Let G be a graph cellularly embedded in a surface S. Given two closed walks c and d in G, we take advantage of the RAM model to describe linear time algorithms to decide if c and d are homotopic in S, either freely or with fixed basepoint. We restrict S to be orientable for the free homotopy test, but allow non-orientable surfaces when the basepoint is fixed. After O(|G|) time preprocessing independent of c and d, our algorithms answer the homotopy test in O(|c|+|d|) time, where |G|, |c| and |d| are the respective numbers of edges of G, c and d. As a byproduct we obtain linear time algorithms for the word problem and the conjugacy problem in surface groups. We present a geometric approach based on previous works by Colin de Verdi\`ere and Erickson.
Computational topology of surfaces has received much attention in the last two decades. Among the notable results we may mention the test of homotopy between two cycles on a surface [DG99], the computation of a shortest cycle homotopic to a given cycle [CE10], or the computation of optimal homotopy and homology bases [EW05]. In their 1999 paper, Dey and Guha announced a linear time algorithm for testing whether two curves on a triangulated surface are freely homotopic. This appeared as a major breakthrough for one of the most basic problem in computational topology. Dey and Guha's approach relies on results by Greendlinger [Gre60] for the conjugacy problem in one relator groups satisfying some small cancellation condition. In the appendix, we show several subtle flaws in the paper of Dey and Guha [DG99] that invalidate their approach and leave little hope for repair. Inspired by the recent work of Colin de Verdière and Erickson [CE10] for computing a shortest cycle in a free homotopy class, we propose a different geometric approach and confirm the results of Dey and Guha for orientable surfaces. As commonly assumed in computational topology, we shall analyse the complexity of our algorithms with the uniform cost RAM model of computation [AHU74]. A notable feature of this model is the ability to manipulate arbitrary integers in constant time per operation and to access an arbitrary memory register in constant time.
In a first part we consider the homotopy test for curves with fixed endpoints drawn in a graph cellularly embedded in a surface S. This test reduces to decide if a loop is contractible in S, i.e., null-homotopic, since a curve c is homotopic to a curve d with fixed endpoints if and only if the concatenation c • d -1 is contractible. The contractibility test was already considered by Dey and Schipper [DS95] using a partial and implicit construction of the universal cover of S. Indeed, a curve is null-homotopic in S if and only if its lift is closed in the universal cover of S. Given a closed curve c, Dey and Schipper detect if c is null-homotopic in O(|c| log g) time, where g is the genus of S. Their implicit construction is relatively complex and does not seem to extend to handle the free homotopy test. Our solution to the contractibility test also relies on a partial construction of the universal cover. We use the more explicit construction of Colin de Verdière and Erickson [CE10, Sec. 3.3 and 4] for tightening paths. It amounts to build a convex region of the universal cover (with respect to some hyperbolic metric) large enough to contain a lift of c. An argument à la Dehn shows that this region can be chosen to have size O(|c|), leading to our first theorem:
Theorem 1 (Contractibility test). Let G be a graph of complexity n cellularly embedded in a surface S, not necessarily orientable. We can preprocess G in O(n) time, so that for any loop c on S represented as a closed walk of k edges in G, we can decide whether c is contractible or not in O(k) time.
We next study the free homotopy test, that is deciding if two cycles c and d drawn in a graph G cellularly embedded in S can be continuously deformed one to the other. By theorem 1, we may assume that none of c and d is contractible. Our strategy is the following. We first build (part of) the cyclic covering of S induced by the cyclic subgroup generated by c in the fundamental group of S. We denote by S c this covering. Assuming that S is orientable, S c is a topological cylinder 1 , and we call any of its non-contractible simple cycles a generator. Since the generators of S c are freely homotopic, their projection on S are freely homotopic to c. Our next task is to extract from S c a canonical generator γ R whose definition only depends on the isomorphism class of S c . To this end, we lift in S c the graph G of S and we endow S c with the corresponding cross-metric introduced by Colin de Verdière and Erickson [CE10]. The set of generators that are minimal for this metric form a compact annulus in S c . We eventually define γ R as the “right” boundary of this annulus. We perform the same operations starting with d instead of c to extract a canonical generator δ R of S d . From standard results on covering spaces [Mas91, §V.6], we know that S c and S d are isomorphic covering spaces if c and d are freely homotopic. It follows that c and d are freely homotopic if and only if γ R and δ R have equal projections on S. Proving that γ R and δ R can be constructed in time proportional to |c| and |d| respectively, we finally obtain:
Theorem 2 (Free homotopy test). Let G be a graph of complexity n cellularly embedded in an orientable surface S. We can preprocess G in O(n) time, so that for any cycles c and d on S represented as closed walks with a total number of k edges in G, we can decide if c and d are freely homotopic in O(k) time.
The word problem in a group presented by generators and relators is to decide if a product of generators and their i
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