DISTRIBUTED HEAVY HITTERS 41

Listing 4.4 DHHE algorithm at any node S_i ∈ I (contd.)

49: upon receive hrequest, IDi from coordinator do

50: for all t∈ID do

51: F req_i ←F req_i∪ {ht, CountM in.getFreq(t)i}

52: if ∃κ∈[g], t∈ Gκ,i then

53: G^κ,i← G^κ,i\ {t}

54: end if

55: end for

56: send message hreply, mi, F reqiito C

57: end upon

58: upon receive hdetection, GIi from coordinator do

59: enqueue GI inJⁱ

60: end upon

Listing 4.5 DHHE algorithm at the coordinator.

1: upon receive hwarning, mi, F reqii from node Si do

2: m←mi

3: GI ←F req_i

4: ID← identiers inF req_i

5: for all S∈ I \ {Si} do

6: send messagehrequest, IDito S_i

7: end for

8: for all received message hreply, m, F reqifrom S do

9: m←m+m_

10: for all hv,fˆv,i ∈F req∧ hv,fˆvi ∈GI do

11: GI ←GI∪ {hv,fˆv+ ˆfv,i}

12: end for

13: end for

14: for all (v,fˆv)∈GI do

15: if fˆ_v < θmthen

16: GI ←GI\ {hv,fˆvi}

17: end if

18: end for

19: if GI 6=∅then

20: returnGI

21: send messagehdetection, GIito each S_ı∈ S

22: end if

23: end upon

FKDHHE Communication Complexity

To Maximise the communication between thesnodes and the coordinator the adversary has to manipulate thesinput streams so that the time needed to locally ll the buers is minimised, and thus the frequency at which nodes communicate with the coordinator is maximised. Theorem 4.4 shows that this is achieved if, for allSi ∈ I, all the heavy hitters appear with the same probability inσ_s.

Communication between the snodes and coordinator is triggered each time a buer becomes full or upon timeout. Indeed, we have to determine the number of items, in expectation, that must be received at any nodeS_i ∈ I to locally ll buerGκ,1≤κ≤g. To analyse the communication cost, we study a generalisation of the coupon collector problem [10,11,41]. The numberTa,n(p)is the number of items that have to be read from a stream with a probability distributionp in order to collectadierent items belonging to[n].

In the following, we denote byUa,nall the sub-sets of items of[n]whose size is exactly equal to a, that is Ua,n ={V ⊆[n]| |V|=a}. Note that we haveU0,n ={∅}. For every V ∈U_a,n, we dene P_V =P

t∈V p_t, withP_∅ = 0.

For each κ = 1, . . . , g−1, we denote with Vκ the set of all the items t inσi whose probability of occurrence pt,i veries θ+ (1−θ)/2^κ < pt,i ≤θ+ (1−θ)/2^κ−1, i.e., the items that can be stored inGκ. We can than denev_k=|V_κ|and we havev_k≤n_k(recall from Section 4.3.1, thatnκ represents the maximal number of items whose probability of occurrence is equal to the lower bound of the range, that isnκ =b1/(θ+ (1−θ)/2^κ)c).

Forκ=g, we denote byV_g the set of all the itemstinσ_iwhose probability of occurrence pt,i veries θ < pt,i ≤ θ+ (1−θ)/2^g−1. We set vg = |Vg|, and we have vg ≤ ng with ng =b1/θc. We also dene pκ,i = (pi)t∈Vκ. Hence, Ua,vκ ={V ⊆Vκ| |V|=a}, and for everyV ∈U_a,vκ,P_V =P

t∈V p_t,i.

Then, for each node Si ∈ I, the expected number of items that need to be recei-ved from a stream σi with a probability distribution pi to ll buer Γκ is given by E[T_νκ,Vκ(p_i)].

Timer Dimensioning The lenght of the timer τ_κ∈[g] is constrained by two opposite needs. A short timer yields a better detection latency but sends the buer more often. On the other end, a long timer reduces the communication but may delay the detection of a distributed heavy hitter. The basic idea is to set a length that matches the expected time to ll the buer, independently of the frequency distribution. In [9]

we prove the following lemma:

Lemma 4.3 (Coupon Collector Expected Value). Let For every n ≥ 1, a = 1, . . . , n, and0< θ≤pt for t∈[n], we have

E[T_a,n(p)]≤H_b1/θc/θ

Thus in accordance with Lemma 4.3, we set for everyκ= 1, . . . , g,τκ=Hb1/θc/θ.

Theorem 4.4 (FKDHHECommunication Cost Upper Bound). The FKDHHE exchanges in average no more than

2ms(logm+ logn) Xg κ=1

νκ

Pτ−1

ı=0 P[T_νκ,nκ(u_κ)> ı] bits. (4.1) whereuκ is the uniform distribution withn items, i.e., ∀t∈Vκ, ut,κ = 1/νκ

4.3. DISTRIBUTED HEAVY HITTERS 43 Proof. From Listings 4.2, 4.4 and 4.5, node Si ∈ I triggers a transmission with the coordinator each time one of its buers is full or upon timeout. Thus, assuming that S_i maintains a single buer Gκ,i, then for mi large, the expected number of times node Si

sends the content ofGκ,i to the coordinator, denoted Cκ,i, veries

C_κ,i ≤ m_i

Finally, we need to determine the number of bits sent each time buerGκ,iis full or upon timeout. From Listing 4.2, the sending of message warning requireslogm_i+ν_κ(logn+ logmi)bits to be transmitted to the coordinator (wherelogmi represents the number of bits needed to code item frequencies, and lognthe one needed to code item identiers).

By Listing 4.5, this triggers a round trip communication between the coordinator and the remainings−1 nodes to collect the frequencies of all potential heavy hitters ofGκ,i

(Line 56 of Listing 4.4 and Line 6 of 4.5). This generates respectivelys−1messages of ν_κlognbits from the coordinator and a message oflogm_i⁰+ν_κ(logm_i⁰+ logn)bits from each node Si⁰ ∈ I 6=Si, wheremi⁰ is the size of σi⁰. As the sum of all the local streams is equal tom, the number of bits sent becauseG^κ,i is full is less than 2sνκ(logm+ logn). Thus, the fact thatC_κ,i must be computed for all the buers at every nodeS_i ∈ I, allows us to complete the proof of the lemma.

Note that whenθ→0, which is the case in the distributed heavy hitters problem, we have τ → ∞, and the denominator in Relation 4.1 tends ton_κ(H_nκ−H_nκ−νκ) [8].

FKDHHE Correctness

We investigate whether malicious nodes can prevent the detection of a heavy hitter. This amounts to showing that, for any given thresholdθ∈(0,1]and approximation parameter ε∈(0,1], the adversary cannot (by nely distributing the occurrences of item tamong the s streams) compel the coordinator to return any items t such that ft <(1−ε)θm and cannot prevent the coordinator from returning all items t such that f_t ≥ θm. We now prove that the FKDHHE algorithm guarantee such properties.

1Notice thatPV_κ may be smaller than1and thenpκ,iis not a probability distribution. To overcome this issue we introduce the item 0with probability of occurrence p0,κ. Item0 can be drawn but does not count towards the number of distinct items collected

Theorem 4.5 (FKDHHE Correctness). The FKDHHE algorithm deterministically and exactly solves the distributed heavy hitters problem ( i.e, δ= 0 and ε= 0).

Proof. The proof consists in showing that the algorithm does not output any items t such thatft< θm(i.e., no false positive) and returns all itemstsuch thatft≥θm(i.e., no false negative).

Let us rst focus on false positives. Suppose by contradiction that the coordinator returns at time j some item t such that ft < θm. Then, by Listing 4.5, this means that the coordinator has received from some nodeS_i ∈ I a messagewarning for which t∈F reqi. By Listing 4.2, this can only happens if nodeSi has identiedtas a potential heavy hitter, that is,pt,i≥θ(see Line 14 of Listing 4.2). Now, upon receipt ofF reqi, the coordinator collects from the others−1nodes the frequency of each itemt⁰ ∈F req_i, and in particulartfrequency, as well as the current size of their input stream (cf., Lines 611 of Listing 4.5). By Lines 15 and 16, the coordinator removes fromGI all the items whose frequency is less thanθm, and in particular item t. Thus the coordinator cannot return tat timej.

A false negative means that the coordinator does not return a true global heavy hitter. Suppose that there existstsuch that f_t≥θmand t is not returned. Thus there exists at least one streamσi such thatpt,i≥θ. By Lines 1524 of Listing 4.2,tis inserted in the buerG^κ that matches its occurrence probability pt,i (if not already present). By Lines 29 and 42,tis sent to the coordinator after at mostH_b1/θc/θreading. By applying an argument similar to the above case, we get a contradiction with the assumption of the case.

To summarize, Theorem 4.5 has shown that FKDHHE accurately tracks distributed heavy hitters, even if they are hidden in distributed streams. In addition, we have provided with Theorem 4.4 an upper bound on the communication cost between the s nodes and the coordinator. This bound is reached when each buer is fed with items that all occur with the same probability.

DHHE Approximation

The following lemma shows that using the Count-Min algorithm to estimatepbt,i = ˆft/m, DHHE is a(ε, δ)-approximation of FKDHHE, for anyε∈(0,1)and probability of failure δ ≤1/2. This is achieved by sharpening the bounds obtained in [30] and by relying on Theorem 4.5. In [8] we prove the following lemma:

Lemma 4.6 (Rened Count-Min Accuracy Bounds). For a stream σ, for any ε∈(0,1) and probability of failureδ≤1/2, the Count-Min algorithm with r=dlog(1/δ)erows and c=d2(1−θ)/εθe columns guarantees that

∀t∈σ,



 P

hfˆt−ft≥εft

i≤δ if pt≥θ Ph

fˆ_t−f ≥ε(m−f)i

≤δ otherwise.

Theorem 4.7 (DHHE Approximation). The DHHE algorithm uses O((logn+ logm) log(1/δ) (1/θ−1)/ε+ (logslogn)/θ) bits of space on each node to (ε, δ)-approximate tracking of distributed heavy hitters in arbitrarily distributed input streams.