## Edge-Sets: An Effective Evolutionary Coding of Spanning Trees (2002)

Citations: | 14 - 7 self |

### BibTeX

@MISC{Raidl02edge-sets:an,

author = {Günther R. Raidl and Bryant A. Julstrom},

title = {Edge-Sets: An Effective Evolutionary Coding of Spanning Trees},

year = {2002}

}

### OpenURL

### Abstract

The fundamental design choices in an evolutionary algorithm are its representation of candidate solutions and the operators that will act on that representation. We propose representing spanning trees in evolutionary algorithms for network design problems directly as sets of their edges, and we describe initialization, recombination, and mutation operators for this representation. The operators offer

### Citations

10926 | Computers and Intractability: A Guide to the Theory of NPCompleteness. Freeman and - Garey, Johnson - 1979 |

706 | Data Structures and Algorithms - Aho, Hopcroft, et al. - 1983 |

531 |
Shortest connection networks and some generalisations
- Prim
- 1957
(Show Context)
Citation Context ...ng tree with the smallest possible cost is a minimum spanning tree (MST) on G. When the graph’s edge costs are fixed and the search is unconstrained, the well-known algorithms of Kruskal [1] and Prim =-=[2]-=- identify MSTs in times that are polynomial in the number of nodes, as do more recent algorithms such as that described by Chazelle [3]. 53However, many variants of the MST problem are computationall... |

436 |
On the Shortest Spanning Subtree of a Graph and the Travelling Salesman Problem
- Kruskal
- 1956
(Show Context)
Citation Context ...ges; a spanning tree with the smallest possible cost is a minimum spanning tree (MST) on G. When the graph’s edge costs are fixed and the search is unconstrained, the well-known algorithms of Kruskal =-=[1]-=- and Prim [2] identify MSTs in times that are polynomial in the number of nodes, as do more recent algorithms such as that described by Chazelle [3]. 53However, many variants of the MST problem are c... |

243 |
A Connectionist Machine for Genetic Hillclimbing
- Ackley
- 1987
(Show Context)
Citation Context ...r under constraints. 4 The One-Max-Tree Problem We compare the techniques just described in a steady-state EA for a simple spanning tree problem called One-Max-Tree. In the well-known One-Max problem =-=[50]-=-, a bit string’s fitness is the number of 1’s it contains. In the One-Max-Tree problem [51], a target spanning tree on a complete base graph is specified, and the fitness of any other tree is the numb... |

138 | On Steiner's problem with rectilinear distance - Hanan - 1966 |

112 |
Generating random spanning trees
- BRODER
- 1989
(Show Context)
Citation Context ...terminant computations and reduced its time to O(n 3 ). This might be acceptable for the initialization of an EA’s population, but is too expensive to be the basis of a recombination operator. Broder =-=[49]-=- described a probabilistic method based on a random walk in G. A particle begins at an arbitrary node in G. At each step, it moves over a randomly chosen adjacent edge to one of its neighbors. When th... |

105 |
Genetic algorithms and random keys for sequencing and optimization
- Bean
- 2011
(Show Context)
Citation Context ...amoorthy et al. [18] described another variant of link-and-node-biasing, which they called problem search space, for the degree-constrained minimum spanning tree problem. 2.5 Network Random Keys Bean =-=[35]-=- described random keys to encode permutations; Rothlauf et al. [36, 37] adapted random keys to represent spanning trees and called them network random keys. In this coding, a genotype is a string of r... |

83 | Representations for Genetic and Evolutionary Algorithms - Rothlauf - 2002 |

74 |
A minimum spanning tree algorithm with inverse-Ackermann type complexity
- Chazelle
- 2000
(Show Context)
Citation Context ...unconstrained, the well-known algorithms of Kruskal [1] and Prim [2] identify MSTs in times that are polynomial in the number of nodes, as do more recent algorithms such as that described by Chazelle =-=[3]-=-. 53However, many variants of the MST problem are computationally difficult. Some seek to minimize objective functions other than the simple sum of a tree’s fixed edge weights. For example, in the op... |

71 | Optimum communication spanning trees
- Hu
- 1974
(Show Context)
Citation Context ...blem are computationally difficult. Some seek to minimize objective functions other than the simple sum of a tree’s fixed edge weights. For example, in the optimum communication spanning tree problem =-=[4]-=-, a tree’s cost depends on communication demands between each pair of nodes and on the tree’s structure. In another version of this problem, edge costs are time-dependent [5]. Often, only spanning tre... |

71 |
Neuer beweis eines satzes über permutationen. Archiv Für Mathematik und Physik
- Prüfer
- 1918
(Show Context)
Citation Context ...nstrained minimum spanning tree problem. 562.3 Prüfer Numbers Cayley’s Formula identifies as n n−2 the number of distinct spanning trees on a complete graph with n nodes [15, 23, pp. 98–106]. Prüfer =-=[24]-=- presented a constructive proof of this result: a pair of inverse mappings between spanning trees on n nodes and vectors of length n − 2 over integers labeling the nodes. These vectors are called Prüf... |

56 |
Transitions in geometric minimum spanning trees
- Monma, Suri
- 1992
(Show Context)
Citation Context ...plane and the edge costs are the Euclidean distances between them, the problem is relatively easy. In this case, there always exists an unconstrained minimum spanning tree of degree no more than five =-=[55]-=-. Finding a d-MST in the plane is NP-hard when d = 3 and is conjectured to remain so when d = 4 [56]. Branchand-bound techniques can find exact solutions for problem instances of several hundred point... |

49 |
On two geometric problems related to the traveling salesman problem
- Papadimitriou, Vazirani
- 1984
(Show Context)
Citation Context ...n this case, there always exists an unconstrained minimum spanning tree of degree no more than five [55]. Finding a d-MST in the plane is NP-hard when d = 3 and is conjectured to remain so when d = 4 =-=[56]-=-. Branchand-bound techniques can find exact solutions for problem instances of several hundred points in reasonable computing times [6, 18], and several authors have described effective polynomial-tim... |

45 | Representing trees in genetic algorithms
- Palmer, Kershenbaum
- 1994
(Show Context)
Citation Context ... its edges can be made explicit in time that is O(n). Applied to such genotypes, positional crossover and mutation operators will generate infeasible solutions, requiring again penalization or repair =-=[17, 18]-=-. Abuali et al. [19] described a repair mechanism that applies to spanning trees on sparse as well as complete graphs. Chu et al. [20] used the predecessor coding with both a penalty and repair in a g... |

32 | Algorithmic combinatorics - Even - 1973 |

27 |
Degree-constrained minimum spanning tree
- Narula, Ho
- 1980
(Show Context)
Citation Context ...her version of this problem, edge costs are time-dependent [5]. Often, only spanning trees that satisfy particular constraints are feasible. Examples of such constraints include the degree constraint =-=[6]-=-, which requires that no node in a spanning tree have more than d ≥ 2 incident edges; leaf constraints [7], which specify or bound the number of leaves a spanning tree may have; the diameter constrain... |

26 |
A genetic algorithm for survival network design
- Davis, Orvosh, et al.
- 1993
(Show Context)
Citation Context ...nts. When such a vector represents a spanning tree, the items are the edges of G, and the vector indicates whether each edge is or is not part of the tree. Several researchers, including Davis et al. =-=[13]-=- and Piggott and Suraweera [14], have used characteristic vectors to represent spanning trees in evolutionary algorithms. These EAs have applied positional operators like k-point crossover and positio... |

25 |
The complexity of the capacitated tree problem
- Papadimitriou
- 1978
(Show Context)
Citation Context ...7], which specify or bound the number of leaves a spanning tree may have; the diameter constraint [8], which bounds the longest path in a spanning tree between any two nodes; and capacity constraints =-=[9]-=-, which limit the capacity of edges. Other combinatorial problems can be mapped to spanning tree problems. These include the rectilinear Steiner problem [10], which seeks the shortest tree composed of... |

24 |
Comparison of algorithms for the degree constrained minimum spanning tree
- Krishnamoorthy, Ernst, et al.
- 1999
(Show Context)
Citation Context ... its edges can be made explicit in time that is O(n). Applied to such genotypes, positional crossover and mutation operators will generate infeasible solutions, requiring again penalization or repair =-=[17, 18]-=-. Abuali et al. [19] described a repair mechanism that applies to spanning trees on sparse as well as complete graphs. Chu et al. [20] used the predecessor coding with both a penalty and repair in a g... |

23 | An efficient evolutionary algorithm for the degree-constrained minimum spanning tree problem
- Raidl
- 2000
(Show Context)
Citation Context ... 8 (3,8), (7,1), (4,11), (4,5), 7 (10,4), (9,4), (5,8) } 3 Figure 1: A spanning tree on twelve nodes and an edge-set that represents it. To our knowledge and with the exception of our own recent work =-=[41, 42]-=-, only the following publications have considered EAs for spanning tree problems that represent candidate spanning trees directly as sets of their edges. Li and Bouchebaba [43] proposed crossover and ... |

20 | A network-flow technique for finding low-weight bounded-degree spanning trees
- Fekete, Khuller, et al.
(Show Context)
Citation Context ...xact solutions for problem instances of several hundred points in reasonable computing times [6, 18], and several authors have described effective polynomial-time approximation schemes and heuristics =-=[57, 58]-=-. In general, however, edge costs need not satisfy the triangle inequality. An unconstrained minimum spanning tree may have degree up to n − 1, and when a graph has a high-degree unconstrained MST, id... |

19 | Determinant factorization: a new encoding scheme for spanning trees applied to the probabilistic minimum spanning tree problem - Abuali, Wainwright, et al. - 1995 |

19 | Prer numbers: A poor representation of spanning trees for evolutionary search
- Gottlieb, Julstrom, et al.
- 2001
(Show Context)
Citation Context ...arge transportation problem [12], and a bicriteria network design problem [28]. A recent comparison of codings in EAs for several spanning tree problems demonstrated the inferiority of Prüfer numbers =-=[29]-=-. There are many other mappings like Prüfer’s from strings of n − 2 node labels to spanning trees. Recently, Picciotto [30] and Deo and Micikevicius [31] described several of them. One, called the Blo... |

18 | A weighted coding in a genetic algorithm for the degreeconstrained minimum spanning tree problem
- Raidl, Julstrom
- 2000
(Show Context)
Citation Context ...e bias of this representation toward star-like structures [33]. However, edge weights increase the size of each genotype on a complete graph from n values to n+n(n−1)/2 = n(n+1)/2. Raidl and Julstrom =-=[34]-=- proposed a variant of this coding, called weight-coding, in an EA for the degree-constrained minimum spanning tree problem. In weight-coding, the weights in each genotype are initially selected from ... |

17 |
Pardalos: Minimum concave-cost network flow problems: applications, complexity, and algorithms
- Guisewite, M
- 1990
(Show Context)
Citation Context ...ar Steiner problem [10], which seeks the shortest tree composed of vertical and horizontal line segments that connects a collection of points in the plane, and the fixed-charge transportation problem =-=[11]-=-, which seeks an economical plan for distributing a commodity from sources to destinations. Finally, some spanning tree problems seek to optimize several objective functions simultaneously [12]. Most ... |

17 |
A theorem on trees. Quarterly
- Cayley
(Show Context)
Citation Context ... − 1)/2 and the size of the search space is 2 n(n−1)/2 . However, only a tiny fraction of these genotypes represent feasible solutions, since a complete graph G has only n n−2 distinct spanning trees =-=[15]-=-. Penalizing genotypes that do not represent spanning trees has not been effective [14]. Repair strategies have been more successful [16], but they require additional computation and weaken the coding... |

17 |
Random spanning tree (in
- Guénoche
- 1983
(Show Context)
Citation Context ...andom Prüfer numbers with uniform probabilities; decoding them yields unbiased random spanning trees. This is an effective and efficient mechanism when the underlying graph G is complete. 63Guénoche =-=[47]-=- described a random spanning tree algorithm for general graphs. It is based on the fact that the number of distinct spanning trees in a graph can be found by computing a determinant of size n × n, and... |

16 |
A new genetic algorithm for the optimal communication spanning tree problem
- Li, Bouchebaba
- 2000
(Show Context)
Citation Context ...our own recent work [41, 42], only the following publications have considered EAs for spanning tree problems that represent candidate spanning trees directly as sets of their edges. Li and Bouchebaba =-=[43]-=- proposed crossover and mutation operators based on edges, paths, and subtrees of spanning trees in an EA for the optimum communication spanning tree problem. Li [44] described the tree representation... |

16 |
Network random keys: a tree network representation scheme for genetic and evolutionary algorithms,” Evolutionary Computation,vol.10,no.1
- Rothlauf, Goldberg, et al.
- 2002
(Show Context)
Citation Context ... a steady-state EA for a simple spanning tree problem called One-Max-Tree. In the well-known One-Max problem [50], a bit string’s fitness is the number of 1’s it contains. In the One-Max-Tree problem =-=[51]-=-, a target spanning tree on a complete base graph is specified, and the fitness of any other tree is the number of edges that it shares with the target. An EA that supports meaningful building blocks ... |

14 |
Genetic algorithm for communications network design- an empirical study of the factors that influence performance
- Chou, Premkumar, et al.
- 2001
(Show Context)
Citation Context ...rained minimum spanning tree problem. Berry et al. [21] used it with special variation operators that produced only feasible solutions for the optimum communication spanning tree problem. Chou et al. =-=[22]-=- performed further investigations on this encoding with respect to the degree-constrained minimum spanning tree problem. 562.3 Prüfer Numbers Cayley’s Formula identifies as n n−2 the number of distin... |

14 |
Unranking and ranking spanning trees of a graph
- Colbourn, Day, et al.
- 1989
(Show Context)
Citation Context ...or general graphs. It is based on the fact that the number of distinct spanning trees in a graph can be found by computing a determinant of size n × n, and it requires time that is O(n 5 ). Colbourne =-=[48]-=- modified the algorithm to require fewer determinant computations and reduced its time to O(n 3 ). This might be acceptable for the initialization of an EA’s population, but is too expensive to be the... |

13 |
On Teleprocessing System Design
- Esau, Williams
- 1966
(Show Context)
Citation Context ...which requires that no node in a spanning tree have more than d ≥ 2 incident edges; leaf constraints [7], which specify or bound the number of leaves a spanning tree may have; the diameter constraint =-=[8]-=-, which bounds the longest path in a spanning tree between any two nodes; and capacity constraints [9], which limit the capacity of edges. Other combinatorial problems can be mapped to spanning tree p... |

12 | How to Encode a Tree
- Picciotto
- 1999
(Show Context)
Citation Context ...everal spanning tree problems demonstrated the inferiority of Prüfer numbers [29]. There are many other mappings like Prüfer’s from strings of n − 2 node labels to spanning trees. Recently, Picciotto =-=[30]-=- and Deo and Micikevicius [31] described several of them. One, called the Blob Code, exhibits stronger locality and heritability than do Prüfer numbers, and an EA for the One-Max-Tree problem performe... |

12 |
Edge exchanges in the degree-constrained minimum spanning tree problem
- Savelsbergh, Volgenant
- 1985
(Show Context)
Citation Context ...xact solutions for problem instances of several hundred points in reasonable computing times [6, 18], and several authors have described effective polynomial-time approximation schemes and heuristics =-=[57, 58]-=-. In general, however, edge costs need not satisfy the triangle inequality. An unconstrained minimum spanning tree may have degree up to n − 1, and when a graph has a high-degree unconstrained MST, id... |

11 | A new evolutionary approach to the degree constrained minimum spanning tree problem
- Knowles, Corne
- 2000
(Show Context)
Citation Context ...times its label appears in the tree’s Prüfer number. However, many researchers have pointed out that Prüfer numbers have poor locality and heritability and are thus unsuitable for evolutionary search =-=[17, 25, 26]-=-. Patterns of values in Prüfer numbers do not represent consistent substructures of spanning trees, so the mutation of a single symbol may change many edges in the represented tree, and crossover ofte... |

11 |
Genetic algorithm for solving bicriteria network topology design problem
- Kim, Gen
(Show Context)
Citation Context ...ned minimum spanning tree problem [18, 27], the minimum spanning tree problem with time-dependent edge costs [5], the fixed-charge transportation problem [12], and a bicriteria network design problem =-=[28]-=-. A recent comparison of codings in EAs for several spanning tree problems demonstrated the inferiority of Prüfer numbers [29]. There are many other mappings like Prüfer’s from strings of n − 2 node l... |

11 |
Feasible encodings for GA solutions of constrained minimal spanning tree problems
- Edelson, Gargano
- 2000
(Show Context)
Citation Context ... labels, and the second holds the nodes’ degrees. The tree this pair represents is obtained by connecting the nodes in the specified order and with the specified degrees. In Prüfer-based permutations =-=[40]-=-, a genotype holds indices into a list of multiple copies of the node labels. These indices specify the order in which labels are removed from the list and concatenated to an initially empty Prüfer nu... |

10 |
Spanning tree-based genetic algorithms for bicriteria fixed charge transportation problem
- Gen, Li
- 1999
(Show Context)
Citation Context ...roblem [11], which seeks an economical plan for distributing a commodity from sources to destinations. Finally, some spanning tree problems seek to optimize several objective functions simultaneously =-=[12]-=-. Most such problems are NP-hard, and we can apply exact optimization algorithms only to small instances of them. For larger instances, we turn to heuristic techniques, including evolutionary algorith... |

10 | Pruefernumbers and genetic algorithms: A lesson on how the low locality of an encoding can harm the performance of GAs - Rothlauf, Goldberg |

10 |
The blob code: A better string coding of spanning trees for evolutionary search
- Julstrom
- 2001
(Show Context)
Citation Context ...onger locality and heritability than do Prüfer numbers, and an EA for the One-Max-Tree problem performed significantly better when it encoded spanning trees via the Blob Code than with Prüfer numbers =-=[32]-=-. As in Prüfer numbers, each node’s degree in the spanning tree a string represents via the Blob Code is one more than the number of times the node’s label appears in the string. The Blob Code’s decod... |

8 |
Approach to the degree-constrained minimum spanning tree problem using genetic algorithm
- Zhou, Gen
- 1997
(Show Context)
Citation Context ...istics. Nonetheless, researchers have encoded spanning trees as Prüfer numbers in evolutionary algorithms for a variety of problems. These include the degree-constrained minimum spanning tree problem =-=[18, 27]-=-, the minimum spanning tree problem with time-dependent edge costs [5], the fixed-charge transportation problem [12], and a bicriteria network design problem [28]. A recent comparison of codings in EA... |

7 |
A genetic-based approach to tree network synthesis with cost constraints
- Berry, Murtagh, et al.
- 1994
(Show Context)
Citation Context ...ince a complete graph G has only n n−2 distinct spanning trees [15]. Penalizing genotypes that do not represent spanning trees has not been effective [14]. Repair strategies have been more successful =-=[16]-=-, but they require additional computation and weaken the coding’s locality and heritability. 2.2 The Predecessor Coding A more compact representation of spanning trees is the predecessor or determinan... |

7 | Tree network design with genetic algorithms - an investigation in the locality of the pruefernumber encoding - Goldberg - 1999 |

6 |
A genetic algorithm with feasible search space for minimal spanning trees with time-dependent edge costs
- Gargano, Edelson, et al.
- 1998
(Show Context)
Citation Context ...ion spanning tree problem [4], a tree’s cost depends on communication demands between each pair of nodes and on the tree’s structure. In another version of this problem, edge costs are time-dependent =-=[5]-=-. Often, only spanning trees that satisfy particular constraints are feasible. Examples of such constraints include the degree constraint [6], which requires that no node in a spanning tree have more ... |

6 |
Application of a genetic-based algorithm for optimal design of tree-structured communication networks
- Berry, Murtagh, et al.
- 1995
(Show Context)
Citation Context ...se as well as complete graphs. Chu et al. [20] used the predecessor coding with both a penalty and repair in a genetic algorithm for the degree-constrained minimum spanning tree problem. Berry et al. =-=[21]-=- used it with special variation operators that produced only feasible solutions for the optimum communication spanning tree problem. Chou et al. [22] performed further investigations on this encoding ... |

6 | Weight-biased edge-crossover in evolutionary algorithms for two graph problems
- Julstrom, Raidl
- 2001
(Show Context)
Citation Context ...for edges of lower cost. We have previously investigated empirically the probability that an edge appears in a nearoptimal degree-constrained spanning tree as a function of the edge’s cost-based rank =-=[59]-=-. On a complete graph of 100 nodes with several different cost structures, for example, it was consistently the case that about 98% of the edges in near-optimal trees with d = 3 had ranks less than 30... |

5 |
Encoding graphs for genetic algorithms: An investigation using the minimum spanning tree problem
- Piggott, Suraweera
- 1995
(Show Context)
Citation Context ...nts a spanning tree, the items are the edges of G, and the vector indicates whether each edge is or is not part of the tree. Several researchers, including Davis et al. [13] and Piggott and Suraweera =-=[14]-=-, have used characteristic vectors to represent spanning trees in evolutionary algorithms. These EAs have applied positional operators like k-point crossover and position-by-position mutation. A chara... |

5 | The link and node biased encoding revisited: Bias and adjustment of parameters
- Gaube, Rothlauf
(Show Context)
Citation Context ...t cannot be represented by node weights alone; edge weights are necessary to render every spanning tree reachable. Edge weights also reduce the bias of this representation toward star-like structures =-=[33]-=-. However, edge weights increase the size of each genotype on a complete graph from n values to n+n(n−1)/2 = n(n+1)/2. Raidl and Julstrom [34] proposed a variant of this coding, called weight-coding, ... |

5 | Evolution strategies, network random keys, and the One-Max Tree problem,” in Applications of Evolutionary Computing
- Schindler, Rothlauf, et al.
(Show Context)
Citation Context ...tive only for small or sparse problems. Rothlauf et al. [36] reported good results with this coding on instances of the optimum communication spanning tree problem of up to 26 nodes. Schindler et al. =-=[38]-=- further investigated random network keys in an evolution strategy framework. 2.6 Other Representations Other representations of spanning trees are less often used. In degree-based permutations [39], ... |