Konstrukcije dvoravnina pomoću kombinatoričkih algoritama

scientific master's theses

• Title (english): Biplane constructions by means of combinatorial algorithms
• Author: Ivica Martinjak
• Mentor: Mario-Osvin Pavčević (mentor)
• Committee member: Marin Golub (predsjednik povjerenstva)
• Committee member: Mario-Osvin Pavčević (član povjerenstva)
• Committee member: Juraj Šiftar (član povjerenstva)
• Granter: University of Zagreb; Faculty of Electrical Engineering and Computing; (Department of Applied Mathematics); Zagreb
• Defense date and country: 2007-01-09, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics; Discrete and Combinatorial Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

Dvoravnine su simetrični dizajni s parametrima 2 - (v, k,2) . Do sada je poznato sedamnaest dvoravnina, sa sedam različitih trojki parametara. Pri tome, dvoravnine s 56 i 79 točaka nisu klasificirane, već su poznati samo pojedini primjeri. Dvoravnine su strukture visoke pravilnosti čija veličina prostora rješenja izrazito raste s brojem točaka (kombinatorička eksplozija). Dodatna poteškoća pri konstrukciji ovih struktura je vrlo mali broj struktura (rezultata) u prostoru pretraživanja. Determinističkim algoritmom (iscrpnom pretragom) klasificirane su dvoravnine redova 2, 3, 4 i 7. Pretragom uz dodatno pretpostavljeni uvjet (jedinice na glavnoj dijagonali incidencijske matrice) konstruirane su dvije dvoravnine 2 - (56,11,2) . Prilikom pretrage uz taj uvjet, kod 2 - (37,9,2) uočena je jednoznačna odrenenost broja incidencijskih matrica brojem odgovarajućih vektora iz (k + 2) -og retka. Analogna se pravilnost javlja i na ispitanom uzorku 2 - (56,11,2) . U eksperimentima s heurističkim algoritmima, pokazalo se da konstrukcija pomoću Hussainovih grafova ima manji prostor rješenja nego konstrukcija pomoću matrica incidencije, uz odrenene pretpostavke. Najveći utjecaj na doseg imalo je matematičko pojednostavljenje problema, postavljanje dodatnog uvjeta na strukturu te filtriranje vektora.

Abstract (english)

A biplane is a symmetric 2 − (v, k,2) design. Until now, seventeen biplanes with seven different triple of parameters are known. Out of which, biplanes with 56 and 79 points are not classified but only a few individual examples are known. Biplanes are highly regular structures, the space solution size of which extremely increases with number of points (combinatorial explosion). Additional difficulty in construction of this structure is a very small number of structures (results) in space solution. Biplanes of order 2, 3, 4 and 7 are classified by deterministic algorithm (exhaustive search). By searching using additional presumptive condition (each element on the main diagonal of incidence matrix is 1) two biplanes 2 − (56,11,2) are constructed. By searching using the same condition in the case of 2 − (37,9,2) it is observed that a number of incidence matrix is unambiguously determined by number of appropriate vectors of (k + 2) nd row. An analogous regularity appears on testing sample 2 − (56,11,2) . Experiments with heuristic algorithms showed that construction according to Hussain graphs has smaller space solution than construction with incidence matrix, on certain assumptions. Mathematical simplification of problem, using additional condition on structure and vector filtering had the main impact on achievement.

• Keywords: incidencijska struktura; kombinatorički dizajn; simetrični dizajn; dvoravnina; grupa automorfizama; iscrpna pretraga; heuristički algoritam; genetski algoritam; penjanje uzbrdo; složenost algoritma
• Keywords (english): incidence structure; block design; symmetric design; biplane; automorphism group; exhaustive search; heuristic algorithm; genetic algorithm; hill climbing; complexity of algorithm
• Language: croatian
• URN:NBN: urn:nbn:hr:168:319837
• Promotion: 2007
• Study programme: Title: Postgraduate master programe in computing; Study programme type: university; Study level: postgraduate; Academic / professional title: magistar inženjer računarstva (magistar inženjer računarstva)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Closed access
• Created on: 2020-01-14 12:41:29