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, ..., vℓ8467;) from the root r=v0 to a leaf v ℓ8467; ε B such that g +(v0) ..... g+(vℓ8467-1). w(vℓ8467;)≥w(T) (where g+(vλ)=out-degree of vλ). We shall use this lemma to obtain the following complexity theoretical results:
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1)
Searching in sorted multi-way trees requires Ω(log(n)) time.
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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)).
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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.
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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.
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© 1989 Springer-Verlag Berlin Heidelberg
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Huckenbeck, U. (1989). On paths in search or decision trees which require almost worst-case time. In: van Leeuwen, J. (eds) Graph-Theoretic Concepts in Computer Science. WG 1988. Lecture Notes in Computer Science, vol 344. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-50728-0_59
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DOI: https://doi.org/10.1007/3-540-50728-0_59
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