On paths in search or decision trees which require almost worst-case time

Abstract

We prove the existence of a particular path p in a search or decision tree; this path will symbolize a computation requiring almost worst-case time. The result about the existence of p is the following: Given a finite tree T with the root r and the set B of leaves. Let every b ε B be attached by a weight w(b)≥0 and let w(T) be the sum ε w(b). Then there exists a path p=(v₀, ..., v_ℓ8467;) from the root r=v₀ to a leaf v _ℓ8467; ε B such that g ⁺(v₀) ..... g⁺(v_ℓ8467-1). w(v_ℓ8467;)≥w(T) (where g⁺(v_λ)=out-degree of v_λ). We shall use this lemma to obtain the following complexity theoretical results:

• 1) Searching in sorted multi-way trees requires Ω(log(n)) time.

• 2) Let finding a rotational minimum among j points take θ(j) time units where T belongs to a particular class of sublogarithmic functions. Then the worst-case complexity of the Plane Convex Hull Problem is in Ω(nT(n)).

• 3) The worst-case complexity of the Convex Hull Problem is in Ω(nlogT(n)) if the following operations altogether take θ(T(n)) time units: Taking an oriented straight line $\vec G$ and deciding which of the n input points are on the right of it.

• 4) The worst case complexity of the Sorting Problem is also in Ω(nlogT(n)) if the following operations can be executed within θ(T(n)) time units: Taking a real t and deciding which of the n input reals are smaller than t.

In the Applications 2) – 4) we shall realize that n·T(n), nlog(T(n)) resp. is even a tight bound.