)-approximated by \(\mathsf{ACC}^0\circ\mathsf{THR}\) circuits of \(2^{\log^k n}\) size. Previously, it was unknown whether \(\mathsf{E}^{\mathsf{NP}}\) contains a function that is not \((1/2+1/\sqrt{n})\)-approximated by polynomial-size \(\mathsf{AC}^0[2]\) circuits. An interesting technical ingredient in this paper is a hardness amplification theorem with an Approximate Sum error corrector, borrowed from [AIK'06].
Approximating All-Pair Bounded-Leg Shortest Path and APSP-AF in Truly-Subcubic Time,
with Ran Duan
Summary: Given a directed graph \(G\) in which each edge has length \(d\) and capacity \(f\), we want to answer the following kind of queries: Given \((u,v,f)\), what is the length of the shortest path from \(u\) to \(v\) that only uses edges with capacity at least \(f\)? For \((1+\epsilon)\)-approximating the answers, we design a data structure with \(\tilde{O}(n^{(\omega+3)/2}\epsilon^{-1.5}\log W)\) preprocessing time and \(O(\log\log_{\epsilon^{-1}}(nW))\) query time, where \(W\) is the maximum edge weight.
Manuscripts / In Submission
On the Range Avoidance Problem for Circuits, with Rahul Santhanam and Zhikun Wang
Summary: In the range avoidance problem, we are given a circuit \(C:\{0, 1\}^n\to \{0, 1\}^\ell\) where \(\ell > n\), and we want to find a string \(y\in\{0, 1\}^\ell\) that is not in the range of \(C\). Using Ryan Williams's Algorithmic Method for proving circuit lower bounds, we show that derandomisation of certain data structures would imply an \({\sf FP}^{\sf NP}\) algorithm for solving this problem. As an application of this result, we characterise circuit lower bounds for \({\sf E}^{\sf NP}\) by non-trivial derandomisation algorithms with \({\sf E}^{\sf NP}\) preprocessing.
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