By Igor Chikalov

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

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

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

The booklet comprises certain and approximate algorithms for choice tree optimization, and limits on minimal commonplace time

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

well as techniques from a variety of branches of discrete arithmetic and computing device technology. The thought of purposes include

the research of typical intensity of determination timber for Boolean capabilities from closed periods, the comparability of result of the functionality

of grasping heuristics for general intensity minimization with optimum choice bushes developed by means of dynamic programming algorithm,

and optimization of determination timber for the nook element popularity challenge from machine vision.

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

in try concept, tough set thought, logical research of information and desktop learning.

**Read or Download Average Time Complexity of Decision Trees PDF**

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**Additional resources for Average Time Complexity of Decision Trees **

**Sample text**

Let (I01 , . . , Ik−1 ), . . , (I0r , . . , Ik−1 ) be all possible partitions of the set {1, . . , s} possessing the following conditions: t ∪k−1 j=0 Ij = {1, . . , s}, and for any numbers i, j ∈ Ek , i = j, for t = 1, . . , r, the relation Iit ∩ Ijt = ∅ holds. Deﬁne an attribute gt : A → Ek , t = 1, . . , r, as follows. If a ∈ Qi and i ∈ Ijt , then gt (a) = j. Consider the problem z = (ν , f1 , . . , fn , g1 , . . , gr ) over the information system U = (A, F ∪ {g1 , . . , gr }) where ν : Ekn+r → ω, and ν (δ1 , .

Then a) if B ∈ {O2 , O3 , O7 }, then GB (n) = 0; b) if B ∈ {O1 , O4 , O5 , O6 , O8 , O9 }, then GB (n) = 1; n, if n is odd , c) if B ∈ {L4 , L5 }, then GB (n) = n − 1 , if n is even ; d) if B ∈ {D1 , D2 , D3 }, then GB (n) = n, if n ≥ 3 , 1 , if n ≤ 2 ; e) if the class B does not coincide with any classes mentioned in (a) – (d), then GB (n) = n. 13. These theorems characterize the relation between HB (n) and GB (n) for each closed class of Boolean functions. 2. Let B be a closed class of Boolean functions, and n a natural number.

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 . Then the following conditions hold for each complete path ξ in the decision tree XΨ (z, P, T ): a) Ψ (π(ξ)) ≤ MΨ (z); b) if T π(ξ) is a nonterminal subtable, then N (T π(ξ), P ) ≤ N (T, P )/ max{2, h(π(ξ))} . Proof. For each i ∈ {1, . . , n}, denote σi the minimum number from Ek such that N (T (fi, σi ), P ) = max{N (T (fi, σ), P ) : σ ∈ Ek } .