Lecture Notes CMSC 251 Lecture 6 Divide and Conquer and MergeSort Thursday Feb 12 1998 Read Chapt 1 on MergeSort and Chapt 4 on recurrences Divide and Conquer The ancient Roman politicians understood an important principle of good algorithm design although they were probably not thinking about algorithms at the time You divide your enemies by getting them to distrust each other and then conquer them piece by piece This is called divide and conquer In algorithm design the idea is to take a problem on a large input break the input into smaller pieces solve the problem on each of the small pieces and then combine the piecewise solutions into a global solution But once you have broken the problem into pieces how do you solve these pieces The answer is to apply divide and conquer to them thus further breaking them down The process ends when you are left with such tiny pieces remaining e g one or two items that it is trivial to solve them Summarizing the main elements to a divide and conquer solution are Divide the problem into a small number of pieces Conquer solve each piece by applying divide and conquer recursively to it and Combine the pieces together into a global solution There are a huge number computational problems that can be solved efficiently using divide andconquer In fact the technique is so powerful that when someone first suggests a problem to me the first question I usually ask after what is the brute force solution is does there exist a divide andconquer solution for this problem Divide and conquer algorithms are typically recursive since the conquer part involves invoking the same technique on a smaller subproblem Analyzing the running times of recursive programs is rather tricky but we will show that there is an elegant mathematical concept called a recurrence which is useful for analyzing the sort of recursive programs that naturally arise in divide and conquer solutions For the next couple of lectures we will discuss some examples of divide and conquer algorithms and how to analyze them using recurrences MergeSort The first example of a divide and conquer algorithm which we will consider is perhaps the best known This is a simple and very efficient algorithm for sorting a list of numbers called MergeSort We are given an sequence of n numbers A which we will assume is stored in an array A 1 n The objective is to output a permutation of this sequence sorted in increasing order This is normally done by permuting the elements within the array A How can we apply divide and conquer to sorting Here are the major elements of the MergeSort algorithm Divide Split A down the middle into two subsequences each of size roughly n 2 Conquer Sort each subsequence by calling MergeSort recursively on each Combine Merge the two sorted subsequences into a single sorted list The dividing process ends when we have split the subsequences down to a single item An sequence of length one is trivially sorted The key operation where all the work is done is in the combine stage which merges together two sorted lists into a single sorted list It turns out that the merging process is quite easy to implement The following figure gives a high level view of the algorithm The divide phase is shown on the left It works top down splitting up the list into smaller sublists The conquer and combine phases are shown on the right They work bottom up merging sorted lists together into larger sorted lists 20 Lecture Notes CMSC 251 input 7 5 2 4 7 5 7 5 output 7 5 2 4 1 6 3 0 split 2 4 2 4 1 6 3 0 1 6 1 6 2 4 5 7 5 7 3 0 3 0 1 2 3 4 5 6 7 0 7 5 merge 2 4 2 4 0 1 3 6 1 6 1 6 0 3 3 0 Figure 4 MergeSort MergeSort Let s design the algorithm top down We ll assume that the procedure that merges two sorted list is available to us We ll implement it later Because the algorithm is called recursively on sublists in addition to passing in the array itself we will pass in two indices which indicate the first and last indices of the subarray that we are to sort The call MergeSort A p r will sort the subarray A p r and return the sorted result in the same subarray Here is the overview If r p then this means that there is only one element to sort and we may return immediately Otherwise if p r there are at least two elements and we will invoke the divide andconquer We find the index q midway between p and r namely q p r 2 rounded down to the nearest integer Then we split the array into subarrays A p q and A q 1 r We need to be careful here Why would it be wrong to do A p q 1 and A q r Suppose r p 1 Call MergeSort recursively to sort each subarray Finally we invoke a procedure which we have yet to write which merges these two subarrays into a single sorted array MergeSort MergeSort array A int p int r if p r q p r 2 MergeSort A p q MergeSort A q 1 r Merge A p q r we have at least 2 items sort A p q sort A q 1 r merge everything together Merging All that is left is to describe the procedure that merges two sorted lists Merge A p q r assumes that the left subarray A p q and the right subarray A q 1 r have already been sorted We merge these two subarrays by copying the elements to a temporary working array called B For convenience we will assume that the array B has the same index range A that is B p r One nice thing about pseudocode is that we can make these assumptions and leave them up to the programmer to figure out how to implement it We have to indices i and j that point to the current elements of each subarray We move the smaller element into the next position of B indicated by index k and then increment the corresponding index either i or j When we run out of elements in one array then we just copy the rest of the other array into B Finally we copy the entire contents of B back into A The use of the temporary array is a bit unpleasant but this is impossible to overcome entirely It is one of the shortcomings of MergeSort compared to some of the other efficient sorting algorithms In case you are not aware of C notation the operator i returns the current value of i and then increments this variable by one Merge Merge array A int p int q int r 21 merges A p q with A q 1 r Lecture Notes CMSC 251 array B p r i …