Data Structures, Algorithms And Applications In C++. Pages · · by By Sahni, Sartaj · data structures in c and data structures course, such as CS (T/W/C/S. Data Structures and Algorith. data structure using c notes pdf . Universities Press. COMPUTER SCIENCE. Data Structures,. Algorithms and. Applications IN. • C++. 8 B. Second Edition. SARTAJ SAHNI copyrighted. Handbook of data structures and applications / edited by Dinesh P. Mehta and Sartaj Sahni. p. cm. — (Chapman Although there are several advanced and specialized texts and . Sartaj Sahni is a Distinguished Professor and Chair of Computer and Information Sciences Reference Manual, Addison Wesley,
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Advanced Data Structures presents a comprehensive look at the ideas, eBook (EBL) . Structures (Mehta and Sahni ) is a step in the same direction. PDF | On Jan 1, , Ellis Horowitz and others published Fundamentals of Data Structure in C++. Sartaj Sahni at University of Florida To examine and define a data structure, following the stages below will ensure . the new priority queues with advanced heap implementations as well as and with. Handbook of data structures and applications / edited by Dinesh P. Mehta and Sartaj Sahni. Although there are several advanced and specialized . Sartaj Sahni is a Distinguished Professor and Chair of Computer and Information Sciences.
This happens to industrial programmers as well. If you have been careful about keeping track of your previous work it may not be too difficult to make changes.
It is usually hard to decide whether to sacrifice this first attempt and begin again or just continue to get the first version working. Different situations call for different decisions, but we suggest you eliminate the idea of working on both at the same time. If you do decide to scrap your work and begin again, you can take comfort in the fact that it will probably be easier the second time.
In fact you may save as much debugging time later on by doing a new version now. This is a phenomenon which has been observed in practice. The graph in figure 1. For each compiler there is the time they estimated it would take them and the time it actually took.
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For each subsequent compiler their estimates became closer to the truth, but in every case they underestimated. Unwarrented optimism is a familiar disease in computing. But prior experience is definitely helpful and the time to build the third compiler was less than one fifth that for the first one. Verification consists of three distinct aspects: program proving, testing and debugging.
Each of these is an art in itself. Before executing your program you should attempt to prove it is correct. Proofs about programs are really no different from any other kinds of proofs, only the subject matter is different.
If a correct proof can be obtained, then one is assured that for all possible combinations of inputs, the program and its specification agree. Testing is the art of creating sample data upon which to run your program. If the program fails to respond correctly then debugging is needed to determine what went wrong and how to correct it. One proof tells us more than any finite amount of testing, but proofs can be hard to obtain. Many times during the proving process errors are discovered in the code.
The proof can't be completed until these are changed. This is another use of program proving, namely as a methodology for discovering errors. Finally there may be tools available at your computing center to aid in the testing process. One such tool instruments your source code and then tells you for every data set: i the number of times a statement was executed, ii the number of times a branch was taken, iii the smallest and largest values of all variables.
As a minimal requirement, the test data you construct should force every statement to execute and every condition to assume the value true and false at least once. One thing you have forgotten to do is to document. But why bother to document until the program is entirely finished and correct? Because for each procedure you made some assumptions about its input and output.
If you have written more than a few procedures, then you have already begun to forget what those assumptions were. If you note them down with the code, the problem of getting the procedures to work together will be easier to solve.
The larger the software, the more crucial is the need for documentation. The previous discussion applies to the construction of a single procedure as well as to the writing of a large software system.
Let us concentrate for a while on the question of developing a single procedure which solves a specific task. The design process consists essentially of taking a proposed solution and successively refining it until an executable program is achieved. The initial solution may be expressed in English or some form of mathematical notation.
At this level the formulation is said to be abstract because it contains no details regarding how the objects will be represented and manipulated in a computer. If possible the designer attempts to partition the solution into logical subtasks. Each subtask is similarly decomposed until all tasks are expressed within a programming language.
This method of design is called the top-down approach. Inversely, the designer might choose to solve different parts of the problem directly in his programming language and then combine these pieces into a complete program. This is referred to as the bottom-up approach. Experience suggests that the top-down approach should be followed when creating a program. However, in practice it is not necessary to unswervingly follow the method.
A look ahead to problems which may arise later is often useful. Underlying all of these strategies is the assumption that a language exists for adequately describing the processing of data at several abstract levels. Let us examine two examples of top-down program development. Suppose we devise a program for sorting a set of n given by the following 1 distinct integers. One of the simplest solutions is "from those integers which remain unsorted, find the smallest and place it next in the sorted list" This statement is sufficient to construct a sorting program.
However, several issues are not fully specified such as where and how the integers are initially stored and where the result is to be placed. One solution is to store the values in an array in such a way that the i-th integer is stored in the i-th array position, A i 1 i n.
We are now ready to give a second refinement of the solution: for i 1 to n do examine A i to A n and suppose the smallest integer is at A j ; then interchange A i and A j.
There now remain two clearly defined subtasks: i to find the minimum integer and ii to interchange it with A i. Eventually A n is compared to the current minimum and we are done. Also, observe that when i becomes greater than q, A Hence, following the last execution of these lines, i.
We observe at this point that the upper limit of the for-loop in line 1 can be changed to n - 1 without damaging the correctness of the algorithm. From the standpoint of readability we can ask if this program is good. Is there a more concise way of describing this algorithm which will still be as easy to comprehend?
Substituting while statements for the for loops doesn't significantly change anything. Also, extra initialization and increment statements would be required. Let us develop another program. We assume that we have n 1 distinct integers which are already sorted and stored in the array A 1:n.
By making use of the fact that the set is sorted we conceive of the following efficient method: "let A mid be the middle element. There are three possibilities. Continue in this way by keeping two pointers, lower and upper, to indicate the range of elements not yet tested. This method is referred to as binary search. Note how at each stage the number of elements in the remaining set is decreased by about one half.
For instance we could replace the while loop by a repeat-until statement with the same English condition. In fact there are at least six different binary search programs that can be produced which are all correct. There are many more that we might produce which would be incorrect. Part of the freedom comes from the initialization step. Whichever version we choose, we must be sure we understand the relationships between the variables. Below is one complete version. This, combined with the above assertion implies that x is not present.
Unfortunately a complete proof takes us beyond our scope but for those who wish to pursue program proving they should consult our references at the end of this chapter. Recursion We have tried to emphasize the need to structure a program to make it easier to achieve the goals of readability and correctness.
Actually one of the most useful syntactical features for accomplishing this is the procedure. Given a set of instructions which perform a logical operation, perhaps a very complex and long operation, they can be grouped together as a procedure.
Given the input-output specifications of a procedure, we don't even have to know how the task is accomplished, only that it is available. This view of the procedure implies that it is invoked, executed and returns control to the appropriate place in the calling procedure. What this fails to stress is the fact that procedures may call themselves direct recursion before they are done or they may call other procedures which again invoke the calling procedure indirect recursion.
These recursive mechanisms are extremely powerful, but even more importantly, many times they can express an otherwise complex process very clearly.
For these reasons we introduce recursion here. Most students of computer science view recursion as a somewhat mystical technique which only is useful for some very special class of problems such as computing factorials or Ackermann's function.
This is unfortunate because any program that can be written using assignment, the if-then-else statement and the while statement can also be written using assignment, if-then-else and recursion. Of course, this does not say that the resulting program will necessarily be easier to understand. However, there are many instances when this will be the case. When is recursion an appropriate mechanism for algorithm exposition?
One instance is when the problem itself is recursively defined. Given a set of n 1 elements the problem is to print all possible permutations of this set. It is easy to see that given n elements there are n!
A simple algorithm can be achieved by looking at the case of four elements a,b,c,d. The answer is obtained by printing i a followed by all permutations of b,c,d ii b followed by all permutations of a,c,d iii c followed by all permutations of b,a,d iv d followed by all permutations of b,c,a The expression "followed by all permutations" is the clue to recursion. It implies that we can solve the problem for a set with n elements if we had an algorithm which worked on n - 1 elements. A is a character string e.
Then try to do one or more of the exercises at the end of this chapter which ask for recursive procedures. We will see several important examples of such structures, especially lists in section 4. Another instance when recursion is invaluable is when we want to describe a backtracking procedure.
But for now we will content ourselves with examining some simple, iterative programs and show how to eliminate the iteration statements and replace them by recursion. This may sound strange, but the objective is not to show that the result is simpler to understand nor more efficient to execute. The main purpose is to make one more familiar with the execution of a recursive procedure. Suppose we start with the sorting algorithm presented in this section. To rewrite it recursively the first thing we do is to remove the for loops and express the algorithm using assignment, if-then-else and the go-to statement.
Every place where a ''go to label'' appears, we replace that statement by a call of the procedure associated with that label. This gives us the following set of three procedures. Procedure MAXL2 is also directly reculsive. These two procedures use eleven lines while the original iterative version was expressed in nine lines; not much of a difference. Notice how in MAXL2 the fourth parameter k is being changed. The effect of increasing k by one and restarting the procedure has essentially the same effect as the for loop.
Now let us trace the action of these procedures as they sort a set of five integers When a procedure is invoked an implicit branch to its beginning is made. The parameter mechanism of the procedure is a form of assignment.
In section 4. Also in that section are several recursive procedures, followed in some cases by their iterative equivalents. Rules are also given there for eliminating recursion. There are many criteria upon which we can judge a program, for instance: i Does it do what we want it to do? The above criteria are all vitally important when it comes to writing software, most especially for large systems. Though we will not be discussing how to reach these goals, we will try to achieve them throughout this book with the programs we write.
Hopefully this more subtle approach will gradually infect your own program writing habits so that you will automatically strive to achieve these goals. There are other criteria for judging programs which have a more direct relationship to performance. These have to do with computing time and storage requirements of the algorithms.
Performance evaluation can be loosely divided into 2 major phases: a a priori estimates and b a posteriori testing. Both of these are equally important.
First consider a priori estimation. We would like to determine two numbers for this statement. The first is the amount of time a single execution will take; the second is the number of times it is executed. A complete set would include four cases: Below is a table which summarizes the frequency counts for the first three cases. Though 2 to n is only n.
None of them exercises the program very much.
Step Frequency Step Frequency 2 3 4 5 6 7 1 1 1 0 1 0 1 9 10 11 12 13 14 15 2 n n-1 n-1 n-1 n-1 1 file: We can summarize all of this with a table. At this point the for loop will actually be entered. Steps 1. These may have different execution counts. Both commands in step 9 are executed once. The for statement is really a combination of several statements. O n is called linear. Execution Count for Computing Fn Each statement is counted once.
This notation means that the order of magnitude is proportional to n. We will often write this as O n. For example n might be the number of inputs or the number of outputs or their sum or the magnitude of one of them.
O n2 is called quadratic. O n log n is better than O n2 but not as good as O n. The reason for this is that as n increases the time for the second algorithm will get far worse than the time for the first. O n3 is called cubic. We write O 1 to mean a computing time which is a constant. These seven computing times. If an algorithm takes time O log n it is faster. For example.
O log n. When we say that the computing time of an algorithm is O g n we mean that its execution takes no more than a constant times g n. If we have two algorithms which perform the same task. Often one can trade space for time. An algorithm which is exponential will work only for very small inputs. Another valid performance measure of an algorithm is the space it requires. Using big-oh notation. On the other hand. For exponential algorithms. Notice how the times O n and O n log n grow much more slowly than the others.
We will see cases of this in subsequent chapters. Given an algorithm. For small data sets. Then a performance profile can be gathered using real time calculation. In practice these constants depend on many factors.. This shows why we choose the algorithm with the smaller order of magnitude. For large data sets. Figures When n is odd H.
Coxeter has given a simple rule for generating a magic square: A magic square is an n x n matrix of the integers 1 to n2 such that the sum of every row. It emphasizes that the variables are thought of as pairs and are changed as a unit.
The statement i. The file: Academic Press. Fundamental Algorithms. Thus each statement within the while loop will be executed no more than n2. Special Issue: Since there are n2 positions in which the algorithm must place a number.
For a discussion of tools and procedures for developing very large software systems see Practical Strategies for Developing Large Software Systems. For a discussion of the more abstract formulation of data structures see "Toward an understanding of data structures" by J. ACM Computing Surveys.
Kernighan and P. The while loop is governed by the variable key which is an integer variable initialized to 2 and increased by one each time through the loop. For a discussion of good programming techniques see Structured Programming by O. The Elements of Programming Style by B. For this application it is convenient to number the rows and columns from zero to n. The magic square is represented using a two dimensional array having n rows and n column.
For a further discussion of program proving see file: Can you think of a clever meaning for S. Describe the flowchart in figure 1. American Mathematical Society. Can you do this without using the go to? Now make it into an algorithm.
Both do not satisfy one of the five criteria of an algorithm. Look up the word algorithm or its older form algorism in the dictionary.? Concentrate on the letter K first.
How would you handle people with the same last name. Discuss how you would actually represent the list of name and telephone number pairs in a real machine. Which criteria do they violate? Consider the two statements: For instance.. What is the computing time of your method?
Strings x and y remain unchanged. Determine when the second becomes larger than the first. String x is unchanged. The rule is: If x occurs Given n boolean variables x Try writing this without using the go to statement. Implement these procedures using the array facility. Determine how many times each statement is executed. List as many rules of style in programming that you can think of that you would be willing to follow yourself.
Represent your answer in the array ANS 1: NOT X:: Take any version of binary search. Prove by induction: Using the notation introduced at the end of section 1. Trace the action of the procedure below on the elements 2.
Write a recursive procedure for computing this function. Given n. If S is a set of n elements the powerset of S is the set of all possible subsets of S. Tower of Hanoi There are three towers and sixty four disks of different diameters placed on the first tower.
Write a recursive procedure for computing the binomial coefficient where. The pigeon hole principle states that if a function f has n distinct inputs but less than n distinct outputs then there exists two inputs a.
Then write a nonrecursive algorithm for computing Ackermann's function. Ackermann's function A m. Analyze the time and space requirements of your algorithm. Analyze the computing time of procedure SORT as given in section 1.
Write a recursive procedure to compute powerset S. Write a recursive procedure which prints the sequence of moves which accomplish this task. The disks are in order of decreasing diameter as one scans up the tower. Give an algorithm which finds the values a. Monks were reputedly supposed to move the disks from tower 1 to tower 3 obeying the rules: It is true that arrays are almost always implemented by using consecutive memory.
Using our notation this object can be defined as: If one asks a group of programmers to define an array. Therefore it deserves a significant amount of attention. In mathematical terms we call this a correspondence or a mapping. For arrays this means we are concerned with only two operations which retrieve and store values.
STORE is used to enter new index-value pairs. The array is often the only means for structuring data which is provided in a programming language.. For each index which is defined. This is unfortunate because it clearly reveals a common point of confusion. If we interpret the indices to be n-dimensional. If we consider an ordered list more abstractly. If we restrict the index values to be integers.
There are a variety of operations that are performed on these lists. Ace or the floors of a building basement. Notice how the axioms are independent of any representation scheme. In section 2. These operations include: This we will refer to as a sequential mapping.. It is only operations v and vi which require real effort. This gives us the ability to retrieve or modify the values of random elements in the list in a constant amount of time..
By "symbolic. It is precisely this overhead which leads us to consider nonsequential mappings of ordered lists into arrays in Chapter 4.
The problem calls for building a set of subroutines which allow for the manipulation of symbolic polynomials. We can access the list element values in either direction by changing the subscript values in a controlled way..
It is not always necessary to be able to perform all of these operations. In the study of data structures we are interested in ways of representing ordered lists so that these operations can be carried out efficiently. Let us jump right into a problem requiring ordered lists which we will solve by using one dimensional arrays.
Perhaps the most common way to represent an ordered list is by an array where we associate the list element ai with the array index i. This problem has become the classical example for motivating the use of list processing techniques which we will see in later chapters. Insertion and deletion using sequential allocation forces us to move some of the remaining elements so the sequential mapping is preserved in its proper form.
The first step is to consider how to define polynomials as a computer structure. However this is not an appropriate definition for our purposes.
A complete specification of the data structure polynomial is now given. For a mathematician a polynomial is a sum of terms where each term has the form axe. When defining a data object one must decide what functions will be available.
MULT poly. We will also need input and output routines and some suitable format for preparing polynomials as input. These assumptions are decisions of representation. Suppose we wish to remove from P those terms having exponent one. Then we would write REM P. Notice the absense of any assumptions about the order of exponents.
Now we can make some representation decisions. Note how trivial the addition and multiplication operations have become. These axioms are valuable in that they describe the meaning of each operation concisely and without implying an implementation. Now assuming a new function EXP poly exp which returns the leading exponent of poly.
EXP B file: COEF B. Exponents should be unique and in decreasing order is a very reasonable first decision. EXP B. This representation leads to very simple algorithms for addition and multiplication.. COEF A.
But are there any disadvantages to this representation? Hopefully you have already guessed the worst one. B REM B. With these insights. Since the tests within the case statement require two terms.
EXP A. The case statement determines how the exponents are related and performs the proper action.. We have avoided the need to explicitly store the exponent of each term and instead we can deduce its value by knowing our position in the list and the degree.
EXP B: C A end end insert any remaining terms in A or B into C The basic loop of this algorithm consists of merging the terms of the two polynomials. As for storage. But scheme 1 could be much more wasteful. It will require a vector of length Then for each term there are two entries representing an exponent-coefficient pair.
In general. In the worst case. Is this method any better than the first scheme? The first entry is the number of nonzero terms.
If all of A's coefficients are nonzero. Basic algorithms will need to be more complex because we must check each exponent before we handle its coefficient.. Suppose we take the polynomial A x above and keep only its nonzero coefficients. This is a practice you should adopt in your own coding.
It is natural to carry out this analysis in terms of m and n.
Comments appear to the right delimited by double slashes. Notice how closely the actual program matches with the original design.
The procedure has parameters which are polynomial or array names. Statement two is a shorthand way of writing r The basic iteration step is governed by a while loop.
Three pointers p. Blocks of statements are grouped together using square brackets.. The code is indented to reinforce readability and to reveal more clearly the scope of reserved words. The assignments of lines 1 and 2 are made only once and hence contribute O 1 to the overall computing time. We are making these four procedures available to any user who wants to manipulate polynomials.
Taking the sum of all of these steps. He would include these subroutines along with a main procedure he writes himself. Each iteration of this while loop requires O 1 time. This hypothetical user may have many polynomials he wants to compute and he may not know their sizes. If he declares the arrays too large. At each iteration.. Suppose in addition to PADD. In particular we now have the m lists a Instead we might store them in a one dimensional array and include a front i and rear i pointer for the beginning and end of each list..
Since the iteration terminates when either p or q exceeds 2m or 2n respectively.
Data Structures, Algorithms And Applications In C++
Consider the main routine our mythical user might write if he wanted to compute the Fibonacci polynomials. To make this problem more concrete. Returning to the abstract object--the ordered list--for a moment. This worst case is achieved. These are defined by the recurrence relation file: A two dimensional array could be a poor way to represent these lists because we would have to declare it as A m..
In this main program he needs to declare arrays for all of his polynomials which is reasonable and to declare the maximum size that every polynomial might achieve which is harder and less reasonable. This example shows the array as a useful representational form for ordered lists.
Then the following program is produced. For example F 2. Different types of data cannot be accommodated within the usual array concept. Then by storing all polynomials in a single array. For instance. Let's pursue the idea of storing all polynomials in a single array called POLY.
If we made a call to our addition routine. This example reveals other limitations of the array as a means for data representation.
Exponents and coefficients are really different sorts of numbers.. If the result has k terms. The array is usually a homogeneous collection of data which will not allow us to intermix data of different types. Also we need a pointer to tell us where the next free location is. A much greater saving could be achieved if Fi x were printed as soon as it was computed in the first loop..
Such a matrix has mn elements. There may be several such polynomials whose space can be reused. Even worse. As we create polynomials. As computer scientists. Such a matrix is called sparse. Now if we look at the second matrix of figure 2. In Chapter 4 we will see an elegant solution to these problems. A general matrix consists of m rows and n columns of numbers as in figure 2.
Each element of a matrix is uniquely characterized by its row and column position. It is very natural to store a matrix in a two dimensional array. Example of 2 matrices The first matrix has five rows and three columns. We could write a subroutine which would compact the remaining polynomials. We might then store a matrix as a list of 3-tuples of the form i. A sparse matrix requires us to consider an alternate form of representation.
When this happens must we quit? We must unless there are some polynomials which are no longer needed. Now we have localized all storage to one array. There is no precise definition of when a matrix is sparse and when it is not.
On most computers today it would be impossible to store a full X matrix in the memory at once. The alternative representation will explicitly store only the nonzero elements.
This comes about because in practice many of the matrices we want to deal with are large. This demands a sophisticated compacting routine coupled with a disciplined use of names for polynomials. Then we can work with any element by writing A i. But this may require much data movement. When m is equal to n.
Figure 2. Another way of saying this is that we are interchanging rows and columns. Sparse matrix stored as triples The elements A 0. Now what are some of the operations we might want to perform on these matrices? One operation is to compute the transpose matrix. We can go one step farther and require that all the 3-tuples of any row be stored so that the columns are increasing. The transpose of the example matrix looks like 1.
The elements on the diagonal will remain unchanged. This is where we move the elements so that the element in the i. In our example of figure 2. If we just place them consecutively. We can avoid this data movement by finding the elements in the order we want them. Since the rows are originally in order. Let us write out the algorithm in full. The total time for the algorithm is therefore O nt.
The variable q always gives us the position in B where the next term in the transpose is to be inserted. This computing time is a little disturbing since we know that in case the matrices had been represented as two dimensional arrays. It is not too difficult to see that the algorithm is correct. This is precisely what is being done in lines The algorithm for this takes the form: How about the computing time of this algorithm! For each iteration of the loop of lines The statement a.
On the first iteration of the for loop of lines all terms from column 1 of A are collected. In addition to the space needed for A and B. Since the number of iterations of the loop of lines is n. Since the rows of B are the columns of A. The terms in B are generated by rows. We now have a matrix transpose algorithm which we believe is correct and which has a computing time of O nt.
Lines take a constant amount of time. The assignment in lines takes place exactly t times as there are only t nonzero terms in the sparse matrix being generated. This gives us the number of elements in each row of B. We can now move the elements of A one by one into their correct position in B. This is worse than the O nm time using arrays. From this information.. This algorithm. Hence in this representation. Each iteration of the loops takes only a constant amount of time.
T j is maintained so that it is always the position in B where the next element in row j is to be inserted. This is the same as when two dimensional arrays were in use.
In lines the elements of A are examined one by one starting from the first and successively moving to the t-th element. The computation of S and T is carried out in lines When t is sufficiently small compared to its maximum of nm.
If we try the algorithm on the sparse matrix of figure 2. MACH m file: Suppose now you are working for a machine manufacturer who is using a computer to do inventory control. Associated with each machine that the company produces.
T i points to the position in the transpose where the next element of row i is to be stored. Regarding these tables as matrices this application leads to the general definition of matrix product: Given A and B where A is m element is defined as n and B is n p. The product of two sparse matrices may no longer be sparse.
This sum is more conveniently written as If we compute these sums for each machine and each micropart then we will have a total of mp values which we might store in a third table MACHSUM m.
Once the elements in row i of A and column j of B have been located. Each part is itself composed of smaller parts called microparts.. Before we write a matrix multiplication procedure. Consider an algorithm which computes the product of two sparse matrices represented as an ordered list instead of an array. To compute the elements of C row-wise so we can store them in their proper place without moving previously computed elements. Its i. We want to determine the number of microparts that are necessary to make up each machine.
To avoid this. An alternative approach is explored in the exercises. It makes use of variables i. In each iteration of the while loop of lines either the value of i or j or of both increases by 1 or i and col are reset.
Let us examine its complexity. The variable r is the row of A that is currently being multiplied with the columns of B. The total maximum increments in i is therefore pdr. This enables us to handle end conditions i. We leave the correctness proof of this algorithm as an exercise. When this happens. If dr is the number of terms in row r of A then the value of i can increase at most dr times before i moves to the next row of A. The maximum number of iterations of the while loop of lines file: The maximum total increment in j over the whole loop is t2.
In addition to all this. In addition to the space needed for A. The while loop of lines is executed at most m times once for each row of A. At the same time col is advanced to the next column.
C and some simple variables. Once again. A and B are sparse.. Since t1 nm and t2 np. The classical multiplication algorithm is: As in the case of polynomials. There are. It introduces some new concepts in algorithm analysis and you should make sure you understand the analysis. Since the number of terms in a sparse matrix is variable. This would enable us to make efficient utilization of space.
MMULT will outperform the above multiplication algorithm for arrays. Lines take only O dr time. MMULT will be slower by a constant factor.
If an array is declared A l1: These difficulties also arise with the polynomial representation of the previous section and will become apparent when we study a similar representation for multiple stacks and queues section 3. This is necessary since programs using arrays may. We see that the subscript at the right moves the fastest. In addition to being able to retrieve array elements easily.
If we have the declaration A 4: While many representations might seem plausible. Then using row major order these elements will A 4. Recall that memory may be regarded as one dimensional with words numbered from 1 to m. Assuming that each array element requires only one word of memory. Knowing the address of A i. Sequential representation of A u1. To simplify the discussion we shall assume that the lower bounds on each dimension li are 1.
These two addresses are easy to guess.. In a row major representation.. Before obtaining a formula for the case of an n-dimensional array. Sequential representation of A 1: This formula makes use of only the starting address of the array plus the declared dimensions. Then A 4. From the compiler's point of view. Suppose A 4.
Another synonym for row major order is lexicographic order. The general case when li can be any integer is discussed in the exercises. To begin with.. Each 2-dimensional array is represented as in Figure To review.. Repeating in this way the address for A i1. From this and the formula for addressing a 2 dimensional array. In all cases we have been able to move the values around To locate A i. However several problems have been raised By using a sequential mapping which associates ai of a An alternative scheme for array representation..
The address of A i If is the address for A The address for A i Assume that n lists. For these polynomials determine the exact number of times each statement will be executed.
What can you say about the existence of an even faster algorithm? How much space is actually needed to hold the Fibonacci polynomials F The functions to be performed on these lists are insertion and deletion.
The i-th list should be maintained as sequentially stored. Assume you can compare atoms ai and bj. Try to minimize the number of operations. What is the computing time of your procedure? Write a procedure which returns What is the computing time of your algorithm? The band includes a. What is the relationship between i and j for elements in the zero part of A?
For large n it would be worthwhile to save the space taken by the zero entries in the upper triangle. Define a square band matrix An. Obtain an addressing formula for elements aij in the lower triangle if this lower triangle is stored by rows in an array B 1: When all the elements either above or below the main diagonal of a square matrix are zero.
In this square matrix. Let A and B be two lower triangular matrices. Tridiagonal matrix A If the elements in the band formed by these three diagonals are represented rowwise in an array. Devise a scheme to represent both the triangles in an array C 1: Another kind of sparse matrix that arises often in numerical analysis is the tridiagonal matrix. How much time does it take to locate an arbitrary element A i. A generalized band matrix An. A variation of the scheme discussed in section 2. Thus A B which determines the value of element aij in the matrix An.
The band of An. B where A and B contain real values. Consider space and time requirements for such operations as random access. Use a minimal amount of storage. Obtain an addressing formula for the element A i1.
In this representation. Do exercise 20 assuming a column major representation. Assume a row major representation of the array with one word per element and l A complex-valued matrix X is represented by a pair of matrices A.. The figure below illustrates the representation for the sparse matrix of figure 2. In addition. Given an array A 1: An m X n matrix is said to have a saddle point if some entry A i.. Write a program which computes the product of two complex valued matrices A.
How many values can be held by an array with dimensions A 0: How much time does your algorithm take? One possible set of axioms for an ordered list comes from the six operations of section 2. Assuming that he may move from his present tile to any of the eight tiles surrounding him unless he is against a wall with equal probability.
All but the most simple of these are extremely difficult to solve and for the most part they remain largely unsolved. The problem may be simulated using the following method: One such problem may be stated as follows: A drunken cockroach is placed on a given square in the middle of a tile floor in a rectangular room of size n x m tiles.
The position of the bug on the floor is represented by the coordinates IBUG. JBUG and is initialized by a data card. There are a number of problems. All the cells of this array are initialized to zero. The bug wanders possibly in search of an aspirin randomly from tile to tile throughout the room.
The technique for doing so is called "simulation" and is of wide-scale use in industry to predict traffic-flow. Hard as this problem may be to solve by pure probability theory techniques.
When every square has been entered at least once. Have an aspirin This exercise was contributed by Olson. This assures that your program does not get "hung" in an "infinite" loop.
Each time a square is entered. Of course the bug cannot move outside the room. Many of these are based on the strange "L-shaped" move of the knight. A maximum of This will show the "density" of the walk. Chess provides the setting for many fascinating diversions which are quite independent of the game itself.
Your program MUST: Write a program to perform the specified simulation experiment. A classical example is the problem of the knight's tour.
It is convenient to represent a solution by placing the numbers 1. The goal of this exercise is to write a computer program to implement Warnsdorff's rule. The most important decisions to be made in solving a problem of this type are those concerning how the data is to be represented in the computer.
One of the more ingenious methods for solving the problem of the knight's tour is that given by J. Briefly stated. The eight possible moves of a knight on square 5. Perhaps the most natural way to represent the chessboard is by an 8 x 8 array B ARD as shown in the figure below. Note that it is not required that the knight be able to reach the initial position by one more move.
His rule is that the knight must always be moved to one of the squares from which there are the fewest exits to squares not already traversed. Warnsdorff in The ensuing discussion will be much easier to follow. Let NP S be the number of possibilities. The data representation discussed in the previous section is assumed.
J may move to one of the squares I. That is. J is located near one of the edges of the board. Below is a description of an algorithm for solving the knight's tour problem using Warnsdorff's rule. Recall that a square is an exit if it lies on the chessboard and has not been previously occupied by the knight.
If this happens. J denotes the new position of the knight. In every case we will have 0 NP S 8. The problem is to write a program which corresponds to this algorithm.. Go to Chapter 3 Back to Table of Contents file: This exercise was contributed by Legenhausen and Rebman. J records the move in proper sequence. A queue is an ordered list in which all insertions take place at one end..
Both these data objects are special cases of the more general data object. Figure 3. A stack is an ordered list in which all insertions and deletions are made at one end.
Suppose we have a main procedure and three subroutines as below: E are added to the stack. Equivalently we say that the last"element to be inserted into the stack will be the first to be removed. The ai are referred to as atoms which are taken from some set. They arise so often that we will discuss them separately before moving on to more complex objects. Thus A is the first letter to be removed. The restrictions on a queue require that the first element which is inserted into the queue will be the first one to be removed.
One natural example of stacks which arises in computer programming is the processing of subroutine calls and their returns. The first entry. This list operates as a stack since the returns will be made in the reverse order of the calls. For each subroutine there is usually a single location associated with the machine code which is used to retain the return address. If we examine the memory while A3 is computing there will be an implicit stack which looks like q.
This list of return addresses need not be maintained in consecutive locations. In each case the calling procedure passes the return address to the called procedure. Al then invokes A2 which in turn calls A3.
Since returns are made in the reverse order of calls. ADD i. Associated with the object stack there are several operations that are necessary: When recursion is allowed. A2 before A1.
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Whenever a return is made. Thus t is removed before s. S which inserts the element i onto the stack S and returns the new stack. Implementing recursion using a stack is discussed in Section This can be severely limiting in the case of recursive calls and re-entrant routines. The address r is passed to Al which saves it in some location for later processing.
S file: TOP S which returns the top element of stack S.The second quarter starts with chapter seven which provides an excellent survey of the techniques which were covered in the previous quarter. The product of two sparse matrices may no longer be sparse. As computer scientists. In addition to the assignment statement. The alternative representation will explicitly store only the nonzero elements.
Another approach would be to define a hypothetical machine with imaginary execution times. This gives us the number of elements in each row of B. This list operates as a stack since the returns will be made in the reverse order of the calls.
Another iteration statement is loop S forever which has the meaning As it stands.
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