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Novel characteristics of split trees by use of renewal theory


 
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1. Title Title of document Novel characteristics of split trees by use of renewal theory
 
2. Creator Author's name, affiliation, country Cecilia Ingrid Holmgren; Cambridge University; United Kingdom
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Random Trees, Split Trees, Renewal Theory
 
3. Subject Subject classification 05C05; 05C80; 68W40; 68P10;68R10; 60C05; 68P05
 
4. Description Abstract We investigate characteristics of random split trees introduced by Devroye [SIAM J Comput 28, 409-432, 1998]; split trees include e.g., binary search trees, $m$-ary search trees, quadtrees, median of $(2k+1)$-trees, simplex trees, tries and digital search trees. More precisely: We use 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 data) to a subset of nodes of an infinite tree. One of our main results is a relation between the deterministic number of balls n and the random number of nodes N. In Devroye [SIAM J Comput 28, 409-432, 1998] there is a central limit law for the depth of the last inserted ball so that most nodes are close to depth $\ln n/\mu+O(\ln n)^{1/2})$, where $\mu$ is some constant depending on the type of split tree; we sharpen this result by finding an upper bound for the expected number of nodes with depths $\geq \mu^{-1}\ln n-(\ln n)^{1/2+\epsilon}$ or depths $\leq\mu^{-1}\ln n+(\ln n)^{1/2+\epsilon}$ for any choice of $\epsilon>0$. We also find the first asymptotic of the variances of the depths of the balls in the tree.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Uppsala University
 
7. Date (YYYY-MM-DD) 2012-01-16
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/1723
 
10. Identifier Digital Object Identifier 10.1214/EJP.v17-1723
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 17
 
12. Language English=en en
 
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