## Closing the Gap: Near-Optimal Steiner Trees in Polynomial Time (1994)

Venue: | IEEE Trans. Computer-Aided Design |

Citations: | 43 - 13 self |

### BibTeX

@ARTICLE{Griffith94closingthe,

author = {Jeff Griffith and Gabriel Robins and Jeffrey S. Salowe and Tongtong Zhang},

title = {Closing the Gap: Near-Optimal Steiner Trees in Polynomial Time},

journal = {IEEE Trans. Computer-Aided Design},

year = {1994},

volume = {13},

pages = {1351--1365}

}

### Years of Citing Articles

### OpenURL

### Abstract

The minimum rectilinear Steiner tree (MRST) problem arises in global routing and wiring estimation, as well as in many other areas. The MRST problem is known to be NP-hard, and the best performing MRST heuristic to date is the Iterated 1-Steiner (I1S) method recently proposed by Kahng and Robins. In this paper we develop a straightforward, efficient implementation of I1S, achieving a speedup factor of three orders of magnitude over previous implementations. We also give a parallel implementation that achieves near-linear speedup on multiple processors. Several performance-improving enhancements enable us to obtain Steiner trees with average cost within 0.25% of optimal, and our methods produce optimal solutions in up to 90% of the cases for typical nets. We generalize I1S and its variants to three dimensions, as well as to the case where all the pins lie on k parallel planes, which arises in, e.g., multi-layer routing. Motivated by the goal of reducing the running times of our algorith...

### Citations

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Citation Context ...t set in a graph, it is easy to show that our maximization problem is NP-complete also, even if the independent sets obey the "inheritance property", one of the axioms for matroids (see Corm=-=en et al. [11]-=-). Our independent sets do not necessarily obey the inheritance property or the exchange property for matroids. Nevertheless, we can use a greedy approximation, described in Figure 5, to approximate t... |

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Citation Context ...ohnson [16] establishes that despite this restriction on the solution space, the MRST problem remains NP-complete; this has given rise to numerous heuristics as surveyed by Hwang, Richards and Winter =-=[27]-=-. Figure 2: Hanan's theorem: there always exists an MRST with Steiner points chosen from the intersections of all the horizontal and vertical lines passing through all the points. In solving intractab... |

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Citation Context ... MST maintenance can also be achieved in sub-linear time [14], but such methods seem impractical due to their complicated description and large hidden constants. A similar method was also used by Yao =-=[45]-=- to obtain a sub-quadratic MST algorithm in higher dimensions, but no attempt was made to optimize the number of necessary regions, whereas we also optimize the number of regions. We now show that onl... |

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Citation Context ...hollow dots represent the original pointset P , while solid dots represent the set S of added Steiner points. Research on the MRST problem has been guided by several fundamental results. First, Hanan =-=[21]-=- has shown that there always exists an MRST with Steiner points chosen from the intersections of all the horizontal and vertical lines passing through all the points in P (see Figure 2), and this resu... |

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Citation Context ...es for inclusion into the pointset. This procedure is iterated until no improving candidates can be found (the B1S algorithm parallelizes similarly). We used the Parallel Virtual Machine (PVM) system =-=[5]-=- [43] to control remote processors. PVM is a widely-available software package that allows a heterogeneous network of parallel and serial computers to be used as a single computational resource. The P... |

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Citation Context ...g through all the points. In solving intractable problems, we often seek provably-good heuristics having bounded worstcase error from optimal. Thus, a third important result is the discovery by Hwang =-=[25]-=- that the rectilinear MST is a fairly good approximation to the MRST, with a worst-case performance ratio of cost(MST ) cost(MRST )s3 2 . This implies that any MST-based strategy that improves upon an... |

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Citation Context ... i.e. an efficient tool for producing near-optimal Steiner trees. 1 Recently, Berman and Ramaiyer [7] and Foessmeier, Kaufmann and Zelikovsky [6] [13] have extended the fundamental work of Zelikovsky =-=[46] [47] to y-=-ield a method similar to I1S (specifically, to the "batched" I1S method described below) with performance ratio bounded by 11 8 ; this work settles in the affirmative the longstanding open q... |

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Citation Context ... parallel implementation may be viewed as an important first step toward obtaining a "Steiner engine", i.e. an efficient tool for producing near-optimal Steiner trees. 1 Recently, Berman and=-= Ramaiyer [7] and Foess-=-meier, Kaufmann and Zelikovsky [6] [13] have extended the fundamental work of Zelikovsky [46] [47] to yield a method similar to I1S (specifically, to the "batched" I1S method described below... |

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Citation Context ...nce ratio of exactly 3 2 . This negative result has motivated research into alternate schemes for MRST approximation, with the best performing among these being the Iterated 1-Steiner (I1S) algorithm =-=[28]-=- [38]. I1S always performs strictly better than 3 2 times optimal 1 , and also performs very well in practice, achieving almost 11% average improvement over MST cost, which is on average less than 0:5... |

59 |
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Citation Context ...ce is solvable in polynomial time when the degree bound is fixed at 14 or more (since we have shown how to efficiently find an ordinary MST that meets such maximum degree constraints). Monma and Suri =-=[35]-=- used a similar perturbation argument to prove that for any poinset in the Euclidean plane, there is an MST with maximum degree of 5. 7 A Parallel Implementation Since a typical CAD environment consis... |

48 |
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Citation Context ...own that the problem of finding a degree-bounded MST is NP-complete, even when the degree bound is fixed at 2 (yielding the Traveling Salesman Problem), or at 3 as shown by Papadimitriou and Vazirani =-=[36]-=-. Corollary 6.2 implies that the degree-bounded MST problem in the Manhattan plane is solvable in polynomial time when the degree bound is fixed at 4 or more, since we have shown how to efficiently fi... |

38 |
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Citation Context ...tal and vertical lines passing through all the points in P (see Figure 2), and this result was generalized by Snyder [42] to all higher dimensions. However, a second major result by Garey and Johnson =-=[16]-=- establishes that despite this restriction on the solution space, the MRST problem remains NP-complete; this has given rise to numerous heuristics as surveyed by Hwang, Richards and Winter [27]. Figur... |

37 | New approximation algorithms for the Steiner tree problem
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Citation Context ... an efficient tool for producing near-optimal Steiner trees. 1 Recently, Berman and Ramaiyer [7] and Foessmeier, Kaufmann and Zelikovsky [6] [13] have extended the fundamental work of Zelikovsky [46] =-=[47] to yield -=-a method similar to I1S (specifically, to the "batched" I1S method described below) with performance ratio bounded by 11 8 ; this work settles in the affirmative the longstanding open questi... |

35 |
New algorithms for the rectilinear Steiner tree problem
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Citation Context ...roves upon an initial MST topology will also enjoy a performance ratio of at most 3 2 . This has prompted a large number of Steiner tree heuristics that resemble classic MST construction methods [23] =-=[24]-=- [26] [31] [32], all producing Steiner trees with average cost 7% to 9% smaller than MST cost [27]. Unfortunately, all MST-based MRST constructions were recently shown by Kahng and Robins [29] to have... |

29 | Routing a Multi-Terminal Critical Net: Steiner Tree Construction
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- 1994
(Show Context)
Citation Context ... 1-Steiner method was generalized to arbitrary weighted graphs by Alexander and Robins [1] [2], and is thus a suitable basis of a practical global router which must handle congestion, obstacles, etc. =-=[15]-=-. The performance success of I1S was achieved at the expense of a high time complexity: although a more efficient variant of I1S can be implemented to run within time O(n 2 log n) [28], the computatio... |

26 |
An O(n logn) algorithm for rectilinear minimal spanning trees
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- 1979
(Show Context)
Citation Context ... upon an initial MST topology will also enjoy a performance ratio of at most 3 2 . This has prompted a large number of Steiner tree heuristics that resemble classic MST construction methods [23] [24] =-=[26]-=- [31] [32], all producing Steiner trees with average cost 7% to 9% smaller than MST cost [27]. Unfortunately, all MST-based MRST constructions were recently shown by Kahng and Robins [29] to have a wo... |

23 |
A new global router for row-based layout
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(Show Context)
Citation Context ...nitial MST topology will also enjoy a performance ratio of at most 3 2 . This has prompted a large number of Steiner tree heuristics that resemble classic MST construction methods [23] [24] [26] [31] =-=[32]-=-, all producing Steiner trees with average cost 7% to 9% smaller than MST cost [27]. Unfortunately, all MST-based MRST constructions were recently shown by Kahng and Robins [29] to have a worst-case p... |

15 | Approaching the 5/4-approximation for rectilinear Steiner trees
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- 1994
(Show Context)
Citation Context ... important first step toward obtaining a "Steiner engine", i.e. an efficient tool for producing near-optimal Steiner trees. 1 Recently, Berman and Ramaiyer [7] and Foessmeier, Kaufmann and Z=-=elikovsky [6] [13] have-=- extended the fundamental work of Zelikovsky [46] [47] to yield a method similar to I1S (specifically, to the "batched" I1S method described below) with performance ratio bounded by 11 8 ; t... |

15 |
Three-Dimensional Routing for Multilayer Ceramic Printed Circuit Boards
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(Show Context)
Citation Context ...ize I1S and its variants to three dimensions, as well as to the intermediate case where all pins lie on k parallel planes. This formulation has several applications, including multi-layer routing [9] =-=[20]-=- [22], and the design of buildings [41]. Empirical testing suggests that this approach is effective for three-dimensional Steiner routing, yielding up to 15% average improvement over MST cost in three... |

15 | An exact rectilinear steiner tree algorithm
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- 1993
(Show Context)
Citation Context ...rforms strictly better than 3 2 times optimal 1 , and also performs very well in practice, achieving almost 11% average improvement over MST cost, which is on average less than 0:5% away from optimal =-=[40]-=-. The Iterated 1-Steiner method was generalized to arbitrary weighted graphs by Alexander and Robins [1] [2], and is thus a suitable basis of a practical global router which must handle congestion, ob... |

14 |
A neighborhood improvement algorithm for rectilinear steiner trees
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Citation Context ...t improves upon an initial MST topology will also enjoy a performance ratio of at most 3 2 . This has prompted a large number of Steiner tree heuristics that resemble classic MST construction methods =-=[23]-=- [24] [26] [31] [32], all producing Steiner trees with average cost 7% to 9% smaller than MST cost [27]. Unfortunately, all MST-based MRST constructions were recently shown by Kahng and Robins [29] to... |

14 |
On Optimal Interconnections
- Robins
(Show Context)
Citation Context ...atio of exactly 3 2 . This negative result has motivated research into alternate schemes for MRST approximation, with the best performing among these being the Iterated 1-Steiner (I1S) algorithm [28] =-=[38]-=-. I1S always performs strictly better than 3 2 times optimal 1 , and also performs very well in practice, achieving almost 11% average improvement over MST cost, which is on average less than 0:5% awa... |

13 | On the maximum degree of minimum spanning trees
- Robins, Salowe
- 1994
(Show Context)
Citation Context ...viously known bounds for two and three dimensions were 6 and 26, respectively). Our results and algorithms on degree-bounded minimum spanning trees are of significant independent theoretical interest =-=[39]-=-, and settle several open issues in complexity theory 2 . The rest of the paper is organized as follows. In Section 2 we review the I1S method. Section 3 outlines our more efficient implementation of ... |

12 |
PVM: A Framework for Parallel and
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- 1990
(Show Context)
Citation Context ...or inclusion into the pointset. This procedure is iterated until no improving candidates can be found (the B1S algorithm parallelizes similarly). We used the Parallel Virtual Machine (PVM) system [5] =-=[43]-=- to control remote processors. PVM is a widely-available software package that allows a heterogeneous network of parallel and serial computers to be used as a single computational resource. The PVM sy... |

11 |
Rectilinear Steiner Tree Construction by Local and Global Re
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(Show Context)
Citation Context ...nd parallel versions of B1S, using both the naive O(n 4 log n) 3 Recently, other Steiner heuristics with performance approaching that of I1S were proposed by Borah, Owens, and Irwin [8], Chao and Hsu =-=[10]-=- and by Lewis, Pong and VanCleave [33]. implementation [28] and our new O(n 3 ) implementation, which incorporates the efficient, dynamic MST maintenance as described in Section 3. We observe that the... |

11 |
The 1-Steiner Tree Problem
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(Show Context)
Citation Context ...d) a degree-2 Steiner point is formed and is thus eliminated from the topology. Although a single 1-Steiner point may be found in O(n 2 ) time using complicated techniques from computational geometry =-=[17]-=- [28], such methods suffer from large constants in their time complexities, and are notoriously difficult to implement. Thus, a batched variant of I1S is usually favored, which Algorithm Iterated 1-St... |

10 | Toward a Steiner Engine: Enhanced Serial and Parallel
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- 1993
(Show Context)
Citation Context ...extensive simulation results regarding performance, running times, and parallel speedups. We conclude in Section 9 with directions for further research. A preliminary version of this work appeared in =-=[3] and in [4-=-]. 2 Robins and Salowe [39] investigate the maximum MST degree for higher dimensions and other Lp norms, and relate the maximum MST degree to the so-called "Hadwiger" numbers. 2 Review of th... |

10 |
Data Structures for On-line Updating of Minimum Spanning Trees
- Fredrickson
- 1985
(Show Context)
Citation Context ...O(n 3 ), a substantial savings. An execution example of this method is given in Figure 6, and Figure 7 describes it formally. Note that dynamic MST maintenance can also be achieved in sub-linear time =-=[14]-=-, but such methods seem impractical due to their complicated description and large hidden constants. A similar method was also used by Yao [45] to obtain a sub-quadratic MST algorithm in higher dimens... |

8 |
A New Approach to FPGA Routing Based on Multi-Weighted Graphs
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- 1994
(Show Context)
Citation Context ...1% average improvement over MST cost, which is on average less than 0:5% away from optimal [40]. The Iterated 1-Steiner method was generalized to arbitrary weighted graphs by Alexander and Robins [1] =-=[2]-=-, and is thus a suitable basis of a practical global router which must handle congestion, obstacles, etc. [15]. The performance success of I1S was achieved at the expense of a high time complexity: al... |

8 |
Narrowing the Gap: Near-Optimal Steiner Trees
- Barrera, Griffith, et al.
- 1993
(Show Context)
Citation Context ...imulation results regarding performance, running times, and parallel speedups. We conclude in Section 9 with directions for further research. A preliminary version of this work appeared in [3] and in =-=[4]. 2 Robins-=- and Salowe [39] investigate the maximum MST degree for higher dimensions and other Lp norms, and relate the maximum MST degree to the so-called "Hadwiger" numbers. 2 Review of the Iterated ... |

8 | On performance bounds for a class of rectilinear steiner tree heuristics in arbitrary dimension - Kahng, Robins - 1992 |

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Chameleon: A new multi-layer channel router
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(Show Context)
Citation Context ...eralize I1S and its variants to three dimensions, as well as to the intermediate case where all pins lie on k parallel planes. This formulation has several applications, including multi-layer routing =-=[9]-=- [20] [22], and the design of buildings [41]. Empirical testing suggests that this approach is effective for three-dimensional Steiner routing, yielding up to 15% average improvement over MST cost in ... |

7 |
Three-Dimensional Integrated Circuit Layout
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(Show Context)
Citation Context ...1S and its variants to three dimensions, as well as to the intermediate case where all pins lie on k parallel planes. This formulation has several applications, including multi-layer routing [9] [20] =-=[22]-=-, and the design of buildings [41]. Empirical testing suggests that this approach is effective for three-dimensional Steiner routing, yielding up to 15% average improvement over MST cost in three-dime... |

6 |
On the Exact Location of Steiner Points in General Dimension
- Snyder
- 1992
(Show Context)
Citation Context ...exists an MRST with Steiner points chosen from the intersections of all the horizontal and vertical lines passing through all the points in P (see Figure 2), and this result was generalized by Snyder =-=[42]-=- to all higher dimensions. However, a second major result by Garey and Johnson [16] establishes that despite this restriction on the solution space, the MRST problem remains NP-complete; this has give... |

5 |
An edge-based heuristic for rectilinear steiner trees
- Borah, Owens, et al.
- 1994
(Show Context)
Citation Context ...on of the serial and parallel versions of B1S, using both the naive O(n 4 log n) 3 Recently, other Steiner heuristics with performance approaching that of I1S were proposed by Borah, Owens, and Irwin =-=[8]-=-, Chao and Hsu [10] and by Lewis, Pong and VanCleave [33]. implementation [28] and our new O(n 3 ) implementation, which incorporates the efficient, dynamic MST maintenance as described in Section 3. ... |

5 |
Space Structures: Their Harmony and Counterpoint
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(Show Context)
Citation Context ...ed cube (Figure 11(a)), i.e., a solid obtained by chopping off the corners of a cube, yielding 6 square faces and 8 equilateral triangle faces (Figure 11(b)); this solid is known as a "cuboctahed=-=ron" [34]-=-. The 14 solid regions of this partition are induced by the 14 faces of the cuboctahedron, namely the 6 pyramids with square cross-section (Figure 11(c)) and the 8 pyramids with triangular cross-secti... |

3 | An architecture-independent unified approach to FPGA routing
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- 1993
(Show Context)
Citation Context ...st 11% average improvement over MST cost, which is on average less than 0:5% away from optimal [40]. The Iterated 1-Steiner method was generalized to arbitrary weighted graphs by Alexander and Robins =-=[1]-=- [2], and is thus a suitable basis of a practical global router which must handle congestion, obstacles, etc. [15]. The performance success of I1S was achieved at the expense of a high time complexity... |

3 |
On Optimal Interconnections for VLSI Layout
- Kahng, Robins
- 1994
(Show Context)
Citation Context ...imum rectilinear Steiner tree problem is central to VLSI physical-design phases such as global routing and wiring estimation, where we seek low-cost topologies to interconnect the pins of signal nets =-=[30]-=- [37]: The Minimum Rectilinear Steiner Tree (MRST) problem: Given a set P of n points, find a set S of Steiner points such that the minimum rectilinear spanning tree (MST) over P [ S has minimum cost.... |

2 |
T' oth, New Results
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(Show Context)
Citation Context ...ce an MST degree of 14. Note that the best previously known bound for the maximum MST degree in three-dimensional Manhattan space was 26, as implied by a result from the theory of sphere packing [12] =-=[44]-=-. Our results regarding MST bounds also settle some open questions in complexity theory, since it is known that the problem of finding a degree-bounded MST is NP-complete, even when the degree bound i... |

1 |
Zelikovsky, Fast Approximation Algorithms for the Rectilinear Steiner Tree Problem, Tech. Rep. WSI-93-14, Wilhelm Schickard-Institut fur Informatik
- Foessmeier, Kaufmann, et al.
- 1993
(Show Context)
Citation Context ...ortant first step toward obtaining a "Steiner engine", i.e. an efficient tool for producing near-optimal Steiner trees. 1 Recently, Berman and Ramaiyer [7] and Foessmeier, Kaufmann and Zelik=-=ovsky [6] [13] have exte-=-nded the fundamental work of Zelikovsky [46] [47] to yield a method similar to I1S (specifically, to the "batched" I1S method described below) with performance ratio bounded by 11 8 ; this w... |

1 |
A Remark on
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(Show Context)
Citation Context .... The three-dimensional analog of Hwang's result suggests that the maximum MST/MRST ratio for three dimensions is at most 5 3 (this is a consequence of a more general conjecture for higher dimensions =-=[19]-=-), although there is currently no known proof of this. An example consisting of six points located in the middles of the faces of a rectilinear cube establishes that 5 3 is a lower bound for the worst... |