Problem 1 (K th quantiles)

Let S = {s1, s2, . . . , sn} be a set of distinct real numbers. The K th quantiles of S is a subset of exactly K 1 numbers s 0 1 < s0 2 , . . . , s0 K1 such that the cardinality of

Sj = {s S | sj1 < s sj}

are the same. Design a divide and conquer algorithm to find the K th quantiles of a given set S of N numbers. You may assume that K is a power of 2 and that it is possible to partition S into K equal sets. Your input is the set S, and the value K. S is not sorted.

Examples:

For S = {2, 4, 1, 6, 0, 9}, K = 3 your design should output {1, 4}.

For S = {1, 2, 4, 1, 3, 0, 18, 2}, K = 2 your design should output {1} ((i.e.: the 2 th quantile is the median)).