Many problems can be modelled as finding shortest paths in large graphs. Our result pro- vides new insights into the kinds of structures a graph will need to possess to be amenable to an efficient shortest path algorithm. The fact that the shortest path in navigable formu- las flips variables that are not in the symmetric difference is evidence that our algorithm exploits a property of the reconfiguration graph that is fundamentally new. Any previ- ously known properties that were used to find shortest paths efficiently also rendered the graph too simple, in that any shortest path only flipped the symmetric difference. It will be interesting to see if our results help us understand other large graphs, in particular, the flip graph of triangulations of a convex polygon where the complexity of finding the shortest path is still open.
Chapter 4
Flipping Edge-Labelled
Triangulations: Part I
1
4.1
Introduction
Triangulations of point sets and graphs are rich in mathematical structure [31] and have many practical applications, for example in meshing. This chapter is about reconfiguration of triangulations. We begin by discussing some necessary background, including some things we have already discussed in previous chapters in order to make this chapter self- contained.
In general, reconfiguration means transforming one solution of a problem to another via elementary steps [55]. Reconfiguration versions of several traditional problems have gained recent attention, e.g., maximum independent set [55, 59], and satisfiability of boolean formulas [47].
The basic operation for reconfiguring triangulations, both in the combinatorial and the geometric setting, is the flip operation that removes one edge of the triangulation and adds the other diagonal of the resulting quadrilateral. In order to obtain a new triangulation, there is a constraint on the removed edge. In the geometric setting a triangulation of a point set is a maximal set of non-crossing edges joining pairs of points. The constraint on a flip is that the two faces incident to the removed edge must form a convex quadrilateral. In the combinatorial setting a triangulation is a maximal planar graph with the clockwise
order of edges around each vertex specified. The constraint on a flip is that the other diagonal of the quadrilateral should not already be an edge of the triangulation.
The fundamental property of flips is that they can be used to reconfigure any triangu- lation to any other triangulation that has the same size and—in the geometric case—the same point set. This was proved by Wagner in 1936 for the combinatorial setting (see [15]). For the geometric setting, it is a consequence of Lawson’s result [63] that any triangulation of a point set can be flipped to the Delaunay triangulation, which then acts as a “canonical” triangulation from which every triangulation can be reached.
There is a substantial literature on flipping, see the survey [15]. Flips are useful in practice in mesh generation and optimizing triangulations [7, 37]. They are also used for counting and generating triangulations [20, 31, 73].
A main subject of investigation is the minimum length of a flip sequence, the so-called flip distance, between two triangulations. Most of the work is on worst case bounds. For combinatorial triangulations on n vertices, the current worst case bound is below 6n [17]. For triangulations of general point sets of size n there is a tight worst case bound of Θ(n2) [52]. For triangulations of points in convex position a famous result of Sleator et al. [95] shows that the flip distance is at most 2n−10 and that 2n−10 flips are necessary in some cases. There is a new proof of this result that does not use hyperbolic geometry [86]. Another interesting problem is to find the exact flip distance between two triangulations. The complexity of this problem is open in the combinatorial setting. The problem was only recently shown to be NP-hard [66, 84] (see Chapter 2), and even APX hard [84], for triangulations of point sets. The case of triangulations of a simple polygon is also NP-hard [1]. However, the case of points in convex position (equivalently, the case of a convex polygon) remains stubbornly unresolved, and the hardness reductions used in the aforementioned papers do not seem to apply.
Flips in triangulations of a convex polygon are especially interesting because they cor- respond exactly to rotations in a binary tree [95] and flip distance corresponds exactly to rotation distance, which was first explored in 1982 [29]. There is an easy factor-2 approx- imation algorithm for rotation distance which can be improved to 1.98 for some special cases [30]. There are efficient algorithms to compute lower and upper bounds on the rota- tion distance, but no known guarantees on the gap between these bounds [5,68, 69, 81].
The idea of performing flips in parallel was introduced by Hurtado et al. [53], see also [42]. In the geometric setting, a set of edges may be simultaneously flipped if each edge may be flipped and no two of the edges are incident to the same face. Hurtado et al. showed that O(log n) simultaneous flips are sufficient and sometimes necessary to reconfigure one triangulation of a convex polygon to another. Bose et al. [16] were the first
to explore simultaneous flips in the combinatorial setting—in this case a simultaneous flip may be performed even if some edges in the set cannot be individually flipped.
In this chapter we initiate the study of edge-labelled flips. If the edges of a triangulation are labelled and we perform a flip, the newly added edge is assigned the label of the removed edge. In particular, this means that the set of edge labels is preserved throughout any flip sequence. In general (for point sets) it is not possible to flip between any two edge-labelled triangulations, but this is true for combinatorial triangulations and for triangulations of convex polygons, the settings that we consider in this chapter.
Our initial motivation was to understand the complexity of computing the flip distance between two triangulations of a convex polygon. Is the problem difficult because we do not know which edge flips to which edge? Having this information is the same as having a labelling of the edges.
One result that gives some hope that flip distance might become easier to compute when edge labels are specified is a polynomial time algorithm by Eppstein [38] to compute the flip distance between two triangulations of a point set that contains no empty pentagon. In this case Eppstein shows that each edge of the initial triangulation can only flip to a unique edge of the final triangulation. In other words, there is a unique labelling of edges, which is one ingredient in Eppstein’s efficient algorithm.
On the other hand, there is a situation where finding a minimum length reconfiguration sequence is NP-hard even with the mapping between initial and final elements, namely for independent set reconfiguration in perfect graphs. For further details, see Section 4.2. Our Results. We prove tight Θ(n log n) bounds on the worst-case flip distance between two edge-labelled triangulations of a point set in convex position, and between two edge- labelled combinatorial triangulations. This contrasts with the Θ(n) bounds for unlabelled flips [15,95]. The extra log n factor arises from sorting-related issues. We prove the upper bound by reducing to the problem of sorting a list using an operation that reverses a (possibly non-contiguous) subsequence at a cost proportional to the minimum length of a contiguous sublist containing the subsequence. The special case where the subsequence itself must be contiguous is the “length-weighted” reversals model introduced by Pinter and Skiena [85] for applications in comparative genomics. Our model seems more powerful since the best-known bound for sorting in their model is O(n log2n). Our Ω(n log n) lower bound generalizes theirs. Our result provides an efficient O(log n)-factor approximation algorithm to compute the flip distance between edge-labelled triangulations of a point set in convex position.
Finally, we consider simultaneous flips. We prove that the simultaneous flip distance between two edge-labelled triangulations of a point set in convex position is O(log2n), in
contrast with the Θ(log n) bound for the unlabelled case [16]. We also prove that Ω(log n) simultaneous flips may be required in some cases.