Finding a Duplicate and a Missing Item in a Stream

We consider the following problem in a stream model: Given a sequence a = a1, a2, . . . , am wich each ai ∈ [n] = {1, . . . , n} and m > n, find a duplicate in the sequence, i.e., find some d = ai = al with i = l by using limited s bits of memory and r passes over the input sequence. In one pass an algorithm reads the input sequence a in the order a1, a2, . . . , am. Since m > n, a duplicate exists by the pigeonhole principle. Muthukrishnan [Mu05a], [Mu05b] has posed the following question for the case where m = n + 1: For s = O(log n), is there a solution with a constant number of passes? We have described the problem generalizing Muthukrishnan’s question by taking the sequence length m as a parameter. We give a negative answer to the original question by

• Type: Conference
• Year: 2007
• Where: TAMC
• Authors: Jun Tarui
• Keywords: Input Sequence, Pigeonhole Principle, Stream Model, TAMC 2007