---
_id: '3129'
abstract:
- lang: eng
text: "Let K be a simplicial complex and g the rank of its p-th homology group Hp(K)
defined with ℤ2 coefficients. We show that we can compute a basis H of Hp(K) and
annotate each p-simplex of K with a binary vector of length g with the following
property: the annotations, summed over all p-simplices in any p-cycle z, provide
the coordinate vector of the homology class [z] in the basis H. The basis and
the annotations for all simplices can be computed in O(n ω ) time, where n is
the size of K and ω < 2.376 is a quantity so that two n×n matrices can be multiplied
in O(n ω ) time. The precomputed annotations permit answering queries about the
independence or the triviality of p-cycles efficiently.\r\n\r\nUsing annotations
of edges in 2-complexes, we derive better algorithms for computing optimal basis
and optimal homologous cycles in 1 - dimensional homology. Specifically, for computing
an optimal basis of H1(K) , we improve the previously known time complexity from
O(n 4) to O(n ω + n 2 g ω − 1). Here n denotes the size of the 2-skeleton of
K and g the rank of H1(K) . Computing an optimal cycle homologous to a given 1-cycle
is NP-hard even for surfaces and an algorithm taking 2 O(g) nlogn time is known
for surfaces. We extend this algorithm to work with arbitrary 2-complexes in O(n
ω ) + 2 O(g) n 2logn time using annotations.\r\n"
alternative_title:
- LNCS
author:
- first_name: Oleksiy
full_name: Busaryev, Oleksiy
last_name: Busaryev
- first_name: Sergio
full_name: Cabello, Sergio
last_name: Cabello
- first_name: Chao
full_name: Chen, Chao
id: 3E92416E-F248-11E8-B48F-1D18A9856A87
last_name: Chen
- first_name: Tamal
full_name: Dey, Tamal
last_name: Dey
- first_name: Yusu
full_name: Wang, Yusu
last_name: Wang
citation:
ama: 'Busaryev O, Cabello S, Chen C, Dey T, Wang Y. Annotating simplices with a
homology basis and its applications. In: Vol 7357. Springer; 2012:189-200. doi:10.1007/978-3-642-31155-0_17'
apa: 'Busaryev, O., Cabello, S., Chen, C., Dey, T., & Wang, Y. (2012). Annotating
simplices with a homology basis and its applications (Vol. 7357, pp. 189–200).
Presented at the SWAT: Symposium and Workshops on Algorithm Theory, Helsinki,
Finland: Springer. https://doi.org/10.1007/978-3-642-31155-0_17'
chicago: Busaryev, Oleksiy, Sergio Cabello, Chao Chen, Tamal Dey, and Yusu Wang.
“Annotating Simplices with a Homology Basis and Its Applications,” 7357:189–200.
Springer, 2012. https://doi.org/10.1007/978-3-642-31155-0_17.
ieee: 'O. Busaryev, S. Cabello, C. Chen, T. Dey, and Y. Wang, “Annotating simplices
with a homology basis and its applications,” presented at the SWAT: Symposium
and Workshops on Algorithm Theory, Helsinki, Finland, 2012, vol. 7357, pp. 189–200.'
ista: 'Busaryev O, Cabello S, Chen C, Dey T, Wang Y. 2012. Annotating simplices
with a homology basis and its applications. SWAT: Symposium and Workshops on Algorithm
Theory, LNCS, vol. 7357, 189–200.'
mla: Busaryev, Oleksiy, et al. *Annotating Simplices with a Homology Basis and
Its Applications*. Vol. 7357, Springer, 2012, pp. 189–200, doi:10.1007/978-3-642-31155-0_17.
short: O. Busaryev, S. Cabello, C. Chen, T. Dey, Y. Wang, in:, Springer, 2012, pp.
189–200.
conference:
end_date: 2012-07-06
location: Helsinki, Finland
name: 'SWAT: Symposium and Workshops on Algorithm Theory'
start_date: 2012-07-04
date_created: 2018-12-11T12:01:33Z
date_published: 2012-06-19T00:00:00Z
date_updated: 2021-01-12T07:41:15Z
day: '19'
department:
- _id: HeEd
doi: 10.1007/978-3-642-31155-0_17
external_id:
arxiv:
- '1107.3793'
intvolume: ' 7357'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1107.3793
month: '06'
oa: 1
oa_version: Preprint
page: 189 - 200
publication_status: published
publisher: Springer
publist_id: '3569'
quality_controlled: '1'
scopus_import: 1
status: public
title: Annotating simplices with a homology basis and its applications
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 7357
year: '2012'
...