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Looking for data structures by seymour lipschutz? This can be done by writing Newborn. Age or simply Father. Then every item automatically becomes a element array. Some languages allow the sex of the sixth newborn to be referenced by writing Newborn. Sex[6] or simply Sex[6] Analogously, the age of the father of the sixth newborn may be referenced by writing Newborn. Age[6] or simply Father. Age[6] c Consider the record structure Student in Example 4.
Furthermore, Test becomes a two-dimensional array. In particular, the second test of the sixth student may be referenced by writing Student. For example, one writes Age Father Newborn instead of Newborn. Fathe and First Name Student[8] instead of Student. On the other hand, AGE is not unique. Assuming the record contains nonhomogeneous data, the record may have to be stored in individual variables, one for each of its elementary data items.
This is illustrated in the next two examples. Again, we assume that the arrays are parallel, i. Records with Variable Lengths Suppose an elementary school keeps a record for each student which contains the following data: Name, Telephone Number, Father, Mother, Siblings. Three such records may be as follows:. Here XXXX means that the parent has died or is not living with the student, or that the student has no sibling at the school.
The above is an example of a variable-length record, since the data element Siblings can contain zero or more names. A matrix with one row column may be viewed as a vector and, similarly, a vector may be viewed as a matrix with only one row column. A matrix with the same number n of rows and columns is called a square matrix or an n-square matrix. The diagonal or main diagonal of an n-square matrix A consists of the elements A1l, A22, …, Ann. The next section will review certain algebraic operations associated with vectors and matrices.
Suppose U and V are n-element vectors. Algorithms for matrix addition and matrix scalar multiplication, which are very similar to algorithms for vector addition and scalar multiplication, are left as exercises for the reader. The complexity of a matrix multiplication algorithm is measured by counting the number C of multiplications. The reason that additions are not counted in such algorithms is that computer multiplication takes much more time than computer addition.
The complexity of the above Algorithm 4. The next example gives an important and surprising result in this area. In Algorithm 4. On the other hand, AB can also be obtained from the following, which. Two general types of n-square sparse matrices, which occur in various applications, are pictured in Fig. It is sometimes customary to omit blocks of zeros in a matrix as in Fig. The second matrix, where nonzero entries can only occur on the diagonal or on elements immediately above or below the diagonal, is called a tridiagonal matrix.
The natural method of representing matrices in memory as two- dimensional arrays may not be suitable for sparse matrices. That is, one may save space by storing only those entries which may be nonzero. This is illustrated for triangular matrices in the following example. Other cases will be discussed in the solved problems. Clearly it would be wasteful to store those entries above the main diagonal of A, since we know they are all zero; hence we store only the other entries of A in a linear array B as indicated by the arrows.
Now there are elements in the rows above aJK, and there are K elements in row J up to and including. Write a module for each of the following tasks: a To print each of the years in which no employee was born. Assume is the current year. Each module traverses the array. Find the values in A after each loop.
Thus every element of A will have the value xl, the original value of A[l]. Thus every value in A will move to the next location. Remark: This example illustrates the reason that, in the insertion algorithm, Algorithm 4. Hence 24 elements are moved. Searching, Sorting. Hence the search is unsuccessful. The algorithm follows. Algorithm P4.
Steps 1 through 4. Same as in Algorithm 4. For any value of k, let nk denote the number of those elements in A that will require k comparisons to be located in A. The average-case running time g n is. The number D of interchanges depends also on the data, as well as on the number n of elements. Dividing by 2 gives us our result. Assuming B is stored in column-major order, we use Eq. Suppose we want to store A in a linear array B as indicated by the arrows in Fig.
The third matrix is symmetric. This can be done in the same way as that for triangular matrices described in Example 4. Each lawyer has his own clients. Which data structure is most useful? Suppose there are only one secretary and one telephone number, and suppose there is a single monthly billing of the clients. Also, suppose clients frequently change from one lawyer to another.
Then Fig. Observe that an item is elementary only if it is not followed by an item with a higher level number. An element in a record structure may be an array itself. Instead of storing the three tests separately, we store them in an array. But Name is a group item.
That is, only GPA[l, 3] is printed. In addition, it uses the data structure in Fig. The variables. The procedure follows: Procedure P4. NUMB — 1. Since this involves variable-length data, how might this be done?
This can be accomplished as in Fig. Write a module which a Finds the sum of the array elements b Returns the address of the middle element of the array.
Which algorithm instruction should be changed so that the algorithm sorts the array elements in descending order? Change Step 3 a of Algorithm 4. Since array A contains N elements, the linear search operation would be considered unsuccessful after N iterations. Arrays 4. Write a module which a Finds all pairs of elements whose sum is Pointer Arrays; Record Structures 4. Miscellaneous 4. After how many iterations of the binary search algorithm, Algorithm 4.
Arrays Assume that the data in Table 4. Table 4. Test the program using a , b and c Test the program using a Rogers, b Johnson and c Bailey. Test the program using: a , Quinn, Michael, 2. Test the program using a Parker and b Fox. Test the program using i , ii and iii Compare with Programming Problem 4. Pointer Arrays Assume the data in Table 4. Test the program using a Harris, b Rivers and c Lopez. Test the program using a Klein, b Daniels, c Meth and d Harris. Test the program using 4.
Test the sub programs on the following array elements by inserting item 11 at 5th position and deleting the element present at the 3rd position: 50 17 33 43 5 8 22 19 40 99 4. Figure 5. Frequently, we want to add items to or delete items from a list. One way to store such data is by means of arrays, discussed in Chapter 4. This makes it easy to compute the address of an element in an array.
On the other hand, arrays have certain disadvantages—e. Also, since an array usually occupies a block of memory space, one cannot simply double or triple the size of an array when additional space is required. For this reason, arrays are called dense lists and are said to be static data structures. Thus successive elements in the list need not occupy adjacent space in memory.
This will make it easier to insert and delete elements in the list. Accordingly, if one were mainly interested in searching through data for inserting and deleting, as in word processing, one would not store the data in an array but rather in a list using pointers.
This latter type of data structure is called a linked list and is the main subject matter of this chapter. We also discuss circular lists and two-way lists—which are natural generalizations of linked lists—and their advantages and disadvantages. Each node is pictured with two parts. The left part represents the information part of the node, which may contain an entire record of data items e.
The pointer of the last node contains a special value, called the null pointer, which is any invalid address. In actual practice, 0 or a negative number is used for the null pointer.
A special case is the list that has no nodes. Example 5. Suppose we want an alphabetical listing of the patients. The entry for the last patient Samuels contains the null pointer, denoted by 0. Observe that the names of the lists are also used as the list pointer variables. Such data may be organized as in Fig. Generally speaking, the information part of a node may be a record with more than one data item.
In such a case, the data must be stored in some type of record structure or in a collection of parallel arrays, such as that illustrated in the following example. Observe that 0 is used as the null pointer. Suppose we want to traverse LIST in order to process each node exactly once.
This section presents an algorithm that does so and then uses the algorithm in some applications. Our traversing algorithm uses a pointer variable PTR which points to the node that is currently being processed. The details of the algorithm are as follows. A formal presentation of the algorithm follows. Algorithm 5. The variable PTR points to the node currently being processed. Observe the similarity between Algorithm 5. The similarity comes from the fact that both are linear structures which contain a natural linear ordering of the elements.
Consequently, the algorithm may be preceded by such an initialization step. Since the procedure must traverse the list, it will be very similar to Algorithm 5. Observe that the procedure traverses the linked list in order to count the number of elements; hence the procedure is very similar to the above traversing algorithm, Algorithm 5. Here, however, we require an initialization step for the variable NUM before traversing the list. In other words, the procedure could have been written as follows:.
Call Algorithm 5. First we have to check to see whether we have reached the end of the list; i. The complexity of this algorithm is the same as that of the linear search algorithm for linear arrays discussed in Sec. The following module reads the social security number NNN of an employee and then gives the employee a 5 percent increase in salary.
Read: NNN. The module takes care of the case in which there is an error in inputting the social security number. Recall that with a sorted linear array we can apply a binary search whose running time is proportional to log2 n. On the other hand, a binary search algorithm cannot be applied to a sorted linked list, since there is no way of indexing the middle element in the list.
This property is one of the main drawbacks in using a linked list as a data structure. The following module reads the name EMP of an employee and then gives the employee a 5 percent increase in salary.
Compare with Example 5. Observe that now we can use the second search algorithm, Algorithm 5. Analogously, some mechanism is required whereby the memory space of deleted nodes becomes available for future use. These matters are discussed in this section, while the general discussion of the inserting and deleting of nodes is postponed until later sections.
Together with the linked lists in memory, a special list is maintained which consists of unused memory cells. This list, which has its own pointer, is called the list of available space or the free-storage list or the free pool.
Suppose our linked lists are implemented by parallel arrays as described in the preceding sections, and suppose insertions and deletions are to be performed on our linked lists. Then the available space in the linear array BED may be linked as in Fig.
Furthermore, suppose LIST is initially empty. Suppose some memory space becomes reusable because a node is deleted from a list or an entire list is deleted from a program. Clearly, we want the space to be available for future use. One way to bring this about is to immediately reinsert the space into the free-storage list. This is what we will do when we implement linked lists by means of linear arrays. However, this method may be too time-consuming for the operating system of a computer, which may choose an alternative method, as follows.
The operating system of a computer may periodically collect all the deleted space onto the free-storage list. Any technique which does this collection is called garbage collection. Garbage collection usually takes place in two steps. First the computer runs through all lists, tagging those cells which are currently in use, and then the computer runs through the memory, collecting all untagged space onto the free-storage list.
The garbage collection may take place when there is only some minimum amount of space or no space at all left in the free-storage list, or when the CPU is idle and has time to do the collection. Generally speaking, the garbage collection is invisible to the programmer. Any further discussion about this topic of garbage collection lies beyond the scope of this text.
In such a case, the programmer may then modify the program by adding space to the underlying arrays. Suppose a node N is to be inserted into the list between nodes A and B. The schematic diagram of such an insertion appears in Fig. That is, node A now points to the new node N, and node N points to node B, to which A previously pointed. Thus a more exact schematic diagram of such an insertion is that in Fig. There are also two special cases. Suppose a patient Hughes is admitted to the ward. Since the customer lists are not sorted, we will assume that each new customer is added to the beginning of its list.
Suppose Gordan is a new customer of Kelly. Again we assume that each new node is inserted at the beginning of the list. Accordingly, after the six insertions, F will point to E, which points to D, which points to C, which points to B, which points to A; and A will contain the null pointer. Insertion Algorithms Algorithms which insert nodes into linked lists come up in various situations.
We discuss three of them here. Then the easiest place to insert the node is at the beginning of the list. An algorithm that does so follows. Steps 1 to 3 have already been discussed, and the schematic diagram of Steps 2 and 3 appears in Fig. The schematic diagram of Steps 4 and 5 appears in Fig. Suppose the test score 75 is to be added to the beginning of the geometry list.
We simulate Algorithm 5. Let N denote the new node whose location is NEW. Otherwise, as pictured in Fig. Else: [Insert after node with location LOC.
While traversing, keep track of the location of the preceding node by using a pointer variable SAVE, as pictured in Fig. The formal statement of our procedure follows. Procedure 5. Now we have all the components to present an algorithm which inserts ITEM into a linked list. Suppose Jones is to be added to the list of patients. Suppose we want to copy all or part of a given list, or suppose we want to form a new list that is the concatenation of two given lists.
These algorithms are covered in the problem sections. Suppose node N is to be deleted from the linked list.
The schematic diagram of such a deletion appears in Fig. Accordingly, when performing deletions, one must keep track of the address of the node which immediately precedes the node that is to be deleted. Thus a more exact schematic diagram of such a deletion is the one in Fig. Suppose Green is discharged, so that BED[8] is now empty.
The new list is pictured in Fig. Deletion Algorithms Algorithms which delete nodes from linked lists come up in various situations. We discuss two of them here. An algorithm that does so must check to see if there is a node in the list. If not, i. The following algorithm deletes N from the list. The simplicity of the algorithm comes from the fact that we are already given the location LOCP of the node which precedes node N. Recall that before we can delete N from the list, we need to know the location of the node preceding N.
This procedure is similar to Procedure 5. Remark: The reader may have noticed that Steps 3 and 4 in Algorithm 5. Suppose the patient Green is discharged. We simulate Procedure 5. Then we simulate Algorithm 5. The following are two kinds of widely used header lists: 1 A grounded header list is a header list where the last node contains the null pointer.
Unless otherwise stated or implied, our header lists will always be circular. Accordingly, in such a case, the header node also acts as a sentinel indicating the end of the list. Although our data may be maintained by header lists in memory, the AVAIL list will always be maintained as an ordinary linked list. The data may be organized as a header list as in Fig.
Circular header lists are frequently used instead of ordinary linked lists because many operations are much easier to state and implement using header lists. This comes from the following two properties of circular header lists: 1 The null pointer is not used, and hence all pointers contain valid addresses.
The next example illustrates the usefulness of these properties. The following is such an algorithm when LIST is a circular header list. The two tests which control the searching loop Step 2 in Algorithm 5. The following is such a procedure when LIST is a circular header list.
Observe the simplicity of this procedure compared with Procedure 5. The header node plays an important part in this representation, since it is needed to represent the zero polynomial.
One such representation appears in Fig. Furthermore, given the location LOC of a node N in such a list, one has immediate access to the next node in the list by evaluating LINK[LOC] , but one does not have access to the preceding node without traversing part of the list. This means, in particular, that one must traverse that part of the list preceding N in order to delete N from the list. This section introduces a new list structure, called a two- way list, which can be traversed in two directions: in the usual forward direction from the beginning of the list to the end, or in the backward direction from the end of the list to the beginning.
Furthermore, given the location LOC of a node N in the list, one now has immediate access to both the next node and the preceding node in the list. This means, in particular, that one is able to delete N from the list without traversing any part of the list. Two-Way Header Lists The advantages of a two-way list and a circular header list may be combined into a two-way circular header list as pictured in Fig.
The list is circular because the two end nodes point back to the header node. Observe that such a two-way list requires only one list pointer variable START, which points to the header node. This is because the two pointers in the header node point to the two ends of the list. The data may be organized into a two-way circular header list by simply adding another array BACK which gives the locations of preceding nodes.
Such a structure is pictured in Fig. This subsection discusses a number of operations on LIST. Then we can use Algorithm 5. Here it is of no advantage that the data are organized as a two-way list rather than as a one- way list. Here the main advantage is that we can search for ITEM in the backward direction if we have reason to suspect that ITEM appears near the end of the list.
For example, suppose LIST is a list of names sorted alphabetically. We assume that LIST is a two-way circular header list.
Accordingly, as pictured in Fig. Linked Lists. Here the four list pointers appear in an array CITY. Each procedure uses Algorithm 5. Return b Procedure 1. Write an algorithm which deletes the last node from LIST. The last node can be deleted only when one also knows the location of the next-to-last node. Algorithm P5. Then, for each list: a Change the list pointer variable so that it points to the header node.
Unless otherwise stated, the terms in p x, y, z will be ordered lexicographically. Assign values to LINK so that the linked list contains the ordered sequence of terms.
The two-way list contains this information, whereas with a one-way list we must traverse the list. For example, if we are searching for Walker in an alphabetical listing, it may be quicker to traverse the list backward. Remark: Generally speaking, a two-way list is not much more useful than a one-way list except in special circumstances. Compare with Solved Problem 5. The algorithm is the same as Algorithm P5.
These are calculated node by node. That is, only the array BACK need be calculated. Which other data structure can be used to achieve this?
A stack data structure which stores the elements in Last In First Out format can be used to reverse the contents of LA. Following are the steps to perform the reversal operation: 1.
Perform the above step till the list LA becomes empty. Now, pop the top element form the stack and push it back into the list LA. Perform the above step till the stack becomes empty. Remark: For more information on stack, refer to Chapter 6. Linked Lists 5. Supplementary Problems 5. Header Lists; Two-Way Lists 5. Assign values to an array LINK so that the linked list contains the ordered sequence of terms.
See Solved Problem 5. Observe that the algorithms are now much simpler. Miscellaneous 5. Programming Problems 5. Test the program for each K.
Test the program using a Rogers, b Baker and c Levine. Test the program using a Newman, b Ford, c Rivers and d Hall. Test the program using a Lewis, b Klein and c Parker. Test the program using a Jones, 36, Levine; and b Olsen, 44, Nelson.
Test the program using a Evans, b Smith and c Lewis. Test the program using a Davis, b Jones and c Rubin. Test the program using a Fletcher, , Female, 21 ; and b Nelson, , Male, 19 Remark: Remember to update the header record whenever there is an insertion or a deletion.
Write a program that appends all the elements of LA2 at the end of LA1. Also, print the elements of the combined list. Chapter Six. There are certain frequent situations in computer science when one wants to restrict insertions and deletions so that they can take place only at the beginning or the end of the list, not in the middle. Two of the data structures that are useful in such situations are stacks and queues. A stack is a linear structure in which items may be added or removed only at one end.
Figure 6. Observe that an item may be added or removed only from the top of any of the stacks. A queue is a linear list in which items may be added only at one end and items may be removed only at the other end.
Consider a queue of people waiting at a bus stop, as pictured in Fig. Another example of a queue is a batch of jobs waiting to be processed, assuming no job has higher priority than the others. The notion of recursion is fundamental in computer science. This topic is introduced in this chapter because one way of simulating recursion is by means of a stack structure.
This means, in particular, that elements are removed from a stack in the reverse order of that in which they were inserted into the stack. We emphasize that these terms are used only with stacks, not with other data structures. Example 6. We emphasize that, regardless of the way a stack is described, its underlying property is that insertions and deletions can occur only at the top of the stack.
Consequently, the elements may be popped from the stack only in the reverse order of that in which they were pushed onto the stack. This implementation of the AVAIL list as a stack is only a matter of convenience rather than an inherent part of the structure. In the following subsection we discuss an important situation where the stack is an essential tool of the processing algorithm itself.
This is illustrated as follows. Suppose that while processing some project A we are required to move on to project B, whose completion is required in order to complete project A. Then we place the folder containing the data of A onto a stack, as pictured in Fig. However, suppose that while processing B we are led to project C, for the same reason. Then we place B on the stack above A, as pictured in Fig.
Furthermore, suppose that while processing C we are likewise led to project D. Then we place C on the stack above B, as pictured in Fig. On the other hand, suppose we are able to complete the processing of project D. Then the only project we may continue to process is project C, which is on top of the stack. Hence we remove folder C from the stack, leaving the stack as pictured in Fig.
Similarly, after completing the processing of C, we remove folder B from the stack, leaving the stack as pictured in Fig. Finally, after completing the processing of B, we remove the last folder, A, from the stack, leaving the empty stack pictured in Fig.
An important example of such a processing in computer science is where A is a main program and B, C and D are subprograms called in the order given. Unless otherwise stated or implied, each of our stacks will be maintained by a linear array STACK; a pointer variable TOP, which contains the location of the top element of the stack; and a variable MAXSTK which gives the maximum number of elements that can be held by the stack.
For notational convenience, the array is drawn horizontally rather than vertically. Procedure 6. Note that WWW is now the top element in the stack. Most of these techniques lie beyond the scope of this text. We do illustrate one such technique in the following example.
As pictured in Fig. The advantages of linked lists over arrays have already been underlined. It is in this perspective that one appreciates the use of a linked list for stacks in comparison to that of arrays. The linked representation of a stack, commonly termed linked stack is a stack that is implemented using a singly linked list.
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