Testing Contractibility in Planar Rips Complexes ∗
Erin W. Chambers
Jeff Erickson
Pratik Worah
September 24, 2007
1
Introduction
R affinely onto the convex hull of its vertices in R2 .
The Rips shadow S(P ) is the image of this canonical
projection map. The Rips shadow is a planar region,
possibly with holes, with a piecewise-linear boundary. We define the complexity of the shadow to be the
total number of boundary vertices and edges.
The canonical projection map p naturally induces a
map π1 (p) : π1 (R(P )) → π1 (S(P )) between the fundamental groups of the Rips complex and its shadow.
Our algorithmic results rely heavily on the following
recent result of Chambers et al. [7]:
A fundamental class of problems in computational
topology deals with properties of paths and cycles in
various topological spaces that are invariant under
continuous deformation, or homotopy. For example,
given two paths, can one be continuously deformed
into the other? Given a topological metric space,
what is the shortest cycle that cannot be continuously
contracted to a single point? These and similar problems have been studied extensively for regions of the
plane with holes and graphs embedded on surfaces.
See, for example, [1, 2, 15] and [6, 10, 8]. (Further Theorem 2.1. For any set P of points in the plane,
references omitted due to space.) However, for gen- the induced map π1 (p) : π1 (R(P )) → π1 (S(P )) is an
eral simplicial complexes, even determining whether isomorphism.
two paths are homotopic is undecidable [16].
Note that, unlike Čech complexes and α-shapes
In this paper, we develop algorithms for some basic
[9], the Rips complex and its shadow are not homohomotopy questions in (Vietoris-)Rips complexes. The
topy equivalent in general.
Rips complex of a set of points is a simplicial complex
We develop an efficient algorithm to compute the
that contains a simplex for each subset with diameRips shadow S(P ) of a given set P of n points in
ter less than 1. These complexes were introduced
the plane. Our algorithm relies on two structural reby Leopold Vietoris [19] as the basis of an early hosults, which may be of independent interest. First,
mology theory; they were later independently disalthough the Rips complex R(P ) can have Θ(n2 )
covered by Elaiyhu Rips and popularized by Gromov
edges and Θ(n3 ) triangles in the worst case, the Rips
[14] (who coined the name ‘Rips complex’) as a tool
shadow S(P ), which is the union of those edges and
for studying hyperbolic groups. Ghrist [12], Carlstriangles, always has complexity O(n). Second, there
son [4], and others have proposed Rips complexes as
is a subset of O(n) Rips edges and Rips triangles
a lightweight representation of the topological strucwhose union is the entire the Rips shadow S(P ).
ture of high-dimensional data. A recent example of
this approach is the analysis by Carlsson and oth- Theorem 2.2. Given a set P of n points in the plane,
ers [3, 5, 12] of a large set of nine-dimensional fea- we can construct S(P ) in O((m + n) log n) time,
ture points extracted from digital photographs (the where m is the number of edges in the proximity
“Mumford data set”). Rips complexes of points in the graph of P .
plane have also been used to model coverage problems in sensor networks [13, 18, 17].
Given our construction of the shadow, we also give
an efficient algorithm to determine whether a given
cycle of Rips edges is contractible, or equivalently,
2 Our Results
whether two paths with common endpoints are homotopic, in the Rips complex R(P ). Theorem 2.1
For any planar Rips complex R, there is a canonical implies that testing contractibility in R(P ) is simply
projection map p : R → R2 that maps each simplex in a matter of tracking how many times the projected
cycle p(γ) winds around each hole in S(P ).
∗ Research for all authors is partially supported by NSF grant
We can directly apply an algorithm of Cabello
DMS-0528086; Erin Chambers is additionally supported by an NSF
et al. [2] which checks contractibility of cycles in genGraduate Research Fellowship.
1
eral planar regions to Rips cycles by choosing an arbitrary point in each hole of the Rips shadow, plus one
point outside the outer boundary. However, the special geometric structure of Rips shadows allows us to
develop a faster algorithm.
Our algorithm constructs a spanning tree of the
holes such that any unit-length line segment crosses
a constant number of edges. (Such a spanning tree
does not exist for general√planar
√ regions—consider a
unit square containing a n × n grid of tiny holes.)
The existence of this spanning tree follows from another structural result of independent interest:
[2] S. Cabello, Y. Liu, A. Mantler, and J. Snoeyink. Testing homotopy for paths in the plane. Discrete Comput.
Geom., 31(1):61–81, 2004.
[3] E. Carlsson, G. Carlsson, and V. de Silva. An algebraic
topological method for feature identification. Intl. J.
Comput. Geom. Appl., 16(4):291–314, 2006.
[4] G. Carlsson. Persistent homology and the analysis
of high dimensional data. Presentation at the Symposium on the Geometry of Very Large Data Sets,
Fields Institute for Research in Mathematical Sciences, February 24, 2005.
[5] G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian. On the local behaevior of spaces of natural images. Preprint, 2006.
[6] E. Chambers and S. Cabello. Multiple source shortest
paths in a genus g graph. In Proc. 18th Ann. ACMSIAM Symp. Discrete Algorithms, pages 89–97, 2007.
[7] E. W. Chambers, V. de Silva, J. Erickson, and
R. Ghrist. Rips complexes of planar point sets.
Preprint, 2007.
[8] T. K. Dey and S. Guha. Transforming curves on surfaces. J. Comput. Sys. Sci., 58(2):297–325, 1999.
[9] H. Edelsbrunner. The union of balls and its dual
shape. Discrete Comput. Geom., 13:415–440, 1995.
[10] J. Erickson and S. Har-Peled. Optimally cutting a surface into a disk. Discrete Comput. Geom., 31(1):37–
59, 2004.
[11] E. Fredkin. Trie memory. Commun. ACM, 3(9):490–
499, 1960.
[12] R. Ghrist. Barcodes: The persistent topology of data.
Preprint, 2007.
[13] R. Ghrist and A. Muhammad. Coverage and holedetection in sensor networks via homology. In Proc.
4th Int. Symp. Information Processing in Sensor Networks, page 34, Piscataway, NJ, USA, 2005. IEEE
Press.
[14] M. Gromov. Hyperbolic groups. In S. M. Gersten, editor, Essays in Group Theory, volume 8 of MSRI Publications, pages 75–265. Springer-Verlag, 1987.
[15] J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. Comput. Geom. Theory Appl., 4(2):63–97, 1994.
[16] A. A. Markov. Insolubility of the problem of homeomorphy. In Proceedings of the International Congress
of Mathematics, pages 14–21. Cambridge University
Press, 1958.
[17] A. Muhammad and A. Jadbabaie. Asymptotic stability of switched higher order Laplacians and dynamic coverage. In Hybrid Systems: Computation and
Control, Lecture Notes Comput. Sci., page to appear.
Springer-Verlag, 2007.
[18] A. Muhammad and A. Jadbabaie. Dynamic coverage verification in mobile sensor networks via
switched higher order Laplacians. In Proceedings of
Robotics: Science and Systems, page to appear, 2007.
[19] L. Vietoris.
Über den hoheren Zusammenhang
kompakter Raume und eine Klasse von zusammenhangstreuen Abbildungen. Math. Ann., 97:454–472,
1927.
Theorem 2.3 (No small holes). Any hole in the
Rips shadow of a set
√ of points
√ in the plane has circumradius at least ( 2 − 1)/8 3 ≈ 0.029893.
Our algorithm then follows a standard strategy
used by Cabello et al. [2] and others (e.g [15, 8, 1])
for encoding the homotopy class of paths and cycles
in two-dimensional spaces. We compute a sequence
of line segments, which we call fences, that form a
spanning tree of the holes; we assign each fence an
arbitrary orientation. The crossing word of any cycle γ records the sequence of fences that γ crosses,
along with the direction of each crossing. We can
reduce any crossing word by removing any matching
pairs of the form xx or xx. A cycle is contractible if
and only if its reduced crossing word is empty. In our
algorithm, we compute a set of fences with the special property that any Rips edge crosses O(1) fences,
which leads to the following algorithm.
Theorem 2.4. After O(m log n) preprocessing time,
we can determine whether any given cycle of k Rips
edges is contractible in R(P ), either in O(k log n)
time using O(n) space, or in O(k) time using O(m)
space.
Finally, we give an algorithm to find the shortest
noncontractible cycle in the complex. For every vertex, our algorithm computes the shortest path tree
and then stores reduced crossing words corresponding to those shortest paths in a hashed trie [11].
Theorem 2.5. Given a set P of n points in the plane,
we can compute the shortest non-contractible cycle
in R(P ) in O(n2 log n + mn) expected time.
References
[1] S. Bespamyatnikh. Computing homotopic shortest
paths in the plane. In Proc. 14th Annu. ACM-SIAM
Sympos. Discrete Algorithms, pages 609–617, 2003.
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