A probabilistic analysis of the height of tries and of the complexity of triesort

Summary

We consider binary tries formed by using the binary fractional expansions of X ₁, ...,X _n, a sequence of independent random variables with common density f on [0,1]. For H _n, the height of the trie, we show that either E(H_n)∼21og₂ n or E(H_n)=∞ for all n≧2 according to whether ∫f ²(x)dx is finite or infinite. Thus, the average height is asymptotically twice the average depth (which is ∼log₂ n when ∫f ²(x)dx>∞). The asymptotic distribution of H _n is derived as well.

If f is square integrable, then the average number of bit comparisons in triesort is nlog₂ n+0(n), and the average number of nodes in the trie is 0(n).