--------------------------------------------------------------------- CS 577 (Intro to Algorithms) Lecture notes: Selection Shuchi Chawla --------------------------------------------------------------------- (This document is best viewed with a fixed width font.) Linear-time Selection ===================== Some times instead of sorting an entire list of elements, we may only be interested in (say) what the smallest element is, or what the 10th smallest element is, and so on. We can easily find the smallest element, even the 10th smallest element, in a list in O(n) time, by just scanning the list and keeping track of the smallest (10 smallest) elements so far. This is much faster than first spending O(n log n) time sorting the list and then looking up the required element in the sorted list. Can this be done for every k-th smallest element (for arbitrary k)? For example, suppose we were to find the median of a list of elements, can we do this in linear time? Keeping track of the n/2 smallest elements no longer works, and takes O(n^2) time in fact. This is called the selection problem. Let us try to apply divide and conquer to this problem in the same way as we did for mergesort, and focus on the case of finding a median of the list. Suppose that we divide the list into two equal halves and find the median in each half in some T(n/2) time. Given these medians how do we find the median of the entire list? If these medians were in fact equal, then the same value would also be the median of the entire list. But, if they are not equal, say the first is smaller than the second, then the true median lies among elements in the first half that are larger than the first median and elements in the second half that are smaller than the second median. Now we could recursively proceed to find the median on this smaller list of elements. But this "combine" step at best takes O(n) time. So our overall running time would (at best) be given by an equation of the form T(n) = 2 T(n/2) + n. This again leads us to an O(n log n) running time, and we are doing no better than sorting. We would be able to get some improvement if instead of recursing in both the halves we only have to recurse in one of them. This leads us to our second idea based on the quicksort algorithm. The quicksort algorithm sorts a given list of elements by first picking an appropriate "pivot". It then divides the list into two parts -- one consisting of all elements smaller than the pivot and the other consisting of all elements larger than the pivot. It then recursively sorts these parts and outputs the sorted first part followed by the pivot followed by the sorted second part. We will analyze a randomized version of this algorithm later on in the course. For now, let us try to apply this pivoting idea to the selection problem. Suppose that we pivot the list using the first element. For example, if our list was (5,20,13,2,7,14,8,1,10), then picking 5 as the pivot would divide the list into (2,1) and (20,13,7,14,8,10). Now, if we are interested in finding the 5th smallest element in the original list (the median), then, given that the first part is of length 2, we know that the required element lies in the second part and is, in particular, the 2nd smallest element in that part. Now we can recursively find this 2nd smallest element in the second part. How long does this procedure take? It takes n time to divide the list into two parts, and then T(m) time in the recursive call, where m is the size of the longer of the two parts. What is a good estimate on this value m? Well, in the worst case, we may pick the smallest element in the list as the pivot, in which case m would be n-1. Our recurrence would then look like T(n) = n + T(n-1). This solves to T(n) = O(n^2). So we have not really made any progress. Ideally, it would be great if we could pick a pivot for which m = n/2. Then, our recurrence would look like T(n) = n + T(n/2). You should verify that this indeed solves to T(n) = O(n). So our goal now is to pick a pivot for which the size of the longer part m is n/2. But such an element is the median of the list by definition -- this is the same problem that we are trying to solve in the first place! We will fix this issue by allowing ourselves a little leeway -- instead of using the median as a pivot, we will pick a pivot that partitions the list into *roughly* equal parts, with m being 7n/10. This idea and the algorithm below was invented by Blum, Floyd, Pratt, Rivest and Tarjan in 1972. Here is how we pick the pivot. We partition the list into n/5 groups of 5 elements each. We then find the median in each group. Note that this is a constant time operation per group since each group is of constant size. This gives us a list of n/5 medians. We now proceed to recursively find the median of this list of medians. Call is p. We then use this median of medians p to pivot the original list. Before we analyze the time complexity of this algorithm, we claim that we pick a good pivot -- m is at most 7n/10. Claim: There are at least 3n/10 elements larger than and at least 3n/10 elements smaller than the median of medians p picked above. Proof: p is a median of n/5 medians. Therefore, at least n/10 of the medians (including itself) are less than or equal to p. Since each of these is a median of 5 elements, each of these medians accounts for a total of 3 elements (including itself) that are smaller than p. Therefore, a total of at least 3n/10 elements are smaller than p. The same argument can be made for elements larger than p. Finally, let us consider the running time of this algorithm. We take O(n) time to find the medians of n/5 sublists. We then take T(n/5) time to recursively find a median of medians. This gives us a pivot. We then take O(n) time to partition the list using this pivot. Finally, we take T(7n/10) time to recursively solve the selection problem over one of the partitions of the list. Therefore, we have T(n) = T(n/5) + T(7n/10) + cn for some constant c. There are several ways of solving this recurrence. Here, we use the "recursion tree" method. At the top level, we spend cn time. At the next level, we spend cn/5 time for the first recursive call, and 7cn/10 time for the second recursive call, which gives us a total of 9cn/10 time. Likewise, at the next level, we spend 81cn/100 time, and so on. Our total time spent looks like T(n) = cn + 9cn/10 + 81cn/100 + 729cn/1000 + ... Note that this is a converging series with each term a multiplicative factor smaller than the last. Even if this process continues forever, the total time taken is dominated by the first term and is therefore O(n). This completes our analysis of the algorithm. A few remarks are in order. 1. Note that in analyzing the running time of the above algorithm, one crucial property that we used was that n/5 + 7n/10 < n. That is, the sum of the sizes of the subproblems we solved recursively was strictly smaller than the size of the original problem. This turned out to be important because it implied that the running time of each recursive step would be a constant factor smaller than that of the previous level. Contrast this to the size of the subproblems we obtain for mergesort. This also hints at a metatheorem for recurrences: For constants c and a1, a2, ..., ak such that a1 + ... + ak < 1, the recurrence T(n) = cn + T(a1 n) + T(a2 n) + ... + T(ak n) solves to T(n) = Theta(n). You should prove this theorem as an exercise. 2. What is a lower bound on the selection problem? Note that the number of different possible outputs of the algorithm is n (each element in the list could be the median). This implies that any algorithm in the comparison-based model must take time Omega(log n). However, note also that any algorithm must read the entire list before determining the median (think about why this is the case). Therefore, another (better!) lower bound on finding the median is n. Our algorithm above achieves the best possible running time (within constant factors).