From the Dandelion Code to the Rainbow code: a class of bijective spanning tree representations with linear complexity and bounded locality

There are many ways to represent spanning trees in genetic algorithms (GAs). Among them are Cayley codes, which represent each tree on n vertices as a string of (n-2) integers from the set [1,n]. In 2003, Thompson showed that the Dandelion Code, a Cayley code with high locality, offers consistently better performance in a GA than all other known Cayley codes, including the Pru/spl uml/fer Code and the Blob Code. In this paper, we study the Dandelion Code and its properties. We give linear-time implementations of the decoding and encoding algorithms, and prove that the representation has bounded locality and asymptotically optimal locality, unlike all other known Cayley codes. We then modify the Dandelion Code to create bijective spanning tree representations for graph topologies other than the complete graph. Two variations are described: the bipartite Dandelion Code (for encoding the spanning trees of a complete bipartite graph) and the Rainbow Code (for encoding the spanning trees of a complete layered graph). Both variations inherit the Dandelion Code's desirable properties, and have the potential to outperform existing GA representations for computationally hard transportation problems (including the Fixed Charge Transportation Problem) and multistage transportation problems, particularly on large instances.     

The Dandelion Code: A New Coding of Spanning Trees for Genetic Algorithms

There are many applications where it is necessary to find an optimal spanning tree. For several of these, recent research has suggested the use of genetic algorithms (GAs). Historically, the Pruumlfer code has been one of the most popular representations for spanning trees, due largely to the bijective mapping between genotype and phenotype. However, it is has attracted much criticism for its low locality, a primary reason for its poor performance in GAs. Other representations such as direct encoding and network random keys have been shown to be far more effective. In 2001, an alternative called the Blob code was identified and adapted for use in GAs. It was shown to exhibit significantly higher locality than the Pruumlfer code. For a simple test problem, a GA using the Blob code was found to substantially outperform one using the Pruumlfer code. This paper suggests an alternative called the Dandelion code, which is more efficient and exhibits yet higher locality. Although both direct encoding and NetKeys are shown to give better results on the test problems used in this paper, the Dandelion code should be considered as a strong alternative, particularly for very large networks