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Distributed Chasing of Network Intruders by Mobile Agents
Lélia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial
To cite this version:
Lélia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial. Distributed Chasing of Network Intruders by Mobile Agents. Proceedings of the 13th Colloquium on Structural Information and Communication Complexity (SIROCCO 2006), 2006, Chester, United Kingdom. pp.70–84, �10.1007/11780823_7�.
�hal-00342000�
LeliaBlin
IBISC
UniversityofEvry
91000Evry
lelia.blinlami.univ-evry.fr
Pierre Fraigniaud y
LRI
CNRS &UniversityofParisSud
91405Orsay,Frane
pierrelri.fr
Niolas Nisse y
LRI
UniversityofParisSud
91405Orsay,Frane
nisselri.fr
Sandrine Vial
IBISC
UniversityofEvry
91000Evry
sandrine.viallami.univ-evry.fr
Abstrat
Graph searhing isoneof themostpopulartoolforanalyzing thehase forapowerful
and hostile software agent (alled the "intruder"), by aset of software agents (alled the
"searhers") in anetwork. The existing solutions for the graphsearhing problem suer
however from a serious drawbak: they are mostly entralized and assume a global syn-
hronization mehanismfor the searhers. In partiular: (1) thesearhstrategy for every
networkisomputedbasedontheknowledgeoftheentiretopologyofthenetwork,and(2)
themovesof thesearhersare ontrolledbyaentralizedmehanismthatdeidesat every
stepwhihsearherhastomove,andwhatmovementithastoperform.
Thispaperaddressesthegraphsearhingproblemin adistributed setting. Wedesribe
a distributed protool that enables searhers with logarithmi size memory to lear any
network,inafullydeentralizedmanner. Thesearhstrategyforthenetworkin whihthe
searhersarelaunhedisomputedonlinebythesearhersthemselveswithout knowingthe
topology of the network in advane. It performs in an asynhronous environment, i.e., it
implementstheneessarysynhronization mehanismin adeentralized manner. In every
network,ourprotoolperformsaonnetedstrategyusingatmostk+1searhers,wherek
istheminimumnumberofsearhersrequiredtolearthenetworkinamonotoneonneted
wayusingastrategyomputedin theentralizedandsynhronoussetting.
Keywords: graphsearhing,distributed algorithm,networkseurity.
y
These authors reeivedadditional supportsfrom the projet\PairAPair" of the ACIMasses deDonnees,
fromtheprojet\Fragile"oftheACISeuriteInformatique,andfromtheprojet\GrandLarge"ofINRIA.
Theseauthorsreeivedadditionalsupportsfromtheprojet\ALGOL"oftheACIMassesdeDonnees,and
fromtheprojet\ROM-EO"oftheRNRTprogram.
Graph searhing [26℄ is one of the most popular tool for analyzing the hase for a powerful
and hostileagent,byaset ofsoftwareagents inanetwork. Roughlyspeaking, graphsearhing
involves an intruder and a set of searhers, all movingfrom node to node along the linksof a
network. Theintruderispowerfulinthesensethat itissupposedto move arbitrarilyfast, and
to be permanentlyaware of thepositionsof thesearhers. However, the intruderannot ross
a node oran edge oupied bya searher withoutbeingaught. Conversely,the searhers are
unaware of the positionof the intruder. They are aiming at surrounding the intruder in the
network. The intruderis aught bythe searhers when a searher enters thenode it oupies.
Forinstane,onesearheranathanintruderinapath(bymovingfromoneextremityofthe
path to theother extremity), whiletwo searhers are required to ath an intruder in a yle
(startingfrom the same node,the two searhers move inoppositediretions). Another typial
exampleis the n-node square mesh, inwhih( p
n) searhers are neessary and suÆient for
athinganintruder. Inadditiontonetworkseurity,graphsearhinghasseveralotherpratial
motivations, suh as resuing speleologists in aves [8℄ or deontaminating a set of polluted
pipes[27 ℄. Ithasalso several appliationsto theGraphMinor theoryasitprovidesadynami
approah to theanalysisof stati graphparameters suhas treewidthand pathwidth [6 ℄.
The main question addressed by graph searhing is: given a graph G, what is the searh
number of G? That is, what is the minimum number of searhers, s(G), required to lear
the graph G, i.e., to apture the intruder? This question is motivated by, e.g., the need for
onsuming the minimum amount of omputing resoures of the network at any time, while
learing it. The deision problem orresponding to omputing the searh number of a graph
is NP-hard [26 ℄, and NP-ompleteness follows from [7 , 24 ℄. Computing the searh number is
however polynomialfor trees[25 , 26 ℄, and the orresponding searh strategy an be omputed
in linear time [30 ℄. In fat, the searh number of a graph is known to be roughly equal to
the pathwidth, pw, of the graph, and therefore the searh number of an n-node graph an be
approximatedinpolynomialtime,uptomultipliativefatorO(logn p
logtw)wheretwdenotes
thetreewidth ofthe graph(see [14 ℄,and usethe fatthatpw=twO(logn)).
The graph searhing problemhas given riseto a vast literature(f. Setion 1.2), in whih
several variants of the problem are disussed and solved. Nevertheless, from a distributed
systems point ofview, theexistingsolutions forthegraph searhingproblem(f.,e.g., [25,26 ,
30 ℄) suer from a serious drawbak: they aremostly entralized. In partiular, (1) the searh
strategy for every network is omputed based on the knowledge of the entire topology of the
network, and (2) the moves of the searhers are ontrolled by a entralized mehanism that
deidesatevery stepwhihsearherhastomove,andwhatmovementithastoperform. These
two fats limitthe appliabilityof the solutions. Indeed, as faras networking orspeleology is
onerned,thetopologyofthenetworkisoftenunknown,oritsmapunpreise. Thetopologyan
even evolvewithtime(eitherslowlyasfor, e.g.,Internet, orrapidlyasfor,e.g.,P2P networks).
Moreover, the mobile entities involved in the searh strategy an hardly be ontrolled by a
entral mehanism ditating their ations. All these onstraints make entralized algorithms
inappropriateformany pratialinstanesof thegraphsearhing problem.
This paper addresses the graph searhing problem in a distributed setting, that is the
searhers must omputetheir own searh strategy forthe network in whih they areurrently
running. This distributedomputation must notrequire knowingthe topology ofthe network
in advane (not even its size), and the searhers must at in absene of any global synhro-
nization mehanism,hene they mustbeable to performinafully asynhronousenvironment.
Distributedstrategies have beenproposedfor spei topologies only,suh astrees[2 ℄, hyper-
ubes[16 ℄,andringsandtori[15 ℄. Inthispaper,weaddresstheprobleminarbitrarytopologies.
preisely,theyarelabeledfrom1to theurrent numberk ofsearhers inthenetwork(ifa new
searherhasto join theteam,itwilltakenumberk+1). Otherwisesearhers areall idential,
and runthesame program. The network and thesearhers areasynhronousinthesensethat
everyationofasearhertakesanitebutunpreditableamountoftime. Moreover, motivated
bythefatthattheintrudermodelsapotentiallyhostileagentthat an,e.g.,orruptthenode
memories, the searh strategy must perform independentlyfrom any loal informationstored
at nodes a priori, and even independently from the node IDs. We thus onsider anonymous
networks, i.e., networksinwhih nodesdonothave labels,orthese labelsare notaessibleto
the searhers. The deg(u) edges inident to any node u are labeled from 1 to deg(u), so that
thesearhersandistinguishthedierentedges inidentto anode. Theselabelsarealledport
numbers. Every node of the network hasa whiteboardin whihsearhers an read, erase, and
write symbols. (A whiteboard is modeling a spei zone of the loal node memory that is
reserved forthe purposeof exhanging information between software agents). At every node,
theloalwhiteboardisassumedtobeaessiblebythesearhersinfairmutualexlusion. Sine
the ontent of the whiteboardat every node aessible by theintruderis orruptible,it is the
role of thesearhers toprotet informationstoredat nodes'whiteboards.
The deisionstaken by a searher at a node (movingvia portnumberp, writing the word
w on the whiteboard, et.) is loal and depends onlyon (1) the urrent state of the searher,
and (2) the ontent of the node's whiteboard (pluspossibly (3) the inoming port number, if
thesearher justentered thenode).
The powerful intruder is assumed to be aware of the edge-labeled network topology, and
thus it does not need the whiteboards to navigate. In fat, as mentioned before, when the
intruder enters a node that is not oupied by a searher, then it an modify oreven remove
theontent of theloalwhiteboard.
Allsearhersstartfromthesamenodeu
0
,alledtheentraneofthenetwork,orthehomebase
ofthesearhers. Thisnodeu
0
isalsoasoureofsearhers,inthesensethatiftheurrent team
of searhers realizethatthey are notnumerousenough forlearingthenetwork, thentheyan
ask for a new searher, that will appear at the soure. Initially, one searher spontaneously
appears at the soure. The size of the team will inrease until it beomes large enough to
learthenetwork. Basially,thesearhers areaimingatexpanding alearedzonearoundtheir
homebase u
0
, that is at expanding a onneted sub-network of the network G, ontaining u
0 ,
untilthe whole network is lear. In partiular, asthe entrane u
0
of the network is a ritial
node,ithasto be permanentlyproteted fromtheintruderinthesensethattheintrudermust
never be ableto aess it.
Among all searh strategies, monotone ones playan important role. A monotone strategy
insures that, one an edge has been leared, it willalways remain lear. Monotone strategies
guaranty a polynomialnumber of moves: exatly one move for learing every edge, plus few
moves requiredbythe searhers to set up theirpositionsbeforelearing thenext edge. In the
onnetedsetting (i.e.,thelearedpartof thenetwork isalwaysonneted),theorresponding
graph searhing parameter is alled monotone onneted searh number starting at u
0 (f.,
[2 , 3 ,16 ,15 , 21 ℄),and isdenoted byms(G;u
0 ).
1.1 Our results
We desribe a distributed protool, alled dist searh, that enables the searhers to lear
anyasynhronousnetworkinafullydeentralizedmanner, i.e.,thesearhstrategyisomputed
onlinebythesearhersthemselves,afterbeinglaunhedinthenetworkwithoutanyinformation
about its topology. This is the rst distributed protool that addresses the graph searhing
Thedistributedsearhstrategyperformedbythesearhersinanasynhronousenvironment
uses a number of searhers that is optimal up to a logarithmi fator. Indeed, we prove that
the number of searhers involved inthe strategy omputed byour protool in a network G is
equalto1plustheminimumnumberofsearhersrequiredtolearGbyamonotoneonneted
searh strategystartingat thehomebaseu
0
2V(G), i.e., isequalto ms(G;u
0
)+1. Sine itis
known [21 ℄ that, for any graph G and for any u
0
2 V(G), we have ms(G;u
0
) s(G)dlogne,
we getthat ourprotool usesat mostO(logn)times theoptimalnumberof searhers. Infat,
it isonjetured that ms(G) 2s(G) forall graph G(f. [3 ℄). Ifthisholds, thenourprotool
uses at mosttwietheoptimalnumberofsearhers.
Our protool is spae-eÆient from many respets. First, it requires only O(logk) bits of
memoryforeahoftheksearhersinvolvedinthesearh. Inpartiular,thisamountofmemory
is independent from the size n of the network. Seond, the amount of information stored at
everywhiteboardneverexeedsO(mlogn)bits,wheremisthenumberofedgesofthenetwork.
Toobtainour results,we had to addressseveral problems.
First, sine the network is a priori unknown to the searhers, they have to explore it.
However, this exploration annot be ahieved easily beause of the potential orruption
of thewhiteboards bythe intruder. Our protool insuresthatexploration and searhing
areperformedsomehowsimultaneously,andthatthewhiteboardsoflearednodesremain
permanentlyproteted unlessthereisnoneedtoprotetthestoredinformationanymore.
Seond,asthe searhers asynhronouslyspreadout inthenetwork, they beome rapidly
unawareoftheirrelativepositions. Ourprotoolsynhronizesthesearhersinanontrivial
mannersothat anationbya searheris notruinedbytheation ofanothersearher.
Finally,to obtainspae-eÆientsolutions,ourprotooltakesadvantagefromtheaesses
to the whiteboards, to store and read information useful to the searhers: it maintains
a stak at every whiteboard, and every searher at a node has aess onlyto the top of
a stak stored loally on the urrent node's whiteboard, and to few other variables also
storedon thewhiteboard.
1.2 Related Works
Graph searhing, originated by Parson in [27 ℄, has been extensively studied in the literature
(see[6 ℄forasurvey). VariantsoftheproblemhavebeendenedbyKirousisandPapadimitriou
in [22 , 23 ℄, and by Bienstok and Seymour in [7 ℄. The notion of rusade allowed Bienstok
and Seymour to simplify the proof of LaPaugh [24 ℄ about monotone graph searhing: forany
graph,there existsa minimalsearh strategythat is monotone(i.e., reontamination doesnot
help). Thenotionofonneted searh strategyhasbeenintroduedbyBarriereetal. [2,3℄. [2℄
desribesa linear-time algorithm that omputes minimalmonotone onneted searh strategy
for trees. [3 ℄ proves that, for any tree T, ms(T) 2 s(T) 2 and this bound is tight. [31 ℄
shows that there exist graphsforwhih no minimalonneted searh strategies aremonotone.
Ontheotherhand,[2 ℄provesthatreontaminationdoesnothelpforonnetedsearhin trees.
Several protools for learing some spei networks in distributedsetting have been pro-
posed inthe literature. Flohiniet al. have proposed protools that address the graph sear-
hingproblemintrees[2 ℄,hyperubes[16 ℄,toriand hordalrings[15 ℄. Foreah oftheselasses
of graphs, the authors have designed a protool usingms(G;u
0
)+1 searhers with O(logn)
bits of memory and whiteboards of size O(logn) bits, that monotonously lears the graph in
ol learing an asynhronous network in a monotone onneted way requires ms(G;u
0 )+1
searhers. Moreover, thisremainstrueevenifthetopologyofthenetworkisknowninadvane.
Our problem is also very muh related to graph exploration and mapping. In absene of
whiteboards, it is known that network exploration is impossible using a nite team of nite
automata [20 , 29 ℄. In fat, itis knownthat noniteteam of niteautomatais able toexplore
all graphs, even if these automata are given powerfulommuniation failities (f., e.g., [10 ℄).
However, exploringtreesis relativelyeasy [11 ℄,and a pre-omputedlabelingof thenodeswith
only three dierent labels enables just one nite automaton to explore all graphs [9℄. In the
reent paper of Reingold provingthat SL= L [28 ℄, a log-spae onstrutible universal explo-
rationsequeneexploringalld-regularn-nodegraphsisdesribed. Finally,[4 ,5,19 ℄investigated
explorationof diretedgraphs.
In [12 , 13 ℄, the objetive of the authors is to determine the position of a blakhole in a
network. A blakhole is an harmful node that destroys any agent visitingthat node without
letting any trae. On the other hand, the blakhole annot move. [12 , 13℄ have proved that
+1 agents are neessary and suÆient to nd a blakhole in any network, where is the
maximumdegree of thenetwork.
2 Model, Formal Statement, and Main Result
In thissetion, we speifyourproblem,and we state formally ourmainresult.
2.1 Our problem
We summarize our problem setting. A network is an anonymous edge-labeled graph G. The
deg(u) edges inident to any node u are labeled by distint integers from 1 to deg(u). These
labels are alled port numbers. A searher is a mobile omputingentity that an move along
theedges of thenetwork. At every node ofthenetwork,there isa whiteboard aessible to the
searhersurrentlyoupyingthisnode. Awhiteboardisa zoneofthenode'smemoryreserved
to the searhers to read, write, and erase information. The aess to every whiteboard is
assumedtobeperformedundertheontrolofafairmutualexlusionmehanism. Thedeision
taken byasearherat anodedependsonitsinternalstate,theontent oftheloalwhiteboard,
and the inoming port number. A deision results in either leaving the node through some
port p, or waiting at the node until it has (again) aess to the whiteboard. The searhers
are generated by a unique node u
0
2 V, alled the homebase. The homebase is a soure of
searhers, inthefollowing sense. New searhers an begenerated at the homebase. Fora new
searher to be generated, at least one searher must be oupying the homebase, and alling
fora newsearher. The ithsearhergenerated at thehomebase isgiven labeli. The searhers
are asynhronous in the sensethat every ation of a searher takes a nite but unpreditable
amount of time. When they are launhed ina network, they ignore its topology, and have no
information about it (they even ignore its size). The goal of the searhers is to apture an
"intruder".
The intruder is a maliious mobile omputing entity that an move along the edges of
the network. The intruderis arbitrarily fast, and is assumed to be permanently aware of the
positions of the searhers. It is invisible in the sense that the searhers are unaware of the
positionof the intruder. On the other hand, the intruder knows the topologyof the network
andisassumedtobepermanentlyawareofthepositionsofthesearhers. Theintruderisaught
if it meets a searher at a node oralong an edge. The intruder hasthe ability to orrupt the
nodes, inludingthe ontent oftheir whiteboards.
