Permutation | EPFL Graph Search

In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set {1, 2, 3}, namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of

From Trees to Barcodes and Back Again:A Combinatorial, Probabilistic and Geometric Study of a Topological Inverse Problem

Adélie Eliane Garin

In this thesis, we investigate the inverse problem of trees and barcodes from a combinatorial, geometric, probabilistic and statistical point of view.Computing the persistent homology of a merge tree yields a barcode B. Reconstructing a tree from B involves gluing the branches back together. We are able to define combinatorial equivalence classes of merge trees and barcodes that allow us to completely solve this inverse problem. A barcode can be associated with an element in the symmetric group, and the number of trees with the same barcode, the tree realization number, depends only on the permutation type. We compare these combinatorial definitions of barcodes and trees to those of phylogenetic trees, thus describing the subtle differences between these spaces. The result is a clear combinatorial distinction between the phylogenetic tree space and the merge tree space.The representation of a barcode by a permutation not only gives a formula for the tree realization number, but also opens the door to deeper connections between inverse problems in topological data analysis, group theory, and combinatorics.Based on the combinatorial classes of barcodes, we construct a stratification of the barcode space. We define coordinates that partition the space of barcodes into regions indexed by the averages and the standard deviations of birth and death times and by the permutation type of a barcode. By associating to a barcode the coordinates of its region, we define a new invariant of barcodes.These equivalence classes define a stratification of the space of barcodes with n bars where the strata are indexed by the symmetric group on n letters and its parabolic subgroups.We study the realization numbers computed from barcodes with uniform permutation type (i.e., drawn from the uniform distribution on the symmetric group) and establish a fundamental null hypothesis for this invariant. We show that the tree realization number can be used as a statistic to distinguish distributions of trees by comparing neuronal trees to random barcode distributions.

EPFL2022

Some coloring and walking problems in graphs

Benjamin Leroy-Beaulieu

Graph theory is an important topic in discrete mathematics. It is particularly interesting because it has a wide range of applications. Among the main problems in graph theory, we shall mention the following ones: graph coloring and the Hamiltonian circuit problem. Chapter 1 presents basic definitions of graph theory, such as graph coloring, graph coloring with color-classes of bounded size b, and Hamiltonian circuits and paths. We also present online algorithms and online coloring. Chapter 2 starts with some general remarks about online graph covering with sets of bounded sizes (such as online bounded coloring): we give a simple method for transforming an online covering algorithm into an online bounded covering algorithm, and to derive the performance ratio of the bounded algorithm from the performance ratio of the unbounded algorithm. As will be shown in later chapters, this method often leads to optimal results. Furthermore, some basic preliminary results on online graph covering with sets of bounded size are given: for every graph, the performance ratio is bounded above by 1/2 + b/2 and for b = 2, this bound is optimal. In the second part, online coloring of co-interval graphs is studied. Based on two industrial applications, two different versions of this problem are discussed. In the case where the intervals are presented in increasing order of their left ends, we show that the performance ratio is 1 in the unbounded case and 2 - 1/b in the bounded case. In the case where the intervals may be presented in any order, we show that the performance ratio is at most 3 in the bounded case. Chapter 3 deals with online coloring of permutation and comparability graphs. First, we give a tight analysis of the First-Fit algorithm on bipartite permutation graphs and we show that its performance ratio is O(√n), even for some simple presentation orders. For both classes of graphs, we show that the performance ratio is bounded above by (χ+1)/2 in the unbounded case and that the performance ratio of First-Fit is equal to 1/2 + b/2 in the bounded case. In the second part of this chapter, we study cocoloring of permutation graphs. We show that the performance ratio is n/4 + 1/2 and we give better bounds in some more restricted cocoloring problems. Chapter 4 deals with an application of online coloring: the online Track Assignment Problem. Depending on the assumptions that are made, the Track Assignment Problem can be reduced to coloring permutation or overlap graphs online. We show that when a permutation graph is presented on a latticial plane, from west to east, then the performance ratio is exactly 2 - (min{b,k})-1, where k is the best known upper bound on the bounded chromatic number. We also show that, when a permutation graph is presented on a latticial plan, starting from the origin and growing, simultaneously or not, towards west and east, then the performance ratio is exactly 2 - 1/χ. We also show that online coloring overlap graphs does not have a performance ratio bounded by a constant, even if the overlap graph is bipartite and presented in increasing order of the intervals left ends. In this special case, we show that First-Fit has a tight performance ratio of O(√n). We consider coloring overlap graphs online where the intervals have a bounded size between 1 and a given number M. In this case, we show that the performance ratio can be bounded above by 2√M if M ≤ M0, and by log M (⎡log M / log log M⎤ + 1) if M > M0, M0 being defined by the equation 2√M0 = 3 log(M0). For large values of M, the ratio is O(log2 M / log log M). Chapter 5 is about online coloring of trees, forests and split-graphs. For trees, we show that the performance ratio of online coloring is exactly ½log2(2n) in the unbouded case and at most 1 + ⎣log2(b)⎦/χb in the bounded case. For split-graphs, we show that the performance ratio of online coloring is exactly 1 + 1/χ in the unbounded case and is at most 2 + 1/χb + 3/b in the bounded case. In Chapter 6, we present a class of digraphs: the quasi-adjoint graphs. These are a super class of both the graphs used for a DNA sequencing algorithm in (Blazewicz, Kasprzak, "Computational complexity of isothermic DNA sequencing by hybridization", 2006) and the adjoints. A polynomial recognition algorithm in O(n3), as well as a polynomial algorithm in O(n2 + m2) for finding a Hamiltonian circuit in quasi-adjoint graphs are given. Furthermore, some results about related problems such as finding a Eulerian circuit while respecting some forbidden transitions (a sequence of two consecutive arcs) are discussed.

EPFL2008