The Scenario The Algorithms Neto The Experiments Conclusions

A FIRST ATTEMPT ON THE DISTRIBUTED PRIZE

COLLECTING STEINER TREE PROBLEM

Niccolo’ G. Rossetti

School of Engineering and Natural Sciencesngr1@hi.is

Master Thesis defence - August 2015

The Scenario The Algorithms Neto The Experiments Conclusions

OUTLINE

1 THE SCENARIO

2 THE ALGORITHMSRooted Tree AlgorithmUnrooted Tree AlgorithmMST-Heuristic

3 NETO

4 THE EXPERIMENTS

5 CONCLUSIONS

The Scenario The Algorithms Neto The Experiments Conclusions

OUTLINE

1 THE SCENARIO

2 THE ALGORITHMSRooted Tree AlgorithmUnrooted Tree AlgorithmMST-Heuristic

3 NETO

4 THE EXPERIMENTS

5 CONCLUSIONS

The Scenario The Algorithms Neto The Experiments Conclusions

THE CENTRALIZED SCENARIO

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The Scenario The Algorithms Neto The Experiments Conclusions

DEFINITION OF THE PROBLEM

NET-WORTH

NW (T ′) =∑v∈V ′

p(v)−∑e∈E ′

w(e)

EQUIVALENT IN OPTIMIZATION

GW (T ′) =∑v /∈V ′

p(v) +∑e∈E ′

w(e)

Notwithstanding the simple definition, the problem isNP-complete to solve.

The Scenario The Algorithms Neto The Experiments Conclusions

THE CENTRALIZED ALGORITHMS

THE CLASSICAL GW-ALGORITHM

It is a 2-approx of the optimal solution

OTHER ALGORITHMS

Further studies outperformed the GW-algorithm by using:different formulations of the mathematical problemheuristics

These algorithms are good in a centralized setting but...

The Scenario The Algorithms Neto The Experiments Conclusions

THE DISTRIBUTED SCENARIO

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The Scenario The Algorithms Neto The Experiments Conclusions

OUR DISTRIBUTED MODEL

nodes are independent computing entitiesnodes have only local knowledgenodes can communicate through messagesmessages are assumed to arrive at destination in finitetimeno global clockevery node in the network executes the given algorithmno failures or byzantine behaviourfocus on the message complexity: assume nodecomputation to be instantaneous

The Scenario The Algorithms Neto The Experiments Conclusions

OUTLINE OF MY WORK

To our knowledge, the distributed PCST has never beenaddressed in literature

WHAT I DID

Designed a distributed implementation of the GW-algorithmDesigned an optimal distributed algorithm for the PCST ontreesDesigned a distributed MST-based heuristicImplemented Neto: a Python framework to experimentdistributed algorithmsExperimented the MST-based heuristic with Neto

The Scenario The Algorithms Neto The Experiments Conclusions

OUTLINE

1 THE SCENARIO

2 THE ALGORITHMSRooted Tree AlgorithmUnrooted Tree AlgorithmMST-Heuristic

3 NETO

4 THE EXPERIMENTS

5 CONCLUSIONS

The Scenario The Algorithms Neto The Experiments Conclusions

ROOTED TREE ALGORITHM (RTA)

∀child if prize(child)−weight(child , self ) ≤ 0 then : prune(child)

The Scenario The Algorithms Neto The Experiments Conclusions

RTA (CONTINUED)

FIRST CONTRIBUTION

Also known as Strong Pruning (centralized) forpost-processingContribution: adapt it to the distributed model

COMPLEXITY

Exactly: 2E messages and 2 ∗ height time (if mex time = 1)

NAIVE UNROOTED APPROACH

Repeat RTA on every node as root and select the bestsolutionWorks better in centralized computationIt can be optimized

The Scenario The Algorithms Neto The Experiments Conclusions

UNROOTED TREE ALGORITHM (UTA)

TWO PHASES

1 Every node finds out the NW value of its rooted tree2 Find and build the best solution

The Scenario The Algorithms Neto The Experiments Conclusions

UTA (CONTINUED)

COMPLEXITY

Still linear as in RTA because it is based on a number ofconvergecasts and broadcasts

SECOND CONTRIBUTION

To our knowledge, no previous Strong Pruning for unrootedvariantUTA easily adapts to centralized computationProof of correctness

The Scenario The Algorithms Neto The Experiments Conclusions

MST-BASED HEURISTIC

THE IDEA

With UTA we can solve to optimality the distributed PCSTon treesGiven a general graph, build a tree and let UTA optimize itAny tree could work, which to build?

THE MIMINUM SPANNING TREE (MST)We chose the MST because:

Minimizes one of the components of the PCST definition(the edge costs)Distributed implementations existWorks well with similar problems

The Scenario The Algorithms Neto The Experiments Conclusions

MST-BASED HEURISTIC

THE IDEA

With UTA we can solve to optimality the distributed PCSTon treesGiven a general graph, build a tree and let UTA optimize itAny tree could work, which to build?

THE MIMINUM SPANNING TREE (MST)We chose the MST because:

Minimizes one of the components of the PCST definition(the edge costs)Distributed implementations existWorks well with similar problems

The Scenario The Algorithms Neto The Experiments Conclusions

MST-BASED HEURISTIC (CONTINUED)

The combination of the MST and UTA is an heuristics and itcannot guarantee the quality of the provided result!

We need to verify how it performs through experiments

The Scenario The Algorithms Neto The Experiments Conclusions

OUTLINE

1 THE SCENARIO

2 THE ALGORITHMSRooted Tree AlgorithmUnrooted Tree AlgorithmMST-Heuristic

3 NETO

4 THE EXPERIMENTS

5 CONCLUSIONS

The Scenario The Algorithms Neto The Experiments Conclusions

NETO

WHAT IS NETO FOR?Simulation and theoretical study of distributed algorithmsAlgorithmic level: no TCP/IP etc protocolsPython and its libraries (SimPy and networkx)Event-driven

SIMILAR TO

ComplexNetworkSim uses same tools, but it’s notdesigned for distributed modelPymote works with same model, with focus on wirelessnetworksPymote might be good for us, but it was published whenwe had already Neto working

The Scenario The Algorithms Neto The Experiments Conclusions

NETO (CONTINUED)

THE IDEA

User implements his event-driven algorithm: handlers,helper functionsBuild the graph (networkx)Easily register the Algorithm on nodes (as plug-and-play)Algorithm composition

N.register([my_algorithm(N)])

The Scenario The Algorithms Neto The Experiments Conclusions

NETO (CONTINUED)

HOW IT WORKS INSIDE

Neto nodes implement a dispatcher and incoming queueIf queue is not empty, the dispatcher delivers messages toregistered algorithms in FIFO orderNo preemptive, algorithms decide when to give controlback to dispatcherSimPy decides which node to simulate

THIRD CONTRIBUTION

Neto constitutes our third result as it proved to serve well duringour experiments!

The Scenario The Algorithms Neto The Experiments Conclusions

OUTLINE

1 THE SCENARIO

2 THE ALGORITHMSRooted Tree AlgorithmUnrooted Tree AlgorithmMST-Heuristic

3 NETO

4 THE EXPERIMENTS

5 CONCLUSIONS

The Scenario The Algorithms Neto The Experiments Conclusions

EXPERIMENTAL SETUP

We run the MST-based heuristic against set of instancesfrom literature for which opt is knownThe instances are taken from real (pipeline/cable) scenarioor built to reflect the same topologyUp to thousands of nodes and tens of thousands of edgesWe also created new instances with typical networktopologies (grid and scale-free)

Encouraging results!

The Scenario The Algorithms Neto The Experiments Conclusions

EXPERIMENTAL RESULTS

FIGURE: Probably the worst serie

The Scenario The Algorithms Neto The Experiments Conclusions

OUTLINE

1 THE SCENARIO

2 THE ALGORITHMSRooted Tree AlgorithmUnrooted Tree AlgorithmMST-Heuristic

3 NETO

4 THE EXPERIMENTS

5 CONCLUSIONS

The Scenario The Algorithms Neto The Experiments Conclusions

SUMMARY OF FINDINGS

We defined and studied a novel variant of the PCST1 RTA (adapted Strong Pruning to distributed model)2 UTA (first Strong Pruning alike optimized for unrooted

PCST)3 Neto4 MST-based heuristic is to our knowledge the first alg for the

dPCST and returns solutions comparable to GW

The Scenario The Algorithms Neto The Experiments Conclusions

FUTURE WORKS

FUTURE WORKS ON THE DPCSTHeuristic with other tree builder instead of MSTAnt Colony OptimizationFurther studies on how to parallelize the resolution of theproblem

FUTURE WORKS ON NETO

Python 3 and SimPy 3 compatibilityGUIFurther development on algorithm composition andrequirements

The Scenario The Algorithms Neto The Experiments Conclusions

THANKS EVERYBODY AND IN PARTICULAR TO

Pall Melsted for the suggestions and patient reviewsHelmut Neukirchen for his interesting commentsKristjan Jonasson for his swift helpsMagnus M. Halldorsson in advanceYimir Vigfusson for the incipit and initial part of the thesis