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

This work was supported bu EU projects FP7-ICT-247870NIFTi and FP7-ICT-247525 HUMAVIPS and the Czech project 1M0567 CAK

25-27 July, 2011EMMCVPR

Center for Machine Perception

Czech Technical University in Prague

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

Alexander Shekhovtsov and Václav Hlaváč

[email protected], [email protected]

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Alexander Shekhovtsov & Vaclav Hlavac, 2011

Overview Goals

Main Results

Preliminary results presented on workshop 2010 in Kiev (not entirely correct)

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Mincut in Computer Vision Test Problems (University of Western Ontario)

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Mincut in Computer Vision Discrete Energy Minimization via Mincut

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Mincut Capacitated network

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Mincut Minimum s-t Cut

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Augmenting Path Approach Path Augmentation

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Augmenting Path Approach Residual Network

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Augmenting Path Approach Costs of all cuts are changed by a constant

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Augmenting Path Approach Augment next path

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Augmenting Path Approach Minimum Cut in Residual Network

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Push-Relabel Approach Extended transformation

excess

excess

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Push-Relabel Approach

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Distributed Algorithms

CPU CPU

Memory

CPU

Mem

CPU

MemQuick

Slow

Distributed Sequential

CPU

MemQuick

Slow

Disk

Distributed Parallel

Shared Memory Parallel

CPU

Memory

Sequential

Distributed Model – Divide Computation AND Memory

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Existing Distributed Algorithms Push-Relabel [Goldberg ‘94]

Region Discharge [Delong and Boykov, CVPR‘08]

t

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Existing Distributed Algorithms Adaptive Bottom-up Merging [Liu and J. Sun, CVPR‘10]

t

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Existing Distributed Algorithms Dual Decomposition for Mincut [Strandmark and Kahl, CVPR‘10]

¡ 1

2

2

1 ¡ 3

1 2

2

11

¡ 1

2

2

11

¡ 3

0 21

2

2

¡ 12

¡ 1

2

1¡ ¸1 1+¸1

1¡ ¸2 0+¸2

v v0 v00M N

¡ 1

2

2

11

¡ 3

1

0 21

1

1

0

1

1

2

2

¡ 1v0 v00

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Existing Distributed Algorithms Dual Decomposition for Mincut [Strandmark and Kahl, CVPR‘10]

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New Algorithm New Distance Function The Algorithm Parallel Version Complexity Bound Experimental Confirmation

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New Distance Function

length of the path = number of boundary edges distance = length of a shortest path to the sink

d¤B(u) = 2

d¤B(v) = 0t

Valid Labeling – distance underestimate

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The Algorithm: Augmenting Path Region Discharge

t

1. Augment paths to the sink 2. Augment paths the boundary with label 03. Augment paths the boundary with label 1

...

Relabel the interior vertices

sweep

estimate ofthe shortestway to the sink

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Parallel Version

t

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

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Complexity Bound

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Experimental Confirmation

S-PRD details: highest label first push-relabel (HI-PR), region-gap heuristic, region-relabel heuristic, global gap heuristicS-ARD details: global gap heuristic

S-PRD: Sequential Push-Relabel Region DischargeS-ARD: Sequential Augmenting path Region Discharge

Test problems: 2D grid with regular connectivity and random costs

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Efficient Implementation Standard Heuristics: global relabel and global gap Boundary Relabel Partial Discharges Boundary Search Trees

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Efficient Implementation Global Relabel Heuristic = compute exact distance, O(m) time Global Gap Heuristic (sufficient condition of sink unreachability)

t

… cannot reach sink

cannot be in the sink set of a minimum cut“decided nodes”

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not possible

Efficient Implementation Boundary Relabel

0

2 11

2

1

1

1

12

2

1

1

0

1

21

2

2

1

1

0

31

4 2

2

33

A valid labeling ofboundary nodes

Group nodes by their label within each region

Compute exact distanceon 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.

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Efficient Implementation Partial Discharges


...


execute only this step in sweep 0

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...


execute up to here in sweep 1

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...


execute up to here in sweep 2

Prevents sending the flow in a wrong direction (redundant work)

Boundary Relabel + Partial Discharges effect:

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00

Efficient Implementation Boundary Search Trees

0

0

1

t

0

0

1

t

0

0

1

t1 1 112 00

Residual region network

Search trees of thesink and boundaryvetices

Labels of the innervertices are determinedby their tree root

Tree with higher root cannot overtake lower

(¡ 1)

Saves computation between sweeps, Integrates region-relabel into augmentation.

Find augmenting paths and encode labeling.

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Results Dependence on Partition Sequential Competition Parallel Competition Local Problem Reduction

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Results: Dependence on Partition

BL06-gargoyle-smlcell complex32x45x32x24partitioned by vertex number

LB07-bunny-med3D 6-connected202x199x157sparse datasliced along dimensions

liver.n6.c103D 6-connected170x170x144sliced along dimensions

Stable over partitions

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Results: Sequential CompetitionBK – Augmenting Path [Boykov-Kolmogorov]HIPR – Highest level Push-Relabel [Goldberg-Tarjan, Cherkassky]S-ARD – Sequential Augmenting Path Region DischargeS-PRD – Sequential Push-Relable Region Discharge ([Delong-Boykov], our impl.)

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Results: Sequential CompetitionBK – Augmenting Path [Boykov-Kolmogorov]HIPR – Highest level Push-Relabel [Goldberg-Tarjan, Cherkassky]S-ARD – Sequential Augmenting Path Region DischargeS-PRD – Sequential Push-Relable Region Discharge ([Delong-Boykov], our impl.)

...

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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

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

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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.

s

t

BR

weak sourcestrong source

strong sink

undecided

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Conclusion New distributed algorithm Terminates in at most B2+1 sweeps (few in practice) Sequential Algorithm

1) competitive with sequential solvers2) uses few sweeps (= loads/unloads of regions)3) suitable to run in the limited memory model

Parallel Algorithm 1) competitive with shared memory algorithms2) 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

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

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