Efficiently Decodable Non-Adaptive Threshold Group Testing

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1 Efficiently Decoable Non-Aaptive Threshol Group Testing arxiv:72759v3 [csit] 3 Jan 28 Thach V Bui, Minoru Kuribayashi, Mahi Cheraghchi, an Isao Echizen SOKENDAI The Grauate University for Avance Stuies), Hayama, Kanagawa, Japan bvthach@niiacjp Grauate School of Natural Science an Technology, Okayama University, Okayama, Japan kminoru@okayama-uacjp Abstract Department of Computing, Imperial College Lonon, UK mcheraghchi@imperialacuk National Institute of Informatics, Tokyo, Japan iechizen@niiacjp We consier non-aaptive threshol group testing for ientification of up to efective items in a set of N items, where a test is positive if it contains at least 2 ) u efective items, an negative otherwise The efective items can be ientifie using t = O 2 e u) uulog ) 2 u 2 u u +log ǫ) 2 logn tests with probability ) u ) at least ǫ for any ǫ > or t = O 2 e u) 2 u 2 u log N +ulog u) 2 logn tests with probability t The ecoing time is O 2 logn) poly2 logn) This result significantly improves the best known results for ecoing non-aaptive threshol group testing: ON logn+n log ǫ ) for probabilistic ecoing, where ǫ >, an ON u logn) for eterministic ecoing I INTRODUCTION The goal of combinatorial group testing is to ientify at most efective items among a population of N items usually is much smaller than N) This problem ates back to the work of Dorfman [], who propose using a pooling strategy to ientify efectives in a collection of bloo samples In each test, a group of items are poole, an the combination is teste The result is positive if at least one item in the group is efective an is otherwise negative Damaschke [2] introuce a generalization of classical group testing known as threshol group testing In this variation, the result is positive if the corresponing group contains at least u efective items, where u is a parameter, is negative if the group contains no more than l efective items, where l < u, an is arbitrary otherwise When u = an l =, threshol group testing reuces to classical group testing We note that l is always smaller than the number of efective items Otherwise, every test yiels a negative outcome, then no information is enhance from tests There are two approaches for the esign of tests The first is aaptive group testing in which there are several testing stages, an the esign of each stage epens on the outcomes of the previous stages The secon is non-aaptive group testing NAGT) in which all tests are esigne in avance, an the tests are performe in parallel NAGT is appealing to researchers in most application areas, such as computational an molecular biology [3], multiple access communications [4] an ata steaming [5] cf [6]) The focus of this work is on NAGT In both threshol an classical group testing, it is esirable to minimize the number of tests an, to efficiently ientify the set of efective items ie, have an efficient ecoing algorithm) For both testings, one nees Ω log N) tests to ientify all efective items [6], [7], [8] using aaptive schemes In aaptive schemes, the ecoing algorithm is usually implicit in the test esign The number of tests an the ecoing time are significantly ifferent between classical non-aaptive CNAGT) an non-aaptive threshol group testing NATGT)

2 In CNAGT, Porat an Rothschil [9] first propose explicit nonaaptive constructions using O 2 logn) tests However, there is no efficient sublinear-time) ecoing algorithm associate with their schemes For exact ientification, there are explicit schemes allowing efective items be ientifie using poly, log N) tests in time poly,logn) [], [] the number of tests can be as low as O +o) logn) if false positives are allowe in the reconstruction) To achieve a nearly optimal number of tests in aaptive group testing an with low ecoing complexity, Cai et al [2] propose using probabilistic schemes that nee Olog logn) tests to fin the efective items in time OlogN +log 2 )) N ) threshol group testing, Damaschke [2] showe that the set of positive items can be ientifie with uin tests with up to g false positives an g false negatives, where g = u l is the gap parameter Cheraghchi [3] showe that it is possible to fin the efective items with O g+2 log logn/)) tests, an that this trae-off is essentially optimal Recently, De Marco et al [4] improve this boun to O 3/2 logn/)) tests uner the extra assumption that the number of efective items is exactly, which is rather restrictive in application Although the number of tests has been extensively stuie, there have been few reports that focus on the ecoing algorithm as well Chen an Fu [5] propose schemes base ) u on CNAGT for when g = that can fin the efective items using O ) ) ulog N tests in u u time ON u logn) Chan et al [6] presente a ranomize algorithm with O log ulogn ) tests to ǫ fin the efective items in time ON logn +N log ) given that g = an u = o) The cost of these ǫ ecoing schemes increases with N Our objective is to fin an efficient ecoing scheme to ientify at most efective items in NATGT when g = Contributions: In this paper, we consier the case where g =, ie, l = u u 2), an call this moel u-natgt We first propose an efficient scheme for ientifying at most efective items in t NATGT in time O 2 logn) poly2 logn), where t is the number of tests Our main iea is to create at least a specifie number of rows in the test matrix such that the corresponing test in each row contains exactly u efective items an such that the efective items in the rows are the efective items to be ientifie We map these rows using a special matrix constructe from a isjunct matrix efine later) an its complementary matrix, thereby converting the outcome in NATGT to the outcome in CNAGT The efective items in each row can then be efficiently ientifie Although Cheraghchi [3] an De Marco et al [4] propose nearly optimal bouns on the number of tests, there are no ecoing algorithms associate with their schemes On the other han, the scheme of Chen et al [5] requires an exponential number of tests, which are much larger than the number of tests in our scheme Moreover, the ecoing complexity of their scheme is exponential in the number of items N, which is impractical Chan et al [6] propose a probabilistic approach to achieve a small number of tests, which combinatorially can be better than our scheme However, their scheme is only applicable when threshol u is much smaller than u = o)) an the ecoing complexity remains high, namely ON logn +N log ), where ǫ > is the precision parameter ǫ We present a ivie an conquer scheme which we then instantiate via eterministic an ranomize ecoing Deterministic ecoing is a eterministic scheme in which all efective items can be foun with probability Ranomize ecoing reuces the number of tests; all efective items can be foun t with probability at least ǫ for any ǫ > The ecoing complexity is O 2 logn) poly2 logn) A comparison with existing work is given in Table I II PRELIMINARIES For consistency, we use capital calligraphic letters for matrices, non-capital letters for scalars, an bol letters for vectors All matrix an vector entries are binary Here are some of the notations use: ) N,,x = x,,x N ) T : number of items, maximum number of efective items, an binary representation of N items 2) S = {j,j 2,,j S }: the set of efective items; carinality of S is S 2

3 TABLE I COMPARISON WITH EXISTING WORK Number of tests t) Decoing complexity Decoing type Cheraghchi [3] O 2 log log N ) ) De Marco et al [4] O 2 u u ) ) ) u ulog N Chen et al [5] O ON u logn) Deterministic u u Chan et al [6] O log ulogn ) ) ON logn +N log ) Ranom ǫ ǫ Deterministic ecoing O 2 u ) e u) 2 u 2 u log N +ulog u) 2 t logn O 2 logn) poly2 logn) Deterministic ) Ranomize ecoing O 2 e u) u ) 2 u 2 u ulog +log u ǫ) 2 t logn O 2 logn) poly2 logn) Ranom 3), : operation relate to u-natgt an CNAGT, to be efine later 4) T : t N measurement matrix use to ientify at most efective items in u-natgt, where integer t is the number of tests 5) G = g ij ): h N matrix, where h 6) M = m ij ): a k N +)-isjunct matrix use to ientify at most u efective items in u-natgt an +) efective items in CNAGT, where integer k is the number of tests 7) M = m ij ): the k N complementary matrix of M; m ij = m ij 8) T i,,g i,,m i,,m j : row i of matrix T, row i of matrix G, row i of matrix M, an column j of matrix M, respectively 9) x i = x i,,x in ) T,S i : binary representation of items an set of inices of efective items in row G i, ) iagg i, ) = iagg i,,g in ): iagonal matrix constructe by input vector G i, A Problem efinition We inex the population of N items from to N Let [N] = {,2,,N} an S be the efective set, where S A test is efine by a subset of items P [N],u,N)-NATGT is a problem in which there are at most efective items among N items A test consisting of a subset of N items is positive if there are at least u efective items in the test, an each test is esigne in avance Formally, the test outcome is positive if P S u an negative if P S < u We can moel,u,n)-natgt as follows: At N binary matrixt = t ij ) is efine as a measurement matrix, where N is the number of items an t is the number of tests x = x,,x N ) T is the binary representation vector of N items, where x x j = inicates that item j is efective, an x j = inicates otherwise The jth item correspons to the jth column of the matrix An entry t ij = naturally means that item j belongs to test i, an t ij = means otherwise The outcome of all tests is y = y,,y t ) T, where y i = if test i is positive an y i = otherwise The proceure to get the outcome vector y is calle the encoing proceure The proceure use to ientify efective items from y is calle the ecoing proceure Outcome vector y is y = T x = ef T, x T t, x ef = y ) y t where is a notation for the test operation in u-natgt; namely, y i = T i, x = if N j= x jt ij u, an y i = T i, x = if N j= x jt ij < u for i =,,t Our objective is to fin an efficient ecoing scheme to ientify at most efective items in,u,n)-natgt 3

4 B Disjunct matrices When u =, u-natgt reuces to CNAGT To istinguish CNAGT an u-natgt, we change notation to an use a k N measurement matrix M instea of the t N matrix T The outcome vector y in )) is equal to N M, x j= y = M x = ef = ef x j m j N y ef = M j = 2) M k, x N j= x j=,x j m j = kj y k where is the Boolean operator for vector multiplication in which multiplication is replace with the AND ) operator an aition is replace with the OR ) operator, an y i = M i, x = N j= x j m ij = N j=,x j = m ij for i =,,k The union of r columns of M is efine as follows: r i= M j i = r i= m j i,, r i= m tj i ) T A column is sai to not be inclue in another column if there exists a row such that the entry in the first column is an the entry in the secon column is If M is a + )-isjunct matrix satisfying the property that the union of at most +) columns oes not inclue any remaining column, x can always be recovere from y We nee M to be a + )-isjunct matrix that can be efficiently ecoe, as in [], [], to ientify at most efective items in u-natgt A k N strongly explicit matrix is a matrix in which the entries can be compute in time polyk) We can now state the following theorem: Theorem [, Theorem 6] Let N There exists a strongly explicit k N +)-isjunct matrix with k = O 2 logn) such that for any k input vector, the ecoing proceure returns the set of efective items if the input vector is the union of at most + columns of the matrix in polyk) time C Completely separating matrix We now introuce the notion of completely separating matrices which are use to get efficient ecoing algorithms for, u, N)-NATGT A u, w)-completely separating matrix is efine as follows: Definition An h N matrix G = g ij ) i h, j N is calle a u,w)-completely separating matrix if for any pair of subsets I,J [N] such that I = u, J = w, an I J =, there exists row l such that g lr = for any r I an g ls = for any s J Row l is calle a singular row to subsets I an J When u =, G is calle a w-isjunct matrix This efinition is slightly ifferent from the one escribe by Lebeev [7] It is easy to verify that, if a matrix is a u, w)-completely separating matrix, it is also a u, v)-completely separating matrix for any v w Below we present the existence of such matrices Theorem 2 Given integers u,w < N, there exists a u,w)-completely separating matrix of size h N, where h = u+w)log en ) eu+w) u+w) 2 ew ) u +ulog + u+w u u2u+w) u an e is base of the natural logarithm Proof: An h N matrix G = g ij ) i h, j N is generate ranomly in which each entry g ij is assigne to with probability of p an to with probability of p For any pair of subsets I,J [N] such that I = u, J = w, the probability of a row is not singular is: p u p) w 3) 4

5 Then, the probability that there is no singular row to subsets I an J is: fp) = p u p) w ) h 4) Using union boun, the probability that any pair of subsets I,J [N] such that I = u, J = w oes not have a singular row, ie, the probability that G is not a u,w)-separating matrix, is: ) ) ) ) N u+w N u+w gp,h,u,w,n) = fp) = p u p) w ) h 5) u+w u u+w u To ensure that there exists G which is a u,w)-separating matrix, one nees to fin p an h such that gp,h,u,w,n) < Choose p = u, we have: u+w u u fp) = p u p) w ) h = w) u ) ) u+w h 6) u+w u u exp h w) u ) ) u+w, where expx) = e x 7) u+w u ) )) u exp h e u u 2 u ) ) u w2u+w) = exp h 8) w u+w) 2 ew u+w) 2 We get 7) because x e x for any x > an 8) because ) + n) x n e x x2 for n >, n x n Then we have: ) ) ) u+w ) u N u+w en eu+w) gp,h,u,w,n) = fp) fp) 9) u+w u u+w u ) u+w ) u en eu+w) u ) ) u w2u+w) exp h ) u+w u ew u+w) 2 < ) ) u+w ) u en eu+w) u ) ) u w2u+w) < exp h 2) u+w u ew u+w) 2 h > u+w)log en ) eu+w) u+w) 2 ew ) u +ulog 3) u+w u w2u+w) u We got 9) because n k) en ) k an ) by using 8) From 3), if we choose k h = u+w)log en ) eu+w) u+w) 2 ew ) u +ulog + 4) u+w u w2u+w) u then gp,h,u,w,n) <, ie, there exists a u,w)-completely separating matrix of size h N Suppose that G is an h N u,w)-completely separating matrix If w is set to u, then every h submatrix, which is constructe by its columns, is a u, u)-completely separating matrix This property is strict an makes the number of rows in G is high To reuce the number of rows, we relax this property as follows: each h submatrix, which is constructe by columns of G, is a u, u)-completely separating matrix with high probability The following corollary escribes this iea in etails 5

6 Corollary Let u,,n be any given positive integers such that u < < N For any ǫ >, there exists an h N matrix such that each h submatrix, which is constructe by its columns, is a u, u)-completely separating matrix with probability at least ǫ, where h = 2 2 u 2 an e is base of the natural logarithm e u) u ) u ulog e u +log ǫ Proof: An h N matrix G = g ij ) i h, j N is generate ranomly in which each entry g ij is assigne to with probability of u an to with probability of u Our task is now to prove that each h matrix G, which is constructe by columns of G, is a u, u)-completely separating ) matrix with probability at least ǫ for any ǫ > Specifically, we prove that h = 2 e u) u 2 u 2 u ulog e +log ) u ǫ is sufficient to achieve such G Similar to the proof in Theorem 2, the probability that G is not a u, u)-completely separating matrix at most ǫ is ) u u u ) u ) u h ) u e u u exp h ) u ) u ) ) u 5) ) u ) u ) e u u)+u) exp h 6) u e u) 2 ǫ 7) ) u ) u ) e u exp 2 u 2 h 8) ǫ u e u) ) 2 u h 2 e u) ulog e 2 u 2 u u +log ) 9) ǫ We get 5) because x e x for any x > an n k) en ) k k 6) is erive because + x n ) n) e x x2 for n >, x n This completes our proof n III PROPOSED SCHEME The basic iea of our scheme, which uses a ivie an conquer strategy, is to create at least κ rows, eg, i,i 2,,i κ such that S i = = S iκ = u an S i S iκ = S Then we map these rows by using a special matrix that enables us to convert the outcome in NATGT to the outcome in CNAGT The efective items in each row can then be efficiently ientifie We present a particular matrix that achieves efficient ecoing for each row in the following section A When the number of efective items equals the threshol In this section, we consier a special case in which the number of efective items equals the threshol, ie, x = u Given a measurement matrix M an a representation vector of u efective items x x = u), what we observe is y = ef M x = ef y,,y k ) T Our objective is to recover y = ef M x = ef y,,y k )T from y Then x can be recovere if we choose M as a +)-isjunct matrix escribe in Theorem To achieve this goal, we create a measurement matrix: [ ] M A = ef 2) M where M = m ij ) is a k N + )-isjunct matrix as escribe in Theorem an M = m ij ) is the complement matrix of M, m ij = m ij for i =,,k an j =,,N We note that M can ) 6

7 be ecoe in time polyk) = poly 2 logn) because k = O 2 logn) Let us assume that the outcome vector is z Then we have: [ ] [ M x z = ef A x = ef ef y = 2) M x y] where y = M x = y,,y k ) T an y = M x = y,,y k ) T The following lemma shows that y = M x is always obtaine from z; ie, x can always be recovere Lemma Given integers 2 u < N, there exists a strongly explicit 2k N matrix such that if there are exactly u efective items among N items in u-natgt, the u efective items can be ientifie in time polyk), where k = O 2 logn) Proof: We construct the measurement matrix A in 2) an assume that z is the observe vector as in 2) Our task is to create vector y = M x from z One can get it using the following rules, where l =,2,,k: ) If y l =, then y l = 2) If y l = an y l =, then y l = 3) If y l = an y l =, then y l = We now prove the correctness of the above rules Because y l =, there are at least u efective items in row M l, Then, the first rule is implie If y l =, there are less than u efective items in row M l, Because x = u, y l =, an the threshol is u, there must be u efective items in row M l, Moreover, since M l, is the complement of M l,, there must be no efective item in test l of M Therefore, y l =, an the secon rule is implie If y l =, there are less than u efective items in row M l, Similarly, if y l =, there are less than u efective items in row M l, Because M l, is the complement of M l,, the number of efective items in row M l, or M l, cannot be equal to zero, since either y l woul equal or y l woul equal Since the number of efective items in row M l, is not equal to zero, the test outcome is positive, ie, y l = The thir rule is thus implie Since we get y = M x, M is a +)-isjunct matrix an u, u efective items can be ientifie in time polyk) by Theorem Example: We emonstrate Lemma by setting u = = 2, k = 9, an N = 2 an efining a isjunct matrix M with the first two columns as follows: M =,y =,y =,y = Assume that the efective items are an 2, ie, x = [,,,,,,,,] T ; then the observe vector is z = [y T y T ] T Using the three rules in the proof of Lemma, we obtain vector y We note that y = M M2 = M x Using a ecoing algorithm which is omitte in this example), we can ientify items an 2 as efective items from y 22) 7

8 B Encoing proceure To implement the ivie an conquer strategy, we nee to ivie the set of efective items into small subsets such that efective items in those subsets can be effectively ientifie We efine κ = S u as an integer, an create a h N matrix G containing κ rows, enote as i,i 2,,i κ, with probability at least ǫ such that i) S i = = S iκ = u an ii) S i S iκ = S for any ǫ where S i is the set of inices of efective items in row G i, For example, if N = 6, the efective items are, 2, an 3, an G, =,,,,,), then S = {,3} These conitions guarantee that all efective items will be inclue in the ecoe set To achieve such a G, for any S, a pruning matrix G of size h after removing N columns G x for x [N]\S must be a u, u)-completely separating matrix with high probability From Definition, G is also a u, S u)-completely separating matrix Then, the κ rows are chosen as follows We choose a collection of sets of efective items: P l = {j l )u+,,j lu } for l =,,κ P is a set satisfying P κ l= P l an P = κu S Then we pick the last set as follows: P κ = S \ κ l= P l) P From Definition, for any P l, there exists a row, enote i l, such that g il x = for x P l an g il y = for y S \ P l, where l =,,κ Then, S il = P l an row i l is singular to sets S il an S \ S il for l =,,κ Conition i) thus hols Conition ii) also hols because κ l= S i l = κ l= P l = S The matrix G is specifie in section IV After creating the matrix G, we generate matrix A as in 2) Then the final measurement matrix T of size 2k +)h N is create as follows: G, G, M iagg, ) A iagg, ) M iagg, ) T = G h, A iagg h, ) = G h, M iagg h, ) M iagg h, ) The vector observe using u-natgt after performing the tests given by the measurement matrix T is G, G, x A iagg, ) A x y = T x = G x = h, G h, x A iagg h, ) A x h G, x y M x y y M x y z = = = 24) G h, x y h y h M x h y h z h M x h y h ef where x i = iagg i, ) x, y i = G i, x, y i = M x i = y i,,y ik ) T ef, y i = M x i = y i,,y ik ) T, an z i = [yi T y T i ]T for i =,2,,h We note that x i is the vector representing the efective items corresponing to row G i, If x i = x i,x i2,,x in ) T, S i = {l x il =,l [N]} We thus have S i = x i Moreover, y i = if an only if x i u 8 23)

9 C The ecoing proceure The ecoing proceure is summarize as Algorithm, where y i = y i,,y ik )T is presume to be M x i The proceure is briefly explaine as follows: Line 2 enumerates the h rows of G Line 3 checks if there are at least u efective items in row G i, Lines 4 to 4 calculate y i, an Line 6 checks if all items in G i are truly efective an as them into S Algorithm Decoing proceure for u-natgt Input: Outcome vector y, M,M,T Output: The set of efective items S : S = 2: for i = to h o 3: if y i = then 4: for l = to k o 5: if y il = then 6: y il = 7: en if 8: if y il = an y il = then 9: y il = : en if : if y il = an y il = then 2: y il = 3: en if 4: en for 5: Decoe y i using M to get the efective set G i 6: if G i = u an j G i M j y i then 7: S = S G i 8: en if 9: en if 2: en for 2: Return S D Correctness of the ecoing proceure Our objective is to recover x i from y i an z i for i =,2,,h Line 2 enumerates the h rows of G We have that y i is the inicator that whether there are at least u efective items in row G i, If y i =, it implies that there are less than u efective items in row G i, Since we only focus on row G i, which has exactly u efective items, z i is not consiere if y i = Lines 3 oes this task When y i =, it implies that there are at least u efective items in row G i, If there are exactly u efective items in this row, they are always ientifie as escribe in Lemma Our task now is to prevent accusing false efective items by ecoing y i Lines 4 to 4 calculates y i from z i We o not know that y i is the union of many columns in M, ie, how many efective items are in row G i, Therefore, our task is to ecoe y i using matrix M to get the efective set G i, then valiate whether all items in G i are efective There exists at least κ rows of G such that there are exactly u efective items in each row An all efective items in these rows are the efective items we nee to ientify Therefore, we only consier the case when the number of efective items obtaine from ecoing y i equals to u, ie, G i = u Our task is now to prevent ientifying false efective items, which is escribe in Line 6 There are two sets of 9

10 efective items corresponing to z i : the first one is the true set, which is S i an unknown, an the secon one is G i, which is expecte to be S i but not sure) an G i = u If G i S i, we can always ientify u efective items an the conition in line 6 always hols because of Lemma We nee to consier the case G i S i, ie, there are more than u efective items in row G i, We classify this case into two categories: ) G i \ S i = : in this case, all elements in G i are efective items We o not nee to consier whether j G i M j y i If this conition hols, we receive the true efective items If it oes not hol, we o not take G i into the efective item set 2) G i \S i : in this case, we prove that j G i M j y i oes not hol, ie, none of elements in G i is ae to the efective item set Let pick j G i \S i an j 2 G i \{j } Since S i an M is a +)-isjunct matrix, there exists a row, enote τ, such that m τj =,m τj2 =, an m τx = for x S i In the other han, because G i = u an S i, there is less than u efective items in row τ, ie, y iτ = Because u S i, y iτ = That implies y iτ = However, x G i m τx = ) x Gi\{j} m miτ ) τx = x Gi\{j} m τx = = y iτ Therefore, j G i M j y i Thus, line 6 eliminates all false efective items Line 2 just returns the efective item set S E The ecoing complexity Because T is constructe using G an M, the probability of successful ecoing of y epens on these choices Given an input vector y i, we get the set of efective items from ecoing of M The probability of successful ecoing of y thus epens only on G Since G has κ rows satisfying i) an ii) with probability at least ǫ, all S efective items can be ientifie in h polyk) time using t = h2k+) tests with probability of at least ǫ for any ǫ We summarize the ivie an conquer strategy in the following theorem: Theorem 3 Let 2 u < N be integers an S be the efective set Suppose that an h N matrix G contains κ rows, enote as i,,i κ, such that i) S i = = S iκ = u an ii) S i S iκ = S, where S il is the inex set of efective items in row G il, An suppose that an k N matrix M is a +)-isjunct matrix that can be ecoe in time A Then a 2k+)h N measurement matrix T, as efine in 23), can be use to ientify at most efective items in u-natgt in time Oh A) The probability of successful ecoing epens only on the event that G has κ rows satisfying i) an ii) Specifically, if that event happens with probability at least ǫ, the probability of successful ecoing is also at least ǫ for any ǫ IV COMPLEXITY OF PROPOSED SCHEME We specify the matrix G in Theorem 3 to get the esire number of tests an ecoing complexity for ientifying at most efective items Specifying G leas to two approaches on ecoing: eterministic an ranomize Deterministic ecoing is a eterministic scheme in which all efective items can be foun with probability It is achievable when every its h submatrices, which are constructe by its columns, are u, u)-completely separating matrices Ranomize ecoing reuces the number of tests; all efective items can be foun with probability at least ǫ for any ǫ > It is achievable when each its h submatrix, which is constructe by its columns, is u, u)-completely separating matrix with probability at least ǫ A Deterministic ecoing The following theorem states that there exists a eterministic algorithm for ientifying all efective items by choosing G of size h N to be a u, u)-completely separating matrix in Theorem 2

11 Theorem 4 Let 2 u N There exists a t N matrix such that at most efective items in t u-natgt can be ientifie in time O 2 logn) poly2 logn), where ) 2 u e u) t = O log N 2 u 2 u +ulog ) ) 2 logn u Proof: On the basis of Theorem 3, a t N measurement matrix T is generate as follows: ) Choose an h N )u, u)-completely separating matrix G as in Theorem 2, where u h = 2 e u) 2 u 2 u log en +ulog ) e u + 2) Choose a k N + )-isjunct matrix M as in Theorem, where k = O 2 logn) an the ecoing time of M is polyk) 3) T is efine as in 23) Since G is a h u, u)-completely separating matrix, for any S, an h pruning matrix G, which is create by removing N columns G x for x [N] \ S, is also a u, u)-completely separating matrix with probability From Definition, G is also a u, S u)-completely separating matrix Then, there exists κ rows satisfying i) an ii) as escribe in section III-B From Theorem 3, efective items can be recovere using t = h O 2 logn) tests with probability at least, ie, the probability, in time h polyk) B Ranomize ecoing For ranomize ecoing, G is chosen such that the pruning matrix G of size h create by removing N columns G x of G for x [N]\S is a u, u)-completely separating matrix with probability at least ǫ for any ǫ > This results is an improve number of tests an ecoing time compare to Theorem 4: Theorem 5 Let 2 u N For any ǫ >, at most efective items in u-natgt can be ientifie using ) 2 u e u) t = O ulog 2 u 2 u u +log ) ) 2 logn ǫ tests with probability at least ǫ The ecoing time is t O 2 logn) poly2 logn) Proof: On the basis of Theorem 3, a t N measurement matrix T is generate ) as follows: u ulog e +log u ǫ) ) Choose an h N matrix G as in Corollary, where h = 2 e u) 2 u 2 u 2) Generate a k N + )-isjunct matrix M using Theorem, where k = O 2 logn) an the ecoing time of M is polyk) 3) Define T as 23) Let G be an h N matrix as escribe in Corollary Then for any S, an h pruning matrix G, which is create by removing N columns G x for x [N]\S, is a u, u)-completely separating matrix with probability at least ǫ From Definition, G is also a u, S u)-completely separating matrix Then, there exists κ rows satisfying i) an ii) as escribe in section III-B with probability at least ǫ From Theorem 3, S efective items can be recovere using t = h O 2 logn) tests with probability at least ǫ in time h polyk) V CONCLUSION We introuce an efficient scheme for ientifying efective items in NATGT However, the algorithm works only for g = Extening the results to g > is left for future work Moreover, it woul be interesting to consier noisy NATGT as well, in which erroneous tests are present in the test outcomes

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