Efficient Exploration of Anonymous Undirected Graphs Ralf Klasing - PowerPoint PPT Presentation

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives Efficient Exploration of Anonymous Undirected Graphs Ralf Klasing CNRS – LaBRI – Université de Bordeaux DISPLEXITY 2014 1/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives Outline Introduction 1 Locally Fair Exploration Strategies 2 The Multi-Agent Rotor-Router 3 Conclusions and Perspectives 4 2/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives Outline Introduction 1 Locally Fair Exploration Strategies 2 The Multi-Agent Rotor-Router 3 Conclusions and Perspectives 4 3/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives Graph Exploration Problem A mobile entity, called agent , is placed on some vertex of an unknown anonymous graph. The agent is allowed to traverse edges of the graph. The goal is to perpetually visit all the vertices of the graph. Possible optimization criteria Fast "first exploration" of every vertex Regular/periodic traversal of vertices/edges in the long run Fair traversals of vertices/edges in the long run 4/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives Graph Exploration Motivation network exploration, crawling webs, sampling of nodes and gathering statistics, ranking nodes network maintenance by a software agent work and load balancing problems 5/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives Unknown, anonymous graphs Unknown graph Unknown topology Unknown size Anonymous graph No node labeling (Some extra information on the edges incident to a vertex may be provided.) 6/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives Memory constraint Objective Use agents with a memory of constant size Justifications Simple and cost effective agents Facilitates design and analysis of algorithms 7/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives Objectives of exploration Parameters/properties to consider Time until all nodes have been visited at least once (cover time) Time between two subsequent visits to a node, in the limit (return time) Convergence to some limit frequency of visits to specific nodes/edges 8/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives Considered exploration strategies Probablistic strategies Random Walk Deterministic strategies Basic Walk Rotor-Router (Propp machine, Edge Ant Walk) Survey: [Gąsieniec, Radzik, 2008] 9/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives In this talk Locally Fair Exploration Strategies The Multi-Agent Rotor-Router 10/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives Outline Introduction 1 Locally Fair Exploration Strategies 2 The Multi-Agent Rotor-Router 3 Conclusions and Perspectives 4 11/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives The Problem Exploration of an unknown anonymous graph G = ( V , E ) . Goal : visit all vertices of the graph, regularly traversing its edges. Situation : At each discrete moment of time, the agent is located at a node of the graph, and is provided with only a local view of the adjacent edges of the graph. Question : Which adjacent edge to take next? 12/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives Random Walk oblivious exploration strategy edge used by the agent to exit its current location is chosen with equal probability from among all the edges adjacent to the current node Results (1) Within polynomial time, the walk visits all of the vertices of the graph. (2) Within polynomial time, the walk stabilizes to the steady state, and henceforth all edges are visited with the same frequency. 13/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives The Problem Design local exploration strategies which derandomize a random walk in a graph in an attempt to achieve the above stated properties in the deterministic sense of worst-case performance . The next vertex to be visited should depend only on the values of certain parameters associated with the edges adjacent to the current node. Gives rise to the definition of locally equitable strategies , i.e. strategies, in which at each step the agent chooses from among the adjacent edges the edge which is in some sense the “poorest”, in an effort to make the traversal fair. 14/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives Two natural notions of equity An exploration is said to follow the Oldest-First (OF) strategy if it directs the agent to an unexplored neighboring edge, if one exists, and otherwise to the neighboring edge for which the most time has elapsed since its last traversal, i.e. the edge which has waited the longest. An exploration is said to follow the Least-Used-First (LUF) strategy if it directs the agent to a neighboring edge which has so far been visited by the agent the smallest number of times. 15/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives OF and LUF in symmetric directed graphs The Oldest-First strategy is equivalent to the well-established “ Propp machine ” model: Edges exiting each node have successive labels. The next edge to be traversed is selected by a pointer. After this edge is traversed, the pointer moves on to the edge with the next label, in a cyclic way. Least-Used-First : For any time moment, the number of visits to any two edges outgoing from the same vertex can differ by at most 1. 16/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives Known Results In the directed case, both of the described locally fair exploration stratagies are known to preserve properties (1) and (2) of the random walk. More precisely, for a symmetric directed graph of diameter D , any exploration which follows such a strategy achieves a cover time of O ( D | E | ) and stabilizes to a globally fair traversal of all the edges. [V. Yanovski, I. A. Wagner, and A. M. Bruckstein, Algorithmica , 2003] 17/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives (OF) and (LUF) in undirected graphs Here, we consider the Oldest-First and Least-Used-First strategies when applied to undirected graphs. 18/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives Results Oldest-First (OF) in undirected graphs In some classes of undirected graphs, any exploration which follows the Oldest-First strategy is unfair, with an exponentially large ratio of visits between the most often and least often visited edges. There exist explorations following the Oldest-First strategy which have exponential cover time of 2 Ω( | V | ) in some graph classes. [Cooper et al., 2009] 19/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives Results Least-Used-First (LUF) in undirected graphs Any exploration of an undirected graph which follows the Least-Used-First strategy is fair, achieving uniform distribution of visits to all edges. Any exploration of an undirected graph which follows the Least-Used-First strategy achieves a cover time of O ( D | E | ) , where D denotes the diameter. This bound is tight. [Cooper et al., 2009] 20/48 Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs