# Merging Discrete Morse Vector Fields: A Case of Stubborn Geometric Parallelization
## 3 The Parallelization Theorems
Our interest in this theorem is that it allows to express the global list \(V_{\alpha}(K)\) in terms of generally smaller partial lists \(V_{\alpha}(U[k])\), \(V_{\alpha}(W[k])\), \(V_{\alpha}(U[2k+1])\), and \(V_{\alpha}(W[2k+1])\). The computation of these smaller lists can be further simplified by the observation that the difference between \(V_{\alpha}(U[k])\) and \(V_{\alpha}(U[2k+1])\) is contained in \(V_{\alpha}(U[2k+1]\cap W[2k+1])\). If \(\alpha\) is a uniformly k-local algorithm for some \(k\geq 1\), then  
\[V_{\alpha}(K)\ =\ (V_{\alpha}(U[k])\cup V_{\alpha}(W[k]))\setminus(V_{\alpha}(U [k]\cap W[k]))\setminus(V_{\alpha}(U[2k+1]\cap W[2k+1])).\]  
The formula from Theorem 3.1 is rewritten in terms of the intersection \(U[2k+1]\cap W[2k+1]\). Notice that the restriction to \(V_{\alpha}(U[k]\cap W[k])\) is justified because the \(k\)-borders of \(U\) and \(W\) and thus also \(M\) are contained in \(U[k]\cap W[k]\). We next implement the formula as Algorithm 1 in this simple case of a two-set covering. ```
Inputs: * \(k\geq 1\) * \(\alpha\) uniformly \(k\)-local algorithm * \(K\) regular CW complex * \(D\) data attached to cells in \(K\), specific for use in \(\alpha\) * \(U(1)\), \(U(2)\) a covering of \(K\) by subcomplexes given as subsets of \(K\) Outputs: * \(V=V_{\alpha}(K,D)\) discrete vector field * \(C=C_{\alpha}(K,D)\) critical cells for\(i=1\), \(2\)do * \(\textbf{build}\ U(i)[k]\) * \(\textbf{build}\ U(i)[2k+1]\) * \(\textbf{build}\ D[|U(i)[k]]\) * \(\textbf{evaluate}\ \alpha\): \(U(i[k])\), \(D|U(i)[k]\to A(i,k)\) # preprocessing step to obtain discrete vector fields in enlargements of patches endfor build the union \(V(1,2)=A(1,k)\cup A(2,k)\) build the intersection \(U(1,2,k)=U(1)[k]\cap U(2)[k]\) build the intersection \(U(1,2,2k+1)=U(1)[2k+1]\cap U(2)[2k+1]\) build \(D|U(1,2,k)\) build \(D|U(1,2,2k+1)\) evaluate \(\alpha\): \(U(1,2,k)\), \(D|U(1,2,k)\to A(1,2,k)\) build the difference \(D=A(1,2,k)\setminus A(1,2,2k+1)\) save the difference \(V=V(1,2)\setminus D\) build \(N\) as all cells employed in \(V\) save the difference \(C=K\setminus N\)
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**Algorithm 1** Two-set parallelization from Corollary 1. There is a special geometric situation with a more direct identification of intersections of enlargements. Two subcomplexes \(U\) and \(W\) of \(K\) are said to be in antithetic position if \((U\cap W)[n]=U[n]\cap W[n]\) for all \(n\geq 0\). We can now state the following consequence of Corollary 1. **Corollary 2**. : _Suppose \(\alpha\) is an algorithm that is uniformly k-local for some \(k\geq 1\) and suppose \(U\) and \(W\) are an antithetic pair, then_  
\[V_{\alpha}(K)=(V_{\alpha}U[k]\cup V_{\alpha}W[k])\setminus(V_{\alpha}(U\cap W)[ k]\setminus V_{\alpha}(U\cap W)[2k+1]).\]  
Proof. : The subcomplexes in Corollary 1 now have a description based entirely on the enlargements of covering complexes and their intersections. **Remark 2**. : _In general, building \(S[n]\) in \(K\) for a subcomplex \(S\) and a number \(n\geq 1\) requires building \(x[n]\) for all \(x\) in \(S\) and taking the union. So the time complexity of this operation is proportional to the size of \(S\). Therefore, depending on the geometry of \(K\) and the choices of \(U\) and \(W\), the sizes of the lists the computer needs to work though while building the enlargements can be greatly reduced by working with the intersection \(U\cap W\) and one single enlarging procedure. This is particularly true when the size of the intersection \(U\cap W\) is significantly smaller than the sizes of \(U\) and \(W\). In the case of simplicial trees, which we address later in Section 5, the size of \(U\cap W\) can be arranged to be an order of magnitude smaller than either \(U\) or \(W\) in the Euclidean case, and that order can be made arbitrarily smaller in the case of a tree, depending on the valence of the tree._  
In the rest of the section, we illustrate in the specific context of the algorithm ProcessLowerStars[2] that the general theorems can be improved on in situations with specific \(\alpha\) and specific geometry.