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A Distributed Mincut/Maxflow Algorithm Combining Path Augmentation and Push-Relabel

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Abstract

We propose a novel distributed algorithm for the minimum cut problem. Motivated by applications like volumetric segmentation in computer vision, we aim at solving large sparse problems. When the problem does not fully fit in the memory, we need to either process it by parts, looking at one part at a time, or distribute across several computers. Many mincut/maxflow algorithms are designed for the shared memory architecture and do not scale to this setting. We consider algorithms that work on disjoint regions of the problem and exchange messages between the regions. We show that the region push-relabel algorithm of Delong and Boykov (A scalable graph-cut algorithm for N-D grids, in CVPR, 2008) uses Θ(n 2) rounds of message exchange, where n is the number of vertices. Our new algorithm performs path augmentations inside the regions and push-relabel style updates between the regions. It uses asymptotically less message exchanges, \(O(\mathcal{B}^{2})\), where \(\mathcal{B}\) is the set of boundary vertices. The sequential and parallel versions of our algorithm are competitive with the state-of-the-art in the shared memory model. By achieving a lower amount of message exchanges (even asymptotically lower in our synthetic experiments), they suit better for solving large problems using a disk storage or a distributed system.

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Notes

  1. A maximum preflow can be completed to a maximum flow using the flow decomposition, in O(mlogm) time. Because we are primarily interested in the minimum cut, we do not consider this step or whether it can be distributed.

  2. The number of sequential phases required in a general case is equal to the minimal coloring of the region interaction graph, i.e. 2 for bipartite graph and so on.

  3. An algorithm is said to be Ω(f(n)) if for some numbers c′ and n 0 and all n>=n 0, the algorithm takes at least c′f(n) time on some problem instance. Here we measure complexity in sweeps.

  4. Region-gap-relabel (Delong and Boykov 2008, Fig. 10) seems to contain an error: only vertices above the gap should be processed in step 3.

  5. The worst-case complexity of breadth-first search shortest path augmentation algorithm is just O(m|C|). The tree adaptation step, introduced by Boykov and Kolmogorov (2004) to speed-up the search, does not have a good bound and introduces an additional n 2 factor.

  6. There is a discrepancy with Delong and Boykov (2008, Fig. 4) regarding the results for the basic push-relabel. The main implementation difference is in the order of processing (HIPR versus FILO). It is also possible that their plot is illustrative and is not using the gap heuristic.

  7. Multithreaded maxflow library, http://www.maths.lth.se/matematiklth/personal/petter/cppmaxflow.php.

  8. Strandmark and Kahl (2010) stated their theorem for even integer costs in the case of two-subproblem separator sets. They remarked that a multiple of 4, resp., 8 is needed in the cases of decompositions for 2D and 3D grids. However, this multiplication is unnecessary if we chose to split the cost unevenly but preserving the integrality (like we did in the example).

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Acknowledgements

This work was supported by the EU project FP7-ICT-247870 NIFTi, FP7-ICT-247525 HUMAVIPS and GACR P103/10/0783.

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  1. Center for Machine Perception, Czech Technical University in Prague, Technicka 2, 16627, Praha 6, Czech Republic

    Alexander Shekhovtsov & Václav Hlaváč

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  1. Alexander Shekhovtsov
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  2. Václav Hlaváč
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Correspondence to Alexander Shekhovtsov.

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Shekhovtsov, A., Hlaváč, V. A Distributed Mincut/Maxflow Algorithm Combining Path Augmentation and Push-Relabel. Int J Comput Vis 104, 315–342 (2013). https://doi.org/10.1007/s11263-012-0571-2

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