On Multistage Learning a Hidden Hypergraph

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by  A. G. D'yachkov

On Multistage Learning a Hidden Hypergraph
A. G. D’yachkov, I.V. Vorobyev, N.A. Polyanskii and V.Yu. Shchukin
Lomonosov Moscow State University, Moscow, Russia
Email: [e-mail], [e-mail], [e-mail], [e-mail]
Abstract—Learning a hidden hypergraph is a natural gener- Oneofthemostimportantaspectsofanysearchingstrategy
alization of the classical group testing problem that consists in isits adaptiveness.Analgorithmisnon-adaptiveifallqueries
detectingunknownhypergraph Hun=H(V,E) bycarrying out arecarriedoutinparallel.Analgorithmisadaptiveifthelater
edge-detecting tests. In the given paper we focus our attention
queries may depend on the answers to earlier queries.
only on a specific family F(t,s,ℓ) of localized hypergraphs for
which the total number of vertices |V| = t, the number of By Nna(t,s,ℓ) (Na(t,s,ℓ)) denotethe minimalnumberof
edges |E| 6 s, s ≪ t, and the cardinality of any edge |e| 6 ℓ, queries in an F(t,s,ℓ)-searching non-adaptive (adaptive) al-
ℓ ≪ t. Our goal is to identify all edges of Hun ∈ F(t,s,ℓ) gorithms.IntroducetheasymptoticrateofF(t,s,ℓ)-searching
6 by using the minimal number of tests. We develop an adaptive non-adaptive algorithms:
1 algorithm that matches the information theory bound, i.e., the
0 totalnumberoftestsofthealgorithmintheworstcaseisatmost Rna(s,ℓ), lim log2t ,
2 sℓlog2t(1+o(1)). We also discussa probabilisticgeneralization t→∞Nna(t,s,ℓ)
y of the problem. In similar way we define the asymptotic rate Ra(s,ℓ) of
a Keywords: Group testing problem, learning hidden hy-
F(t,s,ℓ)-searching adaptive algorithms.
M pergraph, capacity, asymptotic rate, cover-free code
Therestofthepaperisorganizedasfollows.InSect.II,we
7 I. INTRODUCTION discuss previously known results and remind the concept of
1 A. Notations and Definitions cover-free codes which is close to the subject. In Sect. III,
we present the main result of the paper and provide the
Let |A| denote the size of a set A, , denotes the equality
] deterministic adaptive algorithm that matches the information
T by definition and [N] , {1,2,...,N} is the set of integers
theory bound. Finally, in Sect. IV we discuss a probabilistic
I from 1 to N. A hypergraph is the pair H , H(V,E) such
. generalizationoftheproblemoflearningahiddenhypergraph.
s that E ⊂2V \∅, where V is the set of vertices and
c
[ E ,{(e ,...e ) : e ⊂V, i∈[s]} II. PREVIOUS RESULTS
1 s i
For the particular case ℓ = 1, the above definitions were
3
is an s-set of edges. A set S ⊂ V is called an independent already introduced to describe the model called designing
v
5 set of H if it does not contain entire edges of H. We denote screening experiments. It is a classical grouptesting problem.
0 bydim(H)thecardinalityofthelargestedge,i.e.,dim(H)= We refer the reader to the monograph [10] for a survey
7 max|ei|. on group testing and its applications. It is quite clear (e.g.,
6 i∈[s]
see [10]) that an F(t,s,1)-searching adaptive algorithm can
0 B. Statement of the problem
achieve the information theory bound, i.e., N(t,s,1) =
.
1 The problem of learning a hidden hypergraph is described slog t(1+o(1)) as t→∞. Therefore, Ra(s,1)=1/s.
0 2
[8] as follows. Suppose there is an unknown (hidden) hyper- If ℓ = 2, then we deal with learning a hidden graph.
6
graph H = H(V,E) whose edges are not known to us, One important application area for such problem is bioinfor-
1 un
: but we know that the unknown hypergraph Hun belongs to matics [9], more specifically, chemical reactions and genome
v
the known family F of hypergraphs having certain specific sequencing.Alonetal.[7],andAlonandAsodi[6]givelower
i
X structure (e.g, F consists of all Hamiltonian cycles on V). and upper bounds on the minimal number of tests for non-
r Our goal is to identify all edges of E by carrying out the adaptive searching algorithms for certain families of graphs,
a
minimal number N of edge-detecting queries Q(S), where such as stars, cliques, matchings. In [9], Boevel et al. study
S ⊆V: Q(S) = 0 if S is independentof H , and Q(S) = 1 the problem of reconstructing a Hamiltonian cycle. In [5],
un
otherwise. Angluin et al. give a suboptimal F(t,s,2)-searching adaptive
Inthegivenpaperwefocusourattentiononlyonthefamily algorithm. More precisely, they prove Ra(s,2)>1/(12s).
F(t,s,ℓ) of hypergraphs introduced in the abstract, namely: For the general case of parameters s and ℓ, Abasi et al.
givenintegerss,ℓandt,suchthats+ℓ<t,thesetofvertices have recently provided [11] a suboptimal F(t,s,ℓ)-searching
V ,[t] and the family F(t,s,ℓ), consists of all hypergraphs adaptive algorithm. In particular, from their proofs it follows
H(V,E) such that dim(H) 6 ℓ and |E| 6 s. Suppose we Ra(s,ℓ) > 1/(2sℓ). This bound differs up to the constant
knowthatthehypergraphH belongstothefamilyF(t,s,ℓ). factor from the information theory upper bound Ra(s,ℓ) 6
un
An algorithm is said to be an F(t,s,ℓ)-searching algorithm 1/(sℓ). In Sect. III we improve the lower bound and provide
if it finds H , i.e. there exists only one hypergraph from an F(t,s,ℓ)-searchingadaptivealgorithmwhich is optimalin
un
F(t,s,ℓ) that fits all answers to the queries. terms of the asymptotic rate.
A. Cover-Free Codes leads to
log t 1
A binary N ×t-matrix 2 6 +o(1), t→∞,
N sℓ
X =kx (j)k, x (j)=0,1, i∈[N], j ∈[t] (1) Therefore, we complete the proof.
i i
The key result of this paper is given as follows.
is called a code of length N and size t. By x and x(j)
i Theorem 2. There exists an adaptive F(t,s,ℓ)-searching
we denote the i-th row and the j-th column of the code X,
algorithmwhichhasatmostsℓlog t(1+o(1))edge-detecting
respectively. 2
queries. In other words, the rate Ra(s,ℓ)=1/(sℓ).
Before we give the well-known definition of cover-free
In order to prove Theorem 2 we provide a deterministic
codes, note that any F(t,s,ℓ)-searching non-adaptive algo-
F(t,s,ℓ)-searching adaptive algorithm.
rithm consisting of N queries can be represented by a binary
Proof of Theorem 2.
N×tmatrixX suchthateachtestcorrespondstotherow,and
Let H = H(V,E) be a hidden hypergraph from the
un
each vertex stands for the column. We put x (j) = 1 if the
i family F(t,s,ℓ). We will call a vertex v ∈V active, if there
j-th vertex is included to the i-th test; otherwise, xi(j)=0. existsatleastoneedgee∈E suchthatv ∈e.By F,F ⊂V,
Definition 1. [1]. A code X is called a cover-free (s,ℓ)- denotethesetofalreadyfoundactivevertices.ByE′,E′ ⊂E,
code (briefly, CF (s,ℓ)-code) if for any two non-intersecting denote the set of already found edges of H , i.e., if e ∈ E
un
sets S, L ⊂ [t], |S| = s, |L| = ℓ, S ∩L = ∅, there exists a and e ⊂ F, then e ∈ E′. Note that a pair (V,E′) can be
row x , i∈[N], for which
i viewed as a partial hypergraph of H . Let S, S ⊂ V, be a
un
x (j)=0for anyj ∈S, query we deal with. Before we start the proposed algorithm,
i
(2) we set F =∅, E′ =∅ and S =V.
x (k)=1for anyk ∈L.
i
Firstly we describe an algorithm depicted as Alg. 2 which
Taking into account the evident symmetry over s and ℓ, we allows us to to find a new active vertex v, i.e., v 6∈ F. The
introduce Ncf(t,s,ℓ) = Ncf(t,ℓ,s) - the minimal length of inputof the algorithmare set F and a query S which contain
CF (s,ℓ)-codesof size t anddefine the asymptotic rate of CF at least one edge e∈E and e6∈E′. We set S′ =S\F and
(s,ℓ)-codes: S′′ =S\S′.AteachfurtherstepweguaranteethatS′contains
a new active vertex. While |S′| > 1, we run the following
log t
Rcf(s,ℓ)=Rcf(ℓ,s),tl→im∞Ncf(t,2s,ℓ). (3) ip.reo.,cSed′u=re.SSp⊔litSup, |SS′|in=to⌈t|wSo′|/e2q⌉uaalnsdiz|eSd|su=bs⌊e|tSs′S|/12⌋a.ndThSe2n,
1 2 1 2
In [2], Dyachkov et al. show that any CF (s,ℓ)-code rep- wecarryoutaqueryS ⊔S′′.IfQ(S ⊔S′′)=1thenitmeans
1 1
resents a F(t,s,ℓ)-searching non-adaptive algorithm, while thatS containsatleast onenew active vertex,since fromthe
1
any F(t,s,ℓ)-searching non-adaptive algorithm corresponds previousstepsoftheprocedurewehaveQ(S′′)=0.Therefore
to both a CF (s,ℓ−1)-code and CF (s−1,ℓ)-code.The best we set S′ =S and repeat the procedure. If Q(S ⊔S′′)=0
1 1
presently known upper and lower bounds on R(s,ℓ) of CF then at least one new active vertex must lie in S , since from
2
(s,ℓ)-codes were presented in [2], [3]. If ℓ > 1 is fixed and the previous steps we also have Q(S ⊔S ⊔S′′) = 1. Thus
1 2
s → ∞, then these bounds lead to the following asymptotic we set S′ = S , S′′ = S ⊔S′′ and repeat the procedure. At
2 1
equality: final (|S′|=1) we know that the unique vertex v of S′ is an
active vertex of H and v 6∈ F. Notice that Alg. 2 can be
(ℓ+1)ℓ+1log s un
2 (1+o(1))>Rna(s,ℓ) seen as a variation of the binary vertex search.
2eℓ−1 sℓ+1
SecondlyweprovideanalgorithmdepictedasAlg.3which
ℓℓ log e allows us to to find all edges E′ composed on already found
≃R (s,ℓ)> 2 (1+o(1)). (4)
cf eℓ sℓ+1 active vertices F. The only input of the algorithm is the set
F. After we find a new active vertex v we can update the set
III. ADAPTIVE LEARNING A HIDDEN HYPERGRAPH
E′ by searching edges containing v. But since |F| 6 sℓ ≪t
By a counting argument, the lower bound is true. we can set E′ = ∅ and run the following procedure over all
Theorem 1. Any F(t,s,ℓ)-searching algorithm has at S such that S ⊂ F and |S| 6 ℓ. If there is no edge e ∈ E′
leastsℓlog2t(1+o(1))edge-detectingqueries.Inotherwords, such that e⊂S, then carry out a query S. If Q(S)=1 then
the rate Ra(s,ℓ)61/(sℓ). we delete all edges e ∈ E′ such that S ⊂ e and add edge
Proof of Theorem 1. e=S to E′. Note thatAlg. 3 representsan exhaustivesearch
Let N be the minimal number of tests in the worst case of edges.
among all adaptive F(t,s,ℓ)-searching algorithms. Then we Thirdly we present an algorithm depicted as Alg. 4 which
have allows us to to find a query S such that S contains at least
2N >|F(t,s,ℓ)|. one edge e∈E and e6∈E′ (as a consequence S contains at
leastoneactivevertexv 6∈F).Theonlyinputofthealgorithm
This inequity along with the asymptotic equality
is the set E′. We initialize A as vertices included to at least
tsℓ one edge e ∈ E′, set B = V \A and S = ∅. One can see
|F(t,s,ℓ)|= (1+o(1)), t→∞,
(ℓ!)ss! that |A| 6 sℓ ≪ t. Then we run the following procedure for
all sets C ⊂ V such that B ⊂ C, and ∄e ∈ E′, e ⊂ C. Data: set F ⊂V
If Q(C) = 1 then we set S = C and exit the procedure. If Result: subset of edges E′ ⊂E
Q(C) = 0 we proceed to the next query C. If we finish the initialization E′ :=∅;
procedure and have S = ∅, then it means we have found all for ∀S ⊂F: 16|S|6ℓ do
edges of H , i.e., E′ =E. Note that Alg. 4 is an exhaustive if ∄e∈E′ : e⊂S then
un
query search. if Q(S) = 1 then
We give a full description of the proposed F(t,s,ℓ)- for ∀e∈E′ : S ⊂e do
searching algorithm by Alg. 1, and this algorithm is based on delete e from E′;
Alg. 2, 3 and 4. We set F = ∅, E′ = ∅ and S = V. While end
Alg. 3 gives a query S 6=∅, we run the following procedure. add edge e=S to E′;
With the help of Alg. 2 we find a new active vertex v and end
add itto F. Thenwe use Alg. 3 to updatethe set ofedgesE′ end
composed on already found active vertices F. After that we end
Algorithm 3: Searching edges
run Alg. 4 to find an appropriatequery S, which then will be
used in Alg. 2.
Now we upper bound the number of tests of Alg. 1 in the Data: subset of edges E′ ⊂E
worst case. Let|V|=t. It is easy to check thatAlg. 2 usesat Result: query S
most ⌈log |S|⌉6⌈log t⌉ tests. One can see that the number initialization A:={v : v ∈e∈E′}; B :=V \A;
2 2
of active vertices in the hidden hypergraph H ∈ F(t,s,ℓ) S :=∅;
un
is at most sℓ. Alg. 3 uses at most F (s,ℓ) tests, while Alg. 4 for ∀C ⊂V: B ⊂C and ∄e∈E′, e⊂C do
1
uses at most F (s,ℓ) tests, where the functionsF and F do if Q(C) = 1 then
2 1 2
not depend on t. We can upper bound the number of cycles S :=C and break “for loop”;
in Alg. 1 by the number of active vertices. Therefore, the end
total number of tests for the given algorithm does not exceed end
sℓ(log t+F (s,ℓ)+F (s,ℓ)+1). Algorithm 4: Searching a query
2 1 2
Data: set of vertices V of H ∈F(t,s,ℓ)
un
round) algorithm. In the given section we will limit ourselves
Result: set of edges E of H
un
initialization E′ :=∅; F :=∅; S :=V; to the consideration of only so called two-stage searching
procedures (see, e.g., [12]). It means we can adapt the tests
while S 6=∅ do
of the second round of testing only one time after we receive
perform Alg. 2, find v 6∈F and F :=F ⊔v;
perform Alg. 3, and find subset of edges E′; the answers to the queries of the first round.
Now we consider a relaxation of the problem of learning
perform Alg. 4, and find query S;
a hidden hypergraph. Suppose we let an algorithm iden-
end
Algorithm 1: Learning a hidden hypergraph tify almost all hypergraphs from the family F(t,s,ℓ). More
formally, if there exist a subfamily F′(t,s,ℓ) ⊂ F(t,s,ℓ)
of cardinality at least (1 − ε)|F(t,s,ℓ)| such that for any
H ∈ F′(t,s,ℓ) the algorithm finds H , then we say that
Data: query S ⊆V, Q(S)=1, and set F ⊂V un un
the algorithm is F(t,s,ℓ)-searching with probability (1−ε).
Result: vertex v ∈V, v 6∈F, and ∃e∈E, v ∈e
ByNna(t,s,ℓ,ε)(Na(t,s,ℓ,ε),N2st(t,s,ℓ,ε))denotethe
initialization S′ :=S\F; S′′ :=S\S′;
minimal number of queries in a F(t,s,ℓ)-searching non-
while |S′|>1 do
adaptive(adaptive,two-stage) algorithmwith probability(1−
split in half S′: S′ =S ⊔S ;
1 2 ε).Introducethecapacityofnon-adaptiveF(t,s,ℓ)-searching
if Q(S ⊔S′′)=1 then
1 algorithms
S′ :=S ;
1
log t
else Cna(s,ℓ), lim 2 .
S′ :=S , S′′ :=S′′⊔S ; ε→0 Nna(t,s,ℓ,ε)
2 1 t→∞
end
InsimilarwaywedefinethecapacitiesCa(s,ℓ)andC2st(s,ℓ)
end
of adaptive F(t,s,ℓ)-searching algorithms and two-stage
Algorithm 2: Searching a new active vertex
F(t,s,ℓ)-searching algorithms, respectively.
One can easily check that Theorem 1 is true for “almost”
IV. CONCEPT OF “ALMOST” LEARNING A HIDDEN concept as well. From definitions it follows
HYPERGRAPH 1
Cna(s,ℓ)6C2st(s,ℓ)6Ca(s,ℓ)= ,
Remind that there are two natural types of algorithms, sℓ
namely non-adaptive and adaptive. A compromise between where the right-hand side equality holds in virtue of Theo-
these two types is usually called a multistage (or multiple rem 2.
Forthecaseℓ=1thedefinitionofthecapacitywasconsid- FormatrixX we willcallallsuch hypergraphsgood.For any
ered in many papers. We refer the reader to the classic result binary (N ×t) matrix X denote the set of good hypergraphs
[4] in model of designing screening experiments. Malyutov by G(X), G(X) ⊂ F′(t,s,ℓ). Notice that we have proved
proved that Cna(s,1)=1/s. |G(X )|>(1−ε)|F′(t,s,ℓ)|.
1
We conjecture that Cna(s,ℓ) = 1/(sℓ). The following Now we reformulate the statement derived in [4].
theorem reinforces the hypothesis. Lemma 1. The capacity of non-adaptive F(t,ℓ,1)-
Theorem 3. The capacity of two-stage F(t,s,ℓ)- searching algorithms Cna(ℓ,1)=1/ℓ.
searching algorithms C2st(s,ℓ)=1/(sℓ). Notice that if we are given with a code X representing
WeproveTheorem3usingtheprobabilisticmethodandthe a non-adaptiveF(t,ℓ,1)-searchingalgorithm with probability
result established in [4] for the case ℓ=1. (1−ε) then we can replace each entry of X by the following
Proof of Theorem 3. rule: 0 → 1, 1 → 0, and get the code Y, which represents
Firstly we note that the subfamily F′(t,s,ℓ) ⊂ F(t,s,ℓ) a non-adaptiveF(t,1,ℓ)-searchingalgorithm with probability
which consists of s pairwise non-intersecting edges of size ℓ, (1−ε).
i.e., Roughlyspeaking,foranygoodhypergraphfromG(X ) it
1
willbesufficienttoapplysnon-adaptiveF(t ,1,ℓ)-searching
H ∈F(t,s,ℓ): H =(e ,...,e ), i
F′(t,s,ℓ)= 1 s , algorithms with probabilities (1−ε) in parallel at the second
(|ei|=ℓ, ei∩ej =∅ for any i6=j) stage of our strategy in order to find s edges in each Vi, ti =
|V |.
has cardinality |F(t,s,ℓ)|(1 + o(1)) as t → ∞. Thus, for i
More formally, let E(N,t,X) be the ensemble of binary
any ε > 0 and for sufficiently large t it is enough to prove
theexistenceoftwo-stageF′(t,s,ℓ)-searchingalgorithmwith (N × t) codes that consists of all possible permutations of
columns of a code X representing a non-adaptive F(t,1,ℓ)-
probability (1−ε).
searching algorithm with probability (1−ε), and each copy
Now we show that there exists a binary matrix (each test
of X is chosen equiprobably with probability 1/t!. Let Y
correspondstotherow,andeachvertexstandsforthecolumn)
be a random matrix of E(N,t,X). For a good hypergraph
correspondingtothefirststageofgrouptestingsuchthatafter
H ∈ G(X ) let we be given with an appropriate partition
carrying out tests of the first stage for almost all hypergraphs 1
from F′(t,s,ℓ) we can find a good partition of vertices into into disjoint sets V1 ⊔ V2··· ⊔ Vs = V = [t] such that
s disjoint sets: V ⊔V ···⊔V = V = [t], such that e ∈ e1 ∈ V1, e2 ∈ V2,...es ∈ Vs. At the second stage of our
1 2 s 1
V , e ∈ V ,...,e ∈ V . Define the ensemble E(N,t,s) strategy for vertices V1 we will apply N tests of Y without
1 2 2 s s
usingverticesV \V , forverticesV we willapplyN tests of
of s-ary (N ×t)-matrices X = ||x (j)||, where each entry 1 2
i
Y without using vertices V \V and so on. Define the event
x (j) is chosen independently and equiprobably from the set 2
i
B(s,ℓ): “all edges of H can be found applying sN tests”.
{1,2,...,s}. Each s-ary symbol x in X is then replaced by
Estimate the probability of the opposite event to B(s,ℓ):
the binary column of length s, which has only one 1 at x-
th position. In other words, the s-ary (N × t)-matrix X is P(B(s,ℓ))6sε.
replaced by the binary (sN ×t)-matrix X , and each s-ary
1
row of X is replaced by the binary (s×n)-layer of X1 such It means that there exists such a code X2 which is a copy
that each column of the layer contains only one 1. Define the (obtained by a column permutation) of X such that for (1−
eventA(s,ℓ):“givenahypergraphH ∈F′(t,s,ℓ),thereexists sε)·|G(X1)| good hypergraphs we find all edges after the
a layer in X such that the answers to all s edge-detecting second stage of our strategy.
1
queriesinthelayerare1’s”.Ifthisconditionholds,thenweare Finally, for any ε > 0 there exists an ascending sequence
presentedwithagoodpartitionintodisjointsets:V ⊔V ···⊔ of t such that for (1−ε)|F(t,s,ℓ)| hypergraphs there exists
1 2
Vs = V = [t], such that e1 ∈ V1, e2 ∈ V2,...,es ∈ Vs. a code X1, which can be used for the first stage of F(t,s,ℓ)-
Denote the cardinality of Vi by ti = |Vi|. Now estimate the searching algorithm with probability (1−ε), and a code X2,
probability of the opposite event to A(s,ℓ): the modificationof which can be appliedfor the second stage
ofthestrategy,suchthatthetotalnumberoftestsissufficiently
N
s! determined by only queries of the code X and is equal to
P A(s,ℓ) = 1− . 2
ssℓ sℓlog t(1+o(1)).
(cid:16) (cid:17) (cid:18) (cid:19) 2
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