Question

Describe an algorithm that, given n integers in the range 0 to k, preprocesses its input and then answers any query?

Answer 1

The question involves preprocessing a set of integers with counting sort so that queries can be answered quickly. Counting sort creates a count array with occurrences of each integer, which then facilitates efficient query handling. This method is especially useful when k (the maximum integer value) is not far greater than n (the number of integers).

Step-by-step explanation:

The question refers to designing an algorithm for preprocessing a set of integers so that any subsequent query can be answered efficiently. This is a typical scenario in computer science, particularly in the field of data structures and algorithms. Given that we have n integers within the range 0 to k, one possible approach to preprocessing these integers is using a technique known as 'counting sort'. This technique is especially efficient when k is not significantly larger than n.

Counting sort algorithm works by counting the occurrences of each unique number in the original array and then using this information to place each number in its correct position in a new sorted array. The general steps are as follows:

1. Create an array count with size k+1 and initialize all values to 0.
2. For each integer in the original array, increment count[integer].
3. Cumulatively sum up the count array; each count[i] will now represent the position of i in the sorted array.
4. Finally, build the sorted array by placing each original number i into its sorted position indicated by count[i] and decrementing count[i] by 1.

The preprocessing phase is the construction of the count array, which can then be used to quickly answer queries such as the rank of a certain number, or to quickly reconstruct a sorted array that can be used to answer range queries.