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

1/48 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

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

2/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Outline

1

Introduction

2

Locally Fair Exploration Strategies

3

The Multi-Agent Rotor-Router

4

Conclusions and Perspectives

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

3/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Outline

1

Introduction

2

Locally Fair Exploration Strategies

3

The Multi-Agent Rotor-Router

4

Conclusions and Perspectives

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

4/48 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

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

5/48 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

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

6/48 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.)

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

7/48 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

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

8/48 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

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

9/48 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]

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

10/48 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

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

11/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Outline

1

Introduction

2

Locally Fair Exploration Strategies

3

The Multi-Agent Rotor-Router

4

Conclusions and Perspectives

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

12/48 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?

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

13/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Random Walk

• blivious 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.

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

14/48 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.

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

15/48 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.

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

16/48 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.

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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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]

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

18/48 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.

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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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 2Ω(|V |) in some graph classes.

[Cooper et al., 2009]

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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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

• f 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]

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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The Oldest-First (OF) Strategy is unfair

v(2)

3

v(2)

5

v(2)

4

v(2)

6

v(1)

7

= v(2)

1

v(2)

7

= v(3)

1

v(2)

2

e(1)

1

e(n)

8

The graph Gn.

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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The Oldest-First (OF) Strategy is unfair

D1 C1 E1 B1 A1

One possible cycle of traversals of a block B.

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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The Oldest-First (OF) Strategy is unfair

By a tedious case-by-case analysis, one shows that the behavior of the agent in successive traversals of a given block follows a cyclic pattern. In all the cases, in the time period during which the edge e8 is traversed 4 times, the edge e1 is traversed 6 times. Hence, for any k ∈ {1, . . . , n}, e(k)

1

is traversed 3

2 times as

• ften as e(k)

8 .

Hence, e(1)

1

is traversed (3

2)n times as often as e(n) 8 .

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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The Least-Used-First (LUF) Strategy

A Worst Case Lower Bound on Cover Time

The graph G first consists of 3nC + 1 nodes organized in a chain of 4-node cycles. The graph G also consists of nK additional nodes forming together with one extreme node of the chain a complete graph on nK + 1 vertices. The graph G with nC = 6 and nK = 4. The worst-case cover time of G is at least nK(nK + 1)/2 · nC = Ω(D |E|).

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

25/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Outline

1

Introduction

2

Locally Fair Exploration Strategies

3

The Multi-Agent Rotor-Router

4

Conclusions and Perspectives

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Model definition

Let G = (V , E) be an undirected connected graph with: n nodes, m edges, diameter D. Rotor-router model (r-r for short) The edges outgoing from each node v are arranged in a fixed cyclic order known as a port ordering. Each node v maintains a pointer which indicates the edge to be traversed by an agent during its next visit to v. Initially the pointer points to an arbitrary neighbor.

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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Rotor-router movement

Agents traverse edges of G. (Agents are indistinguishable.) In every step each of k agents perform: If the agent is located in v, it is directed along the edge indicated by the pointer. The pointer is advanced to the next edge based on the port

• rdering of v.

(If a node contains more than one agent at a node at the start of a round, the agents will be sent out one after another on different edges according to the rotor-router mechanism.) In every step each agent traverses one edge. (The rounds are synchronous.)

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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Example

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

28/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Example

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

28/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Example

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

28/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Example

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

28/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Example

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

28/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Example

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

29/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Work for single agent

The behavior of the rotor-router for a single agent is well understood. History of results for the single rotor-router For any graph G and starting from arbitrary configuration (cyclic

• rdering, pointer)

Agent explores any graph in time O(nm) [Wagner at al. 1999] After the initial stabilisation period of O(nm) the agent keeps repeating the same Eulerian cycle of the directed symmetric version G of graph G [Bhatt et al. 2002] The cycle is established within Θ(Dm) steps in the worst case [Bampas et al. 2009]

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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Work for multiple agents

There is only one paper considering multiple rotor-router. Adding an agent cannot slow down the exploration [Yanovski et al. 2003]. The multi-agent rotor-router eventually visits all edges of the graph a similar number of times [Yanovski et al. 2003]. Many questions in the multi-agent case remain open.

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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Goals

Here we focus on the multiple rotor-router on a cycle. The goal is to analyze the performance of the multiple rotor-router in three scenarios cover time (i.e. the time before each node of the graph is covered by at least one agent) for the worst initial placement

• f the agents (e.g. agents initially located on the same node
• f the ring)

cover time for the best initial placement of the agents (e.g. agents distributed uniformly on the ring) return time (i.e. time between two successive visits in a vertex after sufficiently many steps) We compare times in these scenarios between multiple rotor-router and expected times of multiple random walk. We develop techniques for analyzing the multiple rotor-router that may be useful in the case of general graphs.

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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Results

Model Cover time Return time worst placement best placement k-agent rotor-router Θ(n2/ log k) Θ(n2/k2) Θ(n/k) k random walks (expectations) Θ(n2/ log k) in literature Θ

• n2

k2 log2 k

• Θ(n/k)

in literature

Table: The cover time of the multi-agent roter-router on the ring compared to multiple random walks (for k < n1/11). [Klasing et al., 2013]

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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Techniques for the Multi-agent Rotor-Router

delayed deployments: allowing the occasional stopping of some of the agents without affecting asymptotic cover time (applicable to general graphs) agent domains: we describe states in the evolution of the system in which particular agents cover nearly disjoint, dynamically changing parts of the graph (applicable to the ring) continuous time approximation: asymptotic description of the behavior of the agents on the ring

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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Delayed deployments

Delayed deployments are a tool developed for analyzing the multi-agent rotor-router (in general graphs). In delayed deployment D it is possible to stop agents. Agents that are stopped in vertex v in time interval [t1, t2] remain in v and resume their movements in step t2 + 1. Notation: D(v, t) - number of agents stopped in v in step t. nD

v (t) - number of visits of agents in deployment D in node v

until step t. R[k] - undelayed deployment of k-agent rotor-router.

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The slow-down lemma

Informally the following lemma states that stopping more agents cannot increase the number of visits at any vertex in any step. Lemma Let D1 and D2 be two delayed deployments, such that: ∀v ∈ V ∀t D1(v, t) ≥ D2(v, t). Then: ∀v ∈ V ∀t nD1

v (t) ≤ nD2 v (t).

The lemma implies that adding agents cannot slow down the exploration. Corollary ∀v ∈ V ∀t nR[k−1]

v

(t) ≤ nR[k]

v

(t).

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The slow-down lemma

The following lemma shows a relation between cover time of delayed and undelayed deployments. Lemma (the slow-down lemma) Let R[k] be a k-rotor router system with an arbitrarily chosen initialization, and let D be any delayed deployment of R[k]. Suppose that deployment D covers all the vertices of the graph after T = C(D) rounds, and in at least τ of these rounds, all agents were active in D. Then, the cover time C(R[k]) of the system can be bounded by: τ ≤ C(R[k]) ≤ T.

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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The slow-down lemma

If the deployment D is defined so that agents in D are delayed in at most a constant proportion of the first C(D) rounds, then the above inequalities lead to an asymptotic bound on the value of the undelayed rotor-router, C(R[k]) = Θ(C(D)).

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Agent domains

The domain of agent i after round t is the set Vi(t) of all nodes such that the i-th agent was the last agent visiting the node until round t, inclusive.

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

39/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Agent domains

Example (on the line): V2 V1 V0

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

39/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Agent domains

Example (on the line): V2 V1 V0

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

39/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Agent domains

Example (on the line): V2 V1 V0

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

39/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Agent domains

Example (on the line): V2 V1 V0

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

39/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Agent domains

Example (on the line): V2 V1 V0

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

39/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Agent domains

Example (on the line): V2 V1 V0

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

39/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Agent domains

Example (on the line): V2 V1 V0

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

39/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Agent domains

Example (on the line): V2 V1 V0

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

39/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Agent domains

Example (on the line): V2 V1 V0

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

39/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Agent domains

Example (on the line): V2 V1 V0

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

39/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Agent domains

Example (on the line): V2 V1 V0

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

39/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Agent domains

Example (on the line): V2 V1 V0

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

39/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Agent domains

Example (on the line): V2 V1 V0

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

39/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Agent domains

Example (on the line): V2 V1 The set of nodes is partitioned between the agents into subsets which we call domains. Agents are traversing their domains and during each cycle can capture one node from neighboring domain (or at least one node not belonging to any domain). Agents with smaller domains will visit borders more frequently thus smaller domains will grow. Intuitively the system should converge to domains of equal sizes.

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Agent domains

Lemma If initially every domains has size at least 22k + 4 then Size of a domain will never decrease below 11k After a sufficiently large number of steps the sizes of adjacent domains will differ by at most O(1).

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

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Continuous-time approximation

Domains of the agents are ordered along the ring as V0(t), V1(t), . . . , Vk (V0(t) denotes unexplored nodes and Vi(t) nodes belonging to the domain of i-th agent). Within T rounds, the agent enlarges its domain by about T/(2|Vi(t)|) to the left, and T/(2|Vi(t)|) to the right, thus by about T/|Vi(t)| in total. This movement is counteracted by the moves of the adjacent agents occupying domains Vi−1 and Vi+1 Consequently, we define the continuous-time approximation of the rotor-router through the set of differential equations: dνi(t) dt = 1 νi(t) − 1 2νi−1(t) − 1 2νi+1(t), for 1 ≤ i ≤ k, where νi(t) = |Vi(t)|, for all 1 ≤ i ≤ k.

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Continuous-time approximation

Whereas the above differential model provides the basic intuition for many of the proofs, the main difficulty lies in taking into account the differences between the continuous-time model and the real rotor-router. In particular, we have to consider the position of the agent within its domain, the discrete changes of the domain size in time, and the initial pointer arrangement in the unvisited part

• f the ring.

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

43/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Outline

1

Introduction

2

Locally Fair Exploration Strategies

3

The Multi-Agent Rotor-Router

4

Conclusions and Perspectives

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs

44/48 Introduction Locally Fair Exploration Strategies The Multi-Agent Rotor-Router Conclusions and Perspectives

Locally Fair Exploration Stategies

We have shown that locally fair strategies in undirected graphs can closely imitate random walks. We obtained an exploration which is fair with respect to all edges, and efficient in terms of cover time. The fairness criterion has to be chosen much more carefully than for symmetric directed graphs: Least-Used-First works, but Oldest-First does not.

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Locally Fair Exploration Stategies

It would be interesting to study modified notions of equity, which are inspired by random walks which select the next edge to be traversed with non-uniform probability. For example, it is possible to decrease the general-case bound

• n the cover time of a random walk to O(|V |2 log |V |), by

applying a probability distribution which reflects the degrees

• f the neighbors of the current node (give preference to

neighbours with lower degree).

[S. Ikeda, I. Kubo, and M. Yamashita. Theoretical Computer Science, 2009]

It is an open question whether a similar bound can be obtained in the deterministic sense using a derandomized strategy.

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The Multi-Agent Rotor Router

We have shown that the multi-agent rotor-router and the parallel random walk have similar speed-up characteristics w.r.t. the number of deployed agents, at least in terms of cover time and return time on the ring. This work may also be seen as a step in the direction of understanding and characterizing the behavior of the multi-agent rotor-router in graphs different from the ring. Some of the techniques developed for the ring, in particular the analysis based on delayed deployments, are also applicable in the general case.

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Future work

Understanding the behavior of the rotor-router for other graph classes Work in progress: we have shown a logarithmic lower bound

• n speed up in general graphs.

Maybe it is possible to analyze the rotor-router on grids or trees using domains.

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Overall Perspectives

Perspectives improvement of existing bounds for the exploration problem, in specific networks exploration with multiple agents large-scale, inhomogeneous networks dynamic networks fault-tolerance, robustness exploration in “dangerous” networks

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Thank You

Ralf Klasing Efficient Exploration of Anonymous Undirected Graphs