Though the Geometric TCC gives a topological verification method for coverage, it is not com- putable as we know little about the domain besides what local metric information the finite col- lection of sensors tell us, and the connected components assumptions, which only tell about what
is preserved between scales. In this section we introduce an computational method to inferring coverage that makes use of the geometric result from Section 4.3. To introduce the algorithm and its proof of correctness, we first introduce the assumptions on our input, adding to the previously stated the geometric assumptions.
We begin with a weighted finite point set P from which our input to the TCC algorithm is built. The input is two graphs (G1, G2) over vertex set P , and their respective restrictions to the
vertex set Q, (G1[Q], G2[Q]), where we remind the reader that Q is the subset of the sensors whose
coverage regions intersect the surrounding set B. For technical purposes we assume that the vertex set P , has a point on each connected component of D \ B2α to avoid degenerate inputs where coverage impossible.
Given the two input graphs, we also assume that the k-barycentric decomposition ˇCech com- plexes of the coverage regions at scales α and β, Cα
k(P ) and C β
k(P ) are sandwiched between the
k-clique complexes of the two graphs (definition is found in Subsection 4.2.2). Though this seems like a specific condition, it in fact generalizes the input of De Silva and Ghrist, who used Rips com- plexes built from sensors at two different scales as their coverage algorithm’s input. Specifically, if the domain has Euclidean metric and the two graphs are the intersection graphs of the k-NN metric balls of P at scale α/ϑd and β, then the clique complexes of these graphs are Rα/ϑd(P )
and Rβ(P ), so the k-clique complexes are the k-Rips complexes and the sandwich assumption is a direct result of the Jung’s Theorem corollary, Equation 2.2. Furthermore, we assume that the coverage regions of radius α and β of the sensors form a good cover, allowing for the application of the relative Persistent Nerve Lemma.
In order to apply Alexander Duality, Theorem 4.4, we assume the existence of nested triangu- lations of the k-NN ε-offsets Pkε and Qεk for ε ∈ {α, β}, the strong and weak sensing radii, which are consistent with some triangulation of the compactification of Rd.
Algorithmic Assumptions
3. Each connected component of D \ B2α contains a point in P .
4. The graphs G1, G2 are defined over vertex set P ⊂ D, and subgraphs G1[Q], G2[Q] induced
by restriction to the vertex set Q = {p ∈ P | cov(p, α) ∩ B 6= ∅}. 5. Clqk(G1) ⊆ Ckα(P ) ⊆ C
β
k(P ) ⊆ Clqk(G2).
6. U = {cov(p, ε) | p ∈ P } is a good cover for ε ∈ {α, β}.
7. There is a triangulation K of Rd∪ {∞} and triangulations Lε and Mε of Pkε and Qεk respec-
tively, such that Mε⊂ Lε in K, for ε ∈ {α, β}.
First we will introduce the result which states an Alexander Duality theorem with respect to the sensors’ offsets. By assumption 7, there exists a sufficiently refined triangulation K of Rd∪ {∞} and respective triangulations of Pkα, Qαk, Pkβ, and Qβk. Due to the naturality of the Alexander Duality theorem 4.4 and the fact that relative homology is a invariant under homeomorphisms of pairs allows us to state the following reinterpretation with our spaces.
Corollary 4.9. Given α and β > 0 and P and Q satisfying Assumption 7, the following diagram commutes where the horizontal maps are isomorphisms.
H0(Qαk, Pkα) ∼ = // Hd(Pkα, Qαk) H0(Qβk, Pkβ) ∼ = // Hd(Pβ k, Q β k). (4.6)
The following lemma gives an upper-bound on the rank of the d-dimensional homology map between the pairs of graphs, relating the clique complexes of the graphs to the homology maps between the spaces in Diagram (4.4).
Lemma 4.10. The rank of the map Hd(Clqk(G1, G1[Q]) ,→ Clqk(G2, G2[Q])) induced by inclusion
Proof. Corollary 4.9 with respect to Diagram (4.4) implies that rk(Hd(Pkβ, Qβk) → Hd(Pkα, Qαk)) = rk i∗. By the Universal Coefficient Theorem, we then have rk(Hd(Pkα, Qkα) → Hd(Pkβ, Qβk)) =
rk i∗, and by the relative homology iteration of the Persistent Nerve Lemma for k-barycentric
decompositions, Lemma 4.5, for all ε ≥ 0, H∗(Pkε, Qεk) ∼= H∗(Ckε(P, Q)), so rk(Hd(Ckα(P, Q) ,→
Ckβ(P, Q))) = rk i∗.
By Assumption 5, the inclusion Clqk(G1, G1[Q]) ,→ Clqk(G2, G2[Q]) can be factored as
Clqk(G1, G1[Q]) ,→ Ckα(P, Q) ,→ C β k(P, Q) ,→ Clqk(G2, G2[Q]). It follows that rk(Hd(Clqk(G1, G1[Q]) ,→ Clqk(G2, G2[Q]))) ≤ rk(Hd(Ckα(P, Q) ,→ C β k(P, Q))) = rk i∗.
The next lemma states that there is an upper-bound on the number of connected components of the coverage region of interest, D \ B2α, if we assume that its connected components are not too close together.
Lemma 4.11. Let (D, B) be a pair of spaces satisfying Assumptions 0 and 2 for positive constants α and β ≥ 3α. If P ⊂ D, Q = {p ∈ P | cov(p, α) ∩ B 6= ∅} and the graph G1 satisfied Assumptions
4 − 6 for some constant k ≥ 1, then dim H0(G1[P \ Q]) ≥ dim H0(D \ B2α).
Proof. Assume there exists p, q ∈ P \ Q such that p and q are connected in Clq(G1[P \ Q]), but not
in D \ B2α. By Assumption 2, we have that d(p, q) ≤ 2α and [p] 6= [q] in H0(D \ B2α). However, the
shortest path pq ∈ (D \B)2α, as the distance between p and q is less than 2α, so [p] = [q] in H0(D2α),
which implies that H0(D \ B2α,→ (D \ B)2α) is not injective, a contradiction to Assumption 2.
This leads to the Toplogical Coverage Criterion algorithm, Algorithm 1. Theorem 4.12 proves the validity of this simple algorithm that checks for k-coverage, or tells the user if more sensors are
Algorithm 1 Check if D \ B2α⊆ Pα k 1: procedure k-Coverage(G1, G2, P, Q, k) 2: let c := dim H0(G1[P \ Q]) 3: let r := rk Hd(Clqk(G1, G1[Q]) ,→ Clqk(G2, G2[Q]))
4: if c = r then return True 5: else return False
necessary, i.e. the domain needs to be resampled in the language of geometric inference. The proof of this theorem uses the fact that we can factor the maps between the clique complexes through the ˇCech complexes associated to the pairs (Pkα, Qαk) and (Pkβ, Qαk) by the input assumptions. Each collection of point sets forms a good cover by assumption, so the by the Nerve Theorem there are natural isomorphisms between the homology groups of the offset pairs and the simplicial complex pairs. The proof is completed by reasoning about the ranks of the homology maps in the corresponding sequence.
Theorem 4.12 (Algorithmic TCC). Consider a pair (D, B), a finite point sample P ⊂ D, and constants k, α, β, where 0 < 3α ≤ β, satisfying Assumptions 0–7. If
rk Hd(Clqk(G1, G1[Q]) ,→ Clqk(G2, G2[Q])) = dim H0(G1[P \ Q])
then D \ B2α⊆ Pα k.
Proof. For simplicity, define
a∗ := Hd(Clqk(G1, G1[Q]) ,→ Clqk(G2, G2[Q]))
and set c = |Components(G1[P \ Q])|, m = H0(D \ B2α). By our hypotheses and Lemma 4.10,
rk i∗ ≥ rk a∗ = c. By Lemma 4.11, c ≥ m, and Assumption 1 implies that j∗ is surjective by
Lemma 4.3 so by definition of B surrounding D, m = rk j∗. Thus rk i∗ ≥ rk a∗ = c ≥ m = rk j∗,