Minimum Number Of Flips Research Articles

Flip Distances Between Graph Orientations

Flip graphs are a ubiquitous class of graphs, which encode relations on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon. For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is the minimum number of flips needed to transform one into the other? We consider flip graphs on orientations of simple graphs, where flips consist of reversing the direction of some edges. More precisely, we consider so-called alpha-orientations of a graph G, in which every vertex v has a specified outdegree alpha (v), and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two alpha-orientations of a planar graph G is at most two is NP-complete. This also holds in the special case of perfect matchings, where flips involve alternating cycles. This problem amounts to finding geodesics on the common base polytope of two partition matroids, or, alternatively, on an alcoved polytope. It therefore provides an interesting example of a flip distance question that is computationally intractable despite having a natural interpretation as a geodesic on a nicely structured combinatorial polytope. We also consider the dual question of the flip distance between graph orientations in which every cycle has a specified number of forward edges, and a flip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard. However, if we restrict to flips that only change sinks into sources, or vice-versa, then the problem can be solved in polynomial time. Here we exploit the fact that the flip graph is the cover graph of a distributive lattice. This generalizes a recent result from Zhang et al. (Acta Math Sin Engl Ser 35(4):569–576, 2019).

Distances in Domino Flip Graphs

This article is about measuring and visualizing distances between domino tilings. Given two tilings of a simply-connected square-tiled surface, their distance is the minimum number of flips between them. For a given shape of surface, we're interested in computing the diameters of the flip graphs, meaning the maximal distance between any two of its tilings. Building on work of Thurston and others, we give geometric interpretations of distances that result in formulas for the diameters of the flip graphs of rectangles or Aztec diamonds.

Computing the Flip Distance Between Triangulations

Let $$\mathcal{{T}}$$ be a triangulation of a set $$\mathcal{{P}}$$ of n points in the plane, and let e be an edge shared by two triangles in $$\mathcal{{T}}$$ such that the quadrilateral Q formed by these two triangles is convex. A flip of e is the operation of replacing e by the other diagonal of Q to obtain a new triangulation of $$\mathcal{{P}}$$ from $$\mathcal{{T}}$$ . The flip distance between two triangulations of $$\mathcal{{P}}$$ is the minimum number of flips needed to transform one triangulation into the other. The Flip Distance problem asks if the flip distance between two given triangulations of $$\mathcal{{P}}$$ is at most k, for some given $$k \in \mathbb {N}$$ . It is a fundamental and a challenging problem. We present an algorithm for the Flip Distance problem that runs in time $$\mathcal {O}(n + k \cdot c^{k})$$ , for a constant $$c \le 2 \cdot 14^{11}$$ , which implies that the problem is fixed-parameter tractable. We extend our results to triangulations of polygonal regions with holes, and to labeled triangulated graphs.

Optimal Linear Bernoulli Factories for Small Mean Problems

Suppose a coin with unknown probability p of heads can be flipped as often as desired. A Bernoulli factory for a function f is an algorithm that uses flips of the coin together with auxiliary randomness to flip a single coin with probability f(p) of heads. Applications include perfect sampling from the stationary distribution of certain regenerative processes. When f is analytic, the problem can be reduced to a Bernoulli factory of the form f(p) = Cp for constant C. Presented here is a new algorithm that for small values of Cp, requires roughly only C coin flips. From information theoretic considerations, this is also conjectured to be (to first order) the minimum number of flips needed by any such algorithm. For large values of Cp, the new algorithm can also be used to build a new Bernoulli factory that uses only 80 % of the expected coin flips of the older method. In addition, the new method also applies to the more general problem of a linear multivariate Bernoulli factory, where there are k coins, the kth coin has unknown probability pk of heads, and the goal is to simulate a coin flip with probability C1p1+⋯ + Ckpk of heads.

Improved Fixed-Parameter Algorithms for Minimum-Flip Consensus Trees

In computational phylogenetics, the problem of constructing a consensus tree for a given set of rooted input trees has frequently been addressed. In this article we study the Minimum-Flip Problem : the input trees are transformed into a binary matrix, and we want to find a perfect phylogeny for this matrix using a minimum number of flips, that is, corrections of single entries in the matrix. The graph-theoretical formulation of the problem is as follows: Given a bipartite graph G = ( Vt ∪ Vc , E ), the task is to find a minimum set of edge modifications such that the resulting graph has no induced path with four edges that starts and ends in Vt , where Vt corresponds to the taxa set and Vc corresponds to the character set. We present two fixed-parameter algorithms for the Minimum-Flip Problem , one with running time O (4.83 k + poly ( m , n )) and another one with running time O (4.42 k + poly ( m , n )) for n taxa, m characters, k flips, and poly ( m , n ) denotes a polynomial function in m and n . Additionally, we discuss several heuristic improvements. We also report computational results on phylogenetic data.