I’m working on a algorithms & data structures multi-part question and need an explanation to help me learn.

The context of this question is Divide and Conquer (merge Sort, etc.):

You are working on a busy fruit farm that produces three types of fruit: apples, bananas, and clementines. The fruits are stored in a warehouse, in a row of barrels. Each barrel contains exactly one kind of fruit. Your job is to sort the row of barrels so that the barrels containing the apples come first, the barrels containing the bananas come second, and the barrels containing the clementines come last.

Since space is limited, each time you can only use a special machine to horizontally flip some barrels in an interval chosen by you as a whole, thus reversing their arrangement in the interval. The machine takes linear time in the number of barrels to flip them.

You are given as input an array Fruit[1..n] consisting of A’s, B’s and C’s, where we use A for a barrel of apples, B for bananas, and C for clementines. As described above, the only operation you can use to move the barrels is FLIP(i,j), which flip the barrels in the interval[i..j] horizontally in time O(j?i).

For example, for Fruit[1..4] = CBAB, we can sort them using two operations: FLIP(1,4) resulting in BABC followed by FLIP(1,2).

1. Give an algorithm to sort the fruit barrels. Make the algorithm as fast as you can; analyze its time complexity.

2. Suppose there are k types of fruits, where k is an input parameter. Again, give an algorithm for this general case, and analyze the time complexity.

3. Each barrel has a harvest date. Before you do the sorting, they are already sorted in the warehouse according to their harvest dates, so that the barrels with earliest harvest dates come first. Is your algorithm in Part 1 stable? That is to say, does your algorithm in Part 1 preserve the order of the harvest date in each type of fruits?