Average Time Complexity of Decision Trees by Igor Chikalov

By Igor Chikalov

Decision tree is a frequent kind of representing algorithms and information. Compact facts types

and speedy algorithms require optimization of tree complexity. This booklet is a study monograph on

average time complexity of choice timber. It generalizes numerous recognized effects and considers a few new difficulties.

The e-book includes designated and approximate algorithms for selection tree optimization, and boundaries on minimal ordinary time

complexity of selection timber. tools of combinatorics, likelihood conception and complexity concept are utilized in the proofs as

well as innovations from a variety of branches of discrete arithmetic and machine technological know-how. The thought of functions include

the examine of usual intensity of choice bushes for Boolean services from closed periods, the comparability of result of the functionality

of grasping heuristics for usual intensity minimization with optimum selection bushes built by way of dynamic programming algorithm,

and optimization of selection timber for the nook element reputation challenge from machine vision.

The ebook will be fascinating for researchers engaged on time complexity of algorithms and experts

in try out conception, tough set thought, logical research of knowledge and computer learning.

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N0 0 ) = d¯i for some i ∈ {1, . . , m}. Denote ξ δ the complete ¯ path in the decision tree Φ such that δ¯ ∈ Tz π(ξ δ ). From the definition of the ¯ tree Φ it follows that the terminal node of the path ξ δ belongs to the subtree Γ˜i . Since the decision tree Γi solves the problem zi , the terminal node of the path ξi is assigned with the number νi (δ1i , . . , δni i ). From the definition of ¯ = νi (δ i , . . , δ i ). Therefore, Φ solves the proper decomposition we have ν(δ) 1 ni problem z.

2. For B ∈ {O1 , O4 , O5 , O6 , O8 , O9 }, the relation HB (n) = 1 holds. 3. For B ∈ {S1 , S3 , S5 , S6 , P1 , P3 , P5 , P6 }, the relation 2− HB (n) = 1 2n−1 , 1, if n ≥ 2 , if n = 1 . holds. 4. For B ∈ {L1 , L2 , L3 , C1 , C2 , C3 }, the relation HB (n) = n holds. 5. For B ∈ {L4 , L5 }, the relation HB (n) = n, if n = 2k + 1 , k ≥ 0 , n−1 , if n = 2k, k ≥ 1 holds. 6. For B = C4 , the relation HB (n) = if n = 2k + 1, k ≥ 0 , n, n− 1 2n−1 , if n = 2k, k ≥ 1 holds. 7. 7/ n ≤ HB (n) ≤ n − 1/2n−1, if n = 2k, k ≥ 1.

M. Consider an arbitrary row δ¯ = (δ10 , . . , δn0 0 , . . , δ1m , . . , δnmm ) ∈ ¯ Tz . Let (δ10 , . . , δn0 0 ) = d¯i for some i ∈ {1, . . , m}. Denote ξ δ the complete ¯ path in the decision tree Φ such that δ¯ ∈ Tz π(ξ δ ). From the definition of the ¯ tree Φ it follows that the terminal node of the path ξ δ belongs to the subtree Γ˜i . Since the decision tree Γi solves the problem zi , the terminal node of the path ξi is assigned with the number νi (δ1i , . . , δni i ). From the definition of ¯ = νi (δ i , .

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