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[{"key": "dc.contributor.advisor", "value": "V\u00e4h\u00e4kangas, Antti", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Leirimaa, Elisa", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2020-06-01T12:27:21Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2020-06-01T12:27:21Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2020", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/69350", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa perehdyt\u00e4\u00e4n verkostoihin ja niihin m\u00e4\u00e4riteltyihin virtauksiin. Virtaus on funktio, joka liitt\u00e4\u00e4 jokaiseen verkoston suunnattuun sivuun kokonaislukuarvon ja toteuttaa tietyt ehdot. Ensinn\u00e4kin, jokaiselle suunnatulle sivulle tulee p\u00e4te\u00e4, ett\u00e4 vastakkaiselle suunnatulle sivulle virtauksen arvo on yht\u00e4 suuri mutta eri merkkinen. Toiseksi jokaisesta k\u00e4rjest\u00e4 l\u00e4hde ja nielu poislukien on l\u00e4hdett\u00e4v\u00e4 virtausta yht\u00e4 paljon kuin siihen on saapunut. L\u00e4hde on se k\u00e4rki, josta virtaus l\u00e4htee liikkeelle, ja nielu on k\u00e4rki, johon virtaus lopulta p\u00e4\u00e4tyy. Kolmanneksi virtauksen arvon on oltava jokaisessa suunnatussa sivussa korkeintaan yht\u00e4 suuri kuin vastaava kapasiteettifunktion arvo. Kapasiteettifunktio liitt\u00e4\u00e4 jokaiseen verkoston suunnattuun sivuun kokonaislukuarvon, joka kuvaa kunkin suunnatun sivun suurinta mahdollista virtauksen arvoa. Tutkielmassa osoitetaan lause, joka kertoo, ett\u00e4 verkoston l\u00e4pi kulkevan suurimman mahdollisen virtauksen arvo on sama kuin pienimm\u00e4n leikkauksen kapasiteetin arvo. T\u00e4t\u00e4 Fordin ja Fulkersonin kehitt\u00e4m\u00e4\u00e4 lausetta kutsutaan suurin virtaus - pienin leikkaus -lauseeksi. T\u00e4t\u00e4 lausetta hy\u00f6dynnet\u00e4\u00e4n l\u00e4pi koko tutkielman, ja sen avulla osoitetaan keskeisi\u00e4 verkkoteorian tuloksia, kuten K\u00f6nigin lause, Hallin lause ja Mengerin lause. K\u00f6nigin lauseen mukaan verkon maksimaalisen sovituksen suuruus on sama kuin pienimm\u00e4n peitteen suuruus. Hallin lause antaa v\u00e4ltt\u00e4m\u00e4tt\u00f6m\u00e4n ja toisaalta riitt\u00e4v\u00e4n ehdon sille, ett\u00e4 kaksiosaiselle verkolle l\u00f6ytyy t\u00e4ydellinen sovitus. Mengerin lause puolestaan antaa verkoston suurimman mahdollisen lukum\u00e4\u00e4r\u00e4n erillisi\u00e4 polkuja. N\u00e4m\u00e4 tulokset saadaan osoitettua konstruoimalla verkosta verkosto ja l\u00e4hett\u00e4m\u00e4ll\u00e4 virtausta l\u00e4hteest\u00e4 verkoston l\u00e4pi. N\u00e4in voidaan hy\u00f6dynt\u00e4\u00e4 virtauksille ja verkostoille osoitettuja tuloksia.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2020-06-01T12:27:21Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2020-06-01T12:27:21Z (GMT). No. of bitstreams: 0\n Previous issue date: 2020", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "41", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "l\u00e4hde", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Virtaukset ja niiden sovelluksia", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202006013607", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "verkkoteoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "virtaus", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "verkostot", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "sovitukset", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "nielu", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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