Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0705.3061

Measuring and Localing Homology Classes

Daniel Freedman·Chao Chen
View original source ↗Read source document ↗Export metadata ↓

Abstract

We develop a method for measuring and localizing homology classes. This involves two problems. First, we define relevant notions of size for both a homology class and a homology group basis, using ideas from relative homology. Second, we propose an algorithm to compute the optimal homology basis, using techniques from persistent homology and finite field algebra. Classes of the computed optimal basis are localized with cycles conveying their sizes. The algorithm runs in $O(β^4 n^3 \log^2 n)$ time, where $n$ is the size of the simplicial complex and $β$ is the Betti number of the homology group.

Explore related subjects

cs.CGmath.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Daniel Freedman, Chao Chen. 2007-05-24. Measuring and Localing Homology Classes. https://arxiv.org/abs/0705.3061

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↗