Parallel Tree-Contraction and Fibonacci Numbers

We show a new property of Fibonacci numbers which is related to the analysis of a very simple and natural parallel tree contraction algorithm. We show that the size of the smallest tree which requires t contractions equals exactly the t-th Fibonacci number. This implies the sharp bound on the number of iterations of the tree contraction algorithm. We contribute also to combinatorics of trees. Parallel evaluation of expressions and (related to it) tree contraction are fundamental techniques in the design of parallel algorithms. In this paper we consider a tree contraction algorithm related to the simultaneous substitutions algorithm of parallel evaluation of expressions given by straight-line programs, as given by Rytter in Ry-90]. In this case the tree is the computation graph of the straight-line program, the parallel evaluation of the expression corresponds to the operations of compressing chains and (simulateneoulsy) removing edges leading to the leaves. We introduce a family of Fibonacci-type trees F k , which are the hardest trees for the algorithm. The importance of the algorithms analyzed here follows also from their relation to the parallel evaluation of straight line programs and recognition of context-free languages, see GR-88], see Ry-90] and GR-88]. We believe that the exact analysis of the tree contraction algorithms is an interesting problem related to combinatorics of trees. Our model of parallel computation is a CREW Parallel Random Access Machine (CREW PRAM), see for example GR-88]. The PRAM consists of a number of processors working synchronously and using a common global memory. Throughout the paper log c n is to be understood as dlog c ne, and w=2 as bw=2c.