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Intro Searching Model Results Conclusion Distributed Chasing of Network Intruders elia Blin 1 Pierre Fraigniaud 2 Nicolas Nisse 3 L Sandrine Vial 1 1 Universit e dEvry, France 2 CNRS, Universit e Paris-Sud, France 3 Universit e

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1/36 Intro Searching Model Results Conclusion

Distributed Chasing of Network Intruders

L´ elia Blin1 Pierre Fraigniaud2 Nicolas Nisse3 Sandrine Vial1

1Universit´

e d’Evry, France

2CNRS, Universit´

e Paris-Sud, France

3Universit´

e Paris-Sud, France.

R´ eunion FRAGILE, septembre 06

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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2/36 Intro Searching Model Results Conclusion

Plan

1

Introduction

2

Graph Searching

3

Our model

4

Our Results

5

Conclusion

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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3/36 Intro Searching Model Results Conclusion

Graph Searching

The Problem In a given network, whose edges are contaminated ; a team of searchers aims at clearing the network. We want to find a strategy that clears the netwok using the fewest searchers as possible. Motivations network security, speleological rescue, ...

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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3/36 Intro Searching Model Results Conclusion

Graph Searching

The Problem In a given network, whose edges are contaminated ; a team of searchers aims at clearing the network. We want to find a strategy that clears the netwok using the fewest searchers as possible. Motivations network security, speleological rescue, ...

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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4/36 Intro Searching Model Results Conclusion

Our Goal

Previous Algorithms input : a graph G

  • utput : an optimal search strategy for G

Our Approach : a distributed algorithm input : a graph G and a starting point v0 ∈ V (G)

  • utput : an “optimal” search strategy computed online by

the searchers themselves

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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4/36 Intro Searching Model Results Conclusion

Our Goal

Previous Algorithms input : a graph G

  • utput : an optimal search strategy for G

Our Approach : a distributed algorithm input : a graph G and a starting point v0 ∈ V (G)

  • utput : an “optimal” search strategy computed online by

the searchers themselves

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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5/36 Intro Searching Model Results Conclusion Definitions Connected

Plan

1

Introduction

2

Graph Searching Basic definitions Connected search

3

Our model

4

Our Results

5

Conclusion

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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6/36 Intro Searching Model Results Conclusion Definitions Connected

Search Strategy, Parson. [GTC,1978]

Sequence of three basic operations,. . .

1 Place a searcher at a vertex of the graph ; 2 Move a searcher along an edge of the graph ; 3 Remove a searcher from a vertex of the graph.

. . . that must result in clearing the graph An edge is cleared when it is traversed by a searcher. A clear edge e is recontaminated if it exits a non-guarded path between e and a contaminated edge. We want to minimize the number of searchers. Let s(G) be the search number of the graph G.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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7/36 Intro Searching Model Results Conclusion Definitions Connected

Simple Examples : Path and Ring

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7/36 Intro Searching Model Results Conclusion Definitions Connected

Simple Examples : Path and Ring

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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7/36 Intro Searching Model Results Conclusion Definitions Connected

Simple Examples : Path and Ring

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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7/36 Intro Searching Model Results Conclusion Definitions Connected

Simple Examples : Path and Ring

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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7/36 Intro Searching Model Results Conclusion Definitions Connected

Simple Examples : Path and Ring

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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7/36 Intro Searching Model Results Conclusion Definitions Connected

Simple Examples : Path and Ring

s(Path)=1

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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7/36 Intro Searching Model Results Conclusion Definitions Connected

Simple Examples : Path and Ring

s(Path)=1

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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7/36 Intro Searching Model Results Conclusion Definitions Connected

Simple Examples : Path and Ring

s(Path)=1

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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7/36 Intro Searching Model Results Conclusion Definitions Connected

Simple Examples : Path and Ring

s(Path)=1

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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7/36 Intro Searching Model Results Conclusion Definitions Connected

Simple Examples : Path and Ring

s(Path)=1

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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7/36 Intro Searching Model Results Conclusion Definitions Connected

Simple Examples : Path and Ring

s(Path)=1

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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7/36 Intro Searching Model Results Conclusion Definitions Connected

Simple Examples : Path and Ring

s(Path)=1 s(Ring)=2

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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8/36 Intro Searching Model Results Conclusion Definitions Connected

Drawbacks of Parson’s model :

In the standard settings :

1 the topology of the graph is known ; 2 a search strategy is performed in a sequential synchronous

way ;

3 searchers can be placed anywhere in the graph.

In a real network :

1 searchers have no knowledge about the topology ; 2 networks are asynchronous ; 3 searchers cannot be teleported,

communications must be secure, ⇒ the clear part must induce a connected subgraph.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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9/36 Intro Searching Model Results Conclusion Definitions Connected

Monotone connected graph searching

Monotone connected search strategy connectedness : At any step of the strategy, the clear part must induce a connected subgraph. monotony : No recontamination ever occurs. Once an edge has been cleared, it remains clear until the end. Monotone connected search number Let mcs(G) be the smallest number of searchers required to clear the graph in a monotone connected manner.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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10/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of Connectedness : Example in a tree

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10/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of Connectedness : Example in a tree

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Cost of Connectedness : Example in a tree

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10/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of Connectedness : Example in a tree

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Cost of Connectedness : Example in a tree

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10/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of Connectedness : Example in a tree

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10/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of Connectedness : Example in a tree

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Cost of Connectedness : Example in a tree

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Cost of Connectedness : Example in a tree

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10/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of Connectedness : Example in a tree

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10/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of Connectedness : Example in a tree

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Cost of Connectedness : Example in a tree

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10/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of Connectedness : Example in a tree

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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10/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of Connectedness : Example in a tree

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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10/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of Connectedness : Example in a tree

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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10/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of Connectedness : Example in a tree

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10/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of Connectedness : Example in a tree

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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10/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of Connectedness : Example in a tree

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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10/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of Connectedness : Example in a tree

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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10/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of Connectedness : Example in a tree

mcs(T) ≥ 4 > s(T)

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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11/36 Intro Searching Model Results Conclusion Definitions Connected

Cost of connectedness : Related Work

In terms of number of searchers For any tree T, s(T) ≤ mcs(T) ≤ 2 s(T) − 2 (tight). Barri` ere, Fraigniaud, Santoro and Thilikos. [WG, 2003] For any graph G, s(G) ≤ mcs(G) ≤ (1 + log n) s(G) Fraigniaud and Nisse [LATIN, 2006] Complexity of monotone connected Graph Searching NP-complete in general case.

  • Gustedt. [DAM, 1993]

Linear in case of trees. Barri` ere, Flocchini, Fraigniaud and Santoro. [SPAA, 2002]

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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12/36 Intro Searching Model Results Conclusion Search Network Agents

Plan

1

Introduction

2

Graph Searching

3

Our model Monotone Connected Search Asynchronous Anonymous Network Mobile Agents

4

Our Results

5

Conclusion

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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13/36 Intro Searching Model Results Conclusion Search Network Agents

Our Goal

For any connected, asynchronous, and anonymous network G, and any v0 ∈ V (G), we propose a distributed algorithm that enables clearing G in a connected way, using searchers starting from v0, and initially unaware of G.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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14/36 Intro Searching Model Results Conclusion Search Network Agents

Monotone connected graph searching

Alternative definition v0 ∈ V (G) is the homebase of the searchers. Initially, any searcher is placed at v0. One single operation is allowed : move a searcher along an edge if it does not implie recontamination. Remarks The homebase remains clear during the whole strategy. mcs(G, v0)

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v mcs(G,u)=2

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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15/36 Intro Searching Model Results Conclusion Search Network Agents

Importance of the homebase

u v mcs(G,v)=3

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16/36 Intro Searching Model Results Conclusion Search Network Agents

Asynchronous Network

v v

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Asynchronous Network

v v

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Asynchronous Network

v v

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Asynchronous Network

v v

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Asynchronous Network

v v

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16/36 Intro Searching Model Results Conclusion Search Network Agents

Asynchronous Network

The searchers cannot distinguish one graph from the other. The two red searchers have the same local behaviour. An extra searcher will be called in both cases. v v

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17/36 Intro Searching Model Results Conclusion Search Network Agents

Cost of asynchronicity

More generally There exist graphs G such that, any distributed asynchronous graph searching protocol requires mcs(G) + 1 searchers to clear G in a connected monotone way. [FHL05] Coordinator The extra searcher, the coordinator is used to synchronize the

  • ther searchers.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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18/36 Intro Searching Model Results Conclusion Search Network Agents

Anonymous Network

Unknown unknown topology unknown size (no upper bound) Anonymous No vertex labeling Local edge labeling Local Memory whiteboards are specific zone of local node memory, accessible in fair mutual exclusion.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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19/36 Intro Searching Model Results Conclusion Search Network Agents

Example of an anonymous graph

1 4 3 2 2 3 4 1 2 1 3

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20/36 Intro Searching Model Results Conclusion Search Network Agents

Mobile Agents

Searchers autonomous mobile computing entities with distincs IDs, running the same algorithm, Mealy automata.

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20/36 Intro Searching Model Results Conclusion Search Network Agents

Mobile Agents

The decision of a searcher... leaving a node via some specific port, switching state, writing on the whiteboard, ... is local and depends on : current state, content of the node’s whiteboard, incoming port number.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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21/36 Intro Searching Model Results Conclusion Search Network Agents

Related Work

Polynomial algorithms in specific topologies Trees. Barri` ere, Flocchini, Fraigniaud and Santoro. [SPAA, 2002] Hypercubes. Flocchini, Huang and Luccio. [IPDPS, 2005] Chordal rings, Tori, Meshes... For each of these classes of graphs :

1 mcs(G) + 1 searchers are used ; 2 each searcher possesses O(log n) bits of memory ; 3 the size of the node’s whiteboard is O(log n) bits. L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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21/36 Intro Searching Model Results Conclusion Search Network Agents

Related Work

Polynomial algorithms in specific topologies Trees. Barri` ere, Flocchini, Fraigniaud and Santoro. [SPAA, 2002] Hypercubes. Flocchini, Huang and Luccio. [IPDPS, 2005] Chordal rings, Tori, Meshes... For each of these classes of graphs :

1 mcs(G) + 1 searchers are used ; 2 each searcher possesses O(log n) bits of memory ; 3 the size of the node’s whiteboard is O(log n) bits. L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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22/36 Intro Searching Model Results Conclusion Theorem Algo

Plan

1

Introduction

2

Graph Searching

3

Our model

4

Our Results Theorem Our Algorithm

5

Conclusion

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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23/36 Intro Searching Model Results Conclusion Theorem Algo

Theorem

For any connected, asynchronous, and anonymous network G, and any u0 ∈ V (G), we propose an distributed algorithm that enables clearing G using searchers starting from the homebase u0, and initially unaware of G.

1 It uses at most k = mcs(G, u0) + 1 searchers if

mcs(G, u0) > 1, and k = 1 searcher if mcs(G, u0) = 1 ;

2 Every searcher involved in the search strategy computed

uses O(log k) bits of memory ;

3 During the execution, at most O(m log n) bits of

information are stored at every whiteboard.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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23/36 Intro Searching Model Results Conclusion Theorem Algo

Theorem

For any connected, asynchronous, and anonymous network G, and any u0 ∈ V (G), we propose an distributed algorithm that enables clearing G using searchers starting from the homebase u0, and initially unaware of G.

1 It uses at most k = mcs(G, u0) + 1 searchers if

mcs(G, u0) > 1, and k = 1 searcher if mcs(G, u0) = 1 ;

2 Every searcher involved in the search strategy computed

uses O(log k) bits of memory ;

3 During the execution, at most O(m log n) bits of

information are stored at every whiteboard.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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24/36 Intro Searching Model Results Conclusion Theorem Algo

Properties

the strategy is computed online by the searchers themselves. after the execution of our algorithm, an “optimal” monotone connected search strategy for G is written on the vertices’whiteboards in a distributed way. in the class of graphs with bounded mcs, searchers can be implemented by finite automata.

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25/36 Intro Searching Model Results Conclusion Theorem Algo

Principles of our algorithm

The Algorithm Initially, one searcher stands at v0, k = 1 While the graph is not clear : Try all monotone connected search strategies using k searchers ; If the graph is not clear, call a new searcher ; Predicate At the end of each loop, the k searchers are standing at v0.

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26/36 Intro Searching Model Results Conclusion Theorem Algo

Basic Idea

to order the possible strategies using k searchers ; to try all the strategies in the increasing order ; either a strategy clears the graph → OK ;

  • r after trying all the strategies, the graph remains

contaminated → another searcher is required.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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27/36 Intro Searching Model Results Conclusion Theorem Algo

Valid moves

Two kind of moves are compatible with a monotone connected strategy

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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27/36 Intro Searching Model Results Conclusion Theorem Algo

Valid moves

Two kind of moves are compatible with a monotone connected strategy (1) A searcher at a clear (or guarded) vertex can move through any port.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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Valid moves

Two kind of moves are compatible with a monotone connected strategy (1) A searcher at a clear (or guarded) vertex can move through any port.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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27/36 Intro Searching Model Results Conclusion Theorem Algo

Valid moves

Two kind of moves are compatible with a monotone connected strategy (1) A searcher at a clear (or guarded) vertex can move to help another searcher.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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27/36 Intro Searching Model Results Conclusion Theorem Algo

Valid moves

Two kind of moves are compatible with a monotone connected strategy (1) A searcher at a clear (or guarded) vertex can move to help another searcher and then to clear an edge.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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27/36 Intro Searching Model Results Conclusion Theorem Algo

Valid moves

Two kind of moves are compatible with a monotone connected strategy (1) A searcher at a clear (or guarded) vertex can move to help another searcher and then to clear an edge.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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27/36 Intro Searching Model Results Conclusion Theorem Algo

Valid moves

Two kind of moves are compatible with a monotone connected strategy (2) A searcher at a vertex with only one contaminated port can move to clear an edge.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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27/36 Intro Searching Model Results Conclusion Theorem Algo

Valid moves

Two kind of moves are compatible with a monotone connected strategy (2) A searcher at a vertex with only one contaminated port can move to clear an edge.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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27/36 Intro Searching Model Results Conclusion Theorem Algo

Valid moves

Two kind of moves are compatible with a monotone connected strategy Such a configuration is the result of a failing strategy No moves are possible.

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Order on moves and strategies

Representation of valid moves (i, j, p) denotes : ”searcher i joins searcher j and the smallest searcher follows the port p to clear an edge” (i, i, p) denotes : ”searcher i follows the port p to clear the corresponding edge” The moves are ordered in the lexicographic order. Order on the sequences of valid moves A sequence of valid moves corresponds to a partial monotone connected search strategy. The sequence are ordered in the lexicographic order.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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29/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm

k ≥ 1 being given, the centralized algorithm tries every strategy using k searchers, starting from v0. Initially, all searchers are at v0 ; Valid moves are performing one by one ; the smallest possible move is performed, if no valid move is possible, the last move performed is backtracked. A virtual stack contains the sequence of valid moves that have leaded to the current situation.

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=1 list of searcher(s) : Blue number of free searcher(s) : 0 | | | | | | | | | | | |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=2 list of searcher(s) : Blue < Green number of free searcher(s) : 0 | | | | | | | | | | | |

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The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=2 list of searcher(s) : Blue < Green number of free searcher(s) : 0 | | | | | | | | | | | (B, B, 1) |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=2 list of searcher(s) : Blue < Green number of free searcher(s) : 0 | | | | | | | | | (B, B, 2) | | (B, B, 1) |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=2 list of searcher(s) : Blue < Green number of free searcher(s) : 0 | | | | | | | (G, G, 2) | | (B, B, 2) | | (B, B, 1) |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=2 list of searcher(s) : Blue < Green number of free searcher(s) : 0 | | | | | | | | | (B, B, 2) | | (B, B, 1) |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=2 list of searcher(s) : Blue < Green number of free searcher(s) : 0 | | | | | | | | | | | (B, B, 1) |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=2 list of searcher(s) : Blue < Green number of free searcher(s) : 0 | | | | | | | | | | | |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=2 list of searcher(s) : Blue < Green number of free searcher(s) : 0 | | | | | | | | | | | (B, B, 2) |

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=2 list of searcher(s) : Blue < Green number of free searcher(s) : 0 | | | | | | | | | (G, G, 1) | | (B, B, 2) |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=2 list of searcher(s) : Blue < Green number of free searcher(s) : 0 | | | | | | | (G, G, 2) | | (G, G, 1) | | (B, B, 2) |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=2 list of searcher(s) : Blue < Green number of free searcher(s) : 0 | | | | | | | | | | | |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=3 list of searcher(s) : Blue < Green < Red number of free searcher(s) : 0 | | | | | | | | | | | |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=3 list of searcher(s) : Blue < Green < Red number of free searcher(s) : 0 | | | | | | | | | | | (B, B, 1) |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=3 list of searcher(s) : Blue < Green < Red number of free searcher(s) : 0 | | | | | | | | | (B, B, 2) | | (B, B, 1) |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=3 list of searcher(s) : Blue < Green < Red number of free searcher(s) : 1(R) | | | | | | | (G, G, 2) | | (B, B, 2) | | (B, B, 1) |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=3 list of searcher(s) : Blue < Green < Red number of free searcher(s) : 1(B) | | | | | (R, B, 1) | | (G, G, 2) | | (B, B, 2) | | (B, B, 1) |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=3 list of searcher(s) : Blue < Green < Red number of free searcher(s) : 2(B,R) | | | (B, G, 2) | | (R, B, 1) | | (G, G, 2) | | (B, B, 2) | | (B, B, 1) |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=3 list of searcher(s) : Blue < Green < Red number of free searcher(s) : 3(B,G,R) | (B, G, 3) | | (B, G, 2) | | (R, B, 1) | | (G, G, 2) | | (B, B, 2) | | (B, B, 1) |

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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30/36 Intro Searching Model Results Conclusion Theorem Algo

The centralized Algorithm : Example

1 1 2 3 2 1 2 2 1 1 1 3 v0 number of searcher(s) : k=3 list of searcher(s) : Blue < Green < Red number of free searcher(s) : 3(B,G,R) | (B, G, 3) | | (B, G, 2) | | (R, B, 1) | | (G, G, 2) | | (B, B, 2) | | (B, B, 1) | all searchers are free ⇒ the graph is clear

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31/36 Intro Searching Model Results Conclusion Theorem Algo

Proof of the centralized Algorithm

1 only valid moves are performed → only valid strategies

are performed.

2 valid strategies are performed in the lexicographic order

→ all valid strategies are performed.

3 if the graph is clear, the algorithm stops and if all

strategies with k searchers have been tried, the algorithm ask for an extra searcher → our algorithm terminates.

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32/36 Intro Searching Model Results Conclusion Theorem Algo

The distributed Algorithm

We aim to distribute the centralized algorithm Difficulties we had to address

  • nly one move at time : the coordinator acts like a token.

for backtracking : all actions performed by searchers are written in stacks distributed on the whiteboards. Somehow, the virtual stack of the centralized algorithm is distributed on the whiteboards.

  • ne searcher must be able to find another searcher : a

trace of any searcher is written in arrays on the whiteboards.

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Sketch of the distributed Algorithm

searcher 1 ... searcher k ... searcher 2 ... searcher i+1 ... ... ... searcher i searcher j>i ... Go p1 p2 p’1 p’2 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher 1 send Go to 2 ... searcher k ... searcher 2 receive Go from 1 ... searcher i+1 ... ... ... searcher i searcher j>i ... Go p1 p2 p’1 p’2 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher 1 send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... ... searcher i searcher j>i receive Go from i−1 ... Go p1 p2 p’1 p’2 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher j>i searcher 1 receive Go from i send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... Send Go to 1 ... searcher i ... receive Go from i−1 Clear by port p1 Go p1 p2 p’1 p’2 Arrive by p’1 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher j>i send Go to 2 searcher 1 receive Go from i send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... Send Go to 1 ... searcher i receive Go from 1 receive Go from i−1 ... Clear by port p1 Go p1 p2 p’1 p’2 Arrive by p’1 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher j>i send Go to 2 searcher 1 receive Go from i send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... Send Go to 1 ... searcher i receive Go from 1 send Go to 3 receive Go from i−1 ... Clear by port p1 receive Go from i−1 Go p1 p2 p’1 p’2 Arrive by p’1 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher j>i send Go to 2 ... receive Go from i−1 Clear by port p1 searcher 1 receive Go from i send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... Send Go to 1 ... searcher i receive Go from 1 send Go to 3 receive Go from i−1 Send Go to i+1 receive Go from i Go p1 p2 p’1 p’2 Arrive by p’1 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher j>i send Go to 2 ... receive Go from i−1 Clear by port p1 receive Go from j−1 searcher 1 receive Go from i send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... Send Go to 1 ... searcher i receive Go from 1 send Go to 3 receive Go from i−1 Send Go to i+1 receive Go from k−1 Send Go to i+2 Go p1 p2 p’1 p’2 Arrive by p’1 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher j>i send Go to 2 ... receive Go from i−1 Clear by port p1 receive Go from j−1 searcher 1 receive Go from i send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... Send Go to 1 ... searcher i receive Go from 1 send Go to 3 receive Go from i−1 Send Go to i+1 receive Go from i receive Go from k−1 Send Go to i+2 Send Go to j+1 Go p1 p2 p’1 p’2 Arrive by p’1 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher j>i send Go to 2 ... receive Go from i−1 Clear by port p1 receive Go from j−1 searcher 1 receive Go from i send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... Send Go to 1 ... searcher i receive Go from 1 send Go to 3 receive Go from i−1 Send Go to i+1 receive Go from i receive Go from k−1 Send Go to i+2 Send Go to j+1 Back p1 p2 p’1 p’2 Arrive by p’1 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher j>i send Go to 2 ... receive Go from i−1 Clear by port p1 receive Go from j−1 searcher 1 receive Go from i send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... Send Go to 1 ... searcher i receive Go from 1 send Go to 3 receive Go from i−1 Send Go to i+1 receive Go from i Back p1 p2 p’1 p’2 Arrive by p’1 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher j>i send Go to 2 searcher 1 receive Go from i send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... Send Go to 1 ... searcher i receive Go from 1 send Go to 3 receive Go from i−1 Send Go to i+1 receive Go from i ... receive Go from i−1 Clear by port p1 Back p1 p2 p’1 p’2 Arrive by p’1 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher j>i send Go to 2 ... receive Go from i−1 Clear by port p1 searcher 1 receive Go from i send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... Send Go to 1 ... searcher i receive Go from 1 send Go to 3 receive Go from i−1 Back p1 p2 p’1 p’2 Arrive by p’1 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher j>i send Go to 2 ... receive Go from i−1 Clear by port p1 searcher 1 receive Go from i send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... Send Go to 1 ... searcher i receive Go from 1 Back p1 p2 p’1 p’2 Arrive by p’1 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher j>i ... receive Go from i−1 Clear by port p1 searcher 1 receive Go from i send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... Send Go to 1 ... searcher i Back p1 p2 p’1 p’2 Arrive by p’1 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher j>i ... receive Go from i−1 Clear by port p1 searcher 1 send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... ... searcher i Back p1 p2 p’1 p’2 Arrive by p’1 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher j>i ... receive Go from i−1 Clear by port p1 searcher 1 send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... ... searcher i Back p1 p2 p’1 p’2 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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33/36 Intro Searching Model Results Conclusion Theorem Algo

Sketch of the distributed Algorithm

searcher j>i receive Go from i Send Go to 1 searcher 1 send Go to 2 ... searcher k ... searcher 2 send Go to 3 receive Go from 1 ... searcher i+1 ... ... ... searcher i ... receive Go from i−1 Clear by port p2 Arrive by p’2 Go p1 p2 p’1 p’2 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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34/36 Intro Searching Model Results Conclusion Theorem Algo

Proof of the distributed algorithm

We prove the equivalence between the algorithms. I.e, we prove both algorithms perform the same strategies in the same order It is a long case by case proof...

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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35/36 Intro Searching Model Results Conclusion

Plan

1

Introduction

2

Graph Searching

3

Our model

4

Our Results

5

Conclusion

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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36/36 Intro Searching Model Results Conclusion

Further Works

Open problems Is there exist an algorithm using finite automata ? How to reduce the size of whiteboards ? What is the amount of knowledge that is required to clear a graph in a polynomial time ?

L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders