We present family metro maps as a new approach to visualize the relations between multiple related families. A family is represented by a metro line, where the parents are the end nodes and the children the intermediate nodes. We introduce family trees as rooted binary trees and consider the problem of automatically drawing a family metro map for a given family tree without crossing line segments, where the successive nodes of a metro line are placed at equal distances. The inter-node distance may be chosen to depend on the generation of the family, such that metro lines of earlier generations are shown more prominently than metro lines of later generations. Family metro maps can be used to visualize the family relations of the descendants of multiple couples that have common descendants. Graphic Abstract

We present family metro maps as a new approach to visualize the relations between multiple related families. A family is represented by a metro line, where the parents are the end nodes and the children the intermediate nodes. We introduce family trees as rooted binary trees and consider the problem of automatically drawing a family metro map for a given family tree without crossing line segments, where the successive nodes of a metro line are placed at equal distances. The inter-node distance may be chosen to depend on the generation of the family, such that metro lines of earlier generations are shown more prominently than metro lines of later generations. Family metro maps can be used to visualize the family relations of the descendants of multiple couples that have common descendants.

The quartet method is a novel hierarchical clustering approach where, given a set of n data objects and their pairwise dissimilarities, the aim is to construct an optimal tree from the total number of possible combinations of quartet topologies on n, where optimality means that the sum of the dissimilarities of the embedded (or consistent) quartet topologies is minimal. This corresponds to an NP-hard combinatorial optimization problem, also referred to as minimum quartet tree cost (MQTC) problem. We provide details and formulation of this challenging problem, and propose a basic greedy heuristic that is characterized by some appealing insights and findings for speeding up and simplifying the processes of solution generation and evaluation, such as the use of adjacency-like matrices to represent the topology structures of candidate solutions; fast calculation of coefficients and weights of the solution matrices; shortcuts in the enumeration of all solution permutations for a given configuration; and an iterative distance matrix reduction procedure, which greedily merges together highly connected objects which may bring lower values of the quartet cost function in a given partial solution. It will be shown that this basic greedy heuristic is able to improve consistently the performance of popular quartet clustering algorithms in the literature, namely a reduced variable neighbourhood search and a simulated annealing metaheuristic, producing novel efficient solution approaches to the MQTC problem.

We propose a new visualization of family relations, called family metro map , which is inspired by the famous London Underground Map. A family is represented by a metro line, where the parents are the end nodes and the children the intermediate nodes. We introduce family trees as rooted binary trees and consider the problem of automatically drawing a family metro map for a given family tree without crossing line segments, where the successive nodes of a metro line are placed at equal distances. The inter-node distance may be chosen to depend on the generation of the family, such that metro lines of earlier generations are shown more prominently than metro lines of later generations.

Given a set of n data objects and their pairwise dissimilarities, the goal of quartet clustering is to construct an optimal tree from the total number of possible combinations of quartet topologies on n, where optimality means that the sum of the dissimilarities of the embedded (or consistent) quartet topologies is minimal. This corresponds to an NP-hard combinatorial optimization problem, also referred to as minimum quartet tree cost (MQTC) problem. We provide details and formulation of this challenging problem, and propose a basic greedy heuristic that is characterized by a very high speed and some interesting implementation details. The solution approach, though simple, substantially improves the performance of a Reduced Variable Neighborhood Search for the MQTC problem. The latter is one of the most popular heuristic algorithms for tackling the MQTC problem.

The minimum quartet tree cost (MQTC) problem is a graph combinatorial optimization problem where, given a set of \(n \ge 4\) data objects and their pairwise costs (or distances), one wants to construct an optimal tree from the \(3 \cdot {n \atopwithdelims ()4}\) quartet topologies on n, where optimality means that the sum of the costs of the embedded (or consistent) quartet topologies is minimal. The MQTC problem is the foundation of the quartet method of hierarchical clustering, a novel hierarchical clustering method for non tree-like (non-phylogeny) data in various domains, or for heterogeneous data across domains. The MQTC problem is NP-complete and some heuristics have been already proposed in the literature. The aim of this paper is to present a first exact solution approach for the MQTC problem. Although the algorithm is able to get exact solutions only for relatively small problem instances, due to the high problem complexity, it can be used as a benchmark for validating the performance of any heuristic proposed for the MQTC problem.

Given a set of n data objects and their pairwise dissimilarities, the goal of the minimum quartet tree cost (MQTC) problem is to construct an optimal tree from the total number of possible combinations of quartet topologies on n, where optimality means that the sum of the dissimilarities of the embedded (or consistent) quartet topologies is minimal. We provide details and formulation of this novel challenging problem, and the preliminaries of an exact algorithm under current development which may be useful to improve the MQTC heuristics to date into more efficient hybrid approaches.

Simulated Annealing is a meta-heuristic that performs a randomized local search to reach near-optimal solutions of combinatorial as well as continuous optimization problems. In this chapter we show how it can be used to train artificial neural networks by examples. Experimental results indicate that good results can be obtained with little or no tuning.