DISTRIBUTED GRAPH TRANSFORMATIONS

SUPPORTED BY MULTI-AGENT SYSTEMS

Adam Sędziwy, PhDAGH University of Science and

Technology Cracow, Poland

1. Objectives2. Graph transformations. The

complexity3. Faciliating graph transformations4. The role of an agent system5. Case study6. Examples of applications7. Summary

OUTLINE

Graph models are widely used for describing a range of systems. For that reason we focus on:

Developing the eff ective method of graph-based computations

Computations have to be performed in parallel

Computations are intended to be carried out by an agent system

OBJECTIVES

A graph grammar is a generalization of a string grammar notion:

The example:

Terminal and nonterminal symbols are replaced by terminal and nonterminal vertices (context-free grammar) or graphs (context grammar)

GRAPH GRAMMARS

The sample graph production:

Q: how to embed R in G-X ? A: Embedding rule is associated with each production.1. Edges {i,X} where i≠a , should be removed2. The (former) edge {a,X} should be replaced with {a,Y} and

{a,e}

GRAPH GRAMMARS (CONT.)

The basic problem: exponential complexity of parsing and membership problem

A graph grammar's parsing complexity vs expressiveness

If some restrictions are imposed on a grammar then the complexity reduction is possible (edNLC, ETPL(k), … )

COMPLEXITY ISSUES

A graph should be decomposed into complementary subgraphs. – Q: How?– A: It depends on a problem: equally sized subgraphs, minimized number of connections, minimized redundancy,...

Processing is performed in parallel with some data exchange.

The desirable property: centralized graph grammar rules apply to be used in a distributed environment

FACILITATING GRAPH COMPUTATIONS –

"WHAT IS TO BE DONE”?

Decomposition based on a vertex replication

GRAPH DECOMPOSITION – VERTEX REPLICATION

Motivation: applicable to subproblems requiring an autonomous activity (decision making) rather than executing a deterministic algorithm

Particular subtasks (e.g., local graph transformations) are performed by agents

An agent's knowledge is represented by a graph structure (data) and a grammar (action)

Agents have to cooperate to achieve their goals

AGENT SYSTEM ON A DISTRIBUTED GRAPH

Centralized graph decomposition

The scenario of a local production

A node’s incorporation

Confl icts

CASE STUDY

The graph G in a centralized and ditributed form

CASE STUDY - DECOMPOSITION

The agent A1 aims at applying the production P to G1

A1 has to obtain an exclusive access to the vertex v

CASE STUDY – THE LOCAL PRODUCTION

Replicas of v have to be removed from other subgraphs incorporation of the vertex v to G1

To preserve the overall consistency the agent A1 requests other agentsto lock relevant vertices (e.g., u,w) while incoroprating v

The source of confl icts!

CASE STUDY - INCORPORATION

A1 determines a set HA (L) of other agents hosting vertices of L

A1 follows 2PC:PHASE 1: A1 requests agents A iHA(L) to block vertices of L

A i responds with AGREEMENT=YES | NO:BLOCKED | NO:NONEXIST

PHASE 2: [AGREEMENT=YES] A1 sends COMMIT; A i responds with a

relevant part of L [AGREEMENT=NO:BLOCKED] A1 sends ABORT to HA(L) and

restarts with a random delay [AGREEMENT=NO:NONEXIST] A1 sends ABORT to HA(L) and

tries to determine HA(L) again

CASE STUDY – INCOROPRATION (CONT.)

CASE STUDY – INCOROPRATION (CONT.)

The agent A1 has incorporated the vertex v

Embedding rules: {p,q} is replaced by {p,a} {p,r} is replaced by {p,b} {u,v} is replaced by {u,a} {w,v} is replaced by {w,b} {q,v}, {r,v} are removed

CASE STUDY - PRODUCTION

P

A border node v may be shared by all complementary graphs (pessimistic case), i.e., O(N).

Pessimistically, the number of vertices to be locked is:

The additional issue: a number of exchanged messages

Solution: the another approach to the graph decomposition – the graph slashing

CONFLICTS

Slashed graphs concept. A single edge is shared by exactly two agents

GRAPH DECOMPOSITION – SLASHING

𝑛𝐿=𝑂 (𝑑𝑚𝑎𝑥 )

CAD system for the architectural design

Separate design processes and graph models for the interior and exterior

Both processes may clash on shared elements (e.g., windows): coordination required

EXAMPLE 1

The control in smart lighting systems

The lighting control is based on sensor data (reflecting an environment state)

Lamps’ performance is controlled by relevant graph grammar productions triggered by changes in an environment

EXAMPLE 2

The graph-based representation (GBR) of systems is the suitable formal approach to model a range of systems.

GBR may be easily decomposed becoming an environment for an agent system deployment

A decomposition method depends on a problem. Following factors should be considered: Statistical properties of an agent system’s behavior

(avoiding conflicts) A number of messages required for completing atomic

operations

SUMMARY

1. L. Kotulski, A. Sędziwy, B. Strug: Heterogeneous graph grammars synchronization in CAD systems supported by hypergraph representations of buildings , Expert Systems with Applications, Elsevier, http://dx.doi.org/10.1016/j.eswa.2013.07.043

2. I. Wojnicki, S. Ernst, L. Kotulski, A. Sędziwy: Advanced Street Lighting Control , Expert Systems with Applications, Elsevier, http://dx.doi.org/10.1016/j.eswa.2013.07.044

3. A. Sędziwy: Eff ective Graph Representation for Agent-Based Distributed Computing , Lecture Notes in Computer Science, Vol. 7327, pp 638-647, Springer, 2012

4. L. Kotulski, A. Sędziwy: Parallel graph transformations supported by replicated complementary graphs , Lecture Notes in Computer Science, Vol. 6594, pp 254-264, Springer, 2011

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