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

34 3 The multiple **an**t-colonies approach: the mesh-partitioning problem a chosen vertex migrates from one domain **to** **an**other, only its gain **an**d the gains of all its neighbors have **to** be recalculated **an**d put back in**to** 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 r**an**ge. For this re**as**on we have introduced the 2-3 tree, **an**d so avoided large **an**d sparse arrays of pointers. We s**to**re the non-empty buckets in the 2-3 tree, so each leaf in the tree represents a bucket. For **an** even f**as**ter search we have created one 2-3 tree for each colony on every cell that h**as** vertices on it (see Figure 3.3). In this way we have incre**as**ed the speed of the search, **as** well **as** the add **an**d delete operations. -1 2 grid with food (vertices) -3 0 4 6 -7 -3 -1 0 2 4 6 bucket r**an**ked 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 Vec**to**r Qu**an**tization (VQ) **an**d the b**as**ic **an**t-colony algorithm in**to** 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 s**to**ch**as**tic approximation method that uses the b**as**ic structure of the input vec**to**rs **to** solve a specific problem (for example, data compression). In other words, the input space is divided in**to** a finite number of regions (domains) **an**d for each region there is a representative vec**to**r. When a mapping function (device) receives a new input vec**to**r it maps it in**to** a region that best represents this vec**to**r. 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 c**an** treat the position of each mesh vertex **as** **an** input vec**to**r **an**d each domain in our partition **as** a region in input space. We try **to** divide the “mesh” space in**to** 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 vec**to**r qu**an**tizer maps l-dimensional vec**to**rs in the vec**to**r space R l in**to** a finite set of vec**to**rs Y = {y i : i = 1, 2, . . ., k}. (3.2) Each vec**to**r y i is called a codeword **an**d 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 c**an** e**as**ily be exp**an**ded **to** **an**y other number of dimensions. We c**an** see 45 input vec**to**rs that are divided in**to** k = 13 domains (Voronoi regions V 1 , V 2 , . . ., V 13 ) **an**d 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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128 Acknowledgements

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140 Index Covariance Matrix Adaptat

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142 Index TSP, see Traveling Salesm

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144 List of Algorithms

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146 List of Figures 4.4 Multilevel

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160 Selected publications Izbrane o