1. This work was supported bu EU projects FP7-ICT-247870 NIFTi and FP7-ICT-247525 HUMAVIPS and the Czech project 1M0567 CAK 25-27 July, 2011 EMMCVPR Center for Machine Perception Czech Technical University in Prague A Distributed Mincut/Maxflow Algorithm Combining Path Augmentation and Push-Relabel Alexander Shekhovtsov and V á clav Hlav á č shekhole@fel.cvut.cz, hlavac@fel.cvut.cz TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A

2. 2/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Overview  Goals  Main Results Preliminary results presented on workshop 2010 in Kiev (not entirely correct)

3. 3/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Mincut in Computer Vision  Test Problems (University of Western Ontario)

4. 4/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Mincut in Computer Vision  Discrete Energy Minimization via Mincut

5. 5/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Mincut  Capacitated network

6. 6/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Mincut  Minimum s-t Cut

7. 7/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Augmenting Path Approach  Path Augmentation

8. 8/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Augmenting Path Approach  Residual Network

9. 9/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Augmenting Path Approach  Costs of all cuts are changed by a constant

10. 10/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Augmenting Path Approach  Augment next path

11. 11/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Augmenting Path Approach  Minimum Cut in Residual Network

12. 12/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Push-Relabel Approach  Extended transformation excess

13. 13/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Push-Relabel Approach

14. 14/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Push-Relabel Approach

15. 15/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Push-Relabel Approach

16. 16/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Push-Relabel Approach

17. 17/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Push-Relabel Approach

18. 18/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Distributed Algorithms CPU Memory CPU Mem CPU Mem Quick Slow Distributed Sequential CPU Mem Quick Slow Disk Distributed Parallel Shared Memory Parallel CPU Memory Sequential  Distributed Model – Divide Computation AND Memory

19. 19/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Existing Distributed Algorithms  Push-Relabel [Goldberg ‘ 94]  Region Discharge [Delong and Boykov, CVPR ‘ 08]

20. 20/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Existing Distributed Algorithms  Adaptive Bottom-up Merging [Liu and J. Sun, CVPR ‘ 10]

21. 21/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Existing Distributed Algorithms  Dual Decomposition for Mincut [Strandmark and Kahl, CVPR ‘ 10]

22. 22/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Existing Distributed Algorithms  Dual Decomposition for Mincut [Strandmark and Kahl, CVPR ‘ 10]

23. 23/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 New Algorithm  New Distance Function  The Algorithm  Parallel Version  Complexity Bound  Experimental Confirmation

24. 24/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 New Distance Function length of the path = number of boundary edges distance = length of a shortest path to the sink Valid Labeling – distance underestimate

25. 25/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 The Algorithm: Augmenting Path Region Discharge 1. Augment paths to the sink 2. Augment paths the boundary with label 0 3. Augment paths the boundary with label 1... Relabel the interior vertices sweep estimate of the shortest way to the sink

26. 26/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Parallel Version parallel sweep Conflicts are resolved by canceling one of the flows (similar to asynchronous parallel push-relabel [Goldberg and Tarjan 88]) all regions discharged concurrently

27. 27/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Complexity Bound

28. 28/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Experimental Confirmation S-PRD details: highest label first push-relabel (HI-PR), region-gap heuristic, region-relabel heuristic, global gap heuristic S-ARD details: global gap heuristic S-PRD: Sequential Push-Relabel Region Discharge S-ARD: Sequential Augmenting path Region Discharge Test problems: 2D grid with regular connectivity and random costs

29. 29/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Efficient Implementation  Standard Heuristics: global relabel and global gap  Boundary Relabel  Partial Discharges  Boundary Search Trees

30. 30/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Efficient Implementation  Global Relabel Heuristic = compute exact distance, O(m) time  Global Gap Heuristic (sufficient condition of sink unreachability) … cannot reach sink cannot be in the sink set of a minimum cut “ decided nodes ”

31. 31/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 not possible Efficient Implementation  Boundary Relabel A valid labeling of boundary nodes Group nodes by their label within each region Compute exact distance on the auxiliary graph Add possible links (red) Improve labeling, knowing only the information on the boundary? do not know how the vertices linked inside Result: a (better) valid labeling. In the limit of small regions = global relabel.

32. 32/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Efficient Implementation  Partial Discharges 1. Augment paths to the sink 2. Augment paths the boundary with label 0 3. Augment paths the boundary with label 1... Relabel the interior vertices execute only this step in sweep 0

33. 33/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Efficient Implementation  Partial Discharges 1. Augment paths to the sink 2. Augment paths the boundary with label 0 3. Augment paths the boundary with label 1... Relabel the interior vertices execute up to here in sweep 1

34. 34/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Efficient Implementation  Partial Discharges 1. Augment paths to the sink 2. Augment paths the boundary with label 0 3. Augment paths the boundary with label 1... Relabel the interior vertices execute up to here in sweep 2 Prevents sending the flow in a wrong direction (redundant work) Boundary Relabel + Partial Discharges effect:

35. 35/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Efficient Implementation  Boundary Search Trees Residual region network Search trees of the sink and boundary vetices Labels of the inner vertices are determined by their tree root Tree with higher root cannot overtake lower Saves computation between sweeps, Integrates region-relabel into augmentation. Find augmenting paths and encode labeling.

36. 36/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Results  Dependence on Partition  Sequential Competition  Parallel Competition  Local Problem Reduction

37. 37/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Results: Dependence on Partition BL06-gargoyle-sml cell complex 32x45x32x24 partitioned by vertex number LB07-bunny-med 3D 6-connected 202x199x157 sparse data sliced along dimensions liver.n6.c10 3D 6-connected 170x170x144 sliced along dimensions Stable over partitions

38. 38/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Results: Sequential Competition BK – Augmenting Path [Boykov-Kolmogorov] HIPR – Highest level Push-Relabel [Goldberg-Tarjan, Cherkassky] S-ARD – Sequential Augmenting Path Region Discharge S-PRD – Sequential Push-Relable Region Discharge ([Delong-Boykov], our impl.)

39. 39/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Results: Sequential Competition BK – Augmenting Path [Boykov-Kolmogorov] HIPR – Highest level Push-Relabel [Goldberg-Tarjan, Cherkassky] S-ARD – Sequential Augmenting Path Region Discharge S-PRD – Sequential Push-Relable Region Discharge ([Delong-Boykov], our impl.)...

40. 40/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Results: Parallel Competition (4 CPUs) DD – Dual Decomposition [ Strandmark and Kahl ’ 10] 2/4 Regions RPR – Region Push-Relable [Delong and Boykov ‘ 08] impl. by Sameh Khamis not applicable reduced graph best parameters P-ARD speed-up over BK: 0.8 – 4, robust method, few sweeps (message exchanges) shared memory Adaptive Bottom-up Merging [Liu and J. Sun ‘ 10] achieves near linear speed-up over BK

41. 41/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Local Problem Reduction  How many vertices can be decided optimally by considering regions separately? 16 regions 64 regions 2D stereo problems are largely decided locally. Other problems are significantly harder.

42. 42/9 Alexander Shekhovtsov & Vaclav Hlavac, 2011 Conclusion  New distributed algorithm  Terminates in at most B 2 +1 sweeps (few in practice)  Sequential Algorithm 1) competitive with sequential solvers 2) uses few sweeps (= loads/unloads of regions) 3) suitable to run in the limited memory model  Parallel Algorithm 1) competitive with shared memory algorithms 2) uses few sweeps (= rounds of message exchange) 3) suitable for execution on a computer cluster  Implementation can be specialized for regular grids (less memory/faster)  (?) no good worst case complexity bound in terms of elementary operations Thanks for you comments