Exact analysis of three tree contraction algorithms

Abstract

We analyze the exact numbers of iterations in three basic tree contraction algorithms. These numbers are bounded by c log₂(n) + c′, for some constants c, c′. We show that the best constant coefficient at log₂(n) for the two tree contraction algorithms given in [10, 11] is 1/log₂φ, where φ is the golden ratio. φ ≈ 1.618 and 1/log₂φ ≈ 1.44. For the rake-compress algorithm of Miller and Reif the best coefficient is shown to be 1/log₂λ, where λ is a real solution of the equation λ³= λ + 1. λ ≈ 1.32 and 1/log₂λ ≈ 2.46. Consequently, the algorithms from [10, 11] make about twice less iterations compared with the Miller and Reif algorithm. Although all three algorithms use similar operations their behaviours are different and they have to be analyzed separately. The proof of the lower bound for c is rather simple, however the proof of the upper bound (matching the lower bound) is more involved. It required a big number of computer experiments to guess some useful properties of the solutions of complicated recurrence equations (similar to dynamic programming recurrences). Several types of Fibonacci-like trees (T_k, T ^*_k and P_k) play an important role in the analysis. A contraction of directed acyclic graphs is analyzed similarly as tree-contraction. We contribute here also to the combinatorics of trees.