A note on the MST heuristic for bounded edge-length Steiner Trees with minimum number of Steiner Points

We give a tight analysis of the MST heuristic recently introduced by G.-H. Lin and G. Xue for approximating the Steiner tree with minimum number of Steiner points and bounded edge-lengths. The approximation factor of the heuristic is shown to be one less than the MST number of the underlying space, defined as the maximum possible degree of a minimum-degree MST spanning points from the space. In particular, on instances drawn from the Euclidean (resp. rectilinear) plane, the MST heuristic is shown to have tight approximation factors of 4, respectively 3. Keywords: Approximation algorithms, Steiner trees, MST heuristic, fixed wireless network design, VLSI CAD. 1 Introduction The classical Steiner tree problem is that of finding a shortest tree spanning a given set of terminal points. The tree may use additional points besides the terminals, these points are commonly referred to as Steiner points. In the Minimum number of Steiner Points Tree (MSPT) problem [7,5] one also seeks a tree ..

Highly scalable algorithms for rectilinear and octilinear Steiner trees

problem, which asks for a minimum-length interconnection of a given set of terminals in the rectilinear plane, is one of the fundamental problems in electronic design automation. Recently there has been renewed interest in this problem due to the need for highly scalable algorithms able to handle nets with tens of thousands of terminals. In this paper we give a practical � heuristic for computing near-optimal rectilinear Steiner trees based on a batched version of the greedy triple contraction algorithm of Zelikovsky [21]. Experiments conducted on both random and industry testcases show that our heuristic matches or exceeds the quality of best known RSMT heuristics, e.g., on random instances with more than 100 terminals our heuristic improves over the rectilinear minimum spanning tree by an average of 11%. Moreover, our heuristic has very well scaling runtime, e.g., it can route a 34k-terminals net extracted from a real design in less than 25 seconds compared to over 86 minutes needed by the edge-based heuristic of Borah, Owens, and Irwin [3]. Since our heuristic is graph-based, it can be easily modified to handle practical considerations such as routing obstacles, preferred directions, via costs, and octilinear routing – indeed, experimental results show only a small factor increase in runtime when switching from rectilinear to octilinear routing. I

Highly Scalable Algorithms for Rectilinear and Octilinear Steiner Trees

... problem, which asks for a minimum-length interconnection of a given set of terminals in the rectilinear plane, is one of the fundamental problems in electronic design automation. Recently there has been renewed interest in this problem due to the need for highly scalable algorithms able to handle nets with tens of thousands of terminals. In this paper we give a practical O(n log² n) heuristic for computing near-optimal rectilinear Steiner trees based on a batched version of the greedy triple contraction algorithm of Zelikovsky [21]. Experiments conducted on both random and industry testcases show that our heuristic matches or exceeds the quality of best known RSMT heuristics, e.g., on random instances with more than 100 terminals our heuristic improves over the rectilinear minimum spanning tree by an average of 11%. Moreover, our heuristic has very well scaling runtime, e.g., it can route a 34k-terminals net extracted from a real design in less than 25 seconds compared to over 86 minutes needed by the O(n²) edge-based heuristic of Borah, Owens, and Irwin [3]. Since our heuristic is graph-based, it can be easily modified to handle practical considerations such as routing obstacles, preferred directions, via costs, and octilinear routing – indeed, experimental results show only a small factor increase in runtime when switching from rectilinear to octilinear routing

Selecting Forwarding Neighbors in Wireless Ad Hoc Networks

Broadcasting is a fundamental operation which is frequent in wireless ad hoc networks. A simple broadcasting mechanism, known as flooding, is to let every node retransmit the message to all its 1-hop neighbors when receiving the first copy of the message. Despite its simplicity, flooding is very inefficient and can result in high redundancy, contention, and collision. One approach to reducing the redundancy is to let each node forward the message only to a small subset of 1-hop neighbors that cover all of the node's 2-hop neighbors. In this paper, we propose two practical heuristics for selecting the minimum number of forwarding neighbors: an O(n log n) time algorithm that selects at most 6 times more forwarding neighbors than the optimum, and an O(n log² n) time algorithm with an improved approximation ratio of 3, where n is the number of 1- and 2-hop neighbors. The best previously known algorithm, due to Bronnimann and Goodrich [2], guarantees O(1) approximation in O(n³ log n) time

Selecting Forwarding Neighbors in Wireless Ad Hoc Networks

Broadcasting is a fundamental operation which is frequent in wireless ad hoc networks. A simple broadcasting mechanism, known as flooding, is to let every node retransmit the message to all its 1-hop neighbors when receiving the first copy of the message. Despite its simplicity, flooding is very inefficient and can result in high redundancy, contention, and collision. One approach to reducing the redundancy is to let each node forward the message only to a small subset of 1-hop neighbors that cover all of the node’s 2-hop neighbors. In this paper we propose two practical heuristics for selecting the minimum number of forwarding neighbors: an O(nlog n) time algorithm that selects at most 6 times more forwarding neighbors than the optimum, and an O(nlog2 n) time algorithm with an improved approximation ratio of 3, where n is the number of 1- and 2-hop neighbors. The best previously known algorithm, due to Bronnimann and Goodrich [2], guarantees O(1) approximation in O(n3 log n) time