Finite Length Analysis on Listing Failure Probability of Invertible Bloom Lookup Tables

0.19 MB·English

by  Daichi Yugawa

Finite Length Analysis on Listing Failure
Probability of Invertible Bloom Lookup Tables
Daichi Yugawa∗ and Tadashi Wadayama∗
∗Department of Computer Science and Engineering,
Nagoya Institute of Technology Nagoya, Japan
Email: [email protected], [email protected]
Abstract—The Invertible Bloom Lookup Tables (IBLT) is a finite length performances. Especially, predicting the error
data structure which supports insertion, deletion, retrieval and floor of the listing failure probability is required to guarantee
listingoperationsofthekey-valuepair.TheIBLTcanbeusedto
the accuracy of a listing process. It is known stopping sets
realize efficient set reconciliation for database synchronization.
[11] dominate the finite length performance of LDGM codes
The most notable feature of the IBLT is the complete listing
operationofthekey-valuepairsbasedonthealgorithmsimilarto forerasurechannels.InthecaseoftheIBLT,thestoppingsets
3 the peeling algorithm for low-density generator-matrix (LDGM) have crucial importance as well as the case of LDGM codes.
1 codes. In thispaper, we will present a stoppingset (SS) analysis In this paper, we will present a stopping set analysis for the
0 for the IBLT which reveals finite length behaviors of the listing
IBLT which unveils the finite length behaviors of the listing
2 failure probability. The key of the analysis is enumeration of
n the number of stopping matrices of given size. We derived failure probability.
a a novel recursive formula useful for computationally efficient The outline of this manuscript is organized as follows.
J enumeration. An upper bound on the listing failure probability Section II introducesnotationand definitionsrequiredfor this
1 based on the union bound accurately captures the error floor paper. A brief review of the IBLT is also given. Section III
behaviors. It will be shown that, in the error floor region, the
3 provides an upper bound on the listing failure probability.
dominant SS have size 2. We propose a simple modification on
hash functions, which are called SS avoiding hash functions, for An enumeration method for the number of stopping matrices
]
T preventing occurrences of the SS of size 2. based on a recursive formula is the heart of the efficient
I evaluation of the upper bound. Section IV presents some
s. I. INTRODUCTION results of computer experiments. It will be shown that, in
c The Invertible Bloom Lookup Tables (IBLT) is a recently the error floor region, the stopping sets with size 2 become
[
developed data structure which supports insertion, deletion, dominant.InSectionV,aclassofhashfunctions,SSavoiding

1 retrieval and listing operations of the key-value pairs [7]– hash functions, is proposed to resolve the stopping sets with
v [10]. The IBLT can be seen as a natural extension of the size 2 for lowering the error floor.
3 Bloomfilter[1]–[6]whichcanhandlesetmembershipqueries.
0
The most notable feature of the IBLT is the complete listing
5 II. PRELIMINARIES
operationofthekey-valuepairsbasedonthealgorithmsimilar
7
. to the peelingalgorithm[12] forlow-densitygenerator-matrix
1 A. Bloom Filter
(LDGM) codes.
0
3 The listing operation enable us to use the IBLT for a Before going into details of the IBLT, we here explain the
1 basis of an efficient set reconciliation algorithm with small structure of the original Bloom filter (BF) which is the basis
: amount of communications. Set reconciliation is a process to of the IBLT. Assume that we have a binary array T and
v
i synchronize contents of two sets at two distinct locations and k-hash functions h1,...,hk. The binary array T is initially
X
itcanbe usedforrealizingdatabasesynchronization,memory set to all zero. When an item x comes to insert, we set
r synchronization,andanimplementationoftheBiffcodes[10]. T[h (x)] = 1 for i ∈ [1,k]. The notation [α,β] means
a i
The implementation of the IBLT is fairly simple and it is the set of consecutive integers from α to β. The process is
naturally scalable to multiple servers, which is a desirable called the Insert(x) operation. The set membership query on
feature for data sets of extremely large size. y is the query for checking whether y is in the BF or not.
The paper by Goodrich and Mitzenmacher [7] provides The LookUp(y) operation returns YES if T[h (y)] = 1 for
i
the detailed analysis on the IBLT such as the optimization i ∈ [1,k]; otherwise it returns NO. The operations Insert(x)
of the number of hash functions to minimize the retrieval andLookUp(y)canbecarriedoutinO(k)-time.Notethatthe
failure probability. They also presented asymptotic thresholds LookUp(y) operation may yield false positive; i.e., it returns
for accurate recovery by using the known results on 2-cores YES when y is not in the BF. The minimization of this false
of random hypergraphs. Furthermore, some fault tolerance positiveprobabilityintermsofthenumberofhashfunctionsis
features of the IBLT are extensively studied. animportanttopicofstudiesoftheBF[1][6].Anappropriately
Fordesigningpracticalapplications,itisbeneficialtoknow designedBF providesa highlyspace efficientset membership
not only the asymptotic behavior of listing processes but also query system with reasonably small false positive probability.
B. IBLT and its Operations h when x ∈ {0,1}b obeys the uniform distribution. The m-
i
cellsaresplitintok-subtableseachofsizem/kandeachhash
As in the case of the BF, k-hash functions h ,...,h are
1 k functionuniformlyselectsa cellinasubtable.Inotherwords,
used in the IBLT. Instead of binary array, the IBLT utilizes
the range of h is [(i−1)∗(m/k)+1,i∗(m/k)].
an array of cells T[1],...,T[m]. A cell T[i] consists of three i
fields which are called Count, KeySum, and ValueSum, which
III. UPPERBOUND ONLISTINGFAILUREPROBABILITY
are denoted by T[i].Count,T[i].KeySum,T[i].ValueSum.
In this section, we will derive an upper bound on the
An input to the IBLT is a key-value pair (Key,Value). The
listingfailureprobability.Thelistingfailureeventoccurswhen
count field represents the number of inserted entries. The
a stopping set [11], which is a combinatorial substructure
KeySum (resp. ValueSum) field stores exclusive OR of key
of a matrix, appears. In order to evaluate the listing failure
(resp. value) of inserted entries. The contents of all the cells
probability, we need to enumerate the number of stopping
are initialized to zero at the beginning.
matrices of given size. A stopping matrix is a matrix with
The IBLT allows 4-operations: Insert(x,y), Delete(x,y),
no row of weight one corresponding to the case where no
Get(x) and ListEntries(). The operation Insert(x,y) stores
cells with counter value equal to one exists.
a key-value pair (x,y) into the IBLT. In an insertion pro-
cess, the key x (resp. value y) is added (over F ) to the
2 A. Enumeration of Stopping Matrix
KeySum (resp. ValueSum) filed of T[h (x)] for i ∈ [1,k];
i
namely, T[h (x)].KeySum = T[h (x)].KeySum ⊕ x and The state matrix B of an IBLT can be represented by an
i i
T[h (x)].ValueSum= T[h (x)].ValueSum⊕y. The count m×n binary matrix where m = ℓk. A row of the matrix B
i i
field of T[h (x)] is also incremented as T[h (x)].count = corresponds to a cell and a column corresponds to an entry.
i i
T[h (x)].count + 1 at the same time. The operation The matrix B can be divided into disjoint k-blocks with size
i
Delete(x,y)removesthekey-valuepair(x,y)fromtheIBLT. ℓ×n. If the (s,t)-element of the u-th block of B is one, this
The process is the same as that of Insert(x,y) except for meansthatthet-thentryishashedtothes-thcellbyusingthe
decrementing the counter. The operation Get(x) retrieves the u-th hash function. Suppose that a sub-matrix M′ consisting
valuecorrespondingtothekeyx.Thisoperationisrealizedas ofseveralcolumnsof M havenorowsofweightone.Insuch
follows. If there exists i∈[1,k] satisfyingT[h (x)].Count= a case, ListEntries() fails to list all the entry in this table
i
1, then Get(x) returns T[h (x)].ValueSum. Otherwise, because M′ cannot be resolved in the peeling process. If a
i
Get(x) declares the failure of the operation. binary matrix M′ does not have a row with weight one, M′
is said to be a stopping matrix. The existence of a stopping
The last operation ListEntries() outputs all the key-value
matrix in B is the necessary and sufficient condition for the
pairs in the IBLT by sequentially removing the entries with
failure of a peeling process [11][12].
the counter value equal to one from the table. The de-
tails of the process is as follows. We first look for i ∈ In our case, the state matrix B is divided into k-subblocks
[1,m] satisfying T[i].Count = 1. If there exists i∗ satis- correspondingtosubtables.Itmightbe reasonableto consider
fying the condition T[i∗].Count = 1, the key-value pair a stopping matrix in a subblock before discussing the proba-
(T[i∗].KeySum,T[i∗].ValueSum)isregisteredintotheout- bility of the event that B includes a stopping matrix.
putlistandthenDelete(T[i∗].KeySum,T[i∗].ValueSum)is Let S(ℓ,n) be the set of ℓ×n binary matrices with column
executed.Thisprocessisiterateduntilnocellwiththecounter weight one; i.e.,
value equal to one can be found. It should be remarked that,
S(ℓ,n)=△{(m ,...,m )∈{0,1}ℓ×n|wt(m )=1,i∈[1,n]},
in some cases, ListEntries() fails to list all the entry in the 1 n i
IBLT. This is because a non-empty IBLT can have counter
where wt(·) represents the Hamming weight function. From
values larger than one for i ∈ [1,m]. This failure event is this definition,it is evidentthatthe cardinalityof S(ℓ,n) is ℓn.
called a listingfailure.Itis desirablethatan IBLT isdesigned The number of the stopping matrices in S(ℓ,n) is denoted by
to decreasethe frequencyof the listing failure eventsas small
z(ℓ,n), which can be written as
as possible.
z(ℓ,n)=△ #{M ∈S(ℓ,n) |M is a stopping matrix}. (1)
C. Probabilistic Model
For convention, z(0,0) is defined to be 1.
It is clear that the probability of the listing failure event,
The next recursive formula plays a key role to enumerate
which is called the listing failure probability, depends on
z(ℓ,n) which is required for evaluating an upper bound for
the definition of the probabilistic model for keys and hash
the listing failure probability.
functions. In this paper (except for Section V), we assume
Theorem 1 (Recursive formula on z(ℓ,n)): The following
the following model for keys and hash functions. The hash
recursive relation
functions h ,...,h have domain {0,1}b and the key of
1 k
the entries to be stored are independent random variables min(ℓ,n) ℓ n
uniformly distributed over {0,1}b. The number of entries z(ℓ,n)=ℓn− c! z(l−c,n−c) (2)
(cid:18)c(cid:19)(cid:18)c(cid:19)
are assumed to be n. The hash functions are assumed to be Xc=1
uniform such that h (x) distributes uniformly in the range of holds for ℓ≥1 and n≥1.
i
(Proof)Leta(ℓ,n)bethecardinalityofnon-stoppingmatrices TABLEI
a(ℓ,n)=△ ℓn−z(ℓ,n).Inthe following,we enumeratea(ℓ,n) VALUESOFz(ℓ,n):NUMBEROFSTOPPINGMATRICESINS(ℓ,n)
by using a recursive relation. For given M ∈ S(ℓ,n), a pair
ℓ\n 1 2 3 4 5 6 7 8 9 10
(i,j) ∈ [1,ℓ]×[1,n] is said to be a pivot of M if Mi,j = 1 1 0 1 1 1 1 1 1 1 1 1
and the Hamming weight of the i-th row of M is 1. The set 2 0 2 2 8 22 52 114 240 494 1004
3 0 3 3 21 63 243 969 3657 12987 43959
of pivots of M is denoted by
4 0 4 4 40 124 664 3196 15712 79228 396616
△ 5 0 5 5 65 205 1405 7425 44385 271205 1666925
piv(M)={(i,j)∈[1,ℓ]×[1,n]|(i,j) is a pivot of M}.
6 0 6 6 96 306 2556 14286 100176 691146 4916436
7 0 7 7 133 427 4207 24409 196105 1471519 11773699
Note that M is a stopping matrix if and only if piv(M) is
8 0 8 8 176 568 6448 38424 347712 2775032 24547664
empty. The cardinality of non-stopping matrices a(ℓ,n) can 9 0 9 9 225 729 9369 56961 573057 4794633 46341081
be represented by 10 0 10 10 280 910 13060 80650 892720 7753510 81163900
min(ℓ,n)
a(ℓ,n)= #T(ℓ,n) (3)
i B. Listing Failure Probability and its Bound
Xi=1
The set of all the state matrix is defined as
where
T(ℓ,n) =△ {M ∈S(ℓ,n) |#piv(M)=i}, i∈[0,min(ℓ,n)]. B(ℓ,n,k) =△ {(M1,...,Mk)T |Mi ∈S(ℓ,n),i∈[1,k]}. (10)
i
(4) The cardinality of B(ℓ,n,k) is ℓnk. According to the scenario
This is because the set of non-stopping matrices can be we have discussed in the previous section, we here define a
partitioned into disjoint sets T(ℓ,n) for i ∈ [1,min(ℓ,n)]. In probability space by assigning the equal probability 1/ℓnk to
i
the following, we will try to prove the equality each element in B(ℓ,n,k).
#Ti(ℓ,n) =c!(cid:18)ℓc(cid:19)(cid:18)nc(cid:19)z(l−c,n−c) (5) abiSliutpy,powsheicthhatisPtfh(eℓ,pnr,okb)abrielpitryestehnattsLthisetElinsttirniegsf(a)iluorpeerpartoiobn-
fails to list all the entries in the IBLT. The next theorem
for i ∈ [1,min(ℓ,n)]. Assume that M ∈ Tc(ℓ,n) is given provides an upper bound on Pf(ℓ,n,k).
(c∈[1,min(ℓ,n)]). By gettingrid ofall thecolumnandrows Theorem2(Upperboundonlistingfailureprobability):For
corresponding to piv(M) from M, we obtain an (ℓ−c)× given ℓ ≥ 1,n ≥ 1,k ≥ 1, the listing failure probability
(n−c) matrix M′. Namely we delete the i-th row and the P (ℓ,n,k) can be upper bounded by
f
j-thcolumnfromM if(i,j)∈piv(M).Fromtheassumption
n k
M ∈Tc(ℓ,n),theresultingmatrixM′mustbeastoppingmatrix P (ℓ,n,k)≤ n z(ℓ,i) . (11)
in T0(ℓ−c,n−c). Note that the size of T0(ℓ−c,n−c) is given by f Xi=2(cid:18)i(cid:19)(cid:18) ℓi (cid:19)
z(ℓ−c,n−c).Therefore,thesizeofT(ℓ,n)istheproductofthe
i (Proof) The peeling process of the ListEntries() fails to
numberofpossiblewaystochoosepiv(M)andz(ℓ−c,n−c). recoverall the entries in the IBLT if and only if B ∈B(ℓ,n,k)
Basedonasimplecombinatorialargument,wecanseethatthe
containsa stoppingmatrixasitssub-matrix.Thus, P (ℓ,n,k)
f
numberofpossiblewaystochoosepiv(M)canbeenumerated
can be characterized as
as c! ℓ n . As a result, we have the equality (5). Combining
c c
(3) a(cid:0)nd(cid:1)(cid:0)(5(cid:1)), the claim of the theorem is obtained. Pf(ℓ,n,k)=Pr[B includes a stopping matrix]. (12)
For some special combinations of ℓ and n, z(ℓ,n) has a
For an index set I ∈ 2[1,n], let B be the sub-matrix
simple expression as follows. I
of B consisting of columns of B with indices in I. If
z(ℓ,1) = 0, ℓ≥1 (6) B is a stopping matrix, then the index set I is said to be a
I
z(ℓ,2) = ℓ, ℓ≥1 (7) stoppingset.TheprobabilityPf(ℓ,n,k)canbeupperbounded
as follows:
z(ℓ,3) = ℓ, ℓ≥1 (8)
z(1,n) = 1, n≥1. (9) Pf(ℓ,n,k) = Pr[B includes a stopping matrix]
Theseexpressionscanbeeasilyprovedbasedonthedefinition
of the stopping matrix and of S(ℓ,n). The recursive formula = Pr BI is a stopping matrix
(2)enableustoevaluatethevalueofz(ℓ,n)efficiently.These I∈2[[1,n]\∅ 
simple expressions can be used as boundary conditions for a ≤ Pr[I is a stopping set]. (13)
recursive evaluation process. I∈2X[1,n]\∅
Table I presents the values of z(ℓ,n) for (ℓ,n) ∈ [1,10]2.
The last inequality is due to the union bound. From the
Thesevaluesarecomputedbasedontherecursiveformula(2).
definition of the probability space defined on B(ℓ,n,k), the
Note that S(ℓ,n) contains 1010-matrices when ℓ = n = 10.
probability that I is a stopping set is given by
A naive enumeration scheme generating all the matrices in
S(ℓ,n) may have computational difficulty even for such small z(ℓ,#I) k
Pr[I is a stopping set]= . (14)
parameters.
(cid:18) ℓ#I (cid:19)
By using this equality, we have the following upper bound:
P (ℓ,n,k) ≤ Pr[I is a stopping set]
f
I∈2X[1,n]\∅
n
= Pr[I is a stopping set|#I =i]
Xi=1I∈2X[1,n]\∅
n k
n z(ℓ,i)
= . (15)
(cid:18)i(cid:19)(cid:18) ℓi (cid:19)
Xi=2
In the last equality, we used the fact z(ℓ,1)=0.
IV. COMPUTEREXPERIMENTS
In this section, we will present several results on computer
experiments and on numerical evaluation of the upper bound
presented in the previous section.
In order to examine the tightness of the bound, Figure 1
Fig.1. Comparisonofthelistingfailureprobability:experimentalvaluesand
presents curves of the listing failure probability obtained by upperbound(n=210, k=3, b=32).
computer experiments (dashed line) and of the upper bound
(solid line). These curves are plotted as functions of the
numberof cells m. The numberof entries is n=210 and the
symbolsizeofthekeyisb=32.Incomputerexperiments,the
number of trials is 106. As a hash function, SHA-1[13] was
used. The numberof the hash functionsassumedto be k =3.
We used pseudorandom 32-bit numbers for pseudorandom
key-valuepairs. Itcanbe observedthattheupperboundgives
fairly tight estimation, as the number of cells m increases.
As in the case of LDPC codes, the error curve in Figure 1
exhibitbothwaterfallanderrorfloorphenomenon.Thisresult
indicatesthattheupperboundpreciselycapturestheerrorfloor
behavior of the listing failure probability.
From the upper bound, it is possible to see a tradeoff
between the water fall and error floor. Figure 2 presents the
upperboundsfork ∈[3,6].Thenumberofentriesisn=100.
A curve of the upper bound is plotted as a function of the
number of cells m. We can observe that the listing failure Fig.2. Comparisonoftheupperboundonlistingfailureprobability:3hashes,
probabilities in the error floor region can be decreased as the 4hashes,5hashesand6hashes(n=100).
number of hash functions k increases. On the other hand,
increments of k pushes the water falls to the right.
From the upper bound and some experimental results, we
see that stopping sets of size 2 dominates the error floor
behavior. Figure 3 presents the upper bound, the asymptote
P (ℓ,n,k) defined by
2
k
△ n z(ℓ,2) n 1
P (ℓ,n,k)= = (16)
2 (cid:18)2(cid:19)(cid:18) ℓ2 (cid:19) (cid:18)2(cid:19)ℓk
and the experimental value of the list error probability. The
result suggest that the probability of occurrence of stopping
set of size 2 determines the depth of an error floor.
V. SS AVOIDINGHASH FUNCTION
We have seen that stopping sets of size 2 dominate the
behaviorofthelistfailureprobabilityintheerrorfloorregion.
The stopping sets of size 2 occur when k-hash values for 2-
distinct keys collide; i.e., Fig. 3. Comparison of the listing failure probability: experimental values,
upperboundandasymptote P2(ℓ,n,k)(n=210, k=3, b=32).
(h (a),h (a),...,h (a))=(h (b),h (b),...,h (b)) (17)
1 2 k 1 2 k
for a 6= b. If this type of collision can be prevented, it is the error floor performance without sacrificing the waterfall
expected that the error floor performance can be improved. performance.
The SS avoiding hash function defined here are designed
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Figure 4 presents the results of a computer experiment on
the SS avoiding hash functions. As an bijective map, the
identity map was exploited. The two curves of listing failure
probabilitiesareplotted;thefirstonecorrespondstothecaseof
a conventionalhash function and the second one corresponds
tothecaseoftheSSavoidinghashfunctionwherethesymbol
size of the key is b=3s. In bothcases, the numberof entries
is n = 210 and the number of hash functions is assumed to
be k =3. We can observe that the SS avoiding hash function
reducesthelistingfailureprobabilitiesintheerrorfloorregion.
Furthermore, the upper bound almost captures the error floor
behavior of the listing failure probability in this settings.
VI. CONCLUSION
In this paper, we presented a finite length performance
analysisonthelisting failureprobabilitywhichmaybe useful
for designing a system or an algorithm including the IBLT
as a building component. The recursive formula presented
in Section III will become an useful tool for finite length
analysis. In Section IV, we have seen that the error floor
performance can be improved by increasing the number of
the hash functions but it degrades the waterfall performance.
From the results shown in Section V, we can expect that ap- Fig. 4. Comparison of the listing failure probability: conventional hash
propriately designed SS avoiding hash functions can improve function andSSavoidinghashfunction (n=210, k=3).