Optimal sorting of seven element sets

Kollár, Ľubor

doi:10.1007/bfb0016270

Cited by 5 publications

(4 citation statements)

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“…The canonical treatment is Knuth's (1998, Section 5.3.1). The main focus is to find exactly optimum sorting algorithm for small m, which are now known up to m = 10 (Knuth 1998;Césary 1968;Kollár 1986;AbouEisha, Chikalov, and Moshkov 2016). Finding optimal algorithms for other distributions and small m could be interesting.…”

Section: Our Approach and Resultsmentioning

confidence: 99%

Preference Elicitation as Average-Case Sorting

Peters

Procaccia

2021

AAAI

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Many decision making systems require users to indicate their preferences via a ranking. It is common to elicit such rankings through pairwise comparison queries. By using sorting algorithms, this can be achieved by asking at most O(m log m) adaptive comparison queries. However, in many cases we have some advance (probabilistic) information about the user's preferences, for instance if we have a learnt model of the user's preferences or if we expect the user's preferences to be correlated with those of previous users. For these cases, we design elicitation algorithms that ask fewer questions in expectation, by building on results for average-case sorting. If the user's preferences are drawn from a Mallows phi model, O(m) queries are enough; for a mixture of k Mallows models, log k + O(m) queries are enough; for Plackett-Luce models, the answer varies with the alternative weights. Our results match information-theoretic lower bounds. We also provide empirical evidence for the benefits of our approach.

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Section: Our Approach and Resultsmentioning

confidence: 99%

Preference Elicitation as Average-Case Sorting

Peters

Procaccia

2021

AAAI

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“…Césary [3] proved that, for n = 7 and n = 8, there are no decision trees for sorting n elements whose average depth is equal to ϕ(n)/n!. Kollár [7] found that the minimum average depth of a decision tree for sorting 7 elements is equal to 62416/7!. We find that the minimum average depth of a decision tree for sorting 8 elements is equal to 620160/8!.…”

Section: Introductionmentioning

confidence: 83%

“…Therefore, for n = 2, 3, 4, 5, 6, 9, 10, each decision tree for sorting n elements, which has minimum average depth, has also minimum depth. We extended this result to the cases n = 7 (Kollár in [7] did not consider this question) and n = 8. For n = 2, .…”

Section: Introductionmentioning

confidence: 94%

Decision trees with minimum average depth for sorting eight elements

AbouEisha

Chikalov

Moshkov

2016

Discrete Applied Mathematics

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We prove that the minimum average depth of a decision tree for sorting 8 pairwise different elements is equal to 620160/8!. We show also that each decision tree for sorting 8 elements, which has minimum average depth (the number of such trees is approximately equal to 8.548 × 10 326365 ), has also minimum depth. Both problems were considered by Knuth in [6]. To obtain these results, we use tools based on extensions of dynamic programming which allow us to make sequential optimization of decision trees relative to depth and average depth, and to count the number of decision trees with minimum average depth.

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“…The problem of sorting is usually defined in theoretical studies (see [ 25 , 26 ]) as sorting a series of n distinct elements

from a set that is linearly ordered. The collection {

} has only one permutation

where

.…”

Section: Applicationmentioning

confidence: 99%

Applications of Depth Minimization of Decision Trees Containing Hypotheses for Multiple-Value Decision Tables

Azad

Moshkov

2023

Entropy

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In this research, we consider decision trees that incorporate standard queries with one feature per query as well as hypotheses consisting of all features’ values. These decision trees are used to represent knowledge and are comparable to those investigated in exact learning, in which membership queries and equivalence queries are used. As an application, we look into the issue of creating decision trees for two cases: the sorting of a sequence that contains equal elements and multiple-value decision tables which are modified from UCI Machine Learning Repository. We contrast the efficiency of several forms of optimal (considering the parameter depth) decision trees with hypotheses for the aforementioned applications. We also investigate the efficiency of decision trees built by dynamic programming and by an entropy-based greedy method. We discovered that the greedy algorithm produces very similar results compared to the results of dynamic programming algorithms. Therefore, since the dynamic programming algorithms take a long time, we may readily apply the greedy algorithms.

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Optimal sorting of seven element sets

Cited by 5 publications

References 1 publication

Preference Elicitation as Average-Case Sorting

Preference Elicitation as Average-Case Sorting

Decision trees with minimum average depth for sorting eight elements

Applications of Depth Minimization of Decision Trees Containing Hypotheses for Multiple-Value Decision Tables

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