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Dijkstra meets Steiner: a fast exact goal-oriented Steiner tree algorithm

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Abstract

We present a new exact algorithm for the Steiner tree problem in edge-weighted graphs. Our algorithm improves the classical dynamic programming approach by Dreyfus and Wagner. We achieve a significantly better practical performance via pruning and future costs, a generalization of a well-known concept to speed up shortest path computations. Our algorithm matches the best known worst-case run time and has a fast, often superior, practical performance: on some large instances originating from VLSI design, previous best run times are improved upon by orders of magnitudes. We are also able to solve larger instances of the d-dimensional rectilinear Steiner tree problem for \(d \in \{3, 4, 5\}\), whose Hanan grids contain up to several millions of edges.

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Author information

Authors and Affiliations

  1. Research Institute for Discrete Mathematics, University of Bonn, Bonn, Germany

    Stefan Hougardy, Jannik Silvanus & Jens Vygen

Authors
  1. Stefan Hougardy
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  2. Jannik Silvanus
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  3. Jens Vygen
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Corresponding author

Correspondence to Jannik Silvanus.

Appendices

Appendix 1: Results on graphic DIMACS instances

We present detailed computational results on DIMACS testsets. Our implementation is limited to instances with less than 64 terminals, so we exclude instances with more terminals.

The implementation of our algorithm is written in the C++ programming language and compiled using the GCC 4.8.2 compiler.

The experiments were performed single-threaded on a machine with 3.33 GHz Intel Xeon W5590 CPUs and 144 GB main memory which produced a score of 391.372724 using the DIMACS benchmark code.

For these experiments, we limited the memory consumption of the algorithm to 100 GB and set the time limit to 7200 s. The reported run times do not include the time to read the instance file from disk (Tables 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46).

Table 4 Results on the testset ALUE
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Table 5 Results on the testset ALUT
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Table 6 Results on the testset DIW
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Table 7 Results on the testset DMXA
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Table 8 Results on the testset GAP
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Table 9 Results on the testset LIN
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Table 10 Results on the testset MSM
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Table 11 Results on the testset TAQ
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Table 12 Results on the testset 1R
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Table 13 Results on the testset 2R
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Table 14 Results on the testset ES10FST
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Table 15 Results on the testset ES20FST
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Table 16 Results on the testset ES30FST
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Table 17 Results on the testset ES40FST
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Table 18 Results on the testset ES50FST
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Table 19 Results on the testset ES60FST
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Table 20 Results on the testset TSPFST
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Table 21 Results on the testset Copenhagen14
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Table 22 Results on the testset WRP3
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Table 23 Results on the testset WRP4
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Table 24 Results on the testset vienna-i-advanced
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Table 25 Results on the testset vienna-i-simple
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Table 26 Results on the testset X
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Table 27 Results on the testset P4E
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Table 28 Results on the testset P4Z
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Table 29 Results on the testset P6E
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Table 30 Results on the testset P6Z
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Table 31 Results on the testset B
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Table 32 Results on the testset C
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Table 33 Results on the testset D
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Table 34 Results on the testset E
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Table 35 Results on the testset I080
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Table 36 Results on the testset I160
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Table 37 Results on the testset I320
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Table 38 Results on the testset I640
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Table 39 Results on the testset PUC
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Table 40 Results on the testset SPG-PUCN
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Table 41 Results on the testset SP
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Table 42 Results on the testset MC
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Table 43 Results on the testset csd
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Table 44 Results on the testset goemans
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Table 45 Results on the testset skutella
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Table 46 Results on the testset smc
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Appendix 2: Results on rectilinear instances

For these experiments, the setup is identical to the one described in Appendix 1. For each instance, first, the Hanan grid was computed and written to an instance file describing an instance of the Steiner tree problem in graphs. The latter instance was then solved. The reported run times only include the time to solve the graphic instance, in particular, the time needed to read in the Hanan grid is not included (Tables 47, 48, 49, 50).

Table 47 Results on the testset CARIOCA
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Table 48 Results on the testset CARIOCA
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Table 49 Results on the testset CARIOCA
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Table 50 Results on the testset bonn-3d
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Hougardy, S., Silvanus, J. & Vygen, J. Dijkstra meets Steiner: a fast exact goal-oriented Steiner tree algorithm. Math. Prog. Comp. 9, 135–202 (2017). https://doi.org/10.1007/s12532-016-0110-1

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