Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.17112

The Rhythm of the Tiles: Narrow and Low Temperature Computation and Simulation in Synchronous Tile Self-Assembly

Florent Becker·Phillip Drake·Benjamin Durbin·Nicolas Ollinger·Brent Orlina·Matthew J. Patitz
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we prove that the synchronicity provided by the syncTAM can be leveraged to perform two additional feats impossible in the asynchronous aTAM. The aTAM has long been known to be equivalent to finite automata in the one-dimensional case. For our first set of results, we prove that this still holds when a finite, fixed amount of space is available in the second dimension. Yet, surprisingly, in the synchronous setting, a small, fixed, width is enough to simulate a Turing Machine. This holds even in the non-cooperative setting. For our second set of main results we prove that there exists a single set of syncTAM tiles that can be used to simulate any syncTAM system simply by using those tiles to create a scaled version of the seed structure of the original system and then letting the tiles from that set self-assemble from the scaled seed structure. While it has been shown (FOCS 2012) that such a tile set also exists in the aTAM for simulating all aTAM systems, that "intrinsically universal" (IU) tile set requires cooperativity (and is impossible without cooperativity as shown in SODA 2014). We prove there exists an IU tile set in the syncTAM that works without cooperativity. Furthermore, there exists a tile set that is IU for the class of directed systems (i.e. systems which each have a single terminal assembly) while remaining directed during simulation. This was shown to be impossible in the aTAM (FOCS 2016).

Explore related subjects

cs.CG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Florent Becker, Phillip Drake, Benjamin Durbin, Nicolas Ollinger, Brent Orlina, Matthew J. Patitz. 2026-10-12. The Rhythm of the Tiles: Narrow and Low Temperature Computation and Simulation in Synchronous Tile Self-Assembly. https://arxiv.org/abs/2610.17112

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Symbolic Constraints in Polyhedral Enclosure and Tetrahedral Decomposition in Genus-0 Polyhedra

I present an exploratory, coordinate-free framework for organizing two related problems in polyhedral combinatorics: screening polygonal face multisets for genus-zero enclosure and examining tetrahedral decompositions that preserve the original boundary vertex set. The boundary is summarized using a face-degree excess measure that records how far the surface departs from full triangulation, while standard Euler and incidence relations provide exact constraints and parity-dependent bounds on the external structure. For tetrahedral decompositions, standard incidence relations for a triangulated 3-ball are separated from the additional patterns generated by a restricted boundary-driven decomposition procedure. Within the manually constructed examples, bipyramidal configurations provide an extremal reference class, and departures from bipyramidal structure are associated with changes in attainable tetrahedral states. The examples also suggest a contraction of the realized internal state space as boundary face-degree excess increases, motivating the notions of S-flexibility and incidence-gated transitions. These patterns are presented as empirical structural observations and conjectural directions rather than general classification theorems. The resulting framework provides a symbolic pre-screening workflow that can reject some impossible polygonal face multisets before explicit graph construction or geometric realization, and is implemented in the open-source R package polyenclose.

cs.CG↗
arXiv2026

Updating zigzag representatives efficiently

Computation of zigzag persistence has progressed in recent years, with results showing that complexities of many problems closely align with those in the non-zigzag setting. The major efficiency gap now lies in the updating of zigzag representatives. In this paper, we propose efficient algorithms for updating zigzag representatives based on a recent algorithm for extracting zigzag representatives from a $R=DV$ decomposition of a constructed non-zigzag. The main difficulty for designing our update algorithms lies in the adjacency change occurring in two operations that lengthen or shorten a filtration. Despite the adjacency change, we find that the update can still be done efficiently in quadratic time.

cs.CG↗
arXiv2026

Shaving Two Logs for Discrete and Continuous Fréchet Distance

We revisit the classical computational geometry problem of computing the exact (discrete or continuous) Fréchet distance between two given polygonal curves with $n$ vertices in a constant dimension $d$. 1. We present an $O((n^2/\log^2n)\log^2\log n)$-time deterministic algorithm for the discrete Fréchet distance problem, improving the previous $O((n^2/\log n)\log\log n)$ result by Agarwal, Ben Avraham, Kaplan, and Sharir [SODA'13]. 2. We present an $O((n^2/\log^2n)\log^2\log n)$-time deterministic algorithm for the decision version of the continuous Fréchet distance problem, improving the previous $O((n^2/\log n)\log^2\log n)$ result by Buchin, Buchin, Meulemans, and Mulzer [SODA'14] and the recent $O(n^2/\log^{1+μ}n)$ randomized result by Cheng and Huang [SODA'25] for a small unspecified constant $μ>0$ (which converges to 0 as $d$ increases). 3. Finally, we present an $O((n^2/\log^2 n)\log^{O(1)}\log n)$-time Las Vegas randomized algorithm for the continuous Fréchet distance problem, improving the previous $O(n^2\log^2\log n)$ randomized result by Buchin, Buchin, Meulemans, and Mulzer [SODA'14] and the recent $O(n^2/\log^μn)$-time randomized result claimed by Cheng and Huang [SODA'25]. Since these problems have near-quadratic OV-based conditional lower bounds and shaving of two log factors is the best that is known for similar, simpler problems such as longest common subsequence and edit distance, our results reach the natural limit of the current state of the art (up to $\log\log$ factors).

cs.CG↗