C Data Structures And Algorithms Pdf
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- Introduction to Data Structures and Algorithms with C
- Table of contents
- Data Structures And Algorithms In C Lecture Notes Pdf
Introduction to Data Structures and Algorithms with C
Both of the solutions proposed require considerable time to solve this problem and therefore are not acceptable. However, it is possible, even with a large word list, to solve the problem in a matter of seconds.
An important concept is that, in many problems, writing a working program is not good enough. If the program is to be run on a large data set, then the running time becomes an issue.
Throughout this book we will see how to estimate the running time of a program for large inputs and, more importantly, how to compare the running times of two programs without actually coding them. We will see techniques for drastically improving the speed of a program and for determining program bottlenecks. These techniques will enable us to find the section of the code on which to concentrate our optimization efforts.
Mathematics ReviewThis section lists some of the basic formulas you need to memorize or be able to derive and reviews basic proof techniques. Some other useful formulas, which can all be derived in a similar manner, follow. These are the "geometric series" formulas. We can derive the last formula for in the following manner. Let S be the sum. Another type of common series in analysis is the arithmetic series. Any such series can be evaluated from the basic formula.
The next two formulas pop up now and then but are fairly infrequent. We then need the following formula, which is used far more in computer science than in other mathematical disciplines. The numbers, H N , are known as the harmonic numbers, and the sum is known as a harmonic sum. The error in the following approximation tends to y 0. These two formulas are just general algebraic manipulations. Modular ArithmeticWe say that a is congruent to b modulo n, written a b mod n , if n divides a -b.
Intuitively, this means that the remainder is the same when either a or b is divided by n. Thus, 81 61 1 mod There are a lot of theorems that apply to modular arithmetic, some of which require extraordinary proofs in number theory.
We will use modular arithmetic sparingly, and the preceding theorems will suffice. The P WordThe two most common ways of proving statements in data structure analysis are proof by induction and proof by contradiction and occasionally a proof by intimidation, by professors only. The best way of proving that a theorem is false is by exhibiting a counterexample.
Proof by InductionA proof by induction has two standard parts. The first step is proving a base case, that is, establishing that a theorem is true for some small usually degenerate value s ; this step is almost always trivial.
Next, an inductive hypothesis is assumed. Generally this means that the theorem is assumed to be true for all cases up to some limit k. This proves the theorem as long as k is finite. To do this, we first verify that the theorem is true for the trivial cases.
As a second example, we establish the following theorem. For the inductive hypothesis, assume that the theorem is true for 1 k n. We haveApplying the inductive hypothesis, we obtain Thus, proving the theorem. Proof by CounterexampleThe statement F k k 2 is false.
Proof by ContradictionProof by contradiction proceeds by assuming that the theorem is false and showing that this assumption implies that some known property is false, and hence the original assumption was erroneous. A classic example is the proof that there is an infinite number of primes. To prove this, we assume that the theorem is false, so that there is some largest prime p k. However, none of p 1 , p 2 ,. This is a contradiction, because every number is either prime or a product of primes.
Hence, the original assumption, that p k is the largest prime, is false, which implies that the theorem is true. A Brief Introduction to RecursionMost mathematical functions that we are familiar with are described by a simple formula.
Mathematical functions are sometimes defined in a less standard form. A function that is defined in terms of itself is called recursive. C allows functions to be recursive. Not all mathematically recursive functions are efficiently or correctly implemented by C's simulation of recursion.
The idea is that the recursive function f ought to be expressible in only a few lines, just like a non-recursive function. Figure 1. Lines 1 and 2 handle what is known as the base case, that is, the value for which the function is directly known without resorting to recursion. Line 3 makes the recursive call. There are several important and possibly confusing points about recursion. A common question is: Isn't this just circular logic? The answer is that although we are defining a function in terms of itself, we are not defining a particular instance of the function in terms of itself.
In other words, evaluating f 5 by computing f 5 would be circular. Evaluating f 5 by computing f 4 is not circular--unless, of course f 4 is evaluated by eventually computing f 5.
The two most important issues are probably the how and why questions. In Chapter 3, the how and why issues are formally resolved.
We will give an incomplete description here. It turns out that recursive calls are handled no differently from any others. Thus, a call is made to compute f 3. Therefore, another call is made to compute f 2.
Now, f 0 must be evaluated. This enables the completion of the calculation for f 1 , which is now seen to be 1. Then f 2 , f 3 , and finally f 4 can be determined. All the bookkeeping needed to keep track of pending function calls those started but waiting for a recursive call to complete , along with their variables, is done by the computer automatically.
An important point, however, is that recursive calls will keep on being made until a base case is reached. For instance, an attempt to evaluate f -1 will result in calls to f -2 , f -3 , and so on. Since this will never get to a base case, the program won't be able to compute the answer which is undefined anyway.
Occasionally, a much more subtle error is made, which is exhibited in Figure 1. The error in the program in Figure 1. Obviously, this doesn't give any clue as to what bad 1 actually is. The computer will thus repeatedly make calls to bad 1 in an attempt to resolve its values. Eventually, its bookkeeping system will run out of space, and the program will crash. Generally, we would say that this function doesn't work for one special case but is correct otherwise.
This isn't true here, since bad 2 calls bad 1. Thus, bad 2 cannot be evaluated either. Furthermore, bad 3 , bad 4 , and bad 5 all make calls to bad 2. Since bad 2 is unevaluable, none of these values are either. In fact, this program doesn't work for any value of n, except 0. With recursive programs, there is no such thing as a "special case.
Base cases. You must always have some base cases, which can be solved without recursion. Making progress. For the cases that are to be solved recursively, the recursive call must always be to a case that makes progress toward a base case.
Throughout this book, we will use recursion to solve problems. As an example of a nonmathematical use, consider a large dictionary. Words in dictionaries are defined in terms of other words.
When we look up a word, we might not always understand the definition, so we might have to look up words in the definition. Likewise, we might not understand some of those, so we might have to continue this search for a while.
Table of contents
If you wish, you can read through a seven-page course description. A page topic summary is also available: Algorithms and data structures—topic summary. This is a collection of PowerPoint pptx slides "pptx" presenting a course in algorithms and data structures. Associated with many of the topics are a collection of notes "pdf". Some presentations may be associated with videos "V" and homework questions "Q" , possibly with answers "A". You will note that the section numbering in the notes is paralleled in the top left corner of the slides; thus, anyone watching the slides can follow along in the notes.
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Data Structures And Algorithms In C Lecture Notes Pdf
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