In this section we present two candidate DAGsGwhich achieve thebestpossible lower bound onΠcc(k G)=Ω(N2loglogN/logN) for constant indegree graphs underplausible
conjectures of the (block) depth-robustness of these DAGs. Our first results shows thatany(e,d)-depth-robust DAGGwithe=d=Ω(NloglogN/logN) can be used to construct a new graphG0=superconc(G) s.t.G0hasΠcc(k G0)=Ω(N2loglogN/logN) by overlaying a superconcentrator on top of G. We first recall the definition of a superconcentrator.
Definition 5. A graph GwithO(N)vertices is a superconcentrator if there exists I,O with|I|=|O|=N such that for all S1⊆I,S2⊆O with|S1|=|S2|=k, there are
It is well known that there exists superconcentrators with |I|=|O|=N, constant indegree and O(N) nodes total e.g. [LT82, Pip77]. We now define the overlay of a superconcentrator on a graphG1.
Definition 6. Let G1 be a fixed DAG with N nodes and G2 = (V, E) be a (a
priori fixed) super-concentrator with N inputs I={i1,...,iN} ⊆V and N outputs
O={o1,...,oN}⊆V. We useG=superconc(G1)to denote the graphG=(V,E∪F1∪F2)
whereF1={(oi,oi+1) : 1≤i<N}andF2={(iu,iv) :(u,v)∈E(G1)}.
Theorem 8 ( [ABP17], Theorem 4).IfGis(e,d)-depth robust, thenΠcc(k G)>ed.
Lemma 7. LetG be an (e,d)-depth robust graph. Then for allS with |S|<e2, it follows thatΠcc(k G−S)≥e
2d.
Proof. Observe that ifGis (e,d)-depth robust and|S|<e2, thenG−Sis e2,d -depth robust. Thus,Πcc(k G−S)≥e
2dby Theorem 8.
Conjecture 1. LetGbe a graph withN nodes sampled uniformly at random from the DRSample distribution. Then with high probability,Gis (e,d)-depth robust, where
e=c1NloglogN
logN andd=
c2NloglogN
logN , for some constantsc1,c2>0.
Theorem 9. LetG1 be an (e,d)-depth robust graph withN nodes. Then for G= superconc(G1),Π
k
cc(G)=ΩminNe−24 e2,Nd−24 ed.
Proof. Let P= (P1,...,Pt)∈ Pk(G) be a legal pebbling of G and let {o1,...,oN} denote the output nodes of the super-concentrator. For each node v∈V(G), let
tv be the first timev is pebbled. Notice thattoi< toi+1 sinceGincludes the edge (oi,oi+1) for eachi < N. Partition L into intervals of 2e consecutive output nodes
L1={o1,...,o2e},L2={o2e+1,...,o4e},...and use letvstarti =o2e(i−1)+1denote the first node in intervalLi,vimid=o2e(i−1)+edenote the middle node, andvlasti =o2e(i−1)+2e denote the last node in the intervalLi. We also useLfirsti ={o2e(i−1)+1,...,o2e(i−1)+e} to denote the firstenodes on the intervalLi andLlasti ={o2e(i−1)+e+1,...,o2e(i−1)+2e} to denote the second half of the interval.
We now lower bound P tvi
end i=tvi
start
|Pi|, the cost incurred during pebbling rounds [tvi
start,tviend]. We consider two cases.
In case 1 we have|Pj|≥e2 for allj∈[tvstart,tvmid]. In this case we have tvi end X j=tvi start |Pj|≥ tvi mid X j=tvi start |Pj|≥ tvi mid−tv i start e 2≥ e2 2 , where the last inequality follows from the observation that
tvi mid+1−tv i start ≥2e(i−1)+e+1−2e(i−1)+1=e .
In case 2 there exists some timej∗∈[t vi
start,tvimid] such that|Pj
∗|<e
2. Now we note that we must completely re-pebble all nodes in the setancestorsG−Pj∗[Llasti ] before roundtvi
last. LetS={i1,...,iN} \ancestorsG−Pj∗[L last
not ancestors ofLlast
i in the graphG−Pj∗i.e., the nodes wedon’t necessarilyneed
to repebble. We first claim that|S| ≤ |Pj∗|< e/2. Suppose note, then, sinceGis a
super-concentrator, there are min(|S|,e)>|Pj∗|vertex disjoint paths fromStoLlasti in G. It follows that there is a path from some nodev∈S toLlast
i which avoidsPj∗, but
this implies thatv∈ancestorsG−Pj∗[L last
i ]. Contradiction, by constructionS is disjoint fromancestorsG−Pj∗[L
last
i ]! It follows that|S|≤|Pj∗|.
We now letS1={v∈V(G1) :iv∈S}be the nodes inG1corresponding to the inputs
S⊆V(G). Note that|S1|=|S|. We have tvi end X j=tvi start |Pj|≥ tvi end X j=j∗ |Pj|≥Πcck(G1−S1)≥ ed 2
where the second to last inequality follows because we need to re-pebble every input node that is not inS and weGcontains a copy ofG1overlayed on top of the inputs. The last inequality follows from Lemma 7.
We have show that for each intervalLi we have tvi end X j=tvi start |Pj|≥min e2 2, ed 2 .
Since there are at leastn 2e
such intervals, the total cost of pebblingGis at least
t X j=1 |Pj|≥ bN 2ec X i=1 tvi end X j=tvi start |Pj|≥ N 2e−1 min e2 2, ed 2 .
If Conjecture 1 holds, then we can takee=ΩNlogloglogNNandd=ΩNlogloglogNNso thatΠcc(k G)=ΩN2logloglogN N.
Corollary 4. LetG1 be a graph withN nodes sampled uniformly at random from the
DRSample distribution. If Conjecture 1 holds, thenΠcc(superconc(k G1))=Ω
N2loglogN logN
with high probability.
However, a superconcentrator overlay is not the most practical construction. Before we describe a more practical construction, we first set up some notation.
We now make a conjecture slightly stronger than Conjecture 1 and show it also leads to an asymptoticallycmc-optimal DAG may be easier for practical implement.
Conjecture 2. LetGbe a graph withN nodes sampled uniformly at random from the DRSample distribution. Then with high probability,Gis (e,d,b)-block depth robust, wheree=c1NloglogN
logN ,d=
c2NloglogN
logN , andb= c3logN
loglogN for some constantsc1,c2,c3>0. Reminder of Theorem 5. Let G1 be an (e,d,b)-block depth-robust graph with
N=2n nodes. ThenΠcc(BRG(k G
1))≥min eN2 ,edb32
.
with and letG=BRG(G1) be a graph withV=[2N]. We partition the nodes [N+1,2N] from the second half ofGintob
16
disjoint intervals of length 16N b : J0= N+1,N+16N b ,J1= N+16N b +1,N+ 32N b ,...
and split each interval Ji into Fi=
N+16biN+1,N+16biN+8bN, the first half of the interval, andLi=
h N+16iN b + 8N b +1,N+ 16(i+1)N b i
, the last half of the interval. Similarly, partition the first half ofGinto disjoint intervals of length 4b:I1=
1,4b,I2= b 4+1, b 2
,.... By Lemma 2, eachLi is connected to all of the intervals{Ik}. For any fixedilettstartbe first time the first node inFiis pebbled and lettlastbe the first time the last node inFiis pebbled. Note that either for allj∈[tstart,tlast],|Pj|≥e2, or there exists aj∈[tstart,tlast] with|Pj|<e2.
In the first case, |Pj| ≥e2 for at least b8 steps, so that the cost of pebbling the interval is at least 8eNb . In the second case, there exists aj∈[tstart,tlast] with|Pj|<e2. Observe thatLi has length 8Nb . Thus by Lemma 3,ancestorsG−Pj(Li) is
e 2,d,b
block depth-robust, so by Theorem 8, the cost to repebbleancestorsG−Pj(Li) is at least
ed 2. Hence, the cost to pebble each interval of length 16bN is at least min 8eNb ,ed2. Accounting for each of the 16b intervals, the total pebbling cost is at least min eN2 ,edb
32
.
Corollary 5. LetG1 be a graph withN nodes sampled uniformly at random from
the DRSample distribution. If Conjecture 2 holds, thenΠcc(BRG(k G1))=Ω
N2loglogN logN
with high probability.