Linear-Time Data Dissemination in Dynamic Networks

0.18 MB·

by  Manfred Schwarz

Linear-Time Data Dissemination in Dynamic
Networks ∗
Manfred Schwarz Martin Zeiner Ulrich Schmid
Institute of Computer Institute ofComputer Institute ofComputer
Engineering Engineering Engineering
7 TU Wien TU Wien TU Wien
1 [email protected] [email protected] [email protected]
0
2

r ABSTRACT in the system as fast as possible. In a synchronous dis-
a
M Broadcasting and convergecasting are pivotal services tributed system, the execution proceeds in the form of
lock-step rounds r = 1,2,..., where all processes send
in distributed systems, in particular,in wireless ad-hoc
9 and sensor networks, which are characterized by time- and receive round-r messages and simultaneously exe-
2 varying communication graphs. We study the question cuteacomputingstep,whichalsostartsthenextround.
Communication is unreliable, though: A message ad-
of whether it is possible to disseminate data available
] versary [1] determines which receiver gets a message
C locally at some process to all n processes in sparsely
fromtherespectivesenderinaround: Iteffectivelygen-
D connectedsynchronousdynamicnetworkswithdirected
eratesasequenceG ,G ,... ofdirectedcommunication
links in linear time. Recently, Charron-Bost, Fu¨gger 1 2
. graphs, where G contains a directed edge (p,q) if the
s andNowakprovedanupperboundofO(nlogn)rounds r
c message fromp is receivedby q in roundr. We assume
forthe casewhereeverycommunicationgraphis anar-
[ that the set of nodes and hence n is fixed, whereas the
bitrary directed rooted tree. We present a new formal-
2 ism, which not only facilitates a concise proof of this edges may change over time. Messages may have arbi-
v trary size, i.e., we adhere to the LOCAL model [37].
result, but also allows us to prove that O(n) data dis-
0 The particular question asked in this paper is: How
seminationis possiblewhenthe number ofleavesofthe
0 many rounds are needed until the uid of some node is
rooted trees are bounded by a constant. In the special
8
known to all n nodes, for a certain message adversary?
6 case of rooted chains, only (n−1) rounds are needed.
We will call this quantity the dissemination time, and
0 Our approach can also be adapted for undirected net-
it is obvious that small dissemination times are ben-
. works, where only (n−1)/2 rounds in the case of arbi-
1 eficial for data distribution applications.1 This even
trary chain graphs are needed.
0
includes consensus algorithms like [7, 15, 8, 39], since
7
system-wide agreement obviously requires system-wide
1 1. INTRODUCTION
: datadissemination. Moreover,smalldisseminationtimes
v
We consider a synchronous network of n failure-free arealsointerestingfordataaggregation,whichisapiv-
i
X nodes with unique ids (uids), which are connected by otal task in wireless sensor networks [3]. After all, con-
directed point-to-point links. The nodes execute a de- vergecasting is the dual of broadcasting: By reverting
r
a terministic algorithmfordisseminating somelocaldata the direction of the links and the sequence of commu-
(say, the uids for simplicity), which shall ensure that nication graphs, a successful broadcast becomes a suc-
theuidofatleastonenodebecomesknowntoallnodes cessful convergecast.
The dissemination time obviously depends heavily
∗
ThisworkhasbeensupportedbytheAustrianScience
upon the message adversary, i.e., the actual sequence
Fund (FWF) projects ADynNet (P28182) and RiSE
(S11405).
1It may be argued that adhering to the LOCAL model
is overly simplistic for e.g. wireless settings, where
the CONGEST model (a maximum message size of
O(logn))isusuallyconsideredmoreappropriate. How-
ever, in the light of our past efforts to solve the prob-
lem with arbitrary message size, we consider is close to
hopelesstoimmediatelyaddressthedatadissemination
in the CONGEST model. A solution in the LOCAL
model, however, might pave the way to the latter also,
as it might suggest ideas for how to properly schedule
ACMISBN978-1-4503-2138-9. theprocesses’activitiesinordertoavoidcongestion,cp.
DOI:10.1145/1235 [32].
of communication graphs G ,G ,...: If e.g. G con- source on a (possibly random but) fixed graph to all
1 2 1
tains a star, it is 1, if every graph consists of the same nodes.
two weakly connected components, it is ∞. We are in- Radiobroadcastinginevolvinggraphshasbeeninves-
terested in an upper bound on the dissemination time tigated in [14, 33, 2], [19] focused on token-forwarding
for at least sparsely connected communication graphs. algorithms, and [30, 11] studied push/pull-algorithms
Morespecifically,werestrictourattentiontothecaseof ondynamicgraphs. Floodingalgorithmshavealsobeen
anoblivious messageadversary[1,15],whereG ,G ,... studiedunderseveralmodels,includingedge-Markovian
1 2
is an arbitrary sequence of graphs each drawn from a and related models [5, 13, 12]. In the edge-Markovian
set G of candidate graphs with each G ∈ G containing modelanedgepresentattimetstayspresentwithprob-
some rooted spanning tree. Note that this is actually ability p and disappears with probability 1−p at time
the weakest per-graph restriction that guarantees a fi- t+1andanabsentedgeappearswithprobabilityq and
niteworst-casedisseminationtimeforobliviousmessage remains absentwith 1−q. Whereasthose papers focus
adversaries. onthe broadcastingofasingleitemfromafixedsource
Arelativelysimplepigeonholeargument(seeLemma4.1) to all nodes,[34, 6] also consider k-tokendissemination
yields an upper bound of O(n2) for the dissemination and all-to-allalgorithms. All this workabove considers
time inthis case.2 Recently,Charron-Bost,Fu¨ggerand undirected communication graphs, however.
NowakimprovedthistoO(nlogn). Weconjecturethat
thisboundcanbefurthertightenedtolineartimeO(n).
3. MODEL
Albeit we were not yet able to prove or disprove this,
despite considerable efforts, we establish results in this We consider a set of processes Π = {1,...,n} with
paper that back-up our conjecture. uids, connected by directed point-to-point links. The
Maincontributionsandpaperorganization: Af- processes execute a deterministic full-information pro-
terashortdiscussionofrelatedworkinSection2anda tocol for distributing a unique local piece of informa-
descriptionofoursystemmodelinSection3,weprovide tion (for ease of exposition, the uid) to the other pro-
the following results in Section 4: cesses. The distributed computation proceeds in an
(i) Weintroducetheconceptofinfluenceandcovering infinite number of synchronous lock-step rounds r =
setsandapplythesetechniquesinanovelandvery 1,2,.... Each round r consists of a communication-
conciseproofoftheknownO(nlogn)upperbound closed message exchange, specified by the communica-
for arbitrary directed rooted trees. tion graph G determined by an oblivious message ad-
r
(ii) WeshowthatadisseminationtimeofO(n)canbe versary[1], followedby a simultaneouslocalcomputing
guaranteed for directed rooted trees with a con- step at every process. In a full information protocol,
stant number of leaves. In the case of directed every process sends its complete state in every round:
rooted paths, i.e., directed rooted trees with only If process p receives the state of some different process
one leaf, the dissemination time is only (n − 1) q (reached at the end of round r > 0 resp. the initial
rounds. state for r = 0) in round r+1, it forwards q’s state as
(iii) InSection5,weadaptourapproachtoundirected part of its own state in all following rounds.
networksandshowthat only (n−1)/2 roundsare An execution of our system is just an infinite se-
needed for the dissemination time in the case of quence of rounds. It can be uniquely described by
arbitrary (undirected) chain graphs. an initial configuration C , which is the vector of the
0
Some outlook in Section 7 concludes our paper. initial states (that includes the uid) of every process,
followed by an infinite sequence G of communication
2. RELATED WORK graphs G ,G ,,.... The configuration reached at the
1 2
end of round r, after the computation step, is denoted
Research on broadcasting and gossiping has a long
by C .
history (see [31] for a survey) and many variations of r
Formally, let G be a directed or undirected labeled
theseproblemshavebeenstudied: Fortheclassicaltele-
graph on n vertices and let G = (G )∞ be an infi-
phone problem we refer to [23, 40, 36], and for radio r r=1
nite sequence of such graphs. Moreover, let σ be the
broadcasting to [20, 22] and references therein. In [18] i
finite prefix of G of length i; we may drop the index
theauthorsconsidertherendezvous-communication-model.
if the length is clear from the context. Let In (r) resp.
The widely-used push/pull/push-pull models [21] were p
Out (r)denotethesetofincomingresp.outgoingedges
studiedonseveralgraphclasseslikethecompletegraph[27, p
of node p in G .
38], hypercubes and Erdo˝s-Renyi-graphs [24, 28] ran- r
Thanks to our full information protocol, every node
domgeometricgraphs[26],andpreferential-attachment
hasknowledgeK (r)attheendofroundr (withK (0)
graphs [16]. A list-based quasi-randomized algorithm p p
representing the initial knowledge), which adheres to
has been studied in [4, 17]. They have in common
the following rules:
that they focus on broadcastinga message froma fixed
(1) Initial state: K (0)={p}for all p∈Π (every node
p
2For allthe O(.) terms inthis paper,the constantscan knows only its own uid at the beginning).
be computed. (2) Updating rule: The knowledge K (r +1) process
p
p obtains at the end of round r+1 is its previous Definition 2. The influence set S (r) of process p
p
knowledge K (r) together with the information he at time r is the set of processes that know about p at
p
gets via all incoming edges in G , i.e., time r, i.e., S (r)={q ∈Π: p∈K (r)}.
r+1 p q
K (r+1)=K (r)∪ K (r). 4.1 Influence Sets and Coverings
p p [ q
q: (q,p)∈Inp(r+1) Obviously,therearealwaysninfluencesets,andeach
node canbe elementofmultiple ofthese. Inthe follow-
Subsequently, we will use phrases like“K (t) at time t”
p
ing lemma, we collect some elementary properties of
for integer times t≥0, which means K (r) at (the end
p
influence sets.
of) round r =t for t>0 and K (0) otherwise.
p
The dissemination time, given a sequence of graphs, Lemma 4.1. For all p∈Π and r ≥0, we have:
is the first time all processes learned the uid from a (i) Initial state: S (0)={p}.
p
common process, which is formally defined as follows: (ii) Updating rule: S (r+1) = S (r)∪ {q′ :
p p Sq∈Sp(r)
Definition 1. Given aclass of graphs Gon n nodes (q,q′)∈Outq(r+1)}.
and an infinitesequence of graphs G ∈GN, i.e., Gr ∈G (iii) Given G, the dissemination time BG = min{r :
for r ≥1, the dissemination time in G is defined as maxp|Sp(r)| =n}.
(iv) S (r)⊆S (r+1) for all p and all r.
p p
(v) If p is the root in G ∈ G, p ∈ S (r), and
  r+1 q
BG =minr : \ Kp(r)6=∅. (1) |Sq(r)|<n then |Sq(r+1)|>|Sq(r)|.
p∈Π Proof. Properties (i) – (iv) are an immediate con-
 
The dissemination time of the class G is defined as sequence of the definition. Let us prove (v): Define
X := Π \ S (r) as the nonempty set of nodes which
G q
B = maxB . (2)
G don’t know from q. Since p is the root, for all v ∈ Π
G∈GN
(and in particular for all v ∈ X) there exists a path
In this paper, we restrict our attention to classes of from p to v in G . Thus there must be an edge from
r
graphs where every element is3 a rooted tree (T), its S (r) to X and hence S (r) grows at least by 1.
q q
subclass consisting of directed chains (C), as well as
From Property (v) of this lemma, we obtain directly
their undirected analoga.
the trivial O(n2)-bound on the dissemination time for
Remark 3.1. Following a commonly used definition rootedtreesT: Bythepigeonholeprinciple,aftern(n−
[23, 35, 29], we could also define a dissemination time 2)+1 rounds, one node was at least n−1 times the
root and hence its influence set has size n.
B (p)=min r : K (r)={p} fornodepinthe
G n Tq∈Π q o Whereas influence sets will turn out to be sufficient
first place, i.e., the time when p becomes known to all
forestablishingourresultsonchaingraphs,weneedthe
nodes in a graph sequence G. The dissemination time
extended concept of coverings for dealing with general
(1) is then B = min B (p). Whereas this alterna-
G p∈Π G rooted trees.
tive expression has been usedrarely for dynamic graphs,
it has been employed for static graphs [35, 25]; an anal- Definition 3. For r≥t let
ogon of (2) has also been studied in [35]. On the other
C (r)={S (r) | p∈I(t)}
hand, intheclassictelephoneproblems anditsvariants, I(t) p
one is interested in max B (p). be a class of influence sets, for some given index set
p∈Π G
I(t)⊆Π. It is called covering if S (r)=
Example 3.2. To illustrate the definitions above we SSp(r)∈CI(t)(r) p
Π. Theinfluencesetsthatmakeupacoveringarecalled
give a short example (see Figure 1).
covering sets. The size of a covering is the number of
The dissemination times are B =B (3)=B (5)=
G G G covering sets it consists of, i.e., the size of its index
3.
set I(t). (For sake of simplicity we will drop the index
sometimes in the following if r =t and the index set is
4. ROOTED TREES
clear from the context.)
In this section, we will present our main results on A sequence of coverings (C (r)) with the addi-
I(r) r≥0
rooted trees T. We use influence sets (see [34, Lemma tional property I(r+1)⊆I(r) we denote by C.
3.2. (b)])forthispurpose,whicharearedualtoknowl-
Clearly,atrivialexampleofacoveringisthesetofall
edge sets: While the knowledge set K (r) describes
p influence sets. To exclude such trivial cases, we intro-
which processes node p has already heardof at the end
duce a subclass of coverings that contain no redundant
of roundr, the influence set S (r) describes which pro-
p sets.
cesses have already heard of p:
Definition 4. A strict covering SC (r) is a cov-
3Actually,wecanimmediatelygeneralizeallourresults I(r)
ering with the property that SC(r)\{S} ∀S ∈SC(r) is
to the case where everyelement only contains a rooted
tree etc., as additional edges can only speed-up data not a covering. A unique node is a node that is element
dissemination. of only one covering set.
5
1 2
3 2
σ: 3 3 1
4
5 2 4 5 4
1
K (1):14 K (2):14 K (3):12345
1 1 1
K (1):25 K (2):235 K (3):235
2 2 2
K (1):35 K (2):1345 K (3):12345
3 3 3
K (1):24 K (2):2345 K (3):12345
4 4 4
K (1):5 K (2):35 K (3):1345
5 5 5
Figure 1: example execution with knowledge sets
The followinglemma states someusefulpropertiesof procedure leads to a covering X′ containing no unique
coverings: nodes. Hence it is a strict covering.
(iv) Note that an influence S which does not grow
Lemma 4.2. In the case of rooted trees T, every cov- has the property that for all p ∈ S also all successors
ering satisfies the following properties: of p must be contained in S. Moreover, each set in
(i) A coveringis astrictcovering iffeach coveringset SC(r−1) contains a unique node. Now let P denote
i
contains a unique node. the unique pathfromthe rootto the leafl (1≤i≤ℓ).
i
(ii) In a strict covering every covering set, except pos- We will show that each path P contains unique nodes
i
sibly one, loses at least one of its unique nodes. from at most one non-growing influence set. Let p be
The only covering set that may not lose one of its a unique node of a non-growing influence set S. Then
unique nodes is the one of the root of Gr. S contain all successors of p and those nodes can not
(iii) Let CI(r)(r) bea strict covering at timer. Assume be unique nodes from anothernon-growingset. Onthe
that at time r + t, for some t > 0, there exists other hand, if there is a unique node p′ from another
a covering set S in X := CI(r)(r +t) containing non-growingset S′ on the path fromthe root to p then
no unique node. Then, X′ := X \ {S} is still S′ would contain all successors from p′, hence also p,
a covering and X′ has at most |S| more unique and p would not be a unique node. Consequently there
nodes than X. By repeating this argument, one can be at most ℓ non-growing covering sets.
canreduceX toanewstrictcoveringSCI(r+t)(r+ (vi)Theonlyinfluencesetsofsize1afterround1are
t) with strictly smaller index set I(r+t)⊂I(r). the sets S (1),...,S (1) where the l are the leaves
(iv) Let ℓ be the number of leaves in Gr. In a strict in the treel1G1. But nlℓode li is surely ciontained in the
covering SCI(r−1)(r−1), at most ℓ influence sets influence set of ist predecessor. So take any covering
do not grow in round r. that does not contains the influence sets of leaves but
(v) If a strict covering consists of only one set, then the influence sets of predecessors of leaves.
dissemination has been completed. (vii) Since C(r) is a covering, one set must contain
(vi) Attimet=2,thereisalwaysacoveringconsisting the root. By Lemma 4.1 (v), this set grows.
of covering sets of size at least 2.
(vii) For each covering C (r) there exists a p ∈ I(r) 4.2 Bounds on dissemination time
I(r)
with |S (r+1)|>|S (r)|.
p p Equipped with the properties from Lemma 4.2, we
Proof. The properties (i) and (v) are obvious. can now give a novel, concise proof of the O(nlogn)-
(ii) Let q be a unique node. Then q is a) the root, or bound established in [9, 10].
b)hasapredecessorwhichisonlyinthesameinfluence
Fact 4.3 ([9, Lemma 4] and [10, Lemma 1]). For
asq(andthusuniquetoo),orc)hasapredecessorwhich
the class T of rooted trees, dissemination is completed
isinanotherinfluencesettoo. Incasec)qisnotunique within BT =O(nlogn) rounds.
anymore. In case b) we repeat the argument with the
predecessors of q until we stop in case c) or a). If we Proof. LetC (r)beastrictcoveringofsizex+1,
I(r)
stop in case c), then one of the predecessor of q is not andzbethenumberofuniquenodesinC(r). Lemma4.2(ii)
unique anymore. If we stop in case a) (and not in c)!) ensures that, after t := ⌈z⌉ rounds, there exists an in-
x
thenq andallitspredecessorsremainunique. Thusthe fluence set in C (r+t) with no unique nodes, which
I(r)
only influence set which may not lose one of its unique canberemoved. ThenewstrictcoveringSC (r+t)
I(r+t)
nodes is the set of the root. resulting from this procedure is of size at most x.
(iii)Since S containsno uniquenodesallthesenodes Starting at r =0 and x+1=n and using z ≤n, we
most be contained in other sets too, and thus we can canboundthedisseminationtimebyBT ≤ n n =
Pi=1(cid:6)i(cid:7)
removeS fromX stillhavingacovering. Repeatingthis O(nlogn).
If we restrict ourselves to rooted trees with a fixed Assume now thatwe havealreadyaddedsuccessfully
number of leaves, it is possible to prove that data dis- (L−1) sets to S(r+1) and that by adding S (r+1)
pL
semination can be completed even in linear time. condition (ii) is violated, i.e., there exists k ≤ (L−1)
sets S (r+1),...,S (r+1)∈S(r+1) such that
Theorem 4.4. FortheclassofrootedtreesT with pi1 pik
k−1
exactly k − 1 leaves, data dissemination is BTk−1 ≤ k
k·(n−3)+2 rounds. (cid:12) S (r+1)(cid:12)=r+1+k
(cid:12)(cid:12)[ piℓ (cid:12)(cid:12)
Proof. LetSCbeasequenceofstrictcoveringswith (cid:12)ℓ=1 (cid:12)
(cid:12) (cid:12)
I(r) ⊆ I(r +1) such that at time r = 2 all influence and
sets are of size at least 2 (see Lemma 4.2 (vi)). Let
k
S (r),...,S (r)betheksmallestinfluencesetsinSC
taotp1Ltiemmemraa4n.dp2kl(eitv)spini(re)vedreynrootuentdheatsizleeaostf oSnpie(ro)f.thDeume (cid:12)(cid:12)(cid:12)(cid:12)ℓ[=1Spiℓ(r+1)∪SpL(r+1)(cid:12)(cid:12)(cid:12)(cid:12)=r+1+k.
(cid:12) (cid:12)
grew by at least one, hence This means that S (r+1)⊆ k S (r+1).
pL Sℓ=1 piℓ
By induction hypothesis,
k
s (r)≥2k+(r−1).
X pi k
i=1 (cid:12) S (r)∪S (r)(cid:12)≥r+1+k.
Thus, if 2k+(r−1)= k(n−1)+1 one set must con- (cid:12)(cid:12)(cid:12)ℓ[=1 piℓ pL (cid:12)(cid:12)(cid:12)
tainn elements anddisseminationis done. Solvingthis (cid:12) (cid:12)
k
Thustheset S (r)∪S (r)didnotgrowinround
equation for r yields r =(k(n−3)+2). Sℓ=1 piℓ pL
(r+1), or equivalently, it captures the last (r+1+k)
In particular, in the special case of a directed chain elements of the chain in round (r+1). Firstly, in case
graph(i.e., k =2), one can do even better: The follow- oftheexistenceofS ,alsothissetcontainsexactlythe
p∗
ing theorem shows that the dissemination time is only last (r+1) elements, which gives
n−1 rounds in this case. Since in the constant chain
k
graph, it takes the root n−1 rounds to disseminate its
(cid:12) S (r)∪S (r)∪S (r)(cid:12)=r+1+k
value, this bound is tight. (cid:12)(cid:12)[ piℓ pL p∗ (cid:12)(cid:12)
(cid:12)ℓ=1 (cid:12)
Theorem 4.5. Let C be the class of directed chains. (cid:12) (cid:12)
which is a contradiction to the induction hypothesis.
At the end of round r, there exists a collection S(r) =
Hence – if S exists – we can add all influence sets
{S (r),...,S (r)} of n−r influence sets with p∗
1p.1 |S |≥rp+n−1r for 1≤i≤n−r, and of S(r) but Sp∗ to S(r +1) and the assertion of this
pi theorem holds in this case. On the other hand we are
2. for all 0≤r ≤n−1,
allowedto delete one set from our ’good’ sets, so we do
notaddS (r+1)toS(r+1). The remainingquestion
(cid:12)(cid:12)[Spi(r)(cid:12)(cid:12)≥r+1+|I|−1 ∀ I ⊆[1,n−r]. (3) is: Can thpeLre be an index L′ > L such that again con-
(cid:12)(cid:12)i∈I (cid:12)(cid:12) dition (ii) is violated? So assume that there is such an
Thus, B(cid:12) C ≤n−(cid:12)1 rounds. index and k1 <L′ sets Spi1,...,Spik1 with
Proof. We do it by induction on r. If r = 0, then k1
e(cid:12)(cid:12)SSspi(si∈0h)IoS=ldpis{(p0f}o)(cid:12)(cid:12)ra=rn.d|IWt|.heuAswssi|lSulpms(h0eo)tw|h=atth1tehftoehrienadslltuapctt.eidoOnabhsvsyieopruotsitolhyn-, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ℓ[=1Spiℓ(r+1)∪SpL′(r+1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)=r+1+k1.
also holds for (r+1). Again,duetoinductionhypothesis,thesetSkℓ=11Spiℓ(r)∪
We take successively a set Spi(r +1) (where Spi(r) SpL′(r) captures the last (r +1+k1) elements of the
was contained in S(r)) and add it to S(r +1) iff the chaininround(r+1). Ifk1 ≥k thenwetakethek1 in-
following two conditions hold: fluencesets fromheretogetherwithSpL(r)andSpL′(r)
and obtain
(i) |S (r+1)|≥r+2 and
pi k1
(ii) inequality (3) holds for all collection of sets of (cid:12) S (r)∪S (r)∪S (r)(cid:12)=r+1+k
S(r+1) and S (r+1). (cid:12)(cid:12)[ piℓ pL′ pL (cid:12)(cid:12) 1
pi (cid:12)ℓ=1 (cid:12)
(cid:12) (cid:12)
Note that condition (i) holds for all but at most one which is a contradiction. If k <k we take the k influ-
1
set from S(r): A set of size m does not grow iff it cap- ence sets from above together with S (r) and S (r)
pL pL′
tures the last m elements of the chain. Since due to againyieldingacontradiction. Sothetheoremisproven.
inequality (3) all sets of size (r+1) are pairwise differ-
ent only one of these sets can be completely at the end Butnotonlytreeswithafew numberofleavesadmit
of the chain, hence at most one set does not grow. If linear-time data-dissemination (see Theorem 4.4), also
such a set exists we denote it by S . trees with only a few inner nodes do:
p∗
Theorem 4.6. FortreeswithℓleaveswehaveBTℓ ≤ three differentgraphsG(1),G(2),G(3) where eachgraph
(n−ℓ)(n−1)+2−max(n,2(n−ℓ)). In particular, in will be applied for multiple rounds.
treeswith onlyk inner nodes (i.e., (n−k)leaves), data- The first graph, G(1), is the simple chain rooted in
dissemination is linear. In fact, BTn−k ≤ k(n−1)+ the process 1 and edges i→i+1.
2−max(n,2k). The tree G(2) is rootedinn andcontainsedges(n→
Proof. After the first round we can find a cover- 1),(n → n−1). Furthermore (i → i+1) for 1 ≤ i ≤
ing of size (n − ℓ) sets (by taking all influence sets ⌊n⌋−1, and (i→i−1) for n−1≥i≥⌊n⌋+2.
2 2
except those of the leaves), all of size at least 2. By Finally,G(3) isrootedin⌊n⌋withedgesi→i+1for
2
Lemma4.2(vii),inallfollowingroundsatleastofthem ⌊n⌋≤i≤n−1andedgesi→i+1for1≤i≤⌊n⌋−1.
2 2
must grow. Furthermore there is an edge n→1.
The execution (fig. 2) is constructed in the following
Remark 4.7. Notethatincaseofthestargraph(i.e.,
way:
n−1 leaves and 1 inner node) this theorem indeed gives
dissemination time of 1. • G =G(1) for 1≤r ≤⌊n−1⌋,
r 2
Turning backto generalrootedtreesT, the following • G =G(2) for ⌊n−1⌋+1≤r ≤n−2, and
theorem presents a lower bound on the dissemination r 2
time. It reveals that, in the worst-case, it takes more • G =G(3) for n−1≤r ≤⌈3n−1⌉−2.
r 2
time than in the case of chain graphs.
In this sequence, the first time an influence set has size
Theorem 4.8. For the class of rooted trees T, BT ≥ n is the last round. Hence ⌈3n−1⌉−2 is a lower bound
⌈3n−1⌉−2 rounds. 2
2 for broadcasting in directed trees.
Proof. We will construct a specific sequence G of
graphswithB =⌈3n−1⌉−2. Thissequenceconsistsof
G 2
n−1
for 1≤r ≤⌊ ⌋:S (r)={i,...,min(r+i,n)}
i
2
n−1 n−1 n
for ⌊ ⌋<r≤n−2:S (r)={i,...,⌊ ⌋+i} for i≤ ,
i
2 2 2
n n−1 n−1 n
S (r)={max(⌊ ⌋+1,i−(r−⌊ ⌋)),...,n,1,...,r−⌊ ⌋} for i>
i
2 2 2 2
3n−1 n −n−1 n+1 n
for n−2<r ≤⌈ ⌉−2:S (r)={max(⌊ ⌋+1,i+2+⌊ ⌋)),...,n,1,...,⌈ ⌉−2} for i> ,
i
2 2 2 2 2
n−1 3n+1
S (r)={i,...,⌊ ⌋+i+r−(n−2)} for i≤⌈ ⌉−2−r,
i
2 2
n 3n+1
S (r)={i,...,n,1,...,r−(n−2)} for ≥i>⌈ ⌉−2−r)
i
2 2
• for all 0≤r ≤(n−1)/2
5. UNDIRECTED TREES (cid:12) S (r)(cid:12)≥2r+1+|I|−1 ∀I ⊆[1,n−2r]. (4)
(cid:12)[ pi (cid:12)
(cid:12) (cid:12)
In undirected graphs, dropping the direction of the (cid:12)i∈I (cid:12)
(cid:12) (cid:12)
edges speeds-up data dissemination. In the case where
Gistheclassofundirectedandconnectedgraphs,after Thus, BCu ≤⌈(n−1)/2⌉ rounds.
n−1 rounds,evenall-to-alldisseminationis completed
Notethatthisboundisalsotight,astheconstantchain
(see [34][Proposition3.1]). But what canbe saidabout
graph reveals.
the dissemination time? The following theorem shows
that dissemination is twice as fast as all-to-all dissemi- Proof. The proof runs along the same lines as the
nationinundirectedchains,androughlyatleast3/2as prooffortheanalogousresultforrootedchainsanduses
fast as in directed rooted trees. induction on r. For r =0 it is obvious true.
Now let’s do the induction step: Again, we take suc-
Theorem 5.1. LetCu betheclassofundirectedchains. cessively a set Spi(r+1) (where Spi(r) was contained
At the end of round r, there exists a set S(r) of n−2r in S(r)) and add it to S(r + 1) iff the following two
influence sets Sp1(r),...,Spn−2r(r) with conditions hold:
• |S |≥2r+1 for 1≤i≤n−2r, and (i) |S (r+1)|≥2r+3 and
pi pi
1 ⌊n−21⌋ n n−2 ⌊n⌋ ⌈3n2−1⌉−2
     2 
σ1:  ...n2 1  ⌊...1n2⌋ ⌊nn2...⌋−+11 ⌊n−21⌋+1  ⌊⌊n2n2...⌋⌋+−11n−1
Figure 2: example execution
(ii) inequality (4) holds for all collection of sets of (2r+1+k ) elements of the same end of the chain in
1
S(r+1) and S (r+1). round (r+1). If k ≥ k then we take the k influence
pi 1 1
sets from here together with S (r) and S (r) and
Here condition (i) holds for all but at most two sets pL pL′
obtain
from S(r): A set grows by exactly 1 iff it is located
at one of the ends of the chain. Since all sets of size k1
(cid:12) S (r)∪S (r)∪S (r)(cid:12)=2r+1+k
(2r + 1) are pairwise different at most two such sets (cid:12)(cid:12)[ piℓ pL′ pL (cid:12)(cid:12) 1
grow by at most 1. If such sets exists we denote them (cid:12)ℓ=1 (cid:12)
(cid:12) (cid:12)
by Sp∗, i∈{1,2}. which is a contradiction. If k1 <k we take the k influ-
i
Assume now thatwe havealreadyaddedsuccessfully ence sets from above together with S (r) and S (r)
pL pL′
(L−1) sets to S(r+1) and that by adding SpL(r+1) again yielding a contradiction.
condition (ii) is violated, i.e., there exists k ≤ (L−1) Thirdly,In caseofabsenceof setsS we areallowed
p∗
sets Spi1(r+1),...,Spik(r+1)∈S(r+1) such that to delete two sets from S(r). Since by the argument
above at one end at most one set violates condition(ii)
k
we are done. So the theorem is proven.
(cid:12) S (r+1)∪S (r+1)(cid:12)≤2r+k+2.
(cid:12)(cid:12)[ piℓ pL (cid:12)(cid:12)
(cid:12)ℓ=1 (cid:12) 6. ACKNOWLEDGMENTS
(cid:12) (cid:12)
By induction hypothesis,
We would like to thank our colleague Kyrill Winkler
k for several useful discussions on this topic.
(cid:12) S (r)∪S (r)(cid:12)≥2r+1+k.
(cid:12)(cid:12)[ piℓ pL (cid:12)(cid:12) 7. OUTLOOK
(cid:12)ℓ=1 (cid:12)
(cid:12) (cid:12)
Thus the set k S (r)∪S (r) grew only by 1 in We presented a number of lower and upper bounds
Sℓ=1 piℓ pL for the dissemination time in dynamic networks under
round(r+1),orequivalently,itcapturesthe last(2r+
oblivious message adversaries, where the set of admis-
1+k)elementsofoneendofthechaininround(r+1).
sible graphs is restricted to contain trees or fixed sub-
Firstly,incaseoftheexistenceofSp∗ atthesameend,
i structures of trees. For rooted directed trees, the best
alsothis setcontainsexactly the last(2r+1)elements,
upper bound is O(nlog(n)) and the best lower bound
which gives
is Ω(n). Hence, it is still an open question whether the
k dissemination time is indeed linear or not. Besides our
(cid:12)(cid:12)(cid:12)[Spiℓ(r)∪SpL(r)∪Sp∗i(r)(cid:12)(cid:12)(cid:12)=2r+1+k interestinfinallyclosingthisquestion,wearewondering
(cid:12)ℓ=1 (cid:12) whether the worst case dissemination time is somehow
(cid:12) (cid:12)
which is a contradiction to the induction hypothesis. connectedtothemaximumpathlength,diameterorthe
Hence –iftwosetsSp∗ exists–we canaddallinfluence maximumnodedegreeoftheindividualcommunication
setsofS(r)butSp∗ toiS(r+1)andtheassertionofthis graphs or their dynamic transitive closure in the graph
theorem holds in tihis case. sequence. Moreover,it remains to be seen whether our
Secondly, if exactly one set S exists (at the oppo- chain upper bound also holds for undirected trees.
p∗
site end) we are allowed to delete one set from our sets
fulfilling condition (ii), so we do not add S (r + 1)
pL
to S(r +1). If there would be an index L′ > L such
thatagaincondition(ii)isviolated,therewouldbealso
k <L′ sets S (r+1),...,S (r+1) with
1 pi1 pik1
k1
(cid:12) S (r+1)∪S (r+1)(cid:12)=2r+3+k .
(cid:12)(cid:12)[ piℓ pL′ (cid:12)(cid:12) 1
(cid:12)ℓ=1 (cid:12)
(cid:12) (cid:12)
Again, the set k1 S (r)∪S (r) captures the last
Sℓ=1 piℓ pL′
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