⎨
⎪⎩
n(x1)/2−y1
z2−x1(N−x1)−y1−n(x1)/2
pz2+x1(x1−1)−y1−n(x1)/2−z1(1−p)ν(x1)−z2 ifx1≤x2=N, y1+x1(N−x1)+n(x1)/2≤z2≤ν(x1)
0 ow.
(16) After making a global round switch, the number of 0s in the diagonal equals N. For the number of zeros z2 ins2 recall that these 0-entries arise from 1-entries by applying the normalization step. Thus, the number of non-diagonal 0-entriesz2must be at least the num-ber of 1-entries above the diagonal before (i.e.,y1) plus all entries below the diagonal of 1-processes (for performing the global round switch all messages must arrive and below 1-processes we have only 1-entries – see (β)), this number equalsn(x1)/2+x1(N−x1).
Since we had(N−x1)0-processes in states1and each process can send messages to at most(N−1)other processes, we havez2≤ν(x1).
Since all(−1)-entries must be overwritten and since due to (β) no 0-entries exists below 1s in the diagonal, the only choice we have for relevant messages leading to 1-entries (before normalizing) which may arrive or not is to choose them from the(n(x1)/2−y1)0-entries above the diagonal; because of the argument above we have to choosez2−(y1+x1(N− x1)+n(x1)/2)from them. This gives the binomal coefficient.
The total number of messages which must arrive for this state transition is the number of new 1-entries above the diagonal plus the number of(−1)-entries ins1, i.e.,
(z2−x1(N−x1)−y1−n(x1)/2) +(x1(N−1)−z1)
= z2+x1(x1−1)−y1−n(x1)/2−z1; this is the exponent ofp. Finally, computing the exponent of(1−p), we have
(n(x1)/2−y1) − (z2−x1(N−x1)−y1−n(x1)/2)
= n(x1)+x1(N−x1)−z2=ν(x1)−z2
4.0.1 Proof Lower Bound
Now we want to show that our previously defined process is indeed a lower bound. As before this can not be done execution-wise, since there exist schedules such that theα-synchronizer is faster than theα-synchronizer.
Example 8 We give an example of a schedule such that theα-synchronizer is faster than the α-synchronizer. HereA(t)andA(t)denote the matrices as defined in Section2.3.
t: 0 1 2 3 4
As before we will construct a measure-preserving bijectiongon the sets of schedules such that theα-synchronizer under a scheduleEis always slower than theα-synchronizer given the scheduleg(E). Again, the basic idea behind our construction is to map the relevant messages of theα-synchronizer on the relevant messages of theα-synchronizer in such a way that theα-synchronizer is always in front.
Then we have the following theorem:
Theorem 3 There exists a bijectiong:E→Ewith – |En| = |g(E)n| ∀E∈E, ∀nand
– R(E, n)≤R(g(E), n) ∀E∈E,∀n.
Here,R(E, n)denotes the global round number of theα-synchronizer at timengiven the scheduleEandR(E, n)the global round number of theα-synchronizer at timengiven the scheduleE.
Proof Let A(n) = A(E, n) and A(n) = A(g(E), n) denote the matrices of the α-and the α-synchronizer under the schedule E andg(E), resp. Moreover, definemn = mini,jA(n)i,j (i = j) as the minimum ofA(n), andan(mn)as the number of non-diagonal entries ofA(n)that equal the minimummn. Withbn(mn)we denote the number of non-diagonal entries inA(n)less than or equal tomn. Similarly,R(n) = R(E, n)and R(n)=R(g(E), n)are the global round numbers ofA(n)andA(n), respectively.
Now we will inductively construct functionsgn: En→En(n≥1), the limit of which givesg, with the following properties:
(i) gnbijective onEn,
(vii) The function gn is an extension of gn−1, i.e., if gn((x1, . . . , xn−1, xn)) = (y1, . . . , yn−1, yn)thengn−1((x1, . . . , xn−1))=(y1, . . . , yn−1)forn≥2.
Then the functiongdefined byg(E)=limn→∞gn(E≤n)has the stated properties.
Remark 4 It is worthwile to mention that conditions (iv) and (v) imply
i =jA(E, n)i,j+
|Gn| ≤
i =jA(gn(E), n)i,j+ |Gn|.
We start withg1 := idM. Theng1 fulfils obviously(i)−(vi). Letgnwith(i)−(vi) be already defined. We will first construct – in dependence onE≤n(and consequently on A(n)) – the functionsgE≤n:M→M. Then, the functiongn+1is defined as follows:
gn+1(E)=gn+1(E≤n, En+1)=(gn(E≤n), gE≤n(En+1)). (17) For our construction, we have to do a case distinction:
Case AR(n)=R(n)=r:
Given the matrices A(n)andA(n), we define the setsA,A, . . . analogously to the proof of Theorem 1 for the upper bound. Moreover, let B˜n resp. B˜n denote the set of positions ofr-entries inA(n, r+1, r+1)resp.A(n, r+1, r+1)above the diagonal.
Our assumptions imply|A|+|G| ≤ |A|+|G|(due to (v)) and|D|+|G| ≥ |D|+|G| (see (iv): we havemn=r−1). Similar to the upper bound we have due to (iv) and reordering step (1a) where messages are inserted line by line, that the number ofr-processes inAnis greater than or equal to the number ofr-processes inAn; equivalently|C| + |D| ≥ |C| +
|D|(the sizes of the submatricesA(n, r, r)andA(n, r, r)) or equivalently
|A| + |B| + |G| ≤ |A| + |B| + |G| (18) (i.e., the total number of entries in columns inA(n)with(r+1)in the diagonal is less than or equal to the number of entries in columns inA(n)with(r+1)in the diagonal). In fact – by adding a term counting the difference of(r+1)-processes inA(n)andA(n)– we even have
|A| + |B| + |G| +(xn−xn)(N−1)= |A| + |B| + |G| (19) withxnresp.xnthe number ofr-processes inA(n)resp.A(n). Furthermore, we have the relations
|B| = (N−xn)(N−xn−1)
2 − | ˜A| and (20)
|B| ≥ | ˜B| = (N−xn)(N−xn−1)
2 − | ˜A|. (21)
Lethbe any bijection from[N]2to[N]2keeping the diagonal fixed with a) h(D∪G)⊆D∪Gand
b)
B⊆h(B) if|B| ≤ |B|
h(B)⊆B if|B|>|B|, and
c)
B˜⊆h(B˜) if| ˜B| ≤ | ˜B|
h(B)˜ ⊆ ˜B if| ˜B|>| ˜B|.
Now definegE≤nonMas the function induced byh. We have to check properties (i) – (vi) forgn+1.
Conditions (i) and (ii) are obviously fulfilled. Property a) implies immediately (iii).
To verify condition (iv) we distinguish two cases: In casemn+1 =r−1 (i.e, no global round switch occured) we have
bn+1(mn+1)−an+1 (mn+1)≥bn(mn)−an(mn)−(|D| + |G| − |D| − |G|)=0, since due to condition a) at most(|D| + |G| − |D| − |G|)more(r−1)-entries inA(n) can be overwritten byr- or(r+1)-entries than inA(n).
In casemn+1 = r(i.e., inA(n)a global round switch occured) letB−,B−, andG−
denote the number of succesfully transmitted messages in the setsB,B, andG. Hence, bn+1(mn+1) = |Fn| + |Cn| + |Dn| + |Bn| −B−+ |Gn| −G−
(6)=N (N−1)− |An| −B−−G−, an+1(mn+1) = |Fn| + |Cn| + |Dn| + |Bn| −B−
(6)=N (N−1)− |An| − |Gn| −B− and consequently,
bn+1(mn+1)−an+1(mn+1) = |An| − |An| + |G| +B−−B−−G−
≥ |An| − |An| + |G| +min(0,|B| − |B|)− |G| ≥0.
The last inequality holds in case of min(|B|,|B|) = |B|due to condition (v), in case of min(|B|,|B|)= |B|due to Eq.18. Thus also (iv) is valid.
IfR(n+1)=R(n+1), we also have to verify condition (v). In caseR(n+1)=r+1 there are no(r+2)-processes and the condition is fulfilled trivially. On the other hand, i.e, in caseR(n+1)=r, first note that we can write
Gn+1=(Gn∪N)\G−
withN =Gn+1\Gnthe set of ‘new’(r−1)-positions inA(n+1, r, r+1)(i.e., those(r− 1)-entries which are in columns of processes which switched fromr- to(r+1)-processes from A(n)toA(n+1)). Hence,
|Gn+1| = |Gn| + |N| − |G−| ≥ |Gn| +0− |G−|.
Moreover, note that every arriving message related toG− increases the number of(r + 1)-entries.
Additionally, we can decompose the setAn+1in the following way:
An+1=An∪G−∪B−.
But, due to our construction of the lower bound, the correspondig equation forAn+1is An+1=An∪G−∪B−∪L,
whereLare the positions of the new(r+1)-entries (instead ofr-entries) below the diagonal of new(r+1)-processes (see step (3)).
Hence,
:= |An+1| + |Gn+1| − |An+1| − |Gn+1|
= |An| + |G−| + |B−| + |L| + |Gn| + |N| − |G−|
−|An| − |G−| − |B−| − |Gn| − |N| + |G−|
= |An| + |B−| + |L| + |Gn| + |N| − |An| − |B−| − |Gn| − |N|;
due to property b) we can bound the difference|B−| − |B−|by max(0,|B| − |B|)from below, hence
≥ |An| + |Gn| − |An| − |Gn| +max(0,|B| − |B|)+ |L| + |N| − |N|. Observe that the number|L| + |N|of(r+1)- or(r−1)-entries below new(r+ 1)-processes equals xn(x2n−1) − xn+1 (x2n+1 −1) (the first term equals the number of entries of r-processes inA(n)below the diagonal, the second term equals the number of entries of r-processes inA(n+1)below the diagonal). Furthermore, the number|N|can be bounded from above by(xn−xn+1)xn+1(the size of the rectangle below the new(r+1)-processes).
Moreover, usingxn+1≥xn+1, we obtain
N ≤(xn−xn+1)xn+1 ≤ xn(xn−1)
2 −xn+1(xn+1−1) 2
≤ xn(xn−1)
2 −xn+1(xn+1−1)
2 .
Consequently,
≥ |An| + |Gn| − |An| − |Gn| +max(0,|B| − |B|) + xn(xn−1)
2 −xn+1(xn+1−1)
2 −xn(xn−1)
2 +xn+1(xn+1−1)
2 .
In case|B|>|B|we use Eq.19to get
≥ (xn−xn)(N−1)+xn(xn−1)
2 −xn(xn−1) 2
= 1
2(xn−xn)(2N−xn−xn−1)≥0
(this is clear forxn ≤N−1, and forxn = Ncondition (v) is trivially fulfilled). In case
|B| ≤ |B|we get by using Eq.19again
≥ |B| − |B| +(xn−xn)(N−1)+xn(x2n−1)−xn(x2n−1) and by using Eqs.20and21
≥ (N−xn)(N−xn−1)
2 − | ˜A| −(N−xn)(N−xn−1) 2 + | ˜A| +(xn−xn)(N−1)+xn(xn−1)
2 −xn(xn−1) 2
= | ˜A| − | ˜A| ≥0.
The last inequality holds due to (vi). So property (v) is proven.
It remains to show (vi). Property c) of the bijection ensures that in A(n) at most max(| ˜B| − | ˜B|,0)more(r+1)-messages above the diagonal than inA(n)can arrive.
This yields
| ˜An+1| − | ˜An+1| ≥ | ˜An| − | ˜An| −max(| ˜B| − | ˜B|,0)
=
| ˜An| − | ˜An| ≥0 if| ˜B| ≤ | ˜B|
| ˜An| + | ˜B| − | ˜An| − | ˜B| = (N−xn)(N2−xn−1)−(N−xn)(N2−xn−1)≥0 ow.
So the proof is finished in Case A.
Case Br :=R(n)=R(n)−1:We will use the setsA,B,C,D,F,G(as defined in the proof of the upper bound) and define the setsA,B,H,I,K,Janalogously to those in the proof of the upper bound.
Again, denote withM = I∪B∪JandM = B∪D∪G the sets of relevant channels. Choose any bijectionh:[N]2→ [N]2, that keeps the diagonal fixed, with
M⊆h(M) if|M| ≥ |M|
h(M)⊆M if|M| ≥ |M| (22)
and
B∪G⊆h(B∪J) if|B| + |G| ≤ |B| + |J|
h(B∪J)⊆B∪G if|B| + |G|>|B| + |J|. (23) DefinegE≤n as the bijection on Minduced byh. Then conditions (i) and (ii) for gn+1
defined in Eq.17are obviously fulfilled. The factR(n+1)≤R(n)+1 immediately implies (iii).
Let us we verify (iv). Ifmn+1=r+1 (i.e., a global round switch inA(n)occured), then mn+1≥maxA(n+1)and we are done. Otherwise, ifmn+1=r, observe thatmn=r, that BandJcontain all relevantr-entries ofA(n)andBandGcontain all those relevant entries inA(n)which can be overwritten by(r+1). Hence the change inanandbncan be bounded, due to property (23), by
bn+1(mn+1)−an+1(mn+1)
≥bn(mn)−an(mn)−max(|B| + |G| − |B| − |J|,0)
= |B| + |C| + |D| + |F| + |G| − |B| − |J| −max(|B| + |G| − |B| − |J|,0)
=
|B| + |C| + |D| + |F| + |G| − |B| − |J| ≥0 if|B| + |G| ≤ |B| + |J|
|C| + |D| + |F| ≥0 if|B| + |G|>|B| + |J|, and so (iv) is proven.
To finish the proof in Case B, it remains to check conditions (v) and (vi) in case of R(n+1)=R(n+1)=r+1, i.e., a global round switch occured inA(n), but then there are no(r+2)-processes inA(n+1)and so (v) and (vi) are trivially fulfilled.
Case CR(n)+1 < R(n)+1:Here we define gE≤n := idM. The conditions (i) and (ii) are clear, and we can ignore (v) and (vi). FromR(n)+1 < R(n) we obtain R(n+1) < R(n+1)and thus (iii). Moreover, we have maxA(n)≤minA(n)and hence A(n+1)i,j ≤A(n+1)i,j for all 1≤i, j ≤N; this implies (iv).
5 Discussion and Future Work
In the previous sections, we constructed an upper and a lower bound for theα-synchronizer and proved several properties of these approximations. In this section, we will complement our analytical findings by some simulation results.
Based on our constructions of the Markov chains in Sections3and4, the upper and lower bound can be directly determined by solving the linear equation system for the steady state distributionπof the respective Markov chain and by computing the expected round round duration fromπusing F¨uger et al. (2015, Theorem 5). Algorithm 1 and 2 provide the details of these computations.
In Figs. 4(for p fixed) and Fig.5(for N fixed), we compare simulations for the α-synchronizer against the exact values of the bounds given in F¨ugger et al. (2015) and of our new bounds. For the simulations, we performed 30 runs with 100000 time steps each. The results for theα-synchronizer are represented as (very thin) box-whisker-charts in Fig4, whereas in Fig.5we only show the average. Note that the bounds of F¨ugger et al. (2015) are represented by triangles, whereas our bounds are represented by squares. Noticeable is the very good approximation by our new upper bound, which confirms the analytic dominance results obtained in Section3.5.
Our bounds are indeed an improvement of the results of F¨ugger et al. (2015), since we reduced the state-space of the Markov chains from exponential to polynomial size and since our bounds (especially the upper bound) are much better. Nevertheless, we don’t have closed formulas and calculating and solving the Markov chains of our bounds is still expensive: We could solve it algebraically only up ton=4 (whereas for theα-synchronizer we could do it only forn=3 and were not even able to generate the state space forn=4). Numercial solutions are computable also for highern: the upper bound we could solve forn=12 in about 12 seconds, forn=20 it took us about five minutes. The lower bound is more time-consuming: Forn=10 it took us about half a minute, whereas forn=12 the solution was computed in 2,2 minutes. To be more precise, the size of the state spaces of the Markov chain and time and memory consumption of our algorithms computing the bounds are summarized in the following table.
Fig. 4 Monte-Carlo simulation results for theα-synchronizer (represented as boxplots) compared against the lower () and upper () bound given in F¨ugger et al. (2015), our new upper bound, and our new lower bound. As usual, the (orange) box of the boxplots represents the interval from the 25% to the 75% quantile, whereas the fences mark the minimum and the maximum
Fig. 5 Monte-Carlo simulation results (forN=5 andN=10 the means represented as•) and exact round duration (forN = 3) for theα-synchronizer compared against the lower () and upper () bound given in F¨ugger et al. (2015), our new upper bound, and our new lower bound
Upper bound Lower bound
Number of states N (N2−1)/2 (N4−N3+5N2−11N+12)/6
Memory (N6) (N8)
Time O(N9) O(N12)
For example, forn=12 the Markov chain describing the upper bounds has 858 states, whereas the Markov chain for the lower bound has 3268 states.
So future challenges are to find bounds which are also easily computable for higher values of nand which are still of good quality. Furthermore, we plan to investigate the asymptotic behavior for largen. The present paper reveals that this is not a priori hope-less, in the sense that the improved understanding of the underlying Markov chain already allowed us to reduce the exponential state space to polynomial and to compute consider-ably improved upper and lower bounds. It may well be the case that the additional insights gained in the analysis of the modified synchronizer algorithms will eventually allow us to come up with computationally better tractable bounds.
Moreover, in F¨ugger et al. (2015) the authors considered variants of theα-synchronizer by allowing processes to actively forget parts of their accumulated knowledge. It is also part of our future work to adapt the ideas of the upper and the lower bound of theα-synchronizer to construct bounds for their case II (when processes forget their knowledge at a local round switch).
6 Glossary of Notation
Ri(t ) local round number of processi p. 4
R(t ) global round number p. 4
Ki,j(t ) knowledge of processifrom processj p. 4
P probability matrix of successful transmission p. 4
λ=λ(N, p) expected round duration p. 5
A(t ), A(i)(t ) matrix of the Markov Chain (after computation step (i)) p. 6 a(t ), a(i)(t ) matrix of the normalized Markov Chain p. 6 md(i)(t ) minimum of the diagonal ofA(i)(t ) p. 6 w(i)(t ) number ofmd(i)(t )-entries in the diagonal ofA(i)(t ) p. 6
(x)= x1−x2 p. 9
ν(x)= (N−1)(N−x) p. 9
ν= N (N−1) p. 9
M set of messagepatterns,∼= {0,1}N2−N p. 12
E set of schedules,=MN p. 12
An= {positions of non-diagonal(r+1)-entries inA(n)} p. 13 Gn= {positions of(r−1)-entries inA(n, r, r+1)} p. 13 Bn= {positions ofr-entries inA(n, r+1, r+1)
andA(n, r, r+1)} p. 13
Fn= {positions ofr-entries inA(n, r+1, r)} p. 13 Cn= {positions of non-diagonalr-entries inA(n, r, r)} p. 13
Dn= {positions of(r−1)-entries inA(n, r, r)} p. 13
Hn= {positions of non-diagonal(r+2)-entries inA(n)} p. 16 In= {positions of(r+1)-entries inA(n, r+2, r+2)
andA(n, r+1, r+2)} p. 16
Kn= {positions of(r+1)-entries inA(n, r+2, r+1)} p. 16 Jn= {positions ofr-entries inA(n, r+1, r+2)} p. 16
n(x)= (N−x)(N−x−1) p. 23
μ(x)= (N−1)x p. 23
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