CS 171: Introduction to Computer Science II Mergesort

 We’ll implement our own Deque (doubly- ended queue) !  You’ve been given a singly-linked list implementation  But removeLast() is too slow   Goal:  Implement using an iterable doubly-linked list  Due:  What would you like?

 Elementary sorting (quadratic cost)  Bubble sort  Selection sort  Insertion sort  Recursion (review) and analysis  Advanced sorting  MergeSort  QuickSort

Basic idea – Divide an array into two halves – Sort each half – Merge the two sorted halves into a sorted array

 A divide and conquer approach using recursion  Partition the original problem into two sub-problems  Solve each sub-problem using recursion (sort)  Combine the results to solve the original problem (merge)

 Merge Two Sorted Arrays  A key step in mergesort  Assume subarrays a[lo..mid] (left half) and a[mid+1...hi] (right half) are sorted  Copy a[] to an auxiliary array aux[]  Merge the two halves of aux[] to a[] ▪ Such that it contains all elements and remains sorted

Merging Two Sorted Arrays  Start from the first elements of each half  Compare and copy the smaller element to a[]  Increment index for the array that had the smaller element  Continue  If we reach the end of either half  Copy the remaining elements of the other half  What’s the cost of merging N elements?

First base case encountered

Return, and continue

Merge

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 MergeSort  Recursive Algorithm (top-down)  Improvements  Non-recursive algorithm (bottom-up)  Runtime analysis of recursive algorithms  QuickSort

 Recursion overhead for tiny subarrays  Merging cost even when the array is already sorted  Requires additional memory space for auxiliary array  QuickSort will address this later

 MergeX.java  Use insertion sort for small subarrays  improve the running time by 10 to 15 percent.  Test whether array is already in order  reduce merging cost for sorted subarrays  Eliminate the copy to the auxiliary array  Switches the role of the input array and the auxiliary array at each level ▪ one sorts an input array and puts the sorted output in the auxiliary array ▪ one sorts the auxiliary array and puts the sorted output in the given array

 MergeSort  Recursive Algorithm (top-down)  Improvements  Non-recursive algorithm (bottom-up)  Runtime analysis of recursive algorithms  QuickSort

Recursive MergeSort (top-down)

Non-Recursive MergeSort (bottom-up)

 MergeSort  Recursive Algorithm (top-down)  Improvements  Non-recursive algorithm (bottom-up)  Runtime analysis of recursive algorithms  QuickSort

 Mergesort  Recursively sort 2 half-arrays  Merge 2 half-arrays into 1 array

 Mergesort  Recursively sort 2 half-arrays  Merge 2 half-arrays into 1 array (cost: N)  Suppose cost function for sorting N items is T(N)  T(N) = 2 * T(N/2) + N  T(1) = 0

 Mergesort  Recursively sort 2 half-arrays  Merge 2 half-arrays into 1 array (cost: N)  Suppose cost function for sorting N items is T(N)  T(N) = 2 * T(N/2) + N  T(1) = 0  Big O cost: N lg(N)

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