arXiv subjects

Matthew Weidner

Publications and source records attributed to Matthew Weidner.

9 recordsLinked to original sources

arXiv2023

The Art of the Fugue: Minimizing Interleaving in Collaborative Text Editing

Most existing algorithms for replicated lists, which are widely used in collaborative text editors, suffer from a problem: when two users concurrently insert text at the same position in the document, the merged outcome may interleave the inserted text passages, resulting in corrupted and potentially unreadable text. The problem has gone unnoticed for decades, and it affects both CRDTs and Operational Transformation. This paper defines maximal non-interleaving, our new correctness property for replicated lists. We introduce two related CRDT algorithms, Fugue and FugueMax, and prove that FugueMax satisfies maximal non-interleaving. We also implement our algorithms and demonstrate that Fugue offers performance comparable to state-of-the-art CRDT libraries for text editing.

cs.DC↗

arXiv2023

For-Each Operations in Collaborative Apps

Conflict-free Replicated Data Types (CRDTs) allow collaborative access to an app's data. We describe a novel CRDT operation, for-each on the list of CRDTs, and demonstrate its use in collaborative apps. Our for-each operation applies a given mutation to each element of a list, including elements inserted concurrently. This often preserves user intention in a way that would otherwise require custom CRDT algorithms. We give example applications of our for-each operation to collaborative rich-text, recipe, and slideshow editors.

cs.DC↗

arXiv2022

Collabs: A Flexible and Performant CRDT Collaboration Framework

A collaboration framework is a distributed system that serves as the data layer for a collaborative app. Conflict-free Replicated Data Types (CRDTs) are a promising theoretical technique for implementing collaboration frameworks. However, existing frameworks are inflexible: they are often one-off implementations of research papers or only permit a restricted set of CRDT semantics, and they do not allow app-specific optimizations. Until now, there was no general framework that lets programmers mix, match, and modify CRDTs. We solve this with Collabs, a CRDT-based collaboration framework that lets programmers implement their own CRDTs, either from-scratch or by composing existing building blocks. Collabs prioritizes both semantic flexibility and performance flexibility: it allows arbitrary app-specific CRDT behaviors and optimizations, while still providing strong eventual consistency. We demonstrate Collabs's capabilities and programming model with example apps and CRDT implementations. We then show that a collaborative rich-text editor using Collabs's built-in CRDTs can scale to over 100 simultaneous users, unlike existing CRDT frameworks and Google Docs. Collabs also has lower end-to-end latency and server CPU usage than a popular Operational Transformation framework, with acceptable CRDT metadata overhead.

cs.DC↗

arXiv2020

Composing and Decomposing Op-Based CRDTs with Semidirect Products

Operation-based Conflict-free Replicated Data Types (CRDTs) are eventually consistent replicated data types that automatically resolve conflicts between concurrent operations. Op-based CRDTs must be designed differently for each data type, and current designs use ad-hoc techniques to handle concurrent operations that do not naturally commute. We present a new construction, the semidirect product of op-based CRDTs, which combines the operations of two CRDTs into one while handling conflicts between their concurrent operations in a uniform way. We demonstrate the construction's utility by using it to construct novel CRDTs, as well as decomposing several existing CRDTs as semidirect products of simpler CRDTs. Although it reproduces common CRDT semantics, the semidirect product can be viewed as a restricted kind of operational transformation, thus forming a bridge between these two opposing techniques for constructing replicated data types.

cs.DC↗

arXiv2019

On Decoding Cohen-Haeupler-Schulman Tree Codes

Tree codes, introduced by Schulman, are combinatorial structures essential to coding for interactive communication. An infinite family of tree codes with both rate and distance bounded by positive constants is called asymptotically good. Rate being constant is equivalent to the alphabet size being constant. Schulman proved that there are asymptotically good tree code families using the Lovasz local lemma, yet their explicit construction remains an outstanding open problem. In a major breakthrough, Cohen, Haeupler and Schulman constructed explicit tree code families with constant distance, but over an alphabet polylogarithmic in the length. Our main result is a randomized polynomial time decoding algorithm for these codes making novel use of the polynomial method. The number of errors corrected scales roughly as the block length to the three-fourths power, falling short of the constant fraction error correction guaranteed by the constant distance. We further present number theoretic variants of Cohen-Haeupler-Schulman codes, all correcting a constant fraction of errors with polylogarithmic alphabet size. Towards efficiently correcting close to a constant fraction of errors, we propose a speculative convex optimization approach inspired by compressed sensing.

cs.IT↗

arXiv2018

On Conjectural Rank Parities of Quartic and Sextic Twists of Elliptic Curves

We study the behavior under twisting of the Selmer rank parities of a self-dual prime-degree isogeny on a principally polarized abelian variety defined over a number field, subject to compatibility relations between the twists and the isogeny. In particular, we study isogenies on abelian varieties whose Selmer rank parities are related to the rank parities of elliptic curves with $j$-invariant 0 or 1728, assuming the Shafarevich-Tate conjecture. Using these results, we show how to classify the conjectural rank parities of all quartic or sextic twists of an elliptic curve defined over a number field, after a finite calculation. This generalizes previous results of Hadian and Weidner on the behavior of $p$-Selmer ranks under $p$-twists.

math.NT↗

arXiv2018

Pseudocharacters of Classical Groups

A $GL_d$-pseudocharacter is a function from a group $\Gamma$ to a ring $k$ satisfying polynomial relations which make it "look like" the character of a representation. When $k$ is an algebraically closed field, Taylor proved that $GL_d$-pseudocharacters of $\Gamma$ are the same as degree-$d$ characters of $\Gamma$ with values in $k$, hence are in bijection with equivalence classes of semisimple representations $\Gamma \rightarrow GL_d(k)$. Recently, V. Lafforgue generalized this result by showing that, for any connected reductive group $H$ over an algebraically closed field $k$ of characteristic 0 and for any group $\Gamma$, there exists an infinite collection of functions and relations which are naturally in bijection with $H^0(k)$-conjugacy classes of semisimple representations $\Gamma \rightarrow H(k)$. In this paper, we reformulate Lafforgue's result in terms of a new algebraic object called an FFG-algebra. We then define generating sets and generating relations for these objects and show that, for all $H$ as above, the corresponding FFG-algebra is finitely presented. Hence we can always define $H$-pseudocharacters consisting of finitely many functions satisfying finitely many relations. Next, we use invariant theory to give explicit finite presentations of the FFG-algebras for (general) orthogonal groups, (general) symplectic groups, and special orthogonal groups. Finally, we use our pseudocharacters to answer questions about conjugacy vs. element-conjugacy of representations, following Larsen.

math.RT↗

arXiv2018

On cycles of pairing-friendly elliptic curves

A cycle of elliptic curves is a list of elliptic curves over finite fields such that the number of points on one curve is equal to the size of the field of definition of the next, in a cyclic way. We study cycles of elliptic curves in which every curve is pairing-friendly. These have recently found notable applications in pairing-based cryptography, for instance in improving the scalability of distributed ledger technologies. We construct a new cycle of length 4 consisting of MNT curves, and characterize all the possibilities for cycles consisting of MNT curves. We rule out cycles of length 2 for particular choices of small embedding degrees. We show that long cycles cannot be constructed from families of curves with the same complex multiplication discriminant, and that cycles of composite order elliptic curves cannot exist. We show that there are no cycles consisting of curves from only the Freeman or Barreto--Naehrig families.

math.NT↗

arXiv2017

Subquadratic time encodable codes beating the Gilbert-Varshamov bound

We construct explicit algebraic geometry codes built from the Garcia-Stichtenoth function field tower beating the Gilbert-Varshamov bound for alphabet sizes at least 192. Messages are identied with functions in certain Riemann-Roch spaces associated with divisors supported on multiple places. Encoding amounts to evaluating these functions at degree one places. By exploiting algebraic structures particular to the Garcia-Stichtenoth tower, we devise an intricate deterministic \omega/2 < 1.19 runtime exponent encoding and 1+\omega/2 < 2.19 expected runtime exponent randomized (unique and list) decoding algorithms. Here \omega < 2.373 is the matrix multiplication exponent. If \omega = 2, as widely believed, the encoding and decoding runtimes are respectively nearly linear and nearly quadratic. Prior to this work, encoding (resp. decoding) time of code families beating the Gilbert-Varshamov bound were quadratic (resp. cubic) or worse.

cs.IT↗