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1 wwwashekalazizwordpresscom Data Structure Test Paper 3 1(a) Define Data Structure Mention the subject matters of data structure The logical and mathematical model of a particular organization of data is called Data Structure Data structure should be simple enough so that it can easily be processed when necessary and data should bear the relationship with the real world scenario [Source: Seymour Lipschutz] The subject matters of data structure are (i) (ii) (iii) (iv) (v) Various types of data and its physical representation in computer memory Various types of data structures with their logical relationships Various operation of different types of data types Algorithms and techniques for solving problems to get obtain desired result implementing necessary operations on data structures Optimization of time for data processing and memory space requirements (b) Mention the operations of data structure The operations on the data structures are (i) (ii) (iii) (iv) (v) (vi) Insertion: Inserting a new item or data or information into the structure Deletion: Removing an item or data or information from the structure Traversing: If there is collection of data items, then accessing each data item exactly once sometimes called visiting each node or element sequentially Searching: Finding some particular data item or information among the collection of information satisfying specific precondition Sorting: arrangement of data items in specific order like ascending order or descending order of integer contents or names in alphabetical order etc Merging: If there two different list of elements, then both can be combined together to form a new list of elements Here both the list can in sorted order and newly formed combined list should also be in sorted manner We should remember that Sorting and Merging are the operations on data structures in special case [Source: Seymour Lipschutz] (c) What is an algorithm? Explain complexity of an algorithm Page 1 of 7

2 wwwashekalazizwordpresscom The term algorithm comes from or, go and rythm Algorithm is a set of tasks which proceeds sequentially until desired result is obtained or a well defined problem is solved The first algorithm was devised by Abu Musa Al-Zaber Al Kwarizmi to find the result of Greatest Common Division of few integers [Source: Knuth] If we think to find a particular data item in an array using sequential searching ie visiting each element from the beginning and accessing the immediate next element successively until we get out desired element then the time required to find the data item depends on the number of elements in the list sometimes called the size of the data or input size to the searching algorithm If the size is smaller, much time may not be required to catch the desired data object but if the size is large then time will be much more consumed obviously This is called complexity of algorithm We should remember that running time of algorithm and size of the data for operation are two constraints of algorithm complexity and complexity can be expressed as a function of either time or space [Ref: Seymour Lipschutz] Another important fact is that if the desired data item is resided in the very beginning of the list then whatever may be the the size of the list, time requirement is the minimum but if would be in the last element even unavailable then time requirement will be the maximum to finish the traversal These are the best case and worst case respectively But we are generally concerned with the average cases that desired element might be in between the start and end position of the data list and probability to find the item Experts concern about the algorithm modification to optimize the time requirements [Ref: Seymour Lipschutz, Shahani] (d) Explain space time trade-off of algorithms We know that an algorithm is a well defined set of instructions to solve a defined problem Our major concern is to use efficient algorithm Algorithm efficiency depends on two major constraints one is time needed to determine the desired result running the algorithm and another is the memory space required to store and process the data If a computer programmer wants to reduce the time for running the algorithm he can achieve it by increasing the memory space usage On the other hand increasing the time for completing algorithm execution, memory space usage can be reduced This is a matter of optimization This issue is known as time space trade off Example: Let us explain the linear search and binary search first Each data element or particular attribute of each record of a list is compared sequentially from the very beginning of the list towards the last until the desired data or record is not found This is a time consuming process This algorithm requires average time n/2 for n number of records in the list If there are records then times comparisons required on an average On the other hand binary search finds the target in a sorted list dividing the list into two segments almost equally divided and tries to locate the target in one segment where it is possibly located The process is repeated on the targeted segment again dividing segment of the list and this process is further repeated until the desired item is found This process is known as divide and conquer method For records in a list, binary search requires log 2 n = log = 15 times comparisons only This is pretty much time efficient algorithm but the list should be in sorted order Page 2 of 7

3 wwwashekalazizwordpresscom Name Pointer 1 Abbey, Gregory 2 2 Brown, John 4 3 Carey, Mary Davis, Earl 1 5 Ellis, Susan 76 Soc Sec No Name Extra Data Davis, Earl XXXXXXXXX Abbey, Gregory XXXXXXXXX Lane, Alice XXXXXXXXX Brown, John XXXXXXXXX Smith, Mary XXXXXXXXX Auxiliary array sorted alphabetically Main file sorted by social security number We have list of records consists of Social Security Numbers, Names and other extra data If we have to find some one s information in this file searching using social security number using linear search, it will definitely be time consuming That s why a memory space consumption slightly increased by adding a new list or auxiliary array consisting names in alphabetical order ie sorted in ascending order and pointer to locate the same person s information in the main file Now a person can easily be found out using binary search technique in the auxiliary array and more detail information can be found through the pointer in the main file This is a good example of time-space trade off [Source: Seymour Lipschutz] 2(a) Consider the pattern P=aaabb, Construct the table and the corresponding labelled directed graph used in the second pattern matching algorithm Q 0 = Q 1 = a Q 2 = aa Q 3 = aaa Q 4 = aaab Q 5 = aaabb States definition f(, a) = a f(a, a) = aa f(aa, a) = aaa f(aaa, a) = aaa f(aaab, a) = a Page 3 of 7

4 wwwashekalazizwordpresscom f(, b) = f(a, b) = f(aa, b) = f(aaa, b) = aaab f(aaab, b) = aaabb A B Q 0 Q 1 Q 0 Q 1 Q 2 Q 0 Q 2 Q 3 Q 0 Q 3 Q 3 Q 4 Q 4 Q 1 Q 5 State table b a a a a a b b Q 0 Q 1 Q 2 Q 3 Q 4 Q 5 b b State diagram [Source: Seymour Lipschutz] (b) Suppose P and T are strings with lengths R and S, respectively and are stored as arrays with one character per element Write an algorithm to find the index of P in T Algorithm: [Source: Seymour Lipschutz] 1 [Initialize] Set K = 1 and MAX = S R +1 2 Repeat Step 3 to 5 while K MAX 3 Repeat for L = 1 to R [Test each character of P] If P[L] T[K + L 1] then go to Step 5 [End of inner loop] 4 [Success] Set INDEX = K and Exit 5 Set K = K + 1 [End of Step 2 outer loop] 6 [Failure] Set INDEX = 0 7 Exit (c) Let W be the string W= aabcdebecaec (i) INSERT(W, 2, pqrs ); (ii) REPLACE(W, rsb, mnbc ) Page 4 of 7

5 wwwashekalazizwordpresscom (i) (ii) INSERT(W, 2, pqrs ) = apqrsabcdebecaec REPLACE(W, rsb, mnbc ) = aabcdebecaec (d) A text T and pattern P and Q are in memory Write an algorithm which replaces every occurrence of P in T by Q Algorithm: [Source: Seymour Lipschutz] 1 [Find index of P] Set K = INDEX(T, P) 2 Repeat while K 0 (a) [Replace P by Q] Set T = REPLACE(T, P, Q) (b) [Update index] Set K = INDEX(T, P) [End of loop] 3 Write T 4 Exit 3(a) What is array? How linear arrays are represented in memory? An array is a list of data elements of single type either integer or real or character No multiple types of data can be stored in an array An array has fixed size ie the number of elements it can hold is fixed For example, an array is declared with name DATA that consist of N number of integer values First element can be accessed as DATA[1], second as DATA[2] and so on This is a one dimensional array In computer memory, it is represented as follows DATA[1] DATA[2] DATA[3] DATA[3] DATA[N] Page 5 of 7

6 wwwashekalazizwordpresscom (b) What is searching? Write Binary Search algorithm What is the complexity of the Binary search? Searching is the finding the location of a given item among collection of elements Usually data elements are stored in computer memory and we are concerned about finding the location or address of the computer memory where target data item is placed Binary search Algorithm: [Source: Seymour Lipschutz] BINARY(DATA, LB, UB, ITEM, LOC) Here DATA is a sorted array with lower bound LB and upper bound UB, and ITEM is a given item of information The variables BEG, END and MID denote, respectively, the beginning, end and middle location of a segment of elements of DATA This algorithm find the location LOC of ITEM in DATA or sets LOC=NULL 1 [Initialize segment variables] Set BEG=LB, END=UB and MID=INT((BEG+END)/2) 2 Repeat Steps 3 and 4 while BEG END and DATA[MID] ITEM 3 If ITEM < DATA[MID] then Set END=MID-1 Else Set BEG=MID+1 [End of If structure] 4 Set MID=INT((BEG+END)/2) [End of Step 2 loop] 5 If DATA[MID]=ITEM then Set LOC=MID Else Set LOC=NULL [End of If structure] 6 Exit The principle of binary search reduces the size of the searching domain of data collection half in each pass That is why we require at most comparisons f(n) to find the desired item as 2 ( ) > n or equivalently f(n) =< log n + 1 Running time of worst case is approximately equal to log n which is approximately equal to running time of average case For example, if input size is then only 20 comparisons are required to find the required data item 2 > 1000 = Page 6 of 7

7 wwwashekalazizwordpresscom (c) Suppose the following numbers are stored in an array: A=(32, 51, 27, 85, 66, 23, 13, 57) Using Bubble sort algorithm, sort these numbers 32, 51, 27, 85, 66, 23, 13, 57 32, 27, 51, 85, 66, 23, 13, 57 32, 27, 51, 66, 85, 23, 13, 57 32, 27, 51, 66, 23, 85, 13, 57 32, 27, 51, 66, 23, 13, 85, 57 32, 27, 51, 66, 23, 13, 57, 85 32, 27, 51, 66, 23, 13, 57, 85 27, 32, 51, 66, 23, 13, 57, 85 27, 32, 51, 23, 66, 13, 57, 85 27, 32, 51, 23, 13, 66, 57, 85 27, 32, 51, 23, 13, 57, 66, 85 27, 32, 51, 23, 13, 57, 66, 85 27, 32, 23, 51, 13, 57, 66, 85 27, 32, 23, 13, 51, 57, 66, 85 27, 32, 23, 13, 51, 57, 66, 85 27, 23, 32, 13, 51, 57, 66, 85 27, 23, 13, 32, 51, 57, 66, 85 27, 23, 13, 32, 51, 57, 66, 85 23, 27, 13, 32, 51, 57, 66, 85 23, 13, 27, 32, 51, 57, 66, 85 23, 13, 27, 32, 51, 57, 66, 85 13, 23, 27, 32, 51, 57, 66, 85 Page 7 of 7