A linear-time algorithm to construct a rectilinear Steiner minimal tree fork-extremal point sets

Abstract

Abstract. A k-extremal point set is a point set on the boundary of a k-sided rectilinear convex hull. Given a k-extremal point set of size n, we present an algorithm that computes a rectilinear Steiner minimal tree in time O(k~n). For constant k, this algorithm runs in O(n) time and is asymptotically optimal and, for arbitrary k, the algorithm is the fastest known for this problem.

Optimal Interconnection Trees in the Plane

“…If we return to the case of finding a minimum rectilinear Steiner tree T for a set of n terminals on the boundary of a given rectilinearly convex polygon R k with k sides, then stronger results are possible, particularly if k is significantly smaller than n. An algorithm for solving this problem with time complexity O.nk 4 / was presented by Richards and Salowe in [323]. The key to developing such an algorithm is to show that there are only a limited number of possible locations for any full component of T containing at least one Steiner point.…”

“…3.31) and hence decomposes the problem of constructing T into four subproblems. Richards and Salowe [323] show that each of these subproblems can be solved in time O.k 4 C n/, using a [91,92] have shown that the running time can be further improved to O.k 2 n/ by refining the dynamic programming method used for the subproblems. We note that Cheng [89] has also developed a similar algorithm for terminals lying on the boundary of a non-convex rectilinear polygon, showing that a minimum rectilinear Steiner tree T can be constructed in time O.k 3 n/ as long as T lies inside the polygon.…”

Planar Manhattan Local Minimal and Critical Networks

“…But the set of local minimal networks in that case is rather more complicated, see [17]. In the case of the shortest Manhattan networks with so-called rectilinear-convex boundaries, a polynomial algorithm constructing such networks is known, see, for example, the work of Richard and Salowe [30].…”

A fast algorithm for computing optimal rectilinear Steiner trees for extremal point sets

“…Thus, homotopic routing requires the solution to a special MRST problem where the terminal points lie on the boundary of a rectilinear polygon. It is commonly assumed that the input speci es the polygon and the terminal points in cyclic order on the boundary of this polygon 4,12]. This paper describes an e cient algorithm when the terminal points lie on their rectilinear convex hull.…”

“…(P cannot contain a right bend otherwise, there cannot be any i n terior edge properly incident t ò as P has to terminate at an exterior vertex.) Thus, T 12 Case (2): There are three possible topologies: (2.1) b t has degree one, (2.2) b t has degree more than one and b t is not incident to a tree edge perpendicular to f, (2.3) b t is incident t o a t r e e e d g e g perpendicular to f. W e compute two trees T 21 and T 22 for the rst two cases and return the minimal one as T 2 . W e argue that case (2.3) has been handled before.…”