Analysis of digital tries with Markovian dependency

W. Szpankowski* Department of Computer Science Purdue University W. Lafayette. IN 47907 USA Digital tries occur in a variety of computer and communication algorithms including symbolic manipulations, compiling, comparison based searching and sorting, digital retrieval techniques, algorilhms on strings, file systems. codes and communication protocols. It is crucial for all of these applications to design and build robust models of the underlying digital trees. In this paper, we present a complete characterization of tries from the depth viewpoint in a Markovian framework. that is. under the assumption that symbols in a key are Markov-dependent. Our main findings show that asymptotically (Le.• for the number of keys n tending to infinity) the average depth EDlI 1/h l'log n + e', the variance var Dn a log n + e", where h I is the entropy of lhe (Markovian dependent) alphabet, and a is a parameter that becomes zero for symmetric independent model, that is, in this case var D" = 0 (1). We also derive limiting distribution for the depth Dno and in particular, we show that Dn tends to the normal distribution in all cases except the symmetric independent model. lbis finding implies that in general the m-th moment E {(Dn -EDn)r..JvarD" )"' --7 m !/[2",12 (m/2)!] for m even, and it tends to zero for m odd. These results extend all previous analyses since most of them have been limited to independent models. t This research is partially supported by NATO Collaborative granL 00571/89. * This research was priml1l)' done while the author was visiting INRIA in Rocquencourt, France. Support was provided in pan by NATO Collaborative granl 00571/89, in part by NSF granls NCR-8702115 and CCR-89OO305, and in plln by grant ROI LM05118 from the National Library of Medicine.