number of operations

= Exercise 2. Present an O(n) algorithm that sorts n positive integer numbers a1, a2, • An which are known to be bounded by n² - 1 (so 0 ≤ ai ≤ n² − 1, for every i 1,..., n. Use the idea of Radix Sort (discussed in class and presented in Section 8.3 in the textbook). Note that in order to obtain O(n) you have to do Radix Sort by writing the numbers in a suitable base. Recall that the runtime of Radix Sort is O(d(n+k)), where d is the number of digits, and k is the base, so that the number of digits in the base is also k. The idea is to represent each number in a base k chosen so that each number requires only 2 "digits," so d = 2. Explain what is the base that you choose and how the digits of each number are calculated, in other words how you convert from base 10 to the base. Note that you cannot use the base 10 representation, because n² – 1 (which is the largest possible value) requires log₁0 (n² - 1) digits in base 10, which is obviously not constant and therefore you would not…

Given a sorted array of N+2 integers between 0 and N with exactly one duplicate, design a logarithmic time algorithm to find the duplicate. Note: you don't need to write a code. Just use your English words to describe the algorithm

Find the running time for each of the following algorithms. Show work by finding a table of values for each while loop, writing the summations, then solving. Be sure to show work on both the upper bound and lower bound, justify the split, and check that the bounds differ by only a constant factor. Use asymptotic notation to write the answer. c) Func4(n) 1 2 3 4 567∞∞ s = 0; for i 1 to 5n do 8 j← 3i; while (j < i³) do s+ s + i - j; j+5 x j; end end 8 return (s);