## Computing Betti numbers via combinatorial Laplacians (1996)

Venue: | In Proc. 28th Ann. ACM Sympos. Theory Comput |

Citations: | 34 - 1 self |

### BibTeX

@INPROCEEDINGS{Friedman96computingbetti,

author = {Joel Friedman},

title = {Computing Betti numbers via combinatorial Laplacians},

booktitle = {In Proc. 28th Ann. ACM Sympos. Theory Comput},

year = {1996},

pages = {386--391}

}

### Years of Citing Articles

### OpenURL

### Abstract

We use the Laplacian and power method to compute Betti numbers of simplicial complexes. This has a number of advantages over other methods, both in theory and in practice. It requires small storage space in many cases. It seems to run quickly in practice, but its running time depends on a ratio, ν, of eigenvalues which we have yet to fully understand. We numerically verify a conjecture of Björner, Lovász, Vrećica, and ˘ Zivaljević on the chessboard complexes C(4,6), C(5,7), and C(5,8). Our verification suffers a technical weakness, which can be overcome in various ways; we do so for C(4,6) and C(5,8), giving a completely rigourous (computer) proof of the conjecture in these two cases. This brings up an interesting question in recovering an integral basis from a real basis of vectors. 1

### Citations

333 |
Eigenvalues in Riemannian Geometry
- Chavel
- 1984
(Show Context)
Citation Context ...nd λ(∆i) for all i when d ≥ 3, we can give some arguments which suggest how λ1(∆i) changes if we take a fixed simplicial complex and subdivide it more and more finely. For one thing, it is known (see =-=[Cha84]-=-) that for continuous Laplacians acting on the i-forms of d-dimensional manifolds, we have that the λj are infinite and grow like λn ≈ cn 2/d . It is known that certain types of refinements of combina... |

111 |
Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix
- Bachem
- 1979
(Show Context)
Citation Context ...gical space directly in this form. Computing homology at present seems hard; currently algorithms require computing the Smith normal form of the matrix. The latter can be done in polynomial time (see =-=[KB79]-=-), but the current polynomial required is still quite large in degree 1 .Computing Betti numbers, on the other hand, seems much easier. To find bi, one only need compute the ranks of two matrices. Thi... |

99 | An Incremental Algorithm for Betti Numbers of Simplicial Complexes on the 3–Sphere
- Delfinado, Edelsbrunner
- 1995
(Show Context)
Citation Context ...age is used; thus the operations become very expensive. It becomes important to have techniques for computing the Betti numbers which are faster and require less storage. As a step in this direction, =-=[DE93]-=- give an algorithm to compute Betti numbers; it works in almost linear time in the ni, but only works for subtriangulations of R 3 . Their method is based on a standard type of topological induction, ... |

61 | On the second eigenvalue and random walks in random d-regular graphs
- Friedman
- 1991
(Show Context)
Citation Context ...own that if X is a random graph on n vertices of degree d, then λ1(∆0) =d − √ d(2 + o(1)) with high probability in many situations (e.g. d ≥ O(log 2 n)andn large or d fixed and even and n large) (see =-=[Fri91]-=-). It follows that ν ≥ 1 − d −1/2 (2 + o(1)) for most graphs. 7.2 Bounds Based on Worst Case Graphs We can get some bounds on ∆0 just from graph theory bounds. Indeed, from [Fri94], it follows that fo... |

60 | The colored Tverberg’s problem and complexes of injective functions - Zivaljevic, Vrecica - 1992 |

40 |
Finite-difference approach to the Hodge theory of harmonic forms
- Dodziuk
- 1976
(Show Context)
Citation Context ... maps ∂i: Ci →Ci−1, as in equation 2.1, such that ∂i−1 ◦ ∂i = 0 for all i. Endowing each Ci with an inner product, we get maps ∂ ∗ i : Ci−1 →Ci (i.e. the transpose of ∂i), and thus a Laplacian (as in =-=[Dod76]-=-), ∆i: Ci →Ci, for each i, defined by ∆i = ∂i+1∂ ∗ i+1 + ∂ ∗ i ∂i. For each i we define the set of harmonic i-forms to be Hi = {c ∈Ci|∆ic =0}. For chain complexes where each Ci is a finite dimensional... |

40 |
The Theory and Applications of Harmonic Integrals
- Hodge
- 1941
(Show Context)
Citation Context ...Laplacian Our main tool is to use the combinatorial Laplacians (see [Hod41, Eck45, Dod76, DP76]) to compute the Betti numbers. These Laplacians are most easily described via the Hodge theory of Hodge =-=[Hod41]-=-. Recall that the Betti numbers, bi, are the dimensions of the homology groups, Hi =ker(∂i)/im(∂i+1) of the chain complex, ···−→Ci+1 ∂i+1 −→ Ci ∂i −→ Ci−1 −→ · · · −→ C−1 =0, (2.1) where Ci is the spa... |

35 | Chessboard complexes and matching complexes - Björner, Lovasz, et al. - 1994 |

27 | On the Betti numbers of chessboard complexes
- Friedman, Hanlon
- 1998
(Show Context)
Citation Context ... section 5 we give two complexes whose Betti numbers we compute via our algorithm, and we give some computational data about our numerical experiments. In section 6 2 This has lead to the recent work =-=[FH]-=-, where this and part of the [BLV ˘ Z94] conjecture is proven. 3swe describe our findings on the chessboard complexes. In section 7 we describe some provable and some conjectural lower bounds for λ1 a... |

22 |
Riemannian structures and triangulations of manifolds
- Dodziuk, Patodi
- 1976
(Show Context)
Citation Context ...d grow like λn ≈ cn 2/d . It is known that certain types of refinements of combinatorial Laplacians have limits whose eigenvalues converge to those of the continuous Laplacian in a certain sense (see =-=[DP76]-=-). So it makes sense to conjecture that for regular types of refinements (i.e. those where the aspect ratio of sides of a simplex remains bounded with respect to some fixed, smooth metric) we have λn ... |

20 | On the complexity of computing the homology type of a triangulation
- DONALD, CHANG
- 1991
(Show Context)
Citation Context ...nly works for subtriangulations of R 3 . Their method is based on a standard type of topological induction, and for more general simplicial complexes it is not clear how to make this method fast. 1In =-=[DC91]-=- it is claimed that this can be done faster for sparse matrices, but the latest version of this paper still has serious flaws. 2sIn this paper we exploit the combinatorial Laplacian to give a simple a... |

19 |
Computational geometry: a retrospective
- Chazelle
- 1994
(Show Context)
Citation Context ...tion in biology, chemistry, robotics, and scene analysis, involving low dimensional topological spaces, as well as time series analysis and dynamical systems, involving higher dimensional spaces (see =-=[Cha95]-=-). 1sA part of the homology groups are the Betti numbers, the i-th Betti number, bi = bi(X), being the rank of Hi(X). The Betti numbers often have intuitive meanings. For example, b0 is simply the num... |

16 |
Some Cohen-Macaulay complexes and group actions
- Garst
- 1979
(Show Context)
Citation Context ... 560 .65 4 1820 1820 2.25 5 4368 7248 7.41 6 7912 33736 25.68 7 10560 90304 57.83 8 9762 128354 74.69 9 5632 91808 50.96 10 1672 23464 11.45 11 208 2704 1.28 Table 3: CPU Seconds per 1000 iterations. =-=[Gar79]-=-,[BLV ˘ Z94], however it appears in some combinatorial geometric problems, such as the Colored Tverberg’s Problem, as in [ ˘ ZV92] (see also [ABFK92] for applications). The k-connectivity of C(m, n) w... |

15 |
Point selections and weak ɛ-nets for convex hulls
- Alon, Bárány, et al.
- 1992
(Show Context)
Citation Context ...2704 1.28 Table 3: CPU Seconds per 1000 iterations. [Gar79],[BLV ˘ Z94], however it appears in some combinatorial geometric problems, such as the Colored Tverberg’s Problem, as in [ ˘ ZV92] (see also =-=[ABFK92]-=- for applications). The k-connectivity of C(m, n) was a key fact in [ ˘ ZV92] for certain values of m, n, k;in general, k-connectivity (for k ≥ 1) is equivalent to the triviality of π0 and π1 and the ... |

6 |
Probabilistic spaces of Boolean functions of a given complexity: generalities and random k-sat coefficients
- Friedman
- 1992
(Show Context)
Citation Context ...es not vanish; for these complexes we can compute all the Betti numbers (using the known Euler characteristic) as soon as we can detect whether or not a Betti number vanishes. This method was used in =-=[Fri92]-=- to help successfully calculate the Betti numbers of a large family of complexes. We finish by summarizing the rest of this paper. In section 2 we describe the combinatorial Laplacians by way of Hodge... |

3 | Harmonische Funktionen und Randvertanfgaben in einem Komplex - Eckmann |

2 |
Minimum higher eigenvalues of Laplacians on graphs
- Friedman
- 1994
(Show Context)
Citation Context ...n and n large) (see [Fri91]). It follows that ν ≥ 1 − d −1/2 (2 + o(1)) for most graphs. 7.2 Bounds Based on Worst Case Graphs We can get some bounds on ∆0 just from graph theory bounds. Indeed, from =-=[Fri94]-=-, it follows that for a connected graph on n nodes we have λ1(∆) ≥ 2 − 2cos(π/n)=π 2 /n 2 + O(n −4 ) where ∆ is the graph Laplacian. Since ∆0 is just a graph Laplacian, it follows that the same bound ... |

2 |
Error bounds on the power method for determining the largest eigenvalue
- Friedman
- 1995
(Show Context)
Citation Context ...ce of the power method; for example, if λ2 �= λ1 then L<∞ iff the power method converges. It is not hard to estimate: � Lemma 3.1 For any α>0, the probability that L ≥ nα is at most 2/(πα). Proof See =-=[Fri95]-=-. L’s importance can be seen from the next lemma. Lemma 3.2 Let T (vr) ≤ λ 2 1(1−η) for some r and η>0. Then L ≥ η(1−η) −r . Also, if λ1 has multiplicity m and λm+1/λ1 = µ, then L ≥ ηµ −2r . Proof We ... |

2 |
Elements of Algebraic Topology. Benjamin/Cummings
- Munkrees
- 1984
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Citation Context ...sed under taking subsets. It gives rise to a certain topological space, and the homology of any “reasonable” topological space can be computed from those of an “approximating” simplicial complex (see =-=[Mun84]-=-). In this paper we restrict ourselves to input data being a simplicial complex (actually it suffices to list the maximal faces). In applications one is often given a topological space directly in thi... |