A Local Facility Location Algorithm Supervisor: Assaf Schuster Denis Krivitski Technion – Israel Institute of Technology

Outline 1.Introduction 2.Related Work 3.Prerequisites 4.Local Majority Voting 5.Distributed Facility Location 6.Speculative Execution 7.Local ArgMin 8.Experimental Results Outline

Large-Scale Distributed Systems LSD – Large-Scale Distributed Peer-to-Peer file sharing networks –eMule: 2-3 million peers –Skype: 5,801,651 peers (for 4/4/06 16:14) Grid systems –EGEE: 10,000 CPU, 10 Petabytes of storage Wireless sensor networks Introduction

Computing in LSD Systems is Difficult Global synchronization is impossible –Synchronization is need after each iteration The input constantly changes –It is hard to keep a large system static Failures are frequent –If a PC fails once a week, a system with a million PCs will have 2 failures every second And of course scalability is necessary Introduction

Current state of LSD computing Embarrassingly parallel tasks –Many interesting problems are not embarrassingly parallel –Used in current grid systems Data storage and retrieval –No computation here –Used in current peer-to-peer systems Introduction

Desired state of LSD computing We want to be able to solve more elaborate problems: –Data mining –Optimization problems In this research we solve the facility location problem in LSD systems. Introduction

The Facility Location Problem We are given: –A set of facilities –A set of clients –A cost function We need to choose: –Which facilities to open –Which facility serves each client Such that the cost is minimized Introduction

Related Work – Data Mining Most of the distributed data mining algorithms were designed for small systems: –Extensive use of global synchronization –Do not tolerate failures Meta-Learning –No synchronization, and tolerates failures –Result quality decreases with the number of nodes Related Work

Related Work – LSD Computing Approaches Gossip –Random walk based –Asymptotically converges to the exact result with high probability Local Algorithms –Eventually achieves the exact result. We are here Related Work

What is a local algorithm? The output of each node depends only on the data of a group of neighbours Eventual correctness guaranteed Size of group may depend on the problem at hand Prerequisites

Local vs. Centralized LocalCentralized 2 link delays, doesn’t depend on the network size 16 link delays, equals to the network diameter Prerequisites

Local Facility Location Architecture Our contribution Proposed by Wolff and Schuster in ICDM Used in local association rules mining algorithm. Our extension Prerequisites

Majority Voting Each node has a poll with votes for the red or green party. Each node is interested to know which party won the elections. Local Majority Voting

Global constants:  λ – majority threshold  γ – bias Input of node u:  c u – the number of local votes  s u – the number of local red votes  G u – a set of neighbors Output of node u:  true if this inequality holds: false otherwise. The input is free to change An ad-hoc output is always available. Its accuracy gradually increases, and eventually it becomes exact. Local Majority Voting

Distributed Facility Location Global constants:  M – a set of possible facility locations  - facility cost Input of node u :  DB u – a set of clients local to node u  G u – a set of neighbors  - service cost Output of node u :  - a set of open facilities, such that Cost(C u ) is minimal. The input is free to change An ad-hoc output is always available. Its accuracy gradually increases, and eventually it becomes exact. Distributed Facility Location

Finding the optimal solution Facility Location is NP-Hard We use a hill climbing heuristic –In this case, hill climbing provides factor 3 approximation In each step we move, open, or close one facility. We stop when the cost doesn’t improve. Distributed Facility Location

Choosing the next step configuration C0C0 C1C1 C2C2 C3C3 C4C4 C5C5 C0C0 Known to every node Distributed over the whole network The local majority vote algorithm can be used to compare the costs of two configurations. Distributed Facility Location

Comparing two configurations. Each node votes in favor of one or another configuration. A configuration that wins the elections has lower cost than the other. C1C1 C2C2 Number of green votes of node u: Number of red votes of node u: Global constants: Distributed Facility Location

Why the ArgMin? To find the best next step out of k possible options, using majority votes, will require O(k 2 ) comparisons. The local ArgMin algorithm makes it in O(k) comparisons in average. Internals of the ArgMin will be described at the end. Distributed Facility Location

The ArgMin interface Global constants:  B – the bias vector Input of node u :  A u – the addendum vector  G u – a set of neighbors Output of node u :  The index i such that: ArgMin is anytime, its output may change, and like majority vote it never terminates! Distributed Facility Location

Speculative execution If we never finish computing the first step, how can we start the second one? The answer: We make a guess and base on it the next step. If the guess turns to be wrong, we backtrack and recompute. Speculative Execution

Every node speculates 1.Eventually, the first iteration will converge to the exact result, and will be the same in every node. 2.Then, the second iteration will be able to converge, and so on until all iterations are exact. 3.When all iterations are exact, every node will output the exact solution. Speculative Execution

ArgMin internals ArgMin uses majority votes to compare pairs of vector elements. ArgMin is also speculative. In every iteration, each configuration is compared to a pivot Local ArgMin

Experimental Results The number of messages each node sends does not depend on the network size Experimental Results

Majority of the nodes provide exact result even if the input continuously changes. Experimental Results

Conclusions We have described a new facility location algorithm suitable for large-scale distributed systems. The algorithm is scalable, communication efficient, and able to efficiently sustain failures.

The End Special thanks to Ran Wolff for his help in supervision of this research