{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article12923","name":"Successive Minimum Spanning Trees","abstract":"In a complete graph K_n with edge weights drawn independently from a uniform distribution U(0,1) (or alternatively an exponential distribution Exp(1)), let T_1 be the MST (the spanning tree of minimum weight) and let T_k be the MST after deletion of the edges of all previous trees T_i, i gamma_k. Thinking of an edge of weight w as arriving at time t=n w, Kruskal\u2019s algorithm defines forests F_k(t), each initially empty and eventually equal to T_k, with each arriving edge added to the first F_k(t) where it does not create a cycle. Using tools of inhomogeneous random graphs we obtain structural results including that C_1(F_k(t))\/n, the fraction of vertices in the largest component of F_k(t), converges in probability to a function rho_k(t), uniformly for all t, and that a giant component appears in F_k(t) at a time t=sigma_k. We conjecture that the functions rho_k tend to time translations of a single function, rho_k(2k+x) -> rho_infty(x) as k -> infty, uniformly in x in R.\r\nSimulations and numerical computations give estimated values of gamma_k for small k, and support the conjectures stated above.","keywords":["miminum spanning tree","second-cheapest structure","inhomogeneous random graph","optimization in random structures","discrete probability","multi-type branching process","functional fixed point","robust optimization","Kruskal\u2019s algorithm"],"author":[{"@type":"Person","name":"Janson, Svante","givenName":"Svante","familyName":"Janson","email":"mailto:svante.janson@math.uu.se","sameAs":"https:\/\/orcid.org\/0000-0002-9680-2790","affiliation":"Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden"},{"@type":"Person","name":"Sorkin, Gregory B.","givenName":"Gregory B.","familyName":"Sorkin","email":"mailto:g.b.sorkin@lse.ac.uk","sameAs":"https:\/\/orcid.org\/0000-0003-4935-7820","affiliation":"Department of Mathematics, The London School of Economics and Political Science, Houghton Street, London WC2A 2AE, England"}],"position":60,"pageStart":"60:1","pageEnd":"60:16","dateCreated":"2019-09-17","datePublished":"2019-09-17","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Janson, Svante","givenName":"Svante","familyName":"Janson","email":"mailto:svante.janson@math.uu.se","sameAs":"https:\/\/orcid.org\/0000-0002-9680-2790","affiliation":"Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden"},{"@type":"Person","name":"Sorkin, Gregory B.","givenName":"Gregory B.","familyName":"Sorkin","email":"mailto:g.b.sorkin@lse.ac.uk","sameAs":"https:\/\/orcid.org\/0000-0003-4935-7820","affiliation":"Department of Mathematics, The London School of Economics and Political Science, Houghton Street, London WC2A 2AE, England"}],"copyrightYear":"2019","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.APPROX-RANDOM.2019.60","funding":"The work was partly supported by the Knut and Alice Wallenberg Foundation.","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/arxiv.org\/abs\/1310.1777","http:\/\/arxiv.org\/abs\/1310.1777"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6348","volumeNumber":145,"name":"Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX\/RANDOM 2019)","dateCreated":"2019-09-17","datePublished":"2019-09-17","editor":[{"@type":"Person","name":"Achlioptas, Dimitris","givenName":"Dimitris","familyName":"Achlioptas","email":"mailto:optas@soe.ucsc.edu","affiliation":"UC Santa Cruz, California, USA"},{"@type":"Person","name":"V\u00e9gh, L\u00e1szl\u00f3 A.","givenName":"L\u00e1szl\u00f3 A.","familyName":"V\u00e9gh","email":"mailto:L.Vegh@lse.ac.uk","affiliation":"London School of Economics and Political Science, London, UK"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article12923","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6348"}}}