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Stigmergy as an Approach to Metaheuristic Optimization - Computer ...

Stigmergy as an Approach to Metaheuristic Optimization - Computer ...

34 3 The multiple

34 3 The multiple ant-colonies approach: the mesh-partitioning problem a chosen vertex migrates from one domain to another, only its gain and the gains of all its neighbors have to be recalculated and put back into appropriate buckets. In our implementation each bucket is represented by a double-linked list of vertices. Because of the multilevel process, it often happens that the potential gain values are dispersed over a wide range. For this reason we have introduced the 2-3 tree, and so avoided large and sparse arrays of pointers. We store the non-empty buckets in the 2-3 tree, so each leaf in the tree represents a bucket. For an even faster search we have created one 2-3 tree for each colony on every cell that has vertices on it (see Figure 3.3). In this way we have increased the speed of the search, as well as the add and delete operations. -1 2 grid with food (vertices) -3 0 4 6 -7 -3 -1 0 2 4 6 bucket ranked 6 double linked list of vertices Figure 3.3 Representation of 2-3 tree used to speed up bucket sorting. 3.4 The hybrid algorithm We have merged the Vector Quantization (VQ) and the basic ant-colony algorithm into a single algorithm called the Hybrid Multiple Ant-Colony Algorithm (H-MACA) [117]. With the VQ we compute a partition, which is then used as a starting partition for the

3.4 The hybrid algorithm 35 B-MACA. With the B-MACA we refine this partition so that the best possible result is obtained. The VQ method [84] is a stochastic approximation method that uses the basic structure of the input vectors to solve a specific problem (for example, data compression). In other words, the input space is divided into a finite number of regions (domains) and for each region there is a representative vector. When a mapping function (device) receives a new input vector it maps it into a region that best represents this vector. This is a simple example of some kind of compression. Of course this is only one possibility of using this method. We used it as a mapping device for the mesh-partitioning problem. The mesh vertices are usually locally connected to their neighbors. Now we can treat the position of each mesh vertex as an input vector and each domain in our partition as a region in input space. We try to divide the “mesh” space into domains, so that the size (the number of vertices) of each domain is approximately the same, with as few as possible connections between the domains. A vector quantizer maps l-dimensional vectors in the vector space R l into a finite set of vectors Y = {y i : i = 1, 2, . . ., k}. (3.2) Each vector y i is called a codeword and the set of all the codewords is called a codebook. Associated with each codeword y i is a nearest-neighbor region called the Voronoi region [123], which is defined by: V i = { x ∈ R l : ‖x − y i ‖ ≤ ‖x − y j ‖, ∀i ≠ j } . (3.3) The set of Voronoi regions partition the entire space R l such that ( ⋃ k ) ( k ) ⋂ V i = R l ∧ V i = ∅ . (3.4) i=1 i=1 An example of the VQ is shown in Figure 3.4. Here we used a two-dimensional graph (l = 2), but it can easily be expanded to any other number of dimensions. We can see 45 input vectors that are divided into k = 13 domains (Voronoi regions V 1 , V 2 , . . ., V 13 ) and are represented by the codewords y 1 ,y 2 , . . .,y 13 . The VQ consists of the following six steps: Step 1: Set the number of domains k according to a given problem.

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