Motivated by the goal of increasing the performance of FPGA-based designs, we propose new Steiner and arborescence FPGA routing algorithms. Our Steiner tree constructions significantly outperform the best known ones and have provably good performance bounds. Our arborescence heuristics produce routing solutions with optimal source–sink pathlengths, and with wirelength on par with the best exist...

We study the approximability of three versions of the Steiner tree problem. For the rst one where the input graph is only supposed connected, we show that it is not approximable within better than |V \N |−ǫ for any ǫ ∈ (0, 1), where V and N are the vertex-set of the input graph and the set of terminal vertices, respectively. For the second of the Steiner tree versions considered, the one where ...

The Quality of Service Steiner Tree Problem is a generalization of the Steiner problem which appears in the context of multimedia multicast and network design. In this generalization, each node possesses a rate and the cost of an edge with length l in a Steiner tree T connecting the non-zero rate nodes is l ·re, where re is the maximum rate in the component of T −{e} that does not contain the s...

An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error from 120◦. This notion arises in numerical approximations of minimum Steiner trees. We investigate theworst-case relative error of the length of an approximate Steiner tree compared to the shortest tree with the same topology. It ha...

Latency reduction in distributed interactive applications has been studied intensively. Such applications may have stringent latency requirements and dynamic user groups. We focus on application-layer multicast with a centralized approach to the group management. The groups are organized in overlay networks that are created using graph algorithms that create a tree structure for the group. A tr...

Given n terminals in the Euclidean plane and a positive constant, find a Steiner tree interconnecting all terminals with the minimum number of Steiner points such that the Euclidean length of each edge is no more than the given positive constant. This problem is NP-hard with applications in VLSI design, WDM optical networks and wireless communications. In this paper, we show that (a) the Steine...

We prove a limit theorem for the total Steiner k-distance of a random b-ary recursive tree with weighted edges. The total Steiner k-distance is the sum of all Steiner k-distances in a tree and it generalises the Wiener index. The limit theorem is obtained by using a limit theorem in the general setting of the contraction method. In order to use the contraction method we prove a recursion formul...

In this paper, we study the Steiner tree problem with minimum number of Steiner points and bounded edge-length (STPMSPBEL), which asks for a tree interconnecting a given set of n terminal points and a minimum number of Steiner points such that the Euclidean length of each edge is no more than a given positive constant. This problem has applications in VLSI design, WDM optimal networks and wirel...

Among other results, we prove the following theorem about Steiner minimal trees in ddimensional Euclidean space: if two finite sets in R have unique and combinatorially equivalent Steiner minimal trees, then there is a homotopy between the two sets that maintains the uniqueness and the combinatorial structure of the Steiner minimal tree throughout the homotopy.

The Steiner tree problem which is known to be NP-Complete is the following. Given a weighted undirected graph G = (V;E), and a set X V of terminals, the objective is to nd a tree of minimum cost which connects all the terminals. If the graph is directed, in addition to X, we are given a root r 2 V , and the objective is to nd a minimum cost arborescence which connects the root to each of the te...