Min Chen School of Computer Science and Engineering Seoul National University Data Structure: Chapter 2

 Definition of Arrays  Operators for Arrays  Value Assignment  Insertion  Search  Deletion  Ordered Arrays  Search Algorithms ▪ Binary Search ▪ Interpolation Search  Sort Algorithms ▪ Bubble Sort ▪ Insertion Sort ▪ Quick Sort

 An array is a series of elements of the same type placed in contiguous memory locations that can be individually referenced by adding an index to a unique identifier. A[0] …… A[1]A[2]A[3]A[4]A[5]A[6]A[7]

 Memory storage for 2 Dimensional arrays A[0][0] A[4][2] A[0][0] A[0][1] A[0][2] A[0][3] A[0][4] A[0][5] A[0][6] A[0][7] A[1][0] A[1][1] A[1][2] A[1][3] ……

 Can represent 3 Dimensional elements A[0][0][0] A[1][2][0] A[2][3][1]

 Value Assignment  Insertion  Search  Deletion

 Just give the value to the corresponding array element  Example:  Assign the value 21 as the 2 nd element ▪ A[1]=21; A[0] …… A[1]A[2]A[3]A[4]A[5]A[6]A[7] 21

 Steps: 1. Insert the value to the corresponding position 2. Move all the items with higher index values up one position  Example:  Insert the value 21 as the 2 nd element ▪ A[1] <- 21; A[0] …… A[1]A[2]A[3]A[4]A[5]A[6]A[7] 21 A[0] …… A[2]A[3]A[4]A[5]A[6]A[7]A[8] 21

 The searchKey variable holds the value we’re looking for. To search for an item, step through the array, comparing searchKey with each element.  Example  Search value 27  Search value A[i] i=0 i=1 i=2 Output: Found in i=2 A[k] k=0k=1k=2k=3k=4k=5k=6k=7 Output: Can’t find 23

 Steps: 1. Begin with a search 2. Delete the target element 3. move all the items with higher index values down one element to fill in the “hole” left by the deleted element  Example:  Delete the first value i=0i=1i=2i=3i=4i=5

 The data in ordered arrays is stored in ascending (or descending) key order  Benefit: Possible faster way of searching  Shortcoming: Need to be sorted Value is increasing

 Solution of The Guess-a-Number Game  Example:  Search value i=4i=6i=5 62>5662<85 Output: Found in i=5

 Motivated by the phonebook search  In previous example  1 st Round A[6-1]=62 Found!

 Sort as bubble arising  Example:  Sort by increasing Length Swap } Sorted How many iterations are needed for complete sort?

 The simplest sort algorithm  Easy to understand  Easy to be implemented  The least efficient  Two Level Loops  Computational Complexity: i from 1 to N-1 step 1 j from 0 to N-i { Swap( j, j+1 ); }

 Divide the array into two parts  Sorted part  Unsorted part  Insert the elements in the unsorted part into the sorted part in terms of the key value Sorted PartUnsorted Part

 Computational Complexity:  some can be substantially faster than others.  Insertion sort is stable  Require a single temporary variable in addition to the original array

 A Divide-and-Conquer solution 1. Divide the array into two parts by a selected pivot value α ▪ the elements smaller than α in left part ▪ Otherwise in right part 2. Sort the two parts individually as two independent arrays by Quick Sort

Select as the Pivot SmallerLarger Pivot

 How to select the pivot?

 Partitioning operates in linear O(N) time, making N plus 1 or 2 comparisons and fewer than N/2 swaps.  The partitioning algorithm may require extra tests in its inner while loops to prevent the indices running off the ends of the array.  The pivot value for a partition in Quick Sort is the key value of a specific item, called the pivot.

 Definition of Arrays  Operators for Arrays  Insertion  Search  Deletion  Ordered Arrays  Search Algorithms ▪ Binary Search ▪ Interpolation Search  Sort Algorithms ▪ Bubble Sort ▪ Insertion Sort ▪ Quick Sort