LAC optimisationNed (Mohaned) HASSOUNMaster in Intelligent Systems

Supervised by Sharon Wood (Uni. of Sussex) and Michael Ratford (Motorola)

IntroductionProblem DefinitionAssimilation of this problem with an existing oneDefinition of Graph partitioning problem2 important ways of PartitioningPresentation of Multilevel approachGraph partitioning algorithmsResults

Basic Components: CellA cell provides radio signal to a mobile station in a certain areaDifferent sizes: up to 50 km

Theoritical view Real-world situation

Network HierarchyPLMN controls many Mobile Switching Centres (MSC)Mobile Switching Centre:computer that places the calls, and takes and receives data from the subscriber or from PSTN MSC controls 1 to many Location Area (LA)Location Area Cells are grouped into location Area.Identified by Internationally unique code composed of several elements. Among them, Location Area Code assigned by the operatorEach cell stores the appropriate identifier.The assignment is done by operator.

PLMNMSCLocation AreaLocation AreaHierarchical structure: Public Land Mobile Network (PLMN) to a cell.

Hand-overHandover between cells:When a mobile station moves from one cell to another, the signal has to be passed to the station of the new cell. This is known as handover.

Mobiles Location updated in the MSC register to be able to forward any call

Handover between MSC:Most general form of handoverCall forwarded to the new MSC

Two Major ProblemsPing-pong handover: mobile station crosses very frequently the MSC boundaries.Consequence: For incoming calls, the last reported MSC may no longer be valid, the call will fail.

Paging problem:Too many mobiles crossing boundaries provokes overload of the connections between MSCs.

Location Area 1Location Area 2

ObjectivesUpgrade the expensive material (Not really a good idea)ORRearrange cells into LAs, such that:It minimises the number of mobiles near the MSC boundaries, i.e. minimise handovers.Constraint: MSCs can deal with limited number of mobiles. Thus, LAs have limited number of cells.Maximise this limitation to reduce number of LAs.

Data available:- Concentration of mobiles in a cell- Estimation of handovers between cells

Bad solution better solutione.g. Rearrange in England

Problem Identification / SolutionProblem type:NP-complete [Garey et al. 76]: untractable.e.g., 100 cells into 2 groups, 2100 1032 solutions. Combinatorial optimisation problem: all or none type, no step size.Not a continuous optimisation problem: No notion of gradient.

Similar problems:Geometric clustering: cluster groups according to position in space.Not appropriate, geographical info not availableQuadratic Assignment Problem: assign finite number of cells to locationsVery similar, except number of cells in LA not fixedGraph Partitioning Problem: Define problem as a graph modelMost appropriate

K-way graph partitioning problemGiven a weighted graph G = (V, E) A cell i is represented by a vertex vi and its density of mobiles a vertex weight wv(i)Handover between two cells represented by an edgeObjectives: Partition G into k disjoint subsets S1, S2, , Sk with following constraints:Handover minimisation: Minimise the edge cut size, i.e. total weight of edges whose incident vertices belong to different subsets.Partition into k equal size subsets

ComparisonComparison of objectives: It is quite similar.Objectives 2 and 3 are valid if subset size is maximised and k is reduced at maximum

2 Partitioning ApproachesK-way partitioning:Partition at once in k subsetsProblem: very difficult

Recursive Bisection:Most commonly used technique for partitioning It first divides the graph into 2 equal sized subsets.And recursively divide the 2 subsets, until size criterion is satisfied.

Inconvenient: Lack of global informationDifficult to maintain good balancing conditionAdvantage: It simplifies the problemVery popular!

S1S3S2S4S1S2G

Multilevel ParadigmIntroduced by Barnard & Simon (1994).3 phases:Coarsening phase: Coarsen down to a few hundred vertices. The graph G0 is transformed into sequence of smaller graphs G1, G2,, Gm Partitioning phase: Partitioning heuristic used to compute k-way partition of the graph Gm = (Vm, Em)Uncoarsening phase: The partition Sm of Gm is projected back through intermediate partitions Use of a Refinement Method

Advantages:Robust and SimpleSpeed up the search processImprove the solution quality

Coarsening phaseA set of vertices of Gi is combined to form vertex v of Gi+1. Approach: collapsing adjacent vertices.Aim: Finding a matching of a graph, i.e. a set of edges, no two of which incident on the same vertex.Coarsening step with matching method

Coarsening phase (cont.)Edge Contraction:2 Vertices form a vertex with their weight addedEdge between them removedOther edges preserved

Various ways to match vertices:Random matching: vertices are visited in random order. Heavy edge matching: select edge with highest weight [Karypis 95] Light Edge matching: select edge with lowest weight. Important for certain partitioning algorithms like Kernighan-Lin [Bui-Jones 93]

Partitioning phaseNo single method is always best (Hendrickson & Leland, 1994)However, 3 criteria to define best method:Ability to respond to different input scaleSpeed: not a real constraintSolution quality (minimum cut-size with balanced subsets)

Various types of algorithms:Combinatorial methodsStochastic MethodsGain-based MethodsS1S3S2S4

Combinatorial MethodsRely on graph connectivityGroup together vertices that are highly connected

Methods:Spectral algorithm: constructs geometric representationGreedy Graph growing algorithm:Start from empty subsetChoose a random vertex and add it to subsetSelect a neighbouring vertex with highest edge weightUntil subsets maximum size is reached

Starting vertexAdded verticesRemaining verticesCut-size

Stochastic MethodsMethods with decisions based on uncertainty3 methods:Simulated AnnealingRandomly chooses solution and Wanders into search space, uphill or downhillEvaluation: Offers good results, but not better than any other. Long runtime.Genetic AlgorithmEvolutionary-based technique: Combine good solutionsEvaluation: powerful, but limited when graph size is important Ant Colony Optimisation:Very recentBased on emergent behaviour of ants when foragingDrawback: Complex implementation

Move-based MethodsIterative in nature

Kernighan-Lin/Fidduccia-Mattheyses (KL/FM) Start from initial solutionSeries of passesMove bordering vertex to neighbouring subset according to GainGain: benefit of reducing the cut-sizeChoose maximum gain, but allow negative gain to move out of local minimaTo avoid infinite loop, vertex allowed to move once (locking mechanism)At the end, recall the best solution found

Advantage: Simple, flexible at handling various graph sizes and objectivesInconvenient: Short-sighted. But variations exist (e.g. relaxed locking mechanism)

Uncoarsening phasePartition Pm of the coarser graph is projected back to the original graph,

An optimal solution Pi+1 is not anymore optimal after projection Pi , but not far from it.Need to use refinement algorithm: KL/FM Refinement are efficient (Karypis 97)Start from good solutionFew swaps converge to good solution

Implemented Solution

Multilevel paradigm:Coarsening phase:Random/Heavy-edge/Light-Edge matchings Partitioning PhaseGreedy Graph Growing Algorithm (GGG)

Uncoarsening PhaseKL/FM algorithm

Testbed:Data: Graph with 1657 vertices, 24707 edgesOver 100 runs per experiment

Result: MatchingPartition into 10 subsetsHeavy Edge Matching (HEM) provides best resultRandomHEMLEM

Result: Partitioning3 variants of Greedy Graph Growing (GGG) testedFor partitions into 2 to 26 subsetsNo exponential increase: GGG is relatively constantVery reliable

Refinement: KL/FMTest improvement of cut-size during uncoarsening phase for a partition into 10 subsets.

High improvement after each uncoarsening step

Zoom in to a pass: Able to avoid local minima

Refinement: KL/FM (cont.)Weight Evolution:System of balancing relaxation:Allows subsets to have weight overpassing balancing conditionAfter each uncoarsening step, relaxation tightened Result:Explore more solutionsBe flexible: depending on average vertex weightAt the end, all subset weights are under threshold, and maximised

Comparison with librariesComparison with CHACO (Hendrickson 1998) and METIS (Karypis 1997) librariesResult:Algorithm outperforms both libraries in cutsizeWeight balancing are nearly similarBut in term of speed, our software is much slower

ConclusionProblem formulation and study:NP-complete, CombinatorialFrom a real-world problem to a Graph partitioning problemDescription: Cells as vertices, handovers as edgesObjectives: Minimisation of the edge cut size and subset balancing.Multilevel paradigm as a framework for researchEfficacy and Simplicity of KL/FM and GGG over other techniquesImprovements of solution compared to other available libraries

ConclusionVisualisation in geographical coordinates For partitioning into 2 subsets and 10 subsetsWITHOUT knowledge of coordinate position of the cells

THANK YOUAny Question?