By Igor Chikalov

Decision tree is a customary kind of representing algorithms and information. Compact information versions

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

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

The e-book comprises specified and approximate algorithms for determination tree optimization, and limits on minimal usual time

complexity of choice bushes. equipment of combinatorics, likelihood thought and complexity concept are utilized in the proofs as

well as suggestions from numerous branches of discrete arithmetic and desktop technology. The thought of functions include

the learn of typical intensity of choice timber for Boolean services from closed sessions, the comparability of result of the functionality

of grasping heuristics for ordinary intensity minimization with optimum determination bushes developed through dynamic programming algorithm,

and optimization of choice timber for the nook aspect acceptance challenge from laptop vision.

The e-book should be attention-grabbing for researchers engaged on time complexity of algorithms and experts

in attempt concept, tough set conception, logical research of information and desktop learning.

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M. 10) and the inequality M (z) ≥ 2 we have ¯ ¯ max{rid / log2 yid : i ∈ {1, . . , m − 1}} ≤ max{q(i) : i ∈ {0, . . , M (z)}} = 2, M (z) log2 M (z) if 2 ≤ M (z) ≤ 3 , , if M (z) ≥ 4 . 7). Then ¯ ¯ h(π(ξ d ))P (d) h(YU,h (z, P ), P ) = ¯ z d∈T ≤ M (z) + 2H(P ) , M (z) + M (z) log2 M (z) H(P ) if 2 ≤ M (z) ≤ 3 , , if M (z) ≥ 4 . 2 results in correctness of the theorem for M (z) ≥ 2. 4 On Possibility of Problem Decomposition In this section, a possibility of reduction is considered for a problem over 2-valued information system.

For z, such that MΨ (z) = m, limi→∞ H(Pi ) = 0, and limi→∞ hΨ (z, Pi ) = m. 18 2 Bounds on Average Time Complexity of Decision Trees Proof. Let m ∈ ω \ {0}. Define a 2-valued information system U as follows: U = (A, F ) where A = {0, 1, . . , m}, F = {f1 , . . , fm } and fi (a) = 1 , if i = a , 0 , if i = a , for any fi ∈ F and a ∈ A. Assume that Ψ (fi ) = 1 for i = 1, . . , m. Let z = (ν, f1 , . . , fm ) be a diagnostic problem. One can see that z has (m + 1) equivalence classes Q0 = {0}, Q1 = {1}, .

Summing these inequalities by i from 0 to r − 2, we obtain r−2 N (T αi (fji+1 , δji+1 ), P ) ≤ N (T, P ) . 4) i=0 Let us show that for any i ∈ {0, . . , r − 2}, N (T π(ξ), P ) ≤ N (T αi (fji+1 , δji+1 ), P ) . 5) The inequality N (T αi (fjr , δ), P ) ≤ N (T αi(fji+1 , δji+1 ), P ) follows from the choice of the attribute fji+1 (see description of the subprocess XΨ ) and the definition of the number δji+1 . The inequality N (T π(ξ), P ) ≤ N (T αi (fjr , δ), P ) is obvious. 5). The inequality N (T π(ξ), P ) ≤ N (T αr−1 , P ) is obvious.

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