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Novel characteristics of split trees by use of renewal theory
 Electron. J. Probab
, 2012
"... We investigate characteristics of random split trees introduced by Devroye [3]; split trees include for example binary search trees, mary search trees, quadtrees, median of (2k + 1)trees, simplex trees, tries and digital search trees. More precisely: We introduce the use of renewal theory in the s ..."
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We investigate characteristics of random split trees introduced by Devroye [3]; split trees include for example binary search trees, mary search trees, quadtrees, median of (2k + 1)trees, simplex trees, tries and digital search trees. More precisely: We introduce the use of renewal theory in the studies of split trees, and use this theory to prove several results about split trees. A split tree of cardinality n is constructed by distributing n “balls ” (which often represent “key numbers”) in a subset of vertices of an infinite tree. One of our main results is to give a relation between the deterministic number of balls n and the random number of vertices N. In [3] there is a central limit law for the depth of the last inserted ball so that most vertices are close to lnnµ +O lnn, where µ is some constant depending on the type of split tree; we sharpen this result by finding an upper bound for the expected number of vertices with depths ≥ lnnµ + ln0.5+ n or depths ≤ lnnµ +ln0.5+ n for any choice of > 0. We also find the first asymptotic of the variances of the depths of the balls in the tree.
A weakly 1stable limiting distribution for the number of random records and cuttings in split trees
"... We study the number of random records in an arbitrary split tree (or equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the dis ..."
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We study the number of random records in an arbitrary split tree (or equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the distribution of this number. After normalization the distributions are shown to be asymptotically weakly 1stable. This work is a generalization of our earlier results for the random binary search tree in [10], which is one specific case of split trees. Other important examples of split trees include mary search trees, quadtrees, medians of (2k+ 1)trees, simplex trees, tries and digital search trees.
Random records and cuttings in split trees. Extended abstract
 Discrete Math. Theor. Comput. Sci. (DMTCS) Proceedings Series Volume A1, 273–286. MR2508793
, 2008
"... We study the number of records in random split trees on n randomly labelled vertices. Equivalently the number of random cuttings required to eliminate an arbitrary random split tree can be studied. After normalization the distributions are shown to be asymptotically 1stable. This work is a generali ..."
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We study the number of records in random split trees on n randomly labelled vertices. Equivalently the number of random cuttings required to eliminate an arbitrary random split tree can be studied. After normalization the distributions are shown to be asymptotically 1stable. This work is a generalization of our earlier results for the random binary search tree which is one specific case of split trees. Other important examples of split trees include mary search trees, quadtrees, median of (2k + 1)trees, simplex trees, tries and digital search trees.
Contents
, 2007
"... i matematik som framläggs för offentlig granskning den 31 oktober, kl 10.15, sal 64119 ..."
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i matematik som framläggs för offentlig granskning den 31 oktober, kl 10.15, sal 64119
Random Records and Cuttings in Split Trees:
, 2015
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.