《算法导论》第五章-思考题（参考答案）

算法导论（第三版）参考答案：思考题5.1，思考题5.2

Problem 5.1 (Probabilstic counting)

With a $b$-bit counter, we can ordinarily only count up to $2^b−1$. With R. Morris’s probabilistic counting, we can count up to a much larger value at the expense of some loss of precision.

We let a counter value of i represent that a count of $n_i$ for $i=0,1,…,2b−1$, where the $n_i$ form an increasing sequence of nonnegative values. We assume that the initial value of the counter is $0$, representing a count of $n_0=0$. The INCREMENT operation works on a counter containing the value i in a probabilistic manner. If $i=2^b−1$, then the operation reports an overflow error. Otherwise, the INCREMENT operation increases the counter by 1 with probability $1/(n_{i+1}−n_i)$, and it leaves the counter unchanged with probability $1−1/(n_{i+1}−n_i)$.

If we select $n_i=i$ for all $i≥0$, then the counter is an ordinary one. More interesting situations arise if we select, say, $ni=2^{i−1}$ for $i>0$ or $n_i=F_i$ (the ith Fibonacci number - see Section 3.2).

For this problem, assume that n2