Consider the following recurrence for a function T that takes on nonnegative values and is defined on integers ≥ 1:T(n) ≤ 5 if 1 ≤ n ≤ 4 or ≤T(7n/10) + T(n/5) + 3n if n > 4. Prove that T(n) is O(n). You should present a particular constant c and prove that T(n) ≤ c · n for all n ≥ 1.
Motivation: This recurrence actually comes up in the following situation. Say we wish to find the median of n distinct elements, using as few comparisons as possible. Recall that the median is an element with rank the floor of n/2 .More generally, say we wish to find the k-th smallest of n elements. We could sort the n numbers but this takes about n log n comparisons. Instead we do the following which, by the theorem proven in this question, uses O(n) comparisons:
Divide the elements up into groups of 5 (don't worry now about what to do if n isn't divisible by 5); find themedian of each group of 5, using a linear in n number of comparisons in total. This gives us about n/5 median points. Recursively find the median of these median points using about T(n/5) comparisons; call this element b. Using a linear in n number of comparisons, compare every element to b; let S1 be the set of elements ≤ b, and let S2 be the set of elements > b. Because of the way b was chosen, each of S1 and S2 must have at most 7n/10 elements.If k ≤ |S1|, then recursively find the k-th smallest element of S1, otherwise recursively find the (k - |S1|)-th smallest element of S2; this takes at most T(7n/10) comparisons.