By Igor Chikalov

ISBN-10: 3642226604

ISBN-13: 9783642226601

Decision tree is a standard type of representing algorithms and information. Compact facts versions

and quick algorithms require optimization of tree complexity. This ebook is a learn monograph on

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

The ebook comprises certain and approximate algorithms for selection tree optimization, and limits on minimal standard time

complexity of determination bushes. tools of combinatorics, chance idea and complexity concept are utilized in the proofs as

well as techniques from a number of branches of discrete arithmetic and desktop technology. The thought of functions include

the learn of normal intensity of choice bushes for Boolean features from closed periods, the comparability of result of the functionality

of grasping heuristics for common intensity minimization with optimum determination timber built through dynamic programming algorithm,

and optimization of choice timber for the nook aspect reputation challenge from machine vision.

The publication could be attention-grabbing for researchers engaged on time complexity of algorithms and experts

in attempt thought, tough set idea, logical research of information and computing device learning.

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**Example text**

For each complete path ξ in XΨ (z, P, T α), replace the number 0 assigned to its terminal node, with the word απ(ξ). Denote Γ the tree resulted from this replacement. Replace in G the node v with the tree Γ . Proceed to the step (t + 2). 4 This section contains proofs of the upper bounds on the minimum average time complexity of decision trees given in Sect. 2 and Sect. 3. 1. Let U = (A, F ) be a k-valued information system, Ψ a weight function for U , z = (ν, f1 , . . , fn ) a problem over U , P a probability distribution for z, and T a nonterminal subtable of the table Tz .

M, Pi be a uniform probability distribution for the problem zi . One can see that ((z0 , P0 ), (z1 , P1 ), . . , (zm , Pm )) is a proper decomposition of the pair n n 1 1 n−1 n−1 (zm , Pm ), h(z0 , P0 ) = h(zm , Pm ) and h(zi , Pi ) = h(zm , Pm ) for i = 1, . . , m. Using induction hypothesis, we obtain h(z0 , P0 ) = (m + 2)(m − 1) /(2m) and h(zi , Pi ) = [(m + 2)(m − 1)/(2m)](n − 1) for i = 1, . . , m. Let Γi be a decision tree for the problem zi that solves zi and is optimal for zi and Pi , i = 0, .

Then a) Γ˜ is a decision tree for z that solves z; b) h(Γ˜ , P ) ≤ h(Γ, P ); c) h(Γ˜ , P ) ≤ h(Γ, P ) − (N0 (ξ) + N1 (ξ))/N (Tz , P ) if ξ is a reducible path. Proof. One can see from the description of the path reduction operation that Γ˜ is a decision tree for the problem z. Let us show that Γ˜ solves z. Let ξ = v1 , e1 , . . , vt , et , vt+1 where v1 , . . , vt+1 ∈ V (Γ ), e1 , . . , et ∈ E(Γ ), t ≥ 1, and for i = 1, . . , t, the node vi is assigned with an attribute fi . Then there exist natural j and k, j < k ≤ t, such that fj , .

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