Problem 3. (20 Points) (Textbook Exercise 6.17) Your friends have been studying the closing prices of tech stocks, looking for interesting patterns. They've defined something called a rising trend as follows. They have the closing price of a given stock recorded for $n$ days in succession; let these prices be denoted $P[1], P[2],..., P[n]$. A rising trend in these prices is a subsequence of prices $P[i_1], P[i_2],..., P[i_k]$ for days $i_1 < i_2 < ... < i_k$, so that
• $i_1 = 1$, and
• $P[i_j] < P[i_{j+1}]$ for each $j = 1,2,..., k-1$.
Thus, a rising trend is a subsequence of the days - beginning on the first day and not necessarily contiguous - so that the price strictly increases over the days in this subsequence.
They are interested in finding the longest rising trend in a given sequence of prices.
Example: Suppose $n = 7$, and the sequence of the prices is 10, 1, 2, 11, 3, 4, 12. Then the longest rising trend is given by prices on the days 1, 4 and 7. Note that days 2, 3, 5 and 6 consists of increasing prices; but since this sequence does not begin at day 1, it does not fit the definition of a rising trend.
(a) Show that the following algorithm does not correctly return the length of the longest rising trend, by giving an instance on which it fails to return the correct answer.
1: Define $i = 1$ and $L = 1$
2: for $j = 2$ to $n$ do
3: if $P[j] > P[i]$ then
4: Set $i = j$
5: Add 1 to $L$
6: Return $L$
In your example, give the actual length of the longest rising trend, and say what the algorithm above returns.
(b) Give an efficient algorithm that takes a sequence of prices $P[1], P[2],..., P[n]$ and returns the length of the longest rising trend.
