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Representations for Genetic and Evolutionary Algorithms
AI
This paper discusses the analysis and design of search operators specifically for tree structures within Genetic Evolutionary Algorithms (GEAs). It contrasts direct and indirect representations for trees, emphasizing the development of problem-specific operators when using direct representations like NetDir. The work further analyzes edge-set encoding and its implications on search performance, highlighting trade-offs in representation design and operator development.
Lecture Notes in Computer Science, 2004
The edge-set encoding is a direct tree representation which directly represents trees as sets of edges. There are two variants of the edge-set encoding: the edge-set encoding without heuristics, and the edge-set encoding with heuristics. An investigation into the bias of the edge-set encoding shows that the crossover operator of the edge-set encoding without heuristics is unbiased, that means it does not favor particular types of trees. In contrast, the crossover operator with heuristics is biased towards the simple minimum spanning tree (MST) and generates more likely trees that are MST-like. As a result, the performance of the edge-set encoding without heuristics does not depend on the structure of the optimal solution. Using the heuristic crossover operator results only in high genetic algorithm (GA) performance if the optimal solution of the problem is slightly different from the simple MST. However, if the optimal solution is not very similar to the simple MST a GA using the heuristic crossover operator fails and is not able to find the optimal solution. Therefore, it is recommended that the edge-set encoding with heuristics should only be used if it is known a priori that the optimal solution is very similar to the simple MST. If this is not known a priori, other unbiased search operators and representations should be used.
2007 IEEE International Conference on Systems, Man and Cybernetics, 2007
This paper presents a comparative study of six encodings which have been used to represent trees in evolutionary algorithms. The study has been divided into two steps: 1) The encoding methods have been evaluated taking into account the time necessary to perform operations such as decoding, crossover and mutation, the feasibility of solutions after those operations, and the corresponding heritability and locality; 2) The encoding methods have been employed in a genetic algorithm to solve three different instances (with 10, 25 and 50 nodes) of the optimal communication spanning tree problem. Finally, the results obtained with each of the encodings are statistically compared using Kruskal-Wallis non-parametric tests and multiple comparisons. The results of this study provide insight into the properties of current encoding schemes for network design problems.
IEEE Transactions on Evolutionary Computation, 2000
We describe the "edge-window-decoder" strategy (EWD), a decoder-based redundant encoding strategy for treebased combinatorial problems, and explore its performance on three well-known tree-based network optimization problems: the degree constrained minimum spanning tree problem (DCMST), the optimum communication spanning tree problem (OCST), and the quadratic minimum spanning tree problem (q-MST). Each is an NP-hard problem, and in each case the best previously published approach (for obtaining fast solutions on large instances) is an evolutionary algorithm (EA), characterized by a specialized encoding. EWD simply encodes a tree as a list of nodes (in which nodes may be repeated), and a tree is built from this list via a "tree construction routine" (TCR). A particular instantiation of EWD involves choosing a specific TCR, and choosing specific designs for the mutation and crossover operators (which work only at the genotype (list of nodes) level). Different combinations of TCRs and operators are described and explored and found to be suited to different classes of problem. We compare these several instantiations of EWD with other encodings (including the previous best reported for each problem) over several benchmark instances, as well as randomly generated instances. We find that instantiations of EWD generally perform favorably, clearly outperforming the best comparative approach in two of the three problem classes, and close to best in the third. Some analyses of EWD in terms of its locality, heritability, and bias are provided. These indicate that EWD shows high locality and heritability, and an intermediate level of bias toward the MST of the problem under consideration. In particular, the basic EWD encoding strategy is unbiased, but its good performance on a range of tree-based problems appears to be explainable in terms of the bias inherent in the associated operators. Further, experiments show that EWD strategies using one particular biased operator are able to apply a consistent intermediate level of bias "pressure" over time, thus allowing the search to range more freely in the early stages over the less represented areas of the landscape, rather than (as is the case with the other biased encoding/operator combinations compared against) quickly becoming committed to MST-like regions.
Theoretical Computer Science, 2007
Trees are probably the most studied class of graphs in Computer Science. In this thesis we study bijective codes that represent labeled trees by means of string of node labels. We contribute to the understanding of their algorithmic tractability, their properties, and their applications.
Information Processing Letters, 1982
2005
We extend our analysis of repetitive patterns found in genetic programming genomes to tree based GP. As in linear GP, repetitive patterns are present in large numbers. Size fair crossover limits bloat in automatic programming, preventing the evolution of recurring motifs. We examine these complex properties in detail: e.g. using depth v. size Catalan binary tree shape plots, subgraph and subtree matching, information entropy, syntactic and semantic fitness correlations and diffuse introns. We relate this emergent phenomenon to considerations about building blocks in GP and how GP works.
2016
There are many situations in which information has a hierarchical or nested structure like that found in family trees or organization charts. The abstraction that models hierarchical structure is called a tree and this data model is among the most fundamental in computer science. It is the model that underlies several programming languages, including Lisp. Trees of various types appear in many of the chapters of this book. For instance , in Section 1.3 we saw how directories and files in some computer systems are organized into a tree structure. In Section 2.8 we used trees to show how lists are split recursively and then recombined in the merge sort algorithm. In Section 3.7 we used trees to illustrate how simple statements in a program can be combined to form progressively more complex statements. The following themes form the major topics of this chapter: 3 The terms and concepts related to trees (Section 5.2). 3 The basic data structures used to represent trees in programs (Sect...
Lecture Notes in Computer Science, 2005
We consider the problem of coding labelled trees by means of strings of vertex labels and we present a general scheme to define bijective codes based on the transformation of a tree into a functional digraph. Looking at the fields in which codes for labelled trees are utilized, we see that the properties of locality and heritability are required and that codes like the well known Prüfer code do not satisfy these properties. We present a general scheme for generating codes based on the construction of functional digraphs. We prove that using this scheme, locality and heritability are satisfied as a direct function of the similarity between the topology of the functional digraph and that of the original tree. Moreover, we also show that the efficiency of our method depends on the transformation of the tree into a functional digraph. Finally we show how it is possible to fit three known codes into our scheme, obtaining maximum efficiency and high locality and heritability.
Artificial Intelligence, 1983
Citing the confusing statements in the AI literature concerning the relationship between branch and bound (B&B) and heuristic search procedures we present a simple and general form,dation of B&B which shouM help dispel ranch of the confusion. We ilhtstrate the utility of the form ulation by showblg that through it some apparently very different algorithms for searching And~Or trees reveal the speciftc nature of their shnilarities and differences, bz addition to git'ing new insights into the relationships among some AI search algo.rithms, the general formulation also pros'ides suggestions on how existing search procedures may be t,aried to obtain new algorithms.
Annals of Combinatorics, 2001
Leaf-labelled trees are widely used to describe evolutionary relationships, particularly in biology. In this setting, extant species label the leaves of the tree, while the internal vertices correspond to ancestral species. Various techniques exist for reconstructing these evolutionary trees from data, and an important problem is to determine how "far apart" two such reconstructed trees are from each other, or indeed from the true historical tree. To investigate this question requires tree metrics, and these can be induced by operations that rearrange trees locally. Here we investigate three such operations: nearest neighbour interchange (NNI), subtree prune and regraft (SPR), and tree bisection and reconnection (TBR). The SPR operation is of particular interest as it can be used to model biological processes such as horizontal gene transfer and recombination. We count the number of unrooted binary trees one SPR from any given unrooted binary tree, as well as providing new upper and lower bounds for the diameter of the adjacency graph of trees under SPR and TBR. We also show that the problem of computing the minimum number of TBR operations required to transform one tree to another can be reduced to a problem whose size is a function just of the distance between the trees (and not of the size of the two trees), and thereby establish that the problem is fixed-parameter tractable.
Information Sciences, 2009
This paper introduces the oriented-tree network design problem (OTNDP), a general problem of tree network design with several applications in different fields. We also present several adaptations needed by evolutionary algorithms with Cayley-type encodings to tackle the OTNDP. In particular, we present these adaptations in two Cayley-encodings known as Prüfer and Dandelion codes. We include changes in Cayley-encodings to consider rooted trees. We also show how to use a fixed-length encoding for Cayley codes in evolutionary algorithms, and how to guarantee that the optimal solution is included in the search space. Finally, we present several adaptations of the evolutionary algorithm’s operators to deal with Cayley-encodings for the OTNDP. In the experimental part of the paper, we compare the performance of an evolutionary algorithm (implementing the two Cayley-encodings considered) in several OTNDP instances: first, we test the proposed techniques in randomly generated instances, and second, we tackle a real application of the OTNDP: the optimal design of an interactive voice response system (IVR) in a call center.
Manuscript distributed under the terms of the GNU Free Documentation License. 31 pp., 2005
1. Introduction 2. Implicit enumeration for nt terminals (nt >=2) 3. Find optimal binary trees using branch and bound, for nt terminals (nt >=2) 4. Build a tree by stepwise addition (n terminals, n >= 3) 5. Branch swapping 5.a. Introduction 5.b. A tree search strategy using SPR rearrangements of given trees 5.c. A tree search strategy using TBR rearrangements of given trees 6. Ratcheting 7. Tree drifting 8. Tree fusing 9. Static approximation 10. An integrated approach 11. Some quick comments on time complexity References
Modelling, Computation and Optimization in Information Systems and Management Sciences, 2008
A well-known method to represent a partially ordered set P consists in associ- ating to each element of P a subset of a fixed set S = {1,...,k} such that the order relation coincides with subset inclusion. Such an embedding is called a bit-vector encoding of P. Such encodings are economical with space and com- parisons between elements can
2000
Su x trees and su x arrays are classical data structures that are used to represent the set of su xes of a given string, and thereby facilitate the e cient solution of various string processing problems | in particular on-line string searching. Here we investigate the potential of suitably adapted binary search trees as competitors in this context. The su x binary search tree (SBST) and its balanced counterpart, the su x AVL-tree, are conceptually simple, relatively easy to implement, and o er time and space e ciency to rival su x trees and su x arrays, with some distinct advantages | for instance in cases where only a subset of the su xes need be represented.
Mathematics in Computer Science, 2017
We observe that a standard transformation between ordinal trees (arbitrary rooted trees with ordered children) and binary trees leads to interesting succinct binary tree representations. There are four symmetric versions of these transformations. Via these transformations we get four succinct representations of n-node binary trees that use 2n + n/(log n) Θ(1) bits and support (among other operations) navigation, inorder numbering, one of preorder or postorder numbering, subtree size and lowest common ancestor (LCA) queries. While this functionality, and more, is also supported in O(1) time using 2n + o(n) bits by Davoodi et al.'s (Phil. Trans. Royal Soc. A 372 (2014)) extension of a representation by Farzan and Munro (Algorithmica 6 (2014)), their redundancy, or the o(n) term, is much larger, and their approach may not be suitable for practical implementations. One of these transformations is related to the Zaks' sequence (S. Zaks, Theor. Comput. Sci. 10 (1980)) for encoding binary trees, and we thus provide the first succinct binary tree representation based on Zaks' sequence. The ability to support inorder numbering is crucial for the well-known range-minimum query (RMQ) problem on an array A of n ordered values. Another of these transformations is equivalent to Fischer and Heun's (SIAM J. Comput. 40 (2011)) 2d-Min-Heap structure for this problem. Yet another variant allows an encoding of the Cartesian tree of A to be constructed from A using only O(√ n log n) bits of working space.
Applied Mathematics and Computer Science, 2004
The features of an evolutionary algorithm that most determine its performance are the coding by which its chromosomes represent candidate solutions to its target problem and the operators that act on that coding. Also, when a problem involves constraints, a coding that represents only valid solutions and operators that preserve that validity represent a smaller search space and result in a more effective search. Two genetic algorithms for the leaf-constrained minimum spanning tree problem illustrate these observations. Given a connected, weighted, undirected graph G with n vertices and a bound ', this problem seeks a spanning tree on G with at least ' leaves and minimum weight among all such trees. A greedy heuristic for the problem begins with an unconstrained minimum spanning tree on G, then economically turns interior vertices into leaves until their number reaches '. One genetic algorithm encodes candidate trees with Prüfer strings decoded via the Blob Code. The s...
Genetic Programming, 2001
In recent years different genetic programming (GP) structures have emerged. Today, the basic forms of representation for genetic programs are tree, linear and graph structures. In this contribution we introduce a new kind of GP structure which we call Linear-tree. We describe the linear-tree-structure, as well as crossover and mutation for this new GP structure in detail. We compare linear-tree programs with linear and tree programs by analyzing their structure and results on different test problems.
In, 2001
The most important element in the design of a decoder-based evolutionary algorithm is its genotypic representation. The genotypedecoder pair must exhibit efficiency, locality, and heritability to enable effective evolutionary search. Prüfer numbers have been proposed to represent spanning trees in evolutionary algorithms. Several researchers have made extravagant claims for the usefulness of this coding, but others have pointed out that Prüfer numbers, though concise and easy to decode, lack the essential properties of locality and heritability. This conflict motivates our study. We examine the properties of Prüfer numbers and compare Prüfer numbers with other codings in evolutionary algorithms for four problems that involve spanning trees. Our conclusion is definite: Prüfer numbers cause poor performance in evolutionary algorithms and should be avoided.
This article introduces the basic concepts of binary trees, and then works through a series of practice problems with solution code in C/C++ and Java. Binary trees have an elegant recursive pointer structure, so they are a good way to learn recursive pointer algorithms.
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