1,809 research outputs found
Subexponential Algorithms for Rectilinear Steiner Tree and Arborescence Problems
A rectilinear Steiner tree for a set T of points in the plane is a tree which connects T using horizontal and vertical lines. In the Rectilinear Steiner Tree problem, input is a set T of n points in the Euclidean plane (R^2) and the goal is to find an rectilinear Steiner tree for T of smallest possible total length. A rectilinear Steiner arborecence for a set T of points and root r in T is a rectilinear Steiner tree S for T such that the path in S from r to any point t in T is a shortest path. In the Rectilinear Steiner Arborescense problem the input is a set T of n points in R^2, and a root r in T, the task is to find an rectilinear Steiner arborescence for T, rooted at r of smallest possible total length. In this paper, we give the first subexponential time algorithms for both problems. Our algorithms are deterministic and run in 2^{O(sqrt{n}log n)} time
Two-Level Rectilinear Steiner Trees
Given a set of terminals in the plane and a partition of into
subsets , a two-level rectilinear Steiner tree consists of a
rectilinear Steiner tree connecting the terminals in each set
() and a top-level tree connecting the trees . The goal is to minimize the total length of all trees. This problem
arises naturally in the design of low-power physical implementations of parity
functions on a computer chip.
For bounded we present a polynomial time approximation scheme (PTAS) that
is based on Arora's PTAS for rectilinear Steiner trees after lifting each
partition into an extra dimension. For the general case we propose an algorithm
that predetermines a connection point for each and
().
Then, we apply any approximation algorithm for minimum rectilinear Steiner
trees in the plane to compute each and independently.
This gives us a -factor approximation with a running time of
suitable for fast practical computations. The
approximation factor reduces to by applying Arora's approximation scheme
in the plane
Further improvements of Steiner tree approximations
The Steiner tree problem requires to find a shortest tree connecting a given set of terminal points in a metric space. We suggest a better and fast heuristic for the Steiner problem in graphs and in rectilinear plane. This heuristic finds a Steiner tree at most 1.757 and 1.267 times longer than the optimal solution in graphs and rectilinear plane, respectively
Rectilinear Steiner Trees in Narrow Strips
A rectilinear Steiner tree for a set of points in is a
tree that connects the points in using horizontal and vertical line
segments. The goal of Minimal Rectilinear Steiner Tree is to find a rectilinear
Steiner tree with minimal total length. We investigate how the complexity of
Minimal Rectilinear Steiner Tree for point sets inside the strip
depends on the strip width . We
obtain two main results. 1) We present an algorithm with running time
for sparse point sets, that is, point sets where each
rectangle inside the strip contains points. 2) For
random point sets, where the points are chosen randomly inside a rectangle of
height and expected width , we present an algorithm that is
fixed-parameter tractable with respect to and linear in . It has an
expected running time of .Comment: 21 pages, 13 figure
Subexponential Algorithms for Rectilinear Steiner Tree and Arborescence Problems
A rectilinear Steiner tree for a set K of points in the plane is a tree that connects k using horizontal and vertical lines. In the Rectilinear Steiner Tree problem, the input is a set K={z1,z2,…, zn} of n points in the Euclidean plane (R2), and the goal is to find a rectilinear Steiner tree for k of smallest possible total length. A rectilinear Steiner arborescence for a set k of points and a root r ∈ K is a rectilinear Steiner tree T for K such that the path in T from r to any point z ∈ K is a shortest path. In the Rectilinear Steiner Arborescence problem, the input is a set K of n points in R2, and a root r ∈ K, and the task is to find a rectilinear Steiner arborescence for K, rooted at r of smallest possible total length. In this article, we design deterministic algorithms for these problems that run in 2O(√ nlog n) time
Non-Crossing Geometric Steiner Arborescences
Motivated by the question of simultaneous embedding of several flow maps, we consider the problem of drawing multiple geometric Steiner arborescences with no crossings in the rectilinear and in the angle-restricted setting. When terminal-to-root paths are allowed to turn freely, we show that two rectilinear Steiner arborescences have a non-crossing drawing if neither tree necessarily completely disconnects the other tree and if the roots of both trees are "free". If the roots are not free, then we can reduce the decision problem to 2SAT. If terminal-to-root paths are allowed to turn only at Steiner points, then it is NP-hard to decide whether multiple rectilinear Steiner arborescences have a non-crossing drawing. The setting of angle-restricted Steiner arborescences is more subtle than the rectilinear case. Our NP-hardness result extends, but testing whether there exists a non-crossing drawing if the roots of both trees are free requires additional conditions to be fulfilled
Exact algorithms for the Steiner tree problem
In this thesis, the exact algorithms for the Steiner tree problem have been investigated. The Dreyfus-Wagner algorithm is a well-known dynamic programming method for computing minimum Steiner trees in general weighted graphs in time O(3k), where k is the number of the terminals. Firstly, two exact algorithms for the Steiner tree problem will be presented. The first one improves the running time of algorithm to O(2.684k) by showing that the optimum Steiner tree T can be partitioned into T = T1 [ T2 [ T3 in a certain way such that each Ti is a minimum Steiner tree in a suitable contracted graph Gi with less than k 2 terminals. The second algorithm is in time O((2 + )k) for any > 0. Every rectilinear Steiner tree problem admits an optimal tree T which is composed of tree stars. Moreover, the currently fastest algorithm for the rectilinear Steiner tree problem proceeds by composing an optimum tree T from tree star components in the cheapest way. F¨oßmeier and Kaufmann showed that any problem instance with k terminals has a number of tree stars in between 1.32k and 1.38k. We also present additional conditions on tree stars which allow us to further reduce the number of candidate components building the optimum Steiner tree to O(1.337k)
Rectilinear Steiner Tree Construction
The Minimum Rectilinear Steiner Tree (MRST) problem is to find the minimal spanning tree of a set of points (also called terminals) in the plane that interconnects all the terminals and some extra points (called Steiner points) introduced by intermediate junctions, and in which edge lengths are measured in the L1 (Manhattan) metric. This is one of the oldest optimization problems in mathematics that has been extensively studied and has been proven to be NP-complete, thus efficient approximation heuristics are more applicable than exact algorithms.
In this thesis, we present a new heuristic to construct rectilinear Steiner trees (RSTs) with a close approximation of minimum length in Ο(n log n) time. To this end, we recursively divide a plane into a set of sub-planes of which optimal rectilinear Steiner trees (optRSTs) can be generated by a proposed exact algorithm called Const_optRST. By connecting all the optRSTs of the sub-planes, a sub-optimal MRST is eventually constructed.
We show experimentally that for topologies with up to 100 terminals, the heuristic is 1.06 to 3.45 times faster than RMST, which is an efficient algorithm based on Prim’s method, with accuracy improvements varying from 1.31 % to 10.21 %
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