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Partial fillup and search time in LC triesPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2007 (English)In: ACM Transactions on Algorithms, ISSN 1549-6325, E-ISSN 1549-6333, Vol. 3, no 4, 44:1-44:14 p.Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2007. Vol. 3, no 4, 44:1-44:14 p.
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:uu:diva-12746DOI: 10.1145/1290672.1290681OAI: oai:DiVA.org:uu-12746DiVA: diva2:40515
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Available from: 2008-01-11 Created: 2008-01-11 Last updated: 2010-05-17Bibliographically approved

Andersson and Nilsson introduced in 1993 a *level-compressed trie* (for short, LC trie) in which a full subtree of a node is compressed to a single node of degree being the size of the subtree. Recent experimental results indicated a “dramatic improvement” when full subtrees are replaced by “partially filled subtrees.” In this article, we provide a theoretical justification of these experimental results, showing, among others, a rather moderate improvement in search time over the original LC tries. For such an analysis, we assume that *n* strings are generated independently by a binary memoryless source, with *p* denoting the probability of emitting a “1” (and *q* = 1 − *p*). We first prove that the so-called α-fillup level *F*_{n}(α) (i.e., the largest level in a trie with α fraction of nodes present at this level) is concentrated on two values with high probability: either *F*_{n}(α) = *k*_{n} or *F*_{n}(α) = *k*_{n} + 1, where *k*_{n} = log_{1/&sqrt;pq} *n* − |ln (*p/q*)|/2 ln^{3/2} (1&sqrt;*pq*) Φ^{−1} (α) &sqrt; ln *n* + *O*(1) is an integer and Φ(*x*) denotes the normal distribution function. This result directly yields the typical depth (search time) *D*_{n}(α) in the α-LC tries, namely, we show that with high probability *D*_{n}(α) ∼ *C*_{2} log log *n*, where *C*_{2} = 1/|log(1 − *h*/log(1/&sqrt;*pq*))| for *p* ≠ *q* and *h* = −*p*log *p*−*q*log *q* is the Shannon entropy rate. This should be compared with recently found typical depth in the original LC tries, which is *C*_{1}log log *n*, where *C*_{1} = 1/|log(1−*h*/log(1/min{*p*, 1−*p*}))|. In conclusion, we observe that α affects only the lower term of the α-fillup level *F*_{n}(α), and the search time in α-LC tries is of the same order as in the original LC tries.

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