Before revision:  For arbitrary-length intervals, we show any c-approximation to \alpha , and thus also \beta , requires \Omega ] \left( \frac{n^{1/c}{2^c}} space. To this end, we introduce a new communication problem and lower bound its information complexity. In light of the lower bound we}{\pi}}\right) provide algorithms for estimating \alpha for arbitrary-length intervals under a bounded intersection assumption .
After revision:  For arbitrary-length intervals, we show any c-approximation to \alpha , and thus also \beta , requires \Omega ] \left( \frac{n^{1/c}{2^c}} \frac{8\sqrt{3}{\pi}}\right) provide algorithms for estimating \alpha for arbitrary-length intervals under a bounded intersection assumption .