Abstract

Our computational model is a random access machine with n read only input registers each containing c log n bits of information and a read and write memory. We measure the time by the number of accesses to the input registers. We show that for all k there is an epsilon > 0 so that if n is sufficiently large then the elements distinctness problem cannot be solved in time kn with epsilon n bits of read and write memory, that is, there is no machine with this values of the parameters which decides whether there are two different input registers whose contents are identical. We also show that there is a simple decision problem that can be solved in constant time (actually in two steps) using non-deterministic computation, while there is no deterministic linear time algorithm with epsilon n log n bits read and write memory which solves the problem. More precisely if we allow kn time for some fixed constant k, then there is an epsilon > 0 so that the problem cannot be solved with epsilon n log n bits of read and write memory if n is sufficiently large. The decision problem is the following: "Find two different input registers, so that the Hamming distance of their contents is at most c log n".

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