Data Structures and Algorithm Analysis - Computer Science at ...

Sec. 15.7 Optimal Sorting 501While selecting the median in this way is guaranteed to eliminate a fraction ofthe elements (leaving at most ⌈(7n − 5)/10⌉ elements left), we still need to be surethat our recursion yields a linear-time algorithm. We model the algorithm by thefollowing recurrence.T(n) ≤ T(⌈n/5⌉) + T(⌈(7n − 5)/10⌉) + 6⌈n/5⌉ + n − 1.The T(⌈n/5⌉) term comes from computing the median of the medians-of-fives,the 6⌈n/5⌉ term comes from the cost to calculate the median-of-fives (exactly sixcomparisons for each group of five element), and the T(⌈(7n−5)/10⌉) term comesfrom the recursive call of the remaining (up to) 70% of the elements that might beleft.We will prove that this recurrence is linear by assuming that it is true for someconstant r, and then show that T(n) ≤ rn for all n greater than some bound.T(n) ≤ T(⌈ n − 5⌉) + T(⌈7n ⌉) + 6⌈ n 5 10 5 ⌉ + n − 1≤ r( n − 5+ 1) + r(7n + 1) + 6( n 5 105 + 1) + n − 1≤ ( r 5 + 7r10 + 11 3r)n +5 2 + 5≤9r + 22n +103r + 10.2This is true for r ≥ 23 and n ≥ 380. This provides a base case that allows us touse induction to prove that ∀n ≥ 380, T(n) ≤ 23n.In reality, this algorithm is not practical because its constant factor costs are sohigh. So much work is being done to guarantee linear time performance that it ismore efficient on average to rely on chance to select the pivot, perhaps by pickingit at random or picking the middle value out of the current subarray.15.7 Optimal SortingWe conclude this section with an effort to find the sorting algorithm with the absolutefewest possible comparisons. It might well be that the result will not bepractical for a general-purpose sorting algorithm. But recall our analogy earlier tosports tournaments. In sports, a “comparison” between two teams or individualsmeans doing a competition between the two. This is fairly expensive (at least comparedto some minor book keeping in a computer), and it might be worth trading afair amount of book keeping to cut down on the number of games that need to beplayed. What if we want to figure out how to hold a tournament that will give usthe exact ordering for all teams in the fewest number of total games? Of course,we are assuming that the results of each game will be “accurate” in that we assume