The asymptotic lower bound for sorting n elements is established as n*log(n). However, when sorting n elements with k distinct values, the complexity can be reduced to n*log(k). This is achieved by mapping each of the k distinct values to integers from 1 to k. The sorting process involves iterating through the list of n elements, incrementing counts in an auxiliary array for each distinct value. The overall cost includes inspecting each value (n), the cost of mapping values (which remains to be determined), and the cost of outputting the sorted list (k). This method highlights the efficiency gained when the number of distinct values is significantly smaller than the total number of elements, optimizing the sorting process.
