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Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2018.34
URN: urn:nbn:de:0030-drops-99824
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9982/
H. Bshouty, Nader ;
Makhoul, Waseem
On Polynomial Time Constructions of Minimum Height Decision Tree
Abstract
A decision tree T in B_m:={0,1}^m is a binary tree where each of its internal nodes is labeled with an integer in [m]={1,2,...,m}, each leaf is labeled with an assignment a in B_m and each internal node has two outgoing edges that are labeled with 0 and 1, respectively. Let A subset {0,1}^m. We say that T is a decision tree for A if (1) For every a in A there is one leaf of T that is labeled with a. (2) For every path from the root to a leaf with internal nodes labeled with i_1,i_2,...,i_k in[m], a leaf labeled with a in A and edges labeled with xi_{i_1},...,xi_{i_k}in {0,1}, a is the only element in A that satisfies a_{i_j}=xi_{i_j} for all j=1,...,k.
Our goal is to write a polynomial time (in n:=|A| and m) algorithm that for an input A subseteq B_m outputs a decision tree for A of minimum depth. This problem has many applications that include, to name a few, computer vision, group testing, exact learning from membership queries and game theory.
Arkin et al. and Moshkov [Esther M. Arkin et al., 1998; Mikhail Ju. Moshkov, 2004] gave a polynomial time (ln |A|)- approximation algorithm (for the depth). The result of Dinur and Steurer [Irit Dinur and David Steurer, 2014] for set cover implies that this problem cannot be approximated with ratio (1-o(1))* ln |A|, unless P=NP. Moshkov studied in [Mikhail Ju. Moshkov, 2004; Mikhail Ju. Moshkov, 1982; Mikhail Ju. Moshkov, 1982] the combinatorial measure of extended teaching dimension of A, ETD(A). He showed that ETD(A) is a lower bound for the depth of the decision tree for A and then gave an exponential time ETD(A)/log(ETD(A))-approximation algorithm and a polynomial time 2(ln 2)ETD(A)-approximation algorithm.
In this paper we further study the ETD(A) measure and a new combinatorial measure, DEN(A), that we call the density of the set A. We show that DEN(A) <=ETD(A)+1. We then give two results. The first result is that the lower bound ETD(A) of Moshkov for the depth of the decision tree for A is greater than the bounds that are obtained by the classical technique used in the literature. The second result is a polynomial time (ln 2)DEN(A)-approximation (and therefore (ln 2)ETD(A)-approximation) algorithm for the depth of the decision tree of A.
We then apply the above results to learning the class of disjunctions of predicates from membership queries [Nader H. Bshouty et al., 2017]. We show that the ETD of this class is bounded from above by the degree d of its Hasse diagram. We then show that Moshkov algorithm can be run in polynomial time and is (d/log d)-approximation algorithm. This gives optimal algorithms when the degree is constant. For example, learning axis parallel rays over constant dimension space.
BibTeX - Entry
@InProceedings{hbshouty_et_al:LIPIcs:2018:9982,
author = {Nader H. Bshouty and Waseem Makhoul},
title = {{On Polynomial Time Constructions of Minimum Height Decision Tree}},
booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)},
pages = {34:1--34:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-094-1},
ISSN = {1868-8969},
year = {2018},
volume = {123},
editor = {Wen-Lian Hsu and Der-Tsai Lee and Chung-Shou Liao},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9982},
URN = {urn:nbn:de:0030-drops-99824},
doi = {10.4230/LIPIcs.ISAAC.2018.34},
annote = {Keywords: Decision Tree, Minimal Depth, Approximation algorithms}
}
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Keywords: |
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Decision Tree, Minimal Depth, Approximation algorithms |
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Collection: |
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29th International Symposium on Algorithms and Computation (ISAAC 2018) |
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Issue Date: |
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2018 |
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Date of publication: |
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06.12.2018 |
