Improved behaviour of tries by adaptive branching
Information Processing Letters
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
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Algorithms on strings, trees, and sequences: computer science and computational biology
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SIGMETRICS '98/PERFORMANCE '98 Proceedings of the 1998 ACM SIGMETRICS joint international conference on Measurement and modeling of computer systems
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Theoretical Computer Science
Selected papers on analysis of algorithms
Selected papers on analysis of algorithms
Average Case Analysis of Algorithms on Sequences
Average Case Analysis of Algorithms on Sequences
Random Structures & Algorithms - Special issue on analysis of algorithms dedicated to Don Knuth on the occasion of his (100)8th birthday
Some results on tries with adaptive branching
Theoretical Computer Science
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BC '98 Proceedings of the IFIP TC6/WG6.2 Fourth International Conference on Broadband Communications: The future of telecommunications
On the number of full levels in tries
Random Structures & Algorithms
Probabilistic behavior of asymmetric level compressed tries
Random Structures & Algorithms
IP-address lookup using LC-tries
IEEE Journal on Selected Areas in Communications
Comparing integer data structures for 32- and 64-bit keys
Journal of Experimental Algorithmics (JEA)
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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 Fn(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 Fn(α) = kn or Fn(α) = kn + 1, where kn = log1/&sqrt;pq n − |ln (p/q)|/2 ln3/2 (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) Dn(α) in the α-LC tries, namely, we show that with high probability Dn(α) ∼ C2 log log n, where C2 = 1/|log(1 − h/log(1/&sqrt;pq))| for p ≠ q and h = −plog p−qlog q is the Shannon entropy rate. This should be compared with recently found typical depth in the original LC tries, which is C1log log n, where C1 = 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 Fn(α), and the search time in α-LC tries is of the same order as in the original LC tries.