COVERAGE COORDINATE-FREE IN SENSOR NETWORKS WITH
CONTROLLED BOUNDARIES VIA HOMOLOGY
V. DE SILVA AND R. GHRIST
A BSTRACT. We introduce tools from computational homology to verify coverage in an idealized sensor network. Our methods are unique in that, while they
are coordinate-free and assume no localization or orientation capabilities for the
nodes, there are also no probabilistic assumptions. The key ingredient is the theory of homology from algebraic topology. We demonstrate the robustness of these
tools by adapting them to a variety of settings, including static planar coverage, 3d barrier coverage, and time-dependent sweeping coverage. We also give results
on hole repair, error tolerance, optimal coverage, and variable radii. An overview
of implementation is given.
1. I NTRODUCTION
Sensor networks are an increasingly essential and pervasive feature of modern
computation and automation [14]. Within this large topic of active and rapidly
developing research, coverage problems are common. Such problems, involving
gaps or holes in sensor networks, appear in a variety of settings relevant to robotics and networks: environmental sensing, communication and broadcasting,
robot beacon navigation, surveillance, security, and warfare are common application domains. A specic example is as follows. Given a collection of nodes X in a
bounded domain D of the plane, assume that each node can sense, broadcast to, or
otherwise cover a region of xed coverage radius about the node. The most basic
form of coverage problem is the simple query: given the nodes, does the collection
of coverage discs at X cover the domain D?
We provide a sufciency criterion for coverage. We do not answer the problem
of how the nodes should be placed in order to maximize coverage nodes are
assumed to be distributed a priori, yet not according to some xed protocol. In
particular, there are no assumptions about random distributions or densities. The
coverage criterion we introduce is both computable and, at this time, centralized.
We do not here demonstrate how to reduce the homological criteria of this paper
to a distributed computation.
VdS supported by DARPA # SPA 30759.
RG supported by DARPA # HR0011-05-1-0008 and by NSF PECASE Grant # DMS - 0337713.
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V. DE SILVA & R. GHRIST
1.1. Assumptions. We assume a complete absence of localization capabilities. Nodes
can determine neither distance nor direction. Only connectivity data between
nodes is used. The only strong assumption we make is on the fence nodes set up
along the boundary of the domain. This strong degree of control along the boundary is not strictly required (see 6 of this paper and also [11]), but it simplies the
statements and proofs of theorems dramatically.
A1: Nodes X broadcast their unique ID numbers. Each node can detect the
identity of any node within broadcast radius rb .
A2: Nodes have radially symmetric covering domains of cover radius rc
rb / 3.
A3: Nodes X lie in a compact connected domain D R2 whose boundary D
is connected and piecewise-linear with vertices marked fence nodes Xf .
A4: Each fence node v Xf knows the identities of its neighbors on D and
these neighbors both lie within distance rb of v .
To summarize, the sensor data for each node consists of a list of node ID numbers
within signal detection range, as well as a binary ag denoting whether or not it is
a marked fence node.
1.2. Results. We claim that, surprisingly, such coarse coordinate-free data is sufcient to rigorously verify coverage in many instances. One constructs the communication graph whose vertices are the nodes of the network and whose edges
represent signal detection connectivity (at radius rb ). From this graph we build
the Rips complex R: the largest simplicial complex with the corresponding graph
as its 1-d skeleton. By assumption A4 the boundary D can be represented as a
1-dimensional fence cycle F R which is canonically identied with D.
Our results are all based on a certain algebraic-topological invariant of these simplicial complexes homology reviewed in Appendix A. The following is the
principal criterion for coverage we derive in this paper:
Main Theorem: The union of the radius rc discs contains D if there is a nontrivial
element of the relative homology H2 (R, F ) whose boundary is nonvanishing.
See Theorem 3.3 for details. The casual reader is advised to think of this homology
H2 (R, F ) as a vector space which is computed from the network according to some
algorithm. The criterion of the Main Theorem is that, rst, this vector space has
dimension greater than zero, and second, one can nd a good basis element.
In 4-11 we provide several extensions of this result. These include the following:
(1) Criteria for performing hole repair in systems for which the coverage criterion fails;
(2) Criteria for localized coverage in an unbounded network resulting from
querying a cycle in the communication graph;
(3) Criteria for coverage in domains with multiple boundary components;
COVERAGE VIA HOMOLOGY
3
(4) A homological approach to identifying redundant nodes in a cover;
(5) Coverage criteria for systems with varying communication and coverage
radii
(6) Coverage criteria for systems with communication errors and faulty nodes;
(7) Barrier coverage for 3-d systems in a tunnel-like domain;
(8) Pursuit-evasion criteria for time-dependent systems.
Comments on implementation and simulations appear in 12, followed by a discussion.
1.3. Related work. There is a large literature on the subject of static or blanket
coverage; see, e.g., [16, 3, 29] and references therein. In addition, there are variants on these problems involving barrier coverage to separate regions. Dynamic
or sweeping coverage [8] is a common and challenging task with applications
ranging from security to housekeeping.
There are two primary approaches to static coverage problems in the literature.
The rst uses computational geometry tools applied to exact node coordinates.
This typically involves computational geometry [23] and Delaunay triangulations
of the domain [29, 27, 37]. Such approaches are very rigid with regards to inputs:
one must know exact node coordinates and one must know the geometry of the
domain precisely to determine the Delaunay complex.
To alleviate the former requirement, many authors have turned to probabilistic
tools. For example, in [25], the author assumes a randomly and uniformly distributed collection of nodes in a domain with a xed geometry and proves expected
area coverage. Other approaches [28, 36, 26, 22] give probabilistic or percolation
results about coverage and network integrity for randomly distributed nodes. The
drawback of these methods is the need for a uniform distribution of nodes.
More recently, the robotics community has explored how networked sensors and
robots can interact and augment each other: see, e.g., [4, 5, 7, 14] and references
therein. There are several new approaches to networks without localization that
come from researchers in ad hoc wireless networks that are not unrelated to coverage questions. One example is the routing algorithm of [33], which generally
works in practice but is a heuristic method involving heat-ow relaxation. The
papers [6, 17, 31, 34] give methods for localizing an entire network if localization
of a certain portion is known. More recent work of Fekete et al. [15] grows and
merges cycles in a distributed manner to ll up a sufciently well-sampled network to determine boundaries in a coordinate-free network. This is one example
of the work in computational geometry concerning unit disc graphs.
The mathematical tools we introduce for coverage problems homology theory
date roughly from the 1930s. The use of homology as an effective tool in scientic computation is more recent: see, e.g., the textbook of [24] and its references.
Homology has recently been used is several applied contexts, from point cloud
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V. DE SILVA & R. GHRIST
shape representation and high-dimensional data analysis [38, 10], vision [1], applied differential equations [24, 30], and hybrid controls [2]. The reader who is
not familiar with homology theory can nd a brief summary tailored towards the
applications of this paper in the Appendix.
2. T HE R IPS COMPLEX
Given a collection of nodes X in a domain, we wish to determine the global properties of U , the union of coverage domains centered at these nodes. However, we are
constrained to use only communication connectivity data between nodes. Instead
of restricting attention to the graph of pairwise-connectivity data, we complete it
to a higher-dimensional complex. This type of simplicial complex was introduced
by Vietoris in the early history of homology theory [35], and has more recently
been reinterpreted by Rips [19] and used extensively in geometric group theory.
Denition 2.1. Given a set of points X = {x } in a metric space and a xed
> 0, the Rips complex of X , R (X ), is the abstract simplicial complex whose
k -simplices correspond to unordered (k + 1)-tuples of points in X which are pairwise within distance of each other.
Our goal is to compare the topology of the Rips complex R = Rrb (X ) to the union
of covering discs U = Urc (X ). The cover U is necessarily a subset of R2 ; the Rips
complex, in contrast, may have any dimension, depending on clustering of nodes.
It is best to visualize R as a high-dimensional space which oats above the Euclidean plane: cf. Fig. 1. This paper asserts that topological features of R sufce to
conclude geometric properties of U .
F IGURE 1. A collection of sensor nodes generates a cover in the
workspace [bottom]. The Rips complex of the network is an abstract simplicial complex which has no localization or coordinate
data [top]. In the example illustrated, the Rips complex encodes
the communication network as one closed 3-simplex, eleven closed
2-simplices, and seven closed 1-simplices connected as shown. The
holes in this Rips complex reect the holes in the sensor cover,
below.
COVERAGE VIA HOMOLOGY
5
The following lemma demonstrates that the choice of bound for rc in A2 is the
appropriate one.
Lemma 2.2. The convex hull of any collection of nodes in D which form a simplex of R
lies within U .
Proof: Any collection of circular disks which meet at a common point x necessarily
covers the convex hull of x and the centers of the discs. So, it sufces to show that
the balls of radius rc intersect. It also sufces to prove this for a 2-simplex of R
thanks to Hellys theorem [13], which implies that a collection of k 4 convex sets
in R2 has a nonempty common intersection provided only that the same is true for
each subset of size 3.
Therefore, consider a triple of points {xi }3 which span a triangle with side lengths
1
at most rb . We must show that the three discs of radius rc centered on {xi }3 meet
1
at a common point. If the triangle is obtuse (or right-angled), then the midpoint
of the longest side is common to all three discs; hence rc rb /2 sufces. If the
triangle is acute then the largest angle, say 1 at vertex x1 , satises /3 1 /2
and so sin(1 ) 3/2. We can compute the circumradius R of the triangle as
R = x2 x3 /2 sin 1 ,
and hence we deduce R rb / 3 rc . Thus, in this case, the three discs meet at
the circumcenter.
Remark 2.3. The ratio rc rb / 3 is optimal: consider an equilateral triangle of
side length rb .
Unfortunately, the radius-rb Rips complex of a set of nodes in R2 does not always
capture the topology of the union of radius-rc balls centered on these nodes. Fig. 2
gives examples for which the Rips complex fails to capture the topology of the
cover.
3. A HOMOLOGICAL CRITERION FOR COVERAGE
We use the homology of R relative to F to obtain a coverage criterion.
The intuition behind the coverage criterion is very straightforward. Based on
the communication graph alone, it is difcult to see potential holes in coverage.
However, upon completing the graph to the Rips complex R, large holes in coverage would seem to be present in the abstract complex: see Fig. 3. One might guess
that showing there are no such holes in R implies coverage. This condition would
be translated into algebraic topological terms as H1 (R) = 0, or, that any cycle in
the communication graph can be realized as the boundary of a surface built from
2-simplices of R, each of which indicates a coverage region thanks to Lemma 2.2.
We use a slightly different criterion than H1 (R) = 0: one which is more robust
to extensions and which yields stronger information about the actual cover. The
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V. DE SILVA & R. GHRIST
F IGURE 2. [left] The Rips complex has the property that all 2simplices determine triangles in the domain which lie within the
radius rc cover. However, the Rips complex does not capture the
topology of the cover. A contractible union of rc balls can have
Rips complex with nontrivial homology in dimension one [center,
in which R is a quadrilateral], two [right, in which R is the boundary of a solid octahedron], or higher.
fence cycle F is canonically identied with the boundary D. If this cycle is nullhomologous that is, if [F ] = 0 in H1 (R) then the 2-chain which bounds F
gives specic information about the cover. Intuitively, this 2-chain has the appearance of lling in D with triangles composed of projected 2-simplices from R, as in
Fig. 4. When translated into the language of algebraic topology, such a 2-chain is a
relative 2-dimensional homology class, a certain type of generator in H2 (R, F ).
F IGURE 3. In a sensor network with a sufciently large hole in coverage [left], the communication graph [center] has a cycle that cannot be lled in by triangles. The lled in Rips complex [right]
sees this hole, even as an abstract complex devoid of sensor node
location data.
The following simple algebraic lemmas complete the setup.
Lemma 3.1. Any nonzero 1-cycle Z1 (F ) denes a nonzero element of H1 ( D).
COVERAGE VIA HOMOLOGY
7
Proof: By the denition of homology, H1 (F ) = Z1 (F )/B1 (F ). However, B1 (F ) =
(C2 (F )) = 0, since C2 (F ) = 0 in the simplicial category; hence Z1 (F ) = H1 (F ) =
H1 ( D).
Lemma 3.2. A cycle Z1 (F ) is nonzero if and only if it has a nonzero coefcient at
every fence edge.
Proof: If is a cycle, then the coefcient of at any pair of adjacent edges is the
same up to a sign, because has coefcient zero at their common vertex. Since
the boundary is connected, has the same coefcient at every edge of F up to a
sign. The lemma follows immediately.
The following theorem is our principal coverage criterion.
Theorem 3.3. For a set of nodes X in a domain D R2 satisfying assumptions A1-A4,
the sensor cover Uc contains D if there exists [] H2 (R, F ) such that = 0.
For readers who struggle with the homological formalism, the example to keep in
mind is that of a generator [] H2 (R, F ) where triangulates the domain D as
in Fig. 4[right].
F IGURE 4. The coverage criterion is an algebraic-topological formulation of the intuition of lling in the fence cycle F of the communication graph [left] with 2-simplices of the Rips complex R
[center] so as to triangulate the domain D [right].
We note (by Lemma 3.2) that the condition = 0 can easily be evaluated by
picking a single fence edge and testing whether the coefcient of on that edge
is nonzero.
Proof: We consider the simplicial realization map : R R2 which sends vertices
of the abstract complex R to the corresponding node points of X D and which
sends a k -simplex of R to the (potentially singular) k -simplex given by the convex
hull of the vertices implicated. Via A4, takes the pair (R, F ) to (R2 , D); we
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V. DE SILVA & R. GHRIST
therefore construct the following diagram from the long exact sequences:
H2 (R, F )
(1)
// H1 (F ) .
H2 (R2 , D)
// H1 ( D )
Here, acts on a class [] H2 (R, F ) by taking the boundary: [] = []
H1 (F ). It follows from the naturality of the long exact sequence that the diagram
of Eqn. (1) is commutative: = . The homology class [] is the winding
number of about D.
By assumption, = 0; hence, by way of Lemma 3.1, we observe that [] =
[] = 0. By commutativity of Eqn. (1), [] = 0, and thus [] = 0.
Assume that U does not contain D and choose p D U . Since, by Lemma 2.2,
every point in (R) lies within U , we have that : (R, F ) (R2 , D) factors
through the pair (R2 p, D). However, H2 (R2 p, D) = 0, as the following
simple computation shows. Let A = R2 p and B be a small ball about p, so that
A B is an open annulus homotopic to S 1 . Let A = D and B = . Using the
relative Mayer-Vietoris sequence of Eqn. (23), we have
(2) H2 (S 1 ) H2 (R2 p, D) 0 H2 (R2 , D) H1 (S 1 )
Since (R2 , D) deformation retracts to the pair (D, D) xing D, we have that
(3)
H2 (R2 , D) H2 (D, D) H2 (D/ D) H2 (S 2 ) R.
=
=
=
=
Since p D, the homomorphism takes the generator of H2 (R2 , D) to that of
H1 (S 1 ). Eqn. (2) therefore simplies to
(4)
=
0 H2 (R2 p, D) R R
By exactness, H2 (R2 p, D) = 0 and thus [] = 0: contradiction.
Remark 3.4. This is not a sharp criterion. It is clearly possible to have the criterion
always fail for injudicious choice of rc . For example, if rc is much larger than the
bound in Assumption (A3), then there will be many instances of coverage without a homological forcing. This being said, we note that even if one chooses the
minimal acceptable bounds from Assumption (A3), it is still possible to arrange the
points to cover DC without the homological criterion detecting this, as illustrated
in Fig. 5.
4. G ENERATORS
FOR REDUNDANT COVERS
Theorem 3.3 guarantees that the covering discs in fact cover the desired area. For
reasons of power conservation, one would like to know which nodes could be
turned off without impinging upon the coverage integrity. This is an important
COVERAGE VIA HOMOLOGY
9
F IGURE 5. Examples of two covers. The homological criterion
holds for one [left] but not for the other [center], because of a 1cycle in R [right]. Note the fragility of the cover [center] within the
1-cycle: a small perturbation of the nodes creates a hole.
problem with a large literature, see, e.g., [26, 22]. A practical approach to this
problem is implicit in homological methods.
Corollary 4.1. If a homology class in H2 (R, F ) satises the criterion of Theorem 3.3, then
the restriction of U to those nodes which make up the representative sufce to cover D,
for any choice of in the homology class.
Proof. Let U denote the restriction of U to the nodes implicated by the representative . Assume that U does not contain D and choose p D U . Lemma 2.2
implies that (R) U . Thus, : (R, F ) (R2 , D) again factors through the
pair (R2 p, D), which has vanishing homology in dimension two.
The independence of the choice of representative in the homology class is extremely important. If one chooses a minimal generator in the sense that
minimizes the number of 0-simplices within [] then Corollary 4.1 yields a
small subset of nodes which is guaranteed to cover the domain. Existing software
packages for computing homology classes can shrink generators (though without rigor in terms of being truly minimal); hence, this is an implementable strategy.
In 12, we give an example.
5. H OLE REPAIR
Since the result of Theorem 3.3 is merely a criterion, one wishes to implement a
strategy for guaranteeing coverage when the criterion fails. We present an elementary means for doing so via homology, the idea being to compute minimal
generators in H1 (R) so as detect holes. We consider a sensor network in which
all nodes are initially in a power saving mode of low coverage radius rc with the
ability to increase the coverage radii of certain nodes. The following result is most
useful in this setting, where the homological criterion fails, but just barely.
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V. DE SILVA & R. GHRIST
Theorem 5.1. Consider a set of nodes X satisfying assumptions A1-A4. Let = {i }K
1
be a basis of K generators in H1 (R) and let Ni = i for each i, where denotes length
of the generator in terms of the number of nodes implicated. Let U denote the set obtained
from the collection U by enlarging all balls based at nodes in i to balls of radius
rb
.
(5)
rc (i) = csc
2
Ni
Then D U .
Thus, for example, any Rips complex which has one or more holes of size four (as
in Fig. 3[right]), then the coverage region is guaranteed to contain D if we require
rc rb / 2 for the implicated nodes dening where the hole is.
Proof: The quantity rc (i) represents the minimal radius needed to cover a regular
Ni -gon. We claim that this is the limiting case.
Consider the image L1 = (i ) of the loop i in D. This is a (not necessarily
embedded) loop in D. A point x D is enclosed by Li if [Li ] is nonzero in H1 (R2
x) Z (this class is the winding number of the loop about x). We demonstrate
=
that covering each node of i with a ball of radius rc (i) covers any such x. For such
an x it follows that one or more of the Ni edges of L subtends an angle at x of at
least 2/Ni ; for otherwise there would exist rays originating at x which miss (i )
entirely, making Li contractible in R2 x and the winding number zero. Let ab be
such an edge. Taking cosines this inequality becomes
(6)
cos
2
Ni
2
rb
|xa|2 + |xb|2 |ab|2
1
2|xa||xb|
2|xa||xb|
where we use the AM-GM inequality and the fact that |ab| rb for the latter inequality. Since cos(2/Ni ) = 1 2 sin2 (/Ni ) we can rearrange to obtain |xa||xb|
(rc (i))2 . Thus x must lie within distance rc (i) of the nearer of the two nodes a, b, as
required.
We now create a modied complex R obtained from R in the following manner.
For each i, sew in an abstract 2-d disc along the loop i . (If one wishes to remain
in the simplicial category, one can triangulate the disc.) Next, extend the map to
a continuous map : R R2 .
The long exact sequence yields a commutative diagram as in Eqn. (1):
(7)
H2 (R , F )
// H1 (F )
H2 (R2 , D)
i
// H1 (R ) .
// H1 ( D )
i
// H (R2 )
1
Because we have lled in all the generators of H1 (R), we have that H1 (R ) = 0 and
: H2 (R , F ) H1 (F ) is onto. Exactness implies that there exists a generator []
of H2 (R ) with = F .
COVERAGE VIA HOMOLOGY
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Assume by way of contradiction that there exists a point p D U . If [Li ] = 0
H1 (R2 p) for any i, then p U by the argument above. Therefore, assume that
these homology classes vanish for all i. Since the set {i } forms a basis for H1 (R),
there exists a 2-chain in C2 (R) such that = F i ci i for some constants
ci . Applying to these 1-chains yields the equation ( ) = D i ci Li . This
descends to an equation in H1 (R2 p), since p is assumed to be not in U and
( ) U U by Lemma 2.2. We know that [ D] = 0 in H1 (R2 p) since p D.
By assumption that all the winding numbers of Li about p vanish, we have that
[ ( )] = 0 H1 (R2 p). However, C2 (R) and is an algebraic sum of 2simplices in R. At least one such 2-simplex of must therefore satisfy ( ) =
0 H1 (R2 p), implying that p ( ) U U . Contradiction.
It follows from this argument that, if one has the hardware constraint of a xed
coverage radius rc which is larger that the bound rb / 3, then one can get a better coverage criterion, as follows. Let N be the largest integer for which rc
2rb / csc(/N ). Then, build a version of the Rips complex for the network which
has all loops in the network of length less than or equal to N lled in by abstract
2-cells. Coverage is guaranteed if the resulting cell complex has a relative cycle in
H2 with nonvanishing boundary.
6. N ETWORKS
WITHOUT BOUNDARIES
Among the conditions on the sensor networks to which these results apply, Assumptions A3-A4 on the boundary are the least natural for a realistic network.
In many contexts (real and hypothetical) networks are of large enough extent that
boundary phenomena are ignorable. The homological criterion of Theorem 3.3
adapts to networks without boundaries in a number of possible ways: we outline
the simplest such extension here.
Consider a cycle in the communication graph. One approach is to interrogate
the network coverage with respect to this cycle: is the area bounded by this cycles
projection to the plane covered? One must be careful: if the projection of to the
plane is a simple closed curve, then it has a well-dened interior whose coverage
can be queried via a homology computation. Cycles which have lots of selfintersection in the projection to the plane are generally to be avoided in a coverage
querying context. Determining whether a given cycle in the network has a simple
closed image is not trivial. The following simple (and well-known) criterion is
efcacious.
Lemma 6.1. Let be a 1-cycle in R whose span, the largest subcomplex of R
generated by the nodes of is precisely . Then the projection ( ) of to the plane is
a simple closed curve.
Proof: Assume that the images of two edges e1 and e2 of intersect in their interiors, forming an X in the plane. Since the lengths of these edges are no larger
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V. DE SILVA & R. GHRIST
than rb , it follows that at least one segment of this X from e1 and one from e2 have
1
length no more than 2 rb . The triangle inequality implies that two end vertices of
these segments are within rb , forming a new edge of .
Corollary 6.2. For a planar network satisfying A1-A2, choose a cycle with = . If
H2 (R, ) has a generator [] with = 0, then the entire domain bounded by ( ) in R2
lies within the cover U .
Proof: The argument of Theorem 3.3 sufces, thanks to Lemma 6.1.
7. D OMAINS
WITH ARBITRARY PLANAR TOPOLOGY
Assumption A3 restricts the topology of the domain D in two features: connectivity of D and connectivity of D. It is not difcult to eliminate both of these
requirements. If D is disconnected, then each connected component of D can be
treated separately. If D is disconnected, we can succeed if we have some extra
information about the connected components of D.
Theorem 7.1. Consider a set of nodes X satisfying assumptions A1-A4, with A3 modied
as follows:
A3 Nodes X lie in a compact connected domain D R2 whose boundary D is
piecewise-linear with vertices marked fence nodes Xf . There is a partition
+
of Xf into Xf Xf representing those on the outer and inner boundary
components respectively.
The sensor cover Uc contains D if there exists [] H2 (R, F ) such that is nonzero on
the outermost boundary component.
To evaluate the condition on , we can pick any edge on the outermost boundary
component and check whether has a nonzero coefcient at that edge (compare
Lemma 3.2).
Proof. This is a modication of the proof of Theorem 3.3. To start with, we can write
the fence subcomplex as a disjoint union F = F + F where F + is the outermost
fence component, and F is the union of the inner fence components. Similarly
one can write D = + D D for the domain boundary. The condition on is
then equivalent to the assertion that [] = 0 where : H2 (R, F ) H1 (F , F )
=
H1 (F + ) is the boundary map in the long exact sequence for the triple (R, F , F ).
COVERAGE VIA HOMOLOGY
13
This time we have a simplicial realization map : (R, F , F ) (R2 , D, D),
which gives us the following commutative diagram:
H2 (R, F )
(8)
H1 (F + )
// H1 (F , F )
H2 (R2 , D)
H1 ( + D)
// H1 ( D , D )
The equalities on the right of the diagram come from the excision theorem, see
Eqn. (20). Since : H1 (F + ) H1 + D is an isomorphism, the same is true of
: H1 (F , F ) H1 ( D, D).
Suppose there exists [] satisfying the criterion in the theorem, so [] = 0. By
commutativity of Eqn. (1) and since the middle map is an isomorphism, it follows that [] = [] = 0. Now assume, for a contradiction, that there is
some point p D U . Since it lies in D the point p is encircled by the outermost boundary component + D but not by any of the other boundary components. Since p U the composite factors as
(9)
i
H2 (R, F ) H2 (R2 p, D) H2 (R2 , D) H1 ( D, D)
We claim that i : H2 (R2 p, D) H1 ( D, D) is the zero map, which gives
the required contradiction since it implies that [] = 0.
In fact = i is the boundary map in the long exact sequence for the triple
2
(R p, D, D). Consider the following excerpt from that sequence:
(10)
j
H2 (R2 p, D) H1 ( D, D) H1 (R2 p, D)
By exactness, we can prove that = 0 by establishing instead that j is one-toone. This can be read off from the following commutative diagram with exact
rows, coming from the inclusion map of pairs j : ( D, D) (R2 p, D).
(11)
H1 ( D)
pp88
ppp
i
ppp
ppp
0 //
H1 ( D)
H (R2 p)
1
i
// H1 ( D , D )
j
k
// H (R2 p, D )
1
The geometric content here is that the map H1 ( D) H1 (R2 p) is zero, since the
interior boundary cycles do not enclose p, whereas the map H1 ( D) H1 (R2 p)
is onto since the outer boundary cycle does encircle p. It follows that the two maps
labeled i have the same kernel and are both onto. By exactness the map labeled k
is one-to-one and therefore the same is true of j . This is what was required.
It is not enough to have = 0 as before. Consider the situation of Fig. 6, in
which a small interior boundary component is a loop of four edges. Then, one
can generate a relative 2-cycle consisting of the four boundary nodes along with a
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V. DE SILVA & R. GHRIST
single interior node which is properly situated. This, of course, does not cover the
domain.
F IGURE 6. An example of a small internal boundary component
[left] giving rise to a fake relative 2-cycle [right] in the Rips complex.
We leave it to the reader to modify the statements of theorems in the following sections to accommodate the case of domains which for which connectivity or simple
connectivity fail.
8. O PAQUE BOUNDARIES AND COMMUNICATION ERRORS
We have not carefully specied the mechanism by which nodes communicate presence over a distance. From Assumption A1 it follows that communication signals
are picked up purely as a function of distance, permeating the boundary of the domain if necessary. In certain physical situations, these communication signals may
not be capable of boundary penetration (e.g., if they are visually-detected beacons).
One might wish to modify the assumptions with the following opaque boundary
condition: Each node can detect the identity of any node connected by a straight line in
D of length at most rb . One changes the Rips complex to include only those edges
which communicate through unobstructed signals.
This is a particular example of the more general phenomenon of having communication errors of the form where two nodes within communication distance fail
to establish a link. For the most general case, consider a system satisfying A1-A4
with Rips complex R. Dene a Rips complex with omissions, ER, to be any subcomplex of R containing F (we assume perfect control of the fence nodes). This
ER may result as a random error in establishing communication links or, as above,
as a systematic failure to establish links near certain types of boundaries.
Theorem 8.1. Consider a set of nodes X in a domain D R2 satisfying assumptions
A1-A4 with ER a Rips complex with omissions. The sensor cover Uc contains D if there
exists [] H2 (ER, F ) such that = 0.
COVERAGE VIA HOMOLOGY
15
Proof: Since ER R, we have
(12)
H2 (ER, F )
// H1 (F ) .
H2 (R2 , D)
// H1 ( D )
The remainder of the proof follows exactly as in Theorem 3.3.
This result implies that the homological coverage criterion relies on the coarse metric data of Assumption A1 only in the positive sense. The criterion does not use the
fact that a failure to communicate implies a lower bound on the distance between
nodes.
9. VARIABLE R ADII
Assumptions A1-A2 on the radial symmetry of sensors are physically unrealistic:
a more accurate model would incorporate asymmetry and/or variable radii, to
accommodate errors or uctuations in signals. It is possible to apply the homological criterion to systems with asymmetric broadcast domains by using the Rips
complex with omissions of 8. One chooses rb to be an upper bound for the broad
cast signal distance and rc rb / 3. The communication network then establishes
links between certain nodes, but not purely as a function of distance. While this
method is applicable, there is a wastefulness in the bound on rc in terms of the
maximal broadcast distance.
We therefore consider systems whose radii rc and rb vary from node to node, as a
next step toward dealing with asymmetry in sensor networks. Consider the case
where a system of nodes X = {xi } satises a modied set of assumptions:
V1: Nodes X = {xi } broadcast their unique ID numbers. The identity of each
i
node can be detected by any node within its broadcast radius rb .
i
V2: Nodes have radially symmetric covering domains of cover radius rc
i / 3.
rb
V3: Nodes X lie in a compact connected domain D R2 whose boundary D
is connected and piecewise-linear with vertices marked fence nodes Xf .
V4: Each fence node v Xf knows the identities of its neighbors on D and
i
these neighbors both lie within distance rb of v .
We modify the construction of the Rips complex as follows. For any pair of nodes
xi and xj , there is an edge in R if and only if the distance between xi and xj in
j
i
D is less than or equal to the minimum of rb and rb . The full complex R is then
the maximal simplicial complex for the edge set as dened. The fence cycle F is
dened in the same way as before, with vertex set Xf and an edge between each
16
V. DE SILVA & R. GHRIST
pair of adjacent nodes along the fence. We dene the variable-radius cover Uc in
i
this context to be the union of closed discs of radii rc centered at node xi .
Theorem 9.1. For a set of nodes X in a domain D R2 satisfying the variable-radius assumptions V1-V4, the variable-radius cover Uc contains D if there exists [] H2 (R, F )
such that = 0.
Proof. The proof of Theorem 3.3, being topological, is largely independent of the
geometry the of system. The crucial geometric step is in the application of Lemma
2.2. We now verify that the variable-radius version of this lemma holds.
Consider a triple of points {x1 , x2 , x3 } which span a triangle in R with side lengths
ij
12 , 13 , and 23 , where ij min(rd , rd ). We must show that the three discs of
i
radius rb centered on xi meet at a common point (and hence cover the triangle
spanned by x1 , x2 , x3 ).
Consider the continuous function
f (x) = max fi (x) = max
i=1,2,3
i=1,2,3
x xi
.
i
rd
Since f (x) as x the function attains a global minimum, say = f (x0 ).
We must show that 1/ 3.
The minimizer x0 must lie inside the triangle x1 x2 x3 , because any point x outside
the triangle can be perturbed so as to decrease all three distances x xi simultaneously. In more detail this argument shows that x0 lies within the convex hull of
its critical vertices, dened as those vertices xi for which f (x0 ) = fi (x0 ).
There are two cases. If x0 has two critical vertices xi , xj , then x0 lies on the edge xi xj
j
i
and = fi (x0 ) = fj (x0 ) = ij /(rd + rd ) 1/2, which is less than 1/ 3. Otherwise
all three vertices x1 , x2 , x3 are critical. The largest of the three angles ij = xi x0 xj
satises ij 2/3. The interior bisector of this angle meets the edge xi xj at a
point y which divides the edge in the ratio x0 xi : x0 xj or ri : rj . Using
the sine rule for triangle x0 yxi we then have
ij ri
1
ri
sin x0 yxi
ri = x0 xi = y xi
sin(ij /2)
(ri + rj ) sin(/3)
3
giving the required bound.
The proof of the theorem now follows that of Theorem 3.3 precisely.
Of course, the results on minimal generators and Rips complexes with omissions
still apply in this setting as well, as the reader may check.
10. B ARRIER COVERAGE IN 3- D
We consider the following modication of the physical workspace of the nodes.
Let the nodes be points in a 3-d tube of the form D R for D R2 as in A3, and let
COVERAGE VIA HOMOLOGY
17
the fence nodes lie in D{0} and satisfy A4. We dene U R2 R by placing a 3-d
ball of radius rc at each xi X . The problem of barrier coverage is to determine
whether there is a path connecting D {} to D {+} avoiding U : see Fig. 7.
F IGURE 7. Barrier coverage in a 3-d tube means the non-existence
of a path from one end of the tube to the other avoiding 3-d balls
of coverage about the nodes. The vestige of the fence cycle F is a
cycle of nodes about the meridian D {0} (balls of coverage not
drawn along F for reasons of clarity).
We construct a Rips complex as before, connecting nodes if they are within distance rb in D R. From A4 it follows that the fence cycle F is precisely D {0}.
Our homological criterion immediately yields a criterion for barrier coverage.
Theorem 10.1. A collection of nodes in D R satisfying A1-A4 as above has barrier
coverage if there exists [] H2 (R, F ) with = 0.
Proof. We prove a stronger result in the spirit of Corollary 4.1. The proof of Lemma 3.1
holds for the 2-skeleton of the Rips complex: three points determine a plane which
intersects the balls in discs of radius rc . Hence, the simplicial realization map
: R D R takes any 2-cycle to a subset of U , the cover restricted to the
nodes of .
Let : R2 R R denote projection to the second factor. Assume that p : R
D R U is a continuous curve with limx p(x) = . Since every point
in () lies within U , we have that : (, ) (R2 R, D {0}) factors
through the pair (R2 R p, D {0}). However, let A = (R2 R) p and B
be a neighborhood of p, so that A B is an annular tube homotopic to S 1 . Let
A = D {0} and B = . Using Eqn. (23), we have
(13)
H2 (S 1 ) H2 ((R2 R)p, D{0})0 H2 ((R2 R), D{0}) H1 (S 1 )
18
V. DE SILVA & R. GHRIST
Since H2 ((R2 R), D {0}) H2 (D, D) R and is an isomorphism, we
=
=
obtain
(14)
=
0 H2 ((D R) p, D {0}) R R
By exactness, H2 ((R2 R) p, D{0}) = 0 and thus, [] = 0: contradiction.
11. P URSUIT- EVASION
AND MOBILE NODES
Consider a situation in which the node positions are a continuous function of time:
X = Xt D for t = 0...1. Assume that the network is sampled to give a nite
sequence of connectivity graphs {i }N at times 0 = t0 < < tN = 1, as in Fig. 8.
0
We assume the following:
T1 If two nodes are connected at time steps ti and ti+1 , then they remain within
the broadcast radius rb for all ti t ti+1 .
T2 Nodes may go off-line or come on-line, represented by deleting the nodes
in the appropriate graph i .
T3 Fence nodes always remain xed and on-line.
F IGURE 8. A mobile network with xed fence nodes sampled at
ve time segments: can an evader avoid being caught in the timedependent union of coverage discs?
We now address the question of whether there can be a wandering loss of coverage. It may be the case that at no time t [0, 1] does there exist a complete
sensor coverage of the domain; however, the changes may obstruct any sequence
of points from jumping from one hole to the next, avoiding the coverage domain.
Verifying the lack of wandering holes is a particular type of pursuit-evasion problem with relevance to problems in security and defense. Note that this problem
is distinct from the sweeping coverage problem, in which one wants to know
whether the union of the cover sets t U (t) contains D.
COVERAGE VIA HOMOLOGY
19
11.1. A prism complex. We present a homological criterion for guaranteeing no
wandering holes via computing the homology of a certain space derived from the
sequence of Rips complexes Ri .
Denition 11.1. Given a sequence {i } of vertex-labeled communication graphs as
above, dene the stacked Rips complex SR to be the cell complex obtained from
the disjoint union i Ri of the Rips complexes Ri of i by the following operation:
For each k -simplex [v1 , . . . , vk+1 ] of Ri which is also a k -simplex
on the same vertices in Ri+1 , connect these k -simplices by a prism
k [0, 1] with k {0} glued to Ri and k {1} glued to Ri+1 .
We treat the time variable t [0, 1] as an extra dimension and consider the problem
of evasive coverage in D [0, 1]. The complex SR has a natural prism structure:
SR is a 1-parameter family of simplicial Rips complexes indexed by t [0, 1], these
slices being equal to Ri at ti . See Fig. 9. We likewise consider the moving covers
as a 1-parameter family in a 3-dimensional setting. If Ut denotes the radius rc cover
of nodes Xt at time t, embed the time-varying covers into D [0, 1] via Ut D{t}.
The problem of wandering loss of coverage now becomes the question of whether
the complement of the union t Ut in D [0, 1] has a tunnel running from bottom
(t = 0) to top (t = 1).
F IGURE 9. Subsequent Rips complexes [left] are attached via
prisms between matching simplices [center] to capture the topology of the mobile cover [right].
Theorem 11.2. Consider a time-varying set of nodes Xt in a domain D R2 satisfying
assumptions A1-A4 and T1-T3. Then, for any continuous curve p : [0, 1] D, p(t) must
lie in Ut for some 0 t 1 if there exists [] H2 (SR, F [0, 1]) such that () = 0,
where : F [0, 1] F is the projection map.
Proof. As in the proof of Theorem 3.3, we consider a simplicial realization map
: SR R2 [0, 1]. Dene as follows. Given the structure of SR as a family of
Rips complexes Rt indexed by t [0, 1], let send each slice to (Rt ) D {t},
where is the realization map from the proof of Theorem 3.3 and the vertices are
sent to Xt .
20
V. DE SILVA & R. GHRIST
The map takes the pair (SR, F [0, 1]) to (R2 [0, 1], D [0, 1]), yielding the
following diagram:
(15)
H2 (SR, F [0, 1])
// H1 (F [0, 1]) .
H2 (R2 [0, 1], D [0, 1])
// H1 ( D [0, 1])
It follows from assumption T3 and Lemma 3.1 that [] = 0. By commutativity of Eqn. (1), [] = 0.
Assume that there exists a continuous curve p : [0, 1] D [0, 1] of points p(t)
+1
{D {t} Ut }. We claim that (SR) t Ut . Assume that the nodes {xi (t)}k=1
i
span a k -simplex of Rt SR at some xed time t. Then sends this to the convex
hull of these nodes in R2 {t}. From Denition 11.1 and assumption T1, any edge
in Rt implies that the node points implicated by this edge are within distance rb at
time t. An application of Lemma 2.2 then guarantees that the convex hull of these
nodes lies within Ut .
We conclude from this and the existence of the wandering curve p that : (SR, F
[0, 1]) (R2 [0, 1], D [0, 1]) factors through the pair (R2 [0, 1] p, D [0, 1]).
However, this has vanishing H2 , using the same argument as in Theorem 10.1.
Thus, [] = 0: contradiction.
11.2. A simplicial model. In practice, computing with the stacked Rips complex is
inconvenient. The software we use is meant for simplicial complexes, not the more
general prism complex SR. We therefore provide a simple means of reducing the
stacked Rips complex to a simplicial object which is much smaller and simpler to
encode.
Denition 11.3. Given a collection of network graphs {i } as in Denition 11.1,
dene the amalgamated Rips complex to be the space obtained from the disjoint
union i Ri of the Rips complexes Ri of i by the following operation:
For each k -simplex [v1 , . . . , vk+1 ] of Ri which is also a k -simplex
on the same vertices in Ri+1 , identify these simplices.
A few observations are in order. First, the amalgamated Rips complex AR is a
cell complex built from simplices. It is not, properly speaking, a [combinatorial]
simplicial complex since there may be, e.g., more than one 1-simplex connecting
two vertices; hence, cells in this complex are not uniquely dened by their faces.
Second, since the fence nodes are assumed stationary, the fence cycle F is xed in
each Ri and thus is identied to yield a well-dened cycle F AR.
Proposition 11.4. The pair (SR, F [0, 1]) is homotopy equivalent to (AR, F ).
COVERAGE VIA HOMOLOGY
21
Proof: For each i, consider the maximal subcomplex Si Ri which is also a subcomplex of Ri+1 . The prism subcomplex Si [0, 1] SR is a properly embedded
subcomplex; hence the collapse of Si [0, 1] to the simplicial subcomplex Si in
AR is a homotopy equivalence. The amalgamated complex AR is the result of
applying the sequence of collapses to SR, and the subcomplex F [0, 1] SR is
collapsed via projection of the second factor.
This immediately implies the following:
Corollary 11.5. The homological condition of Theorem 11.2 is satised if and only if
H2 (AR, F ) has a generator [] with = 0.
These hypotheses are preferable to those of Theorem 11.2 in that the spaces involved are smaller, simplicial, and there is no condition involving the projection of
the boundary of the generator. For a software package that can handle only true
combinatorial simplicial complexes, there is a simple modication of AR available. Since the homological criterion resides in H2 , one can identify all k -simplices
with the same boundary for k 2. Only the multiple 1-simplices need be distinguished, and these may be handled by inserting additional vertices and rening
the cell structure.
12. C OMPUTATION
Unlike homotopy groups (such as the fundamental group 1 ), homology is computable, and existing software packages make the homological coverage criteria
of this paper implementable for reasonable numbers of nodes. We have used the
open-source package Plex [40], which consists of: (i) C++ code for manipulating
simplicial complexes, written by Patrick Perry; (ii) C++ code for persistent homology calculations, written by Lutz Kettner and Afra Zomorodian, published independently as part of the CGAL project [39]; (iii) a MATLAB front-end and script
library, designed and written by Vin de Silva and Patrick Perry.
Since we use pre-existing code for homology computations, a few remarks are in
order with regards to implementation.
(1) Plex does not automatically compute relative homology. In order to compute homology relative to the fence, we use the following simple procedure. To compute H2 (R, F ), add a disjoint abstract vertex to R and augment this vertex to every simplex in F . This is called placing a cone over
the subcomplex F , and it yields a complex C(R, F ) whose homotopy type
is that of the quotient space R/F . It follows from the Excision Theorem
[20] and homotopy invariance that H (R, F ) H (R/F ) H (C(R, F ))
=
=
for 1; hence, this faithfully captures the homology.
(2) Our exposition of homology in Appendix A phrases everything in terms
of linear algebra on real vector spaces, for clarity and intuition. In general,
homology can be computed with any coefcient ring. The real coefcients
22
V. DE SILVA & R. GHRIST
that we use for intuition are not optimal for computation, since round-off
error can impact computation. To avoid round-off error, we use homology
with coefcients in the eld Z2 . All of our arguments are independent of
the eld coefcients used; hence the criterion is still valid with this assumption.
(3) We compute generators for homology using the persistent homology algorithm, with the interior simplices being processed rst and the cone simplices being processed last. Under this ordering the algorithm is guaranteed to give a unique homology cycle spanning the fence if any exists (although this uniqueness does not seem to be signicant). The cycle can be
read off explicitly from the results of the computation.
Fig. 10 shows a network in a simply-connected domain with 212 nodes which satises the homological coverage criterion of Theorem 3.3. The gure also shows
the image of the Rips complex in R2 under the realization map . A choice of a
simple generator shows that 111 of the nodes may be put in sleep mode without
a loss of coverage. Of necessity, this illustration shows the location of the nodes
within the domain. We stress that the algorithms have no knowledge of this data.
The input to the problem is the network connectivity graph and the fence cycle
in that graph. The generator shown here is the one produced by the homology
computation, with no subsequent optimization. No other geometric data is used.
We do not at this time present a complete analysis of the numerical implementation
of the coverage criterion.
13. C ONCLUSIONS
The applicability of homology theory to sensor networks initiated in this paper
is not as surprising as might at rst appear. Indeed, the two elds share several
features. Problems in both homology and sensor networks have as inputs a large
collection of local objects (simplices, sensors) with local interaction rules (faces,
communication). From this collection (chain complex, sensor network), one seeks
to determine global properties of the system (homology, coverage). The primary
point of departure is that chain complexes carry with them a rich algebraic structure which can be exploited to great effect. We have demonstrated that certain
features of this algebraic structure carry over to answer important questions in
coverage, power conservation, and evasion-detection. This represents a new and
powerful importation of algebraic tools in networks.
13.1. Remarks.
(1) We have not specied communication protocols on the level of hardware,
having concerned ourselves in this paper with the mathematical tools. We
claim, however, that the Rips complex can be built in a distributed fashion
COVERAGE VIA HOMOLOGY
23
F IGURE 10. A typical simulation: [top] the locations of 212 nodes
in D; [center] the image of the Rips complex R projected to D; [bottom] a simple generator of H2 (R, F ) extracts 101 nodes which are
guaranteed to cover D, leaving 111 nodes to be safely put into sleep
mode.
on the hardware level: see [32]. We expect the signal complexity of this operation to be reasonable, since the Rips complex is completely determined
by its 1-skeleton.
(2) In this paper, we have focused on the case where there is complete control
over the fence nodes. In practice, such control may not be available. By endowing nodes with the capability of detecting the boundary of the domain,
it is possible to reconstruct a fence subcomplex F composed of nodes near
the boundary. Since these are not assumed to be well spaced (as in A4) the
24
V. DE SILVA & R. GHRIST
proofs of all the results here are invalid. We demonstrate in [12, 11] how
to recover some of the results of this paper in that more general case via
persistent homology.
(3) We stress that the coverage criterion is not if-and-only-if. It is a rigorous
test to guarantee coverage, and, thus, any system which is just barely
covered will likely fail that test.
(4) The test as given in this paper is centralized: a distributed coverage algorithm is greatly desirable.
13.2. Questions. This paper represents merely the rst step in applications of algebraic topology to sensor networks. We comment on possible and probable extensions below.
(1) What is the computational complexity of the homological criterion as a
function of number of nodes? The most straightforward algorithm for computing homology (using Smith normal form) can be quintic in the number
of simplices. More recent algorithms are much faster, but the subquadratic
algorithm of [9] relies on duality for Euclidean spaces, and is not applicable
for arbitrary simplicial complexes. Our experiments hint at a subquadratic
run-time, and it may be that Rips complexes of planar networks have a
sufciently restricted topology to merit such a claim.
(2) Can one construct an effective homological coverage criterion which is distributed, allowing nodes with limited computational capabilities to compute local homology? What are the demands on the nodes computational
power and memory in such a system? What demands are made on the
communication network in a distributed homology computation?
(3) Can the mobile-network coverage criterion for wandering holes be made
asynchronous? Rather than sampling the entire network at once, subsets of
nodes should sample their connectivity and register their network graph
with a central processor. Does a homological criterion holds for such systems?
(4) By changing the bound in A2 to rc rb , the homological criterion veries
3-coverage in a planar network [a simple exercise]. Is it possible to verify k coverage for any k via homology? One wants to impose as few restrictions
on rc as possible.
(5) In practice, coverage and communication domains are not radially symmetric: elliptical or conical shapes are closer to reality in many cases. Is
it possible to construct a homological coverage criterion for sensors whose
communication and/or coverage domains are not radially symmetric? What
additional capabilities do the sensors require in order to handle such asymmetry?
(6) With the exception of the work in 11, we are working in a setting for which
it is desired that there are more than enough sensors necessary to cover
the domain. In such a sensor-rich environment, it is possible for the Rips
complex to attain a very high dimension. This is highly undesirable for
COVERAGE VIA HOMOLOGY
25
computational reasons. Is there a way to compress the Rips complex in
a preprocessing step without changing the appropriate homology group?
This seems reasonable: a 20-dimensional simplex implies a cluster of nodes,
most of which should be redundant.
(7) If we endow the nodes with additional capabilities, such as the ability to
measure some angular data about neighboring nodes, what global problems can be solved? Problems involving degree computation and target
isolation are solvable with only a very weak form of angular data at the
nodes [18].
(8) The sensor networks of this paper are relatively idealized. Real sensors
and real networks have unavoidable stochastic features. Is it possible to develop a homology theory with stochastic simplices which returns rigorous
coverage criteria in the form of, perhaps, expected homology classes?
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COVERAGE VIA HOMOLOGY
A PPENDIX A. H OMOLOGY
27
BASICS
The mathematical tools we use are by no means novel: with the exception of the
simulations, this paper could have been written in the middle of the previous century. However, as these tools are not in the repertoire of researchers in sensor
networks, we give a brief primer, coupled with the warning that homology theory
takes some work to understand. Those wanting a more complete treatment can
nd it in the excellent text of Hatcher [20].
A.1. Simplicial homology. Homology is an algebraic procedure for counting holes
in topological spaces. There are numerous variants of homology: we use simplicial homology with real coefcients, a theory adapted to simplicial complexes.
Given a set of points V , a k -simplex is an unordered subset {v0 , v1 , . . . , vk } where
vi V and vi = vj for all i = j . The faces of this k -simplex consist of all (k 1)simplices of the form {v0 , . . . , vi1 , vi+1 , . . . , vk } for some 0 i k . A simplicial
complex is a collection of simplices which is closed with respect inclusion of faces.
Triangulated surfaces form a concrete example, where the vertices of the triangulation correspond to V . The orderings of the vertices correspond to an orientation.
Any abstract simplicial complex on a (nite) set of points V has a geometric realization in some Rn .
Let X denote a simplicial complex. Roughly speaking, the homology of X , denoted H (X ), is a sequence of vector spaces {Hk (X ) : k = 0, 1, 2, 3 . . .}, where
Hk (X ) is called the k -dimensional homology of X . The dimension of Hk (X ),
called the k th Betti number of X , is a coarse measurement of the number of different holes in the space X that can be sensed by using subcomplexes of dimension
k.
For example, the dimension of H0 (X ) is equal to the number of connected components of X . These are the types of holes in X that points can detect are two
points connected by a sequence of edges or not? The simplest basis for H0 (X )
consists of a choice of vertices in X , one in each path-component of X . Likewise,
the simplest basis for H1 (X ) consists of loops in X , each of which surrounds a
different hole in X . For example, if X is a graph, then H1 (X ) is a measure of
the number and types of cycles in the graph, this measure being outtted with the
structure of a vector space.
Let X denote a simplicial complex. Dene for each k 0, the vector space Ck (X )
to be the vector space whose basis is the set of oriented k -simplices of X ; that is,
a k -simplex {v0 , . . . , vk } together with an order type denoted [v0 , . . . , vk ] where a
change in orientation corresponds to a change in the sign of the coefcient:
[v0 , . . . , vi , . . . , vj , . . . , vk ] = [v0 , . . . , vj , . . . , vi , . . . , vk ].
For k larger than the dimension of X , we set Ck (X ) = 0. The boundary map is
dened to be the linear transformation : Ck Ck1 which acts on basis elements
28
V. DE SILVA & R. GHRIST
[v0 , . . . , vk ] via
k
(16)
[v0 , . . . , vk ] :=
i=0
(1)i [v0 , . . . , vi1 , vi+1 , . . . , vk ].
This gives rise to a chain complex: a sequence of vector spaces and linear transformations
Ck+1 Ck Ck1 C2 C1 C0
Consider the following two subspaces of Ck : the cycles (those subcomplexes without boundary) and the boundaries (those subcomplexes which are themselves
boundaries).
(17)
k -cycles
k -boundaries
:
:
Zk (X ) = ker( : Ck Ck1 )
Bk (X ) = im( : Ck+1 Ck )
A simple lemma demonstrates that = 0; that is, the boundary of a complex
has empty boundary. It follows that Bk is a subspace of Zk . This has great implications. The k -cycles in X are the basic objects which count the presence of a
hole of dimension k in X . But, certainly, many of the k -cycles in X are measuring
the same hole; still other cycles do not really detect a hole at all they bound a
subcomplex of dimension k + 1 in X .
We say that two cycles and in Zk (X ) are homologous if their difference is a
boundary:
[ ] = [ ]
Bk (X ).
The k -dimensional homology of X , denoted Hk (X ) is the quotient vector space,
(18)
Hk (X ) =
Zk (X )
.
Bk (X )
Specically, an element of Hk (X ) is an equivalence class of homologous k -cycles.
This inherits the structure of a vector space in the natural way: [ ] + [ ] = [ + ]
and c[ ] = [c ] for c R.
By arguments utilizing barycentric subdivision, one may show that the homology
H (X ) is a topological invariant of X : it is indeed an invariant of homotopy type.
Readers familiar with the Euler characteristic of a triangulated surface will not nd
it odd that intelligent counting of simplicies yields an invariant.
For a simple example, the reader is encouraged to contemplate the physical meaning of H1 (X ). Elements of H1 (X ) are equivalence classes of (nite collections of)
oriented cycles in the 1-skeleton of X , the equivalence relation being determined
by the 2-skeleton of X .
COVERAGE VIA HOMOLOGY
29
A.2. Relative homology. The precise version of homology used in our theorems
is a relative homology. Often, one wishes to compute holes modulo some region
of the space.
Let Y X be a subcomplex of X . We dene the relative chains as follows:
Ck (X, Y ) is the quotient space obtained from Ck (X ) by collapsing the subspace
generated by k -simplices in Y . One veries that this quotient is respected by
and that the subspaces dened by the kernel and image are well-dened and satisfy
Bk (X, Y ) Zk (X, Y ) Ck (X, Y ).
It then follows that the relative homology
(19)
Hk (X, Y ) =
Zk (X, Y )
Bk (X, Y )
is well-dened. This homology H (X, Y ) measures holes detected by chains whose
boundaries lie in Y .
It follows from the excision theorem that the relative homology of (X, Y ) is equal
to the regular homology of the quotient space X/Y obtained by identifying all
simplices in Y to a single abstract vertex.
(20)
Hk (X, Y ) Hk (X/Y ) k > 0.
=
A.3. Induced homomorphisms. Is it often remarked that homology is functorial, by
which it is meant that things behave the way they ought. A simple example of this
which is crucial to our applications arises as follows.
Consider two simplicial complexes X and X . Let f : X X be a continuous
simplicial map: f takes each k -simplex of X to a k -simplex of X , where k k .
Then, the map f induces a linear transformation f# : Ck (X ) Ck (X ). It is a
simple lemma to show that f# takes cycles to cycles and boundaries to boundaries;
hence there is a well-dened linear transformation on the quotient spaces
f : Hk (X ) Hk (X )
:
f : [ ] [f# ( )].
This is called the induced homomorphism of f on H . Functoriality means that
(1) the identity map Id : X X induced the identity map on homology; and (2)
the composition of two maps g f induces the composition of the linear transformation: (g f ) = g f .
A.4. Exact sequences. Computing algebraic topological invariants is greatly simplied by the use of exact sequences. A sequence of vector spaces {Vi } connected
by linear transformations i : Vi Vi1 is said to be exact if the kernel of i is
equal to the image of i+1 .
30
V. DE SILVA & R. GHRIST
Given a simplicial complex X with subcomplex Y X , the long exact sequence
of the pair (X, Y ) is
(21)
j
i
i
Hk (Y ) Hk (X ) Hk (X, Y ) Hk1 (Y )
Here, i is the map induced by inclusion i : Y X , j is induced by the quotient
X X/Y , and is the map which takes a relative k -cycle in Hk (X, Y ) and
returns the boundary, , a (k 1)-cycle in Y .
This sequence is exact and is an effective means of computing relative homology
groups. Of equal importance is the Mayer-Vietoris sequence of a space X = A B :
(22)
Hk (A B ) Hk (A) Hk (B ) Hk (A B ) Hk1 (A B )
Here (c) = (c, c) and (c, c ) = c + c , with of a cycle = c c being
[c] = [c ]. Also of relevance to the proofs of this paper is a relative version of
the Mayer-Vietoris sequence:
(23)
Hk (A B, A B ) Hk (A, A ) Hk (B, B ) Hk (A B, A B )
Hk1 (A B, A B )
Here (X, Y ) = (A B, A B ).
It requires no small amount of time, effort, and motivation to become familiar with
homological tools. We hope to have provided the latter.
D EPARTMENT OF M ATHEMATICS , P OMONA C OLLEGE , C LAREMONT CA 91711, USA
D EPARTMENT OF M ATHEMATICS AND C OORDINATED S CIENCE L ABORATORY, U NIVERSITY OF I LLI NOIS , U RBANA IL, 61801