| Summary: | We define a Euclidean tree T = $(V,E)$ to be a weighted tree such that each edge $e in E$ has been assigned a positive integer weight l(e) called the length of e. A placement of T = $(V,E)$ is an assignment $f: V to { bf R} sp2$ (or R$ sp3$) of vertices to points in space satisfying the following: for each edge $e = (u,v) in E$, the Euclidean distance between f(u) and f(v) is equal to l(e). We define the EUCLIDEAN TREE PLACEMENT problem as follows. Given a Euclidean tree T = (V,E) and a subset of m vertices in V labeled $u sb1,u sb2, cdots,u sb{m}$, and given m nonempty sets $S sb1,S sb2, cdots,S sb{m}$, not necessarily disjoint, in R$ sp2$ (or R$ sp3$), find the feasible placements $f: V to { bf R} sp2$ (or R$ sp3$) such that for $1 le i le m$, $f(u sb{i}) in S sb{i}$, or else report no such placement exists. === In this thesis, we study the EUCLIDEAN TREE PLACEMENT problem and give both lower and upper bounds for it. Specifically, we first examine two special decision versions of the problem and show their NP-hardness by reductions from PARTITION. Then we provide a geometric method for placing Euclidean trees in the plane with each of the leaves constrained to a point. We also discuss an algebraic method for doing this. The given constraints on the leaves implicitly impose constraints on all vertices in V. Thus each vertex is associated with a point set (not necessarily connected) in which it can be placed in some feasible placement of the tree. Each such point set is shown to have a finite number of circular arcs on its boundary. Let r be the maximum boundary arc number over all such point sets for a Euclidean tree having at most n vertices. Then, for an input tree with n vertices, the geometric method takes $O(nr sp2 log r)$ time (real RAM), and the algebraic method takes $O(2 sp{4n sp2})$ time (rational RAM). At present r is not known to be bounded by a polynomial in n. Nevertheless, the geometric algorithm seems much more practical to implement than the algebraic algorithm. === In developing the geometric method, we design an $O(m log sp2m)$ algorithm for computing the Minkowski sum of a planar figure, possibly containing holes and bounded by a collection of m circular arcs of various radii, with a circle centered at the origin. This algorithm can be directly used for computing the offset of the figure.
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