ISIT ’ 98 Plenary Lecture Report : Variations on the Theme of ‘ Twenty Questions ’

ly, the group testing problem is as follows: given a universeU of patients, each of whom is either positive or negative, identify the setP of positive patients by an adaptive algorithm that uses tests of the following form: doesS, a subset ofU, contain a positive patient? A good general reference on group testing is [4]. Let us consider the somewhat artificial case in which the number of patients is known to be d. Since there are( ) d n possibilities for the set P, the information theory bound tells us that the worst-case complexity is at least   log ( ) 2 dn . F. Hwang [10] has given a simple algorithm with worst-case complexity bounded above by   log ( ) 2 1 dn d + − . Interestingly, oblivious algorithms for the group testing problem are inherently more expensive than the best adaptive algorithms for the problem. It can be shown that the minimum worst case number of tests for an oblivious group testing algorithm is at least d o n 2 2 2 1 1 ( ( )) log + . However, if one is willing to accept a small probability of error, then a very simple randomized oblivious algorithm performs reasonably well [2]. The algorithm simply constructs a set of N tests randomly, such that, independently for every test S and every patient a, the probability that a S ∈ is 1 1 d + . It is easily shown that the number of tests required to ensure that, with probability greater than or equal to1− ε, every patient is correctly designated as positive or negative, is ( ( ))( ) ln( ) 1 1 1 + + o d e ε , which exceeds the information theory bound by a small constant factor when ε is a constant and d is bounded above by a fractional power ofn. This result can be improved signficantly by placing each patient randomly in exactly N d +1 of the N tests, rather than letting the number of tests containing a given patient be a random variable. This oblivious algorithm is easily extended to an interesting generalization of group testing that we call the Multi-Disease ISIT’98 Plenary Lecture Report . . . continued from front cover