Adaptive group testing for consecutive positives

Abstract

Motivated from an application to DNA library screening, Balding and Torney [The design of pooling experiments for screening a clone map, Fungal Genet. Biol. 21 (1997) 302–307] and Colbourn [Group testing for consecutive positives, Ann. Combin. 3 (1999) 37–41] studied the following group testing for consecutive positives. Suppose V n = { v 1 ≺ v 2 ≺ ⋯ ≺ v n } is a linearly ordered set in which each item has a positive or negative state, and there are at most d positive items, forming a consecutive set under the order ≺ . The object is to determine the set of positive items by means of group testing. To do this, we choose an arbitrary subset P ⊆ V n called a pool, the outcome is positive when there is at least one positive item in this pool, and negative otherwise. Group testing is the processing of selecting pools and testing, to determine exactly which items are positive. This paper considers adaptive group testing by which each pool is examined before the next pool is selected. The minimum number of tests needed to determine the positives for at most d consecutive positive in a linearly ordered set of n items is denoted by M ( C d , n ) . Improving Colbourn's bounds of ( log 2 dn ) - 1 ⩽ M ( C d , n ) ⩽ ( log 2 dn ) + c where c is a constant, we prove that ⌈ log 2 dn ⌉ - 1 ⩽ ⌈ log 2 ( dn + 1 - d ( d - 1 ) / 2 ) ⌉ ⩽ M ( C d , n ) ⩽ ⌈ log 2 d ⌉ + ⌈ log 2 n ⌉ ⩽ ⌈ log 2 dn ⌉ + 1 for n ⩾ d - 1 and d ⩾ 1 . Finally, we prove that M ( C d , n ) = ⌈ log 2 ( dn + 1 - d ( d - 1 ) / 2 ) ⌉ for M ( C 2 s , n ) and M ( C d , n ) for d ⩽ 3 . We conjecture that the equality holds for a general M ( C d , n ) .