Edge Conflicts do not Determine Geodesics in the Associahedron

Abstract

There are no known efficient algorithms to calculate distance in the one-skeleta of associahedra, a problem that is equivalent to finding rotation distance between rooted binary trees or the flip distance between polygonal triangulations. One measure of the difference between trees is the number of conflicting edge pairs, and a natural way of trying to find short paths is to minimize successively this number of conflicting edge pairs using flip operations in the corresponding triangulations. We describe examples that show that the number of such conflicts does not always decrease along geodesics. Thus, a greedy algorithm that always chooses a transformation that reduces conflicts will not produce a geodesic in all cases. Further, for any specified amount, there are examples of pairs of all large sizes showing that the number of conflicts can increase by that amount along any geodesic between the pairs.

An Efficient Sampling Algorithm for Difficult Tree Pairs

“…We note that the examples produced by DPS lie in starkly broader classes of difficult pairs than those specific known earlier examples of Dehornoy [8], Pournin [11], and Cleary and Maio [3]. Those earlier examples give difficult tree pairs of increasingly large sizes but though there are multiple possible examples of increasing size, these numbers do not grow nearly as fast as the set of all possible difficult pairs or as those constructed here.…”

“…In general (see Cleary, Rechnitzer and Wong [2]) there are a sizable number of common edges and one-off edges, resulting on average about at least a 10% reduction in the size of a randomly selected tree pair to a largest difficult remaining tree pair. It is not difficult to construct specific examples of specified size of difficult tree pairs-examples of Dehornoy [8], Pournin [11], and Cleary and Maio [3] are families of difficult pairs but in each case of a restricted type. In many of these very specific cases, analysis to that family of instances can give coincident upper and lower bounds on rotation distance, giving an exact calculation.…”

Strong convexity in flip-graphs

“…Our third main result and second application of Theorem 1.1 is on the problem of computing the distances within F(Σ) when Σ is a topological surface. In the case when Σ is a convex polygon, a popular procedure to estimate the distance of two given triangulations in F(Σ) is based on the number of crossing arc pairs between them [7]. It is known that, in the more general cases of convex punctured polygons [28] and arbitrary topological surfaces [9], performing some flip in one of the triangulations makes the number of such crossings decrease.…”

“…Its geometry is specially interesting because of its relation with the rotation distance of binary trees [32,31]. While its diameter is known exactly [26], the time-complexity of computing distances in this graph remains an important unsolved problem [1,7,18,25]. The flip-graph of a convex polygon coincides with that of a topological disk whose boundary contains the triangulations' vertices.…”

“…In this case, we say that ε is preserved. This property, that is instrumental in a number of results on the geometry of F(Σ) [7,26,31], is obtained by projecting the paths in F(Σ) between two triangulations that contain ε to paths in F ε (Σ) between the same two triangulations via an explicit normalization of any triangulation of Σ into a triangulation that contains ε. It is proven in [9] that the same holds in the case when Σ is an arbitrary topological surface via a similar normalization of its triangulations.…”