Optimal Morse functions and $H(\mathcal{M}^2, \mathbb{A})$ in $\tilde{O}(N)$ time

In this work, we design a nearly linear time discrete Morse theory based algorithm for computing homology groups of 2-manifolds, thereby establishing the fact that computing homology groups of 2-manifolds is remarkably easy. Unlike previous algorithms of similar flavor, our method works with coefficients from arbitrary abelian groups. Another advantage of our method lies in the fact that our algorithm actually elucidates the topological reason that makes computation on 2-manifolds easy. This is made possible owing to a new simple homotopy based construct that is referred to as expansion frames. To being with we obtain an optimal discrete gradient vector field using expansion frames. This is followed by a pseudo-linear time dynamic programming based computation of discrete Morse boundary operator. The efficient design of optimal gradient vector field followed by fast computation of boundary operator affords us near linearity in computation of homology groups. Moreover, we define a new criterion for nearly optimal Morse functions called pseudo-optimality. A Morse function is pseudo-optimal if we can obtain an optimal Morse function from it, simply by means of critical cell cancellations. Using expansion frames, we establish the surprising fact that an arbitrary discrete Morse function on 2-manifolds is pseudo-optimal. Classical Morse Theory [16, 17] analyzes the topology of the Riemannian manifolds by studying critical points of smooth functions defined on it. In the 90’s Robin Forman formulated a completely combinatorial analogue of Morse theory, now known as discrete Morse theory. The fact that Forman’s theory can be formulated in language of graph theory makes it possible to use powerful machinery from modern algorithmics to provide efficient algorithms with rigorous guarantees. It is worth noting that the reader can understand this work without any prior knowledge of Morse theory as long as 1 ar X iv :1 50 5. 02 23 0v 1 [ cs .C G ] 9 M ay 2 01 5 he understands the equivalent graph theory problem. Knowledge of discrete Morse theory is however useful for the more inclined reader who wishes to understand the context and wider range of applicability of this work. In subsection 1.2, we provide a quick overview of the graph theory setting of discrete Morse theory in order to enable the reader to make a quick foray into the core computer science problem at hand. 1 Background and Preliminaries 1.1 Discrete Morse theory Forman provides an extremely readable introduction to discrete Morse theory in [7]. Notation 1. The relation ’≺’ is used to denote the following: τ ≺ σ → τ ⊂ σ & dim τ = dimσ − 1. Notation 2 (The d-(d-1) level of Hasse graph). By the term, d-(d-1) level of Hasse graph H we mean the subset of edges of the Hasse graph that join d-dimensional cofaces to (d-1)-dimensional faces of Hasse graph. Definition 3. Boundary & Couboundary of a simplex σ: We define the boundary and respectively coboundary of a simplex as bd σ = {τ | τ ≺ σ} cbd σ = {ρ |σ ≺ ρ} Definition 4. Discrete Morse Function: Let K denote a finite regular cell complex and let L denote the set of cells of K. A function F : L → R is called a discrete Morse function (DMF) if it usually assigns higher values to higher dimensional cells, with at most one exception locally at each cell. Equivalently, a function F : L → R is a discrete Morse function if for every σm ∈ L we have: (A.) N1(σ) = #{ρ ∈ cbd σ|F(ρ) ≤ F(σ)} ≤ 1 (B.) N2(σ) = #{τ ∈ bd σ |F(τ) ≥ F(σ)} ≤ 1 A cell σ is critical if N1(σ) = N2(σ) = 0; A non-critical cell is a regular cell. Definition 5 (Combinatorial Vector Field). A combinatorial vector field (DVF) V on L is a collection of pairs of cells {〈α, β〉} such that {αm ≺ β(m+1)} and each cell occurs in at most one such pair of V. Definition 6 (Discrete Gradient Vector Field). A pair of cells {αm ≺ β(m+1)} s.t. F(α) ≥ F(β) determines a gradient pair. A discrete gradient vector field (DGVF) V corresponding to a DMF F is a collection of cell pairs α(p) ≺ β(p+1) such that α(p) ≺ β(p+1) ∈ V iff F(β) ≤ F(α).