Review 1 Arrays & Strings Array Array Elements Accessing array elements Declaring an array Initializing an array Two-dimensional Array Array of Structure String Array of Strings Examples

Searching Introduction to Searching External and Internal Searching Types of Searching Linear or sequential search Binary Search Algorithms for Linear Search Algorithms for Binary Search 2

Introduction Information retrieval is one of the most important applications of computers We are given a name and are asked for an associated telephone listing We are given an account number and are asked for the transactions occurring in that account We are given an employee name or number and are asked for the employee records 3

Introduction Information retrieval is one of the most important applications of computers We are given a name and are asked for an associated telephone listing We are given an account number and are asked for the transactions occurring in that account We are given an employee name or number and are asked for the employee records 4

Searching is a process of checking and finding an element from a list of elements If we want to find the presence of an element “data” in A, then we have to search for it The search is successful if data does appear in A and unsuccessful otherwise A question you should always ask when selecting a search algorithm is “How fast does the search have to be?” The reason is that, in general, the faster the algorithm is, the more complex it is. Bottom line: you don’t always need to use or should use the fastest algorithm 5

Key In these examples, we are given one piece of information, which we shall call a key We are asked to find a record that contains other information associated with the key We shall allow both the possibility that there is more than one record with the same key and that there is no record with the given key 6

Records and their keys 7

Analysis Searching for the keys that locate records is often the most time-consuming action in a program therefore, the way records are arranged and the choice of method used for searching can make a substantial difference in the program’s performance For this reason, we should check that how much work is done by each of the algorithms we develop We shall find that counting the number of times that one key is compared with another gives us an excellent measure of the total amount of work that the algorithm will do and of the total amount of computer time it will require when it is run 8

External and Internal Searching The searching problem falls naturally into two cases External Searching If there are many records, perhaps each one quite large, then it will be necessary to store the records in files on disk or tape, external to the computer memory Internal Searching In this case, the records to be searched are stored entirely within the computer memory 9

Types of Searching There are two types of searching techniques: Linear or Sequential Searching Binary Searching Linear or Sequential Searching In linear search, each element of an array is read one by one sequentially It is compared with the desired element A search will be unsuccessful if all the elements are read and the desired element is not found 10

Sequential Search on an Unordered List In this case, we have an unordered list of items/elements We check each and every element in the list sequentially and it is compared with the desired element Algorithm Get the search criterion (key) Get the first record from the file While ( (record != key) and (still more records) ) Get the next record End_while If ( record = key ) Then success Else there is no match in the file End_else 11

Sequential Search on an Ordered List In this case, we have an ordered list of elements We check each and every element in the list sequentially and it is compared with the desired element Algorithm: Get the search criterion (key) Get the first record from the file While ( (record < key) and (still more records) ) Get the next record End_while If ( record = key ) Then success Else there is no match in the file End_else 12

Algorithm for Linear Search Let A be an array of n elements, A[1],A[2],A[3], A[n]. “data” is the element to be searched. Then this algorithm will find the location “loc” of data in A. Set loc = – 1,if the search is unsuccessful. 1. Input an array A of n elements and “data” to be searched and initialize loc = – Initialize i = 0; and repeat through step 3 if (i < n) by incrementing i by one. 3. if (data = A[i]) (a) loc = i (b) GOTO step 4 4. if (loc > 0) (a) Display “data is found and searching is successful” 5. else (a) Display “data is not found and searching is unsuccessful” 6. exit 13

Sequential Search of Ordered vs. Unordered List Let’s do a comparison. If the order was ascending alphabetical on customer’s last names, how would the search for Shahab Ali on the ordered list compare with the search on the unordered list? Unordered list if Shahab Ali was in the list? if Shahab Ali was not in the list? Ordered list if Shahab Ali was in the list? if Shahab Ali was not in the list? 14

Ordered vs. Unordered (cont) How about Mohammad Waqas? Unordered if Mohammad Waqas was in the list? if Mohammad Waqas was not in the list? Ordered if Mohammad Waqas was in the list? if Mohammad Waqas was not in the list? 15

Ordered vs. Unordered (cont) Observation: the search is faster on an ordered list Also, keep in mind that the list has to first be placed in order for the ordered search. Conclusion: the efficiency of these algorithms is roughly the same. So, if we need a faster search, we need a completely different algorithm. How else could we search an ordered file? 16

Binary Search Sequential search is easy to write and efficient for short lists, but a disaster for long ones Imagine trying to find the name “Ahmad Tahir” in a large telephone book by reading one name at a time starting at the front of the book To find any entry in a long list, there are far more efficient methods, provided that the keys in the list are already sorted into order If we have an ordered list, we can use a different strategy The binary search gets its name because the algorithm continually divides the list into two parts 17

Binary Search Binary search is an extremely efficient algorithm when it is compared to linear search Binary search technique searches “data” in minimum possible comparisons First compare the target key with one in the center of the list and then restrict our attention to only the first or second half of the list, depending on whether the target key comes before or after the central one With one comparison of keys we thus reduce the list to half its original size Continuing in this way, at each step, we reduce the length of the list to be searched by half 18

In only twenty steps, this method will locate any requested key in a list containing more than a million elements This approach of course requires that the elements in the list already be in completely order Suppose we have a sorted array of n elements, then apply the following steps to search an element 19

1. Find the middle element of the array (i.e., n/2 is the middle element if the array or the sub-array contains n elements) 2. Compare the middle element with the data to be searched, then there are following three cases (a) If it is a desired element, then search is successful (b) If it is less than desired data, then search only the first half of the array, i.e., the elements which come to the left side of the middle element (c) If it is greater than the desired data, then search only the second half of the array, i.e., the elements which come to the right side of the middle element Repeat the same steps until an element is found or exhaust the search area 20

Algorithm for Binary Search Let A be an array of n elements A[1],A[2],A[3], A[n] “Data” is an element to be searched “mid” denotes the middle location of a segment (or array or sub-array) of the element of A LB and UB is the lower and upper bound of the array which is under consideration 21

1. Input an array A of n elements and “data” to be searched 2. LB = 0, UB = n; mid = int ((LB+UB)/2) 3. Repeat step 4,5 and 6 while (LB <= UB) and (A[mid] ! = data) 4. If (data < A[mid]) (a) UB = mid–1 5. Else (a) LB = mid Mid = int ((LB + UB)/2) 7. If (A[mid]== data) (a) Display “the data found” 8. Else (a) Display “the data is not found” 9. Exit 22

How a Binary Search Works Always look at the center value. Each time you get to discard half of the remaining list. Is this fast ? 23

How Fast is a Binary Search? Worst case: 11 items in the list took 4 tries How about the worst case for a list with 32 items ? 1st try - list has 16 items 2nd try - list has 8 items 3rd try - list has 4 items 4th try - list has 2 items 5th try - list has 1 item 24

How Fast is a Binary Search? (con’t) List has 250 items 1st try items 2nd try - 63 items 3rd try - 32 items 4th try - 16 items 5th try - 8 items 6th try - 4 items 7th try - 2 items 8th try - 1 item List has 512 items 1st try items 2nd try items 3rd try - 64 items 4th try - 32 items 5th try - 16 items 6th try - 8 items 7th try - 4 items 8th try - 2 items 9th try - 1 item 25

What’s the Pattern? List of 11 took 4 tries List of 32 took 5 tries List of 250 took 8 tries List of 512 took 9 tries 32 = 2 5 and 512 =

A Very Fast Algorithm! How long (worst case) will it take to find an item in a list 30,000 items? 2 10 = = = = = = So, it will take only 15 tries! 27

log 2 n Efficiency After 1 bisectionN/2 items After 2 bisectionsN/4 = N/2 2 items... After i bisectionsN/2 i =1 item i = log 2 N 28

Lg n Efficiency We say that the binary search algorithm runs in log 2 n time. (Also written as lg n) Lg n means the log to the base 2 of some value of n. 8 = 2 3 lg 8 = 316 = 2 4 lg 16 = 4 There are no algorithms that run faster than lg n time. 29

Example Suppose we have an array of 7 elements Following steps are generated if we binary search a data = 45 from the above array Step 1: LB = 0; UB = 6 mid = (0 + 6)/2 = 3 A[mid] = A[3] = 30 30

Step 2: Since (A[3] < data) - i.e., 30 < 45 - reinitialise the variable LB, UB and mid LB = mid+1 LB = 4 UB = 6 mid = (4 + 6)/2 = 5 A[mid] = A[5] = 45 31

Step 3: Since (A[5] == data) - i.e., 45 == 45 - searching is successful 32

Summary Introduction to Searching External and Internal Searching Types of Searching Linear or sequential search Binary Search Algorithms for Linear Search Algorithms for Binary Search 33