# Large parallel machines can be extremely slow for small problems

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## Abstract

We consider concurrent-write PRAMs with a large number of processors of unlimited computational power and an infinite shared memory. Our adversary chooses a small number of our processors and gives them a 0–1 input sequence (each chosen processor gets a bit, and each bit is given to one processor). The chosen processors are required to compute the*PARITY* of their input, while the others do not take part in the computation. If*at most q* processors are chosen and*q* ≤1/2 log log*n*, then we show that computing PARITY needs*q* steps in the worst case. On the other hand, there exists an algorithm which computes PARITY in*q* steps (for any*q* <*n*) in this model, thus our result is sharp. Surprisingly, if our adversary chooses*exactly q* of our processors, then they can compute PARITY in [*q*/2] + 2 steps, and in this case we show that it needs at least [*q*2] steps. Our result implies that large parallel machines which are efficient when only a small number of their processors are active cannot be constructed. On the other hand, a result of Ajtai and Ben-Or [1] shows that if we have*n* input bits which contain at most polylog*n* 1's and polynomially many processors which are all allowed to work, then the*PARITY* can be solved in*constant* time.

### Key words

Parallel computation Lower bounds Fault tolerant computation### References

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