The suffix tree — the compacted trie of all the suffixes of a string — is the most important and widely-used data structure in string processing. We consider a natural combinatorial question about suffix trees: for a string S of length n, how many nodes νS(d) can there be at (string) depth d in its suffix tree? We prove ν(n, d) = maxS∈Σn νS(d) is O((n/d) logn), and show that this bound is almos...

A 1984 problem of S. Z. Ditor asks whether there exists a lattice of cardinality א2, with zero, in which every principal ideal is finite and every element has at most three lower covers. We prove that the existence of such a lattice follows from either one of two axioms that are known to be independent of ZFC, namely (1) Martin’s Axiom restricted to collections of א1 dense subsets in posets of ...

 In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)setminus A(L)| leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices ha...

We consider the trade-offs in the choice between depth (a narrow high quality position) and breadth (a wide low quality range). In particular, the extent of depth or breadth in a market can be non-monotonic in the strength of competition.

Despite the potential for rapid evolution, stasis is commonly observed over geological timescales-the so-called "paradox of stasis." This paradox would be resolved if stabilizing selection were common, but stabilizing selection is infrequently detected in natural populations. We hypothesize a simple solution to this apparent disconnect: stabilizing selection is hard to detect empirically once p...

Recent work shows that the memory requirements of bestfirst heuristic search can be reduced substantially by using a divide-and-conquer method of solution reconstruction. We show that memory requirements can be reduced even further by using a breadth-first instead of a best-first search strategy. We describe optimal and approximate breadth-first heuristic search algorithms that use divide-and-c...