Éva Tardos. Cornell University. Spring c Jon Kleinberg andÉva Tardos Much of the course is concerned with techniques for designing algorithms, and. Algorithm Design - John Kleinberg - Éva teshimaryokan.info - Ebook download as PDF File .pdf), Text File .txt) or read book online. best instructor manual. These are a revised version of the lecture slides that accompany the textbook Algorithm Design by Jon Kleinberg and Éva Tardos. Here are the original and.

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Kleinberg, Jon. Algorithm design / Jon Kleinberg, Éva Tardos.—1st ed. p. cm. Includes bibliographical references and index. ISBN (alk. paper). 1. Kleinberg, Jon. Algorithm design / Jon Kleinberg, l~va Tardoslst ed. p. cm. Includes bibliographical references and index. ISBN (alk. paper). 1. Algorithm Analysis. Contribute to davie/CSAlgorithm-Analysis development by creating an account on GitHub.

So the main thing we need to show is the following. Venu Ramasubramanian. But there are only 2n elements total. One way to get an approximate sense of how fast logb n grows is to note that. Priority queues support the addition and deletion of elements from the set.

Problems and Solved Exercises An important feature of the book is the collection of problems. Across all chapters, the book includes over problems, almost a! We view the problems as a crucial component of the book, and they are structured in keeping with our overall approach to the material. Most of them consist of extended verbal descriptions of a problem arising in an application area in computer science or elsewhere out in the world, and part of the problem is to practice what we discuss in the text: We view a complete answer to one of these problems as consisting of all these components: To help with the process of working on these problems, we include in each chapter a section entitled "Solved Exercises," where we take one or more problems and describe how to go about formulating a solution.

The discussion devoted to each solved exercise is therefore significantly longer than what would be needed simply to write a complete, correct solution in other words,. This material can thus be treated either as a review or as new material; by including it, we hope the book can be used in a broader array of courses, and with more flexibility in the prerequisite knowiedge that is assumed. In keeping with the approach outlined above, we develop the basic algorithm design techniques by drawing on problems from across many areas of computer science and related fields.

To mention a few representative examples here, we include fairly detailed discussions of applications from systems and networks caching, switching, interdomain routing on the Internet , artificial.

Pedagogical Features and Supplements In addition to the Problems and solved exercises. It begins with an informal overview of what it means for a problem to be computationally tractable. It then discusses growth rates of functions and asymptotic analysis more formally. In cases where extensions to the problem or further analysis of the algorithm is pursued.

The remainder of Chapter 1 discusses a list of five "representative problems" that foreshadow topics from the remainder of the course. As noted earlier. In particular.

We begin immediately with the Stable Matching Problem. To reflect this style. Chapter-by-Chapter Synopsis Chapter I starts by introducing some representative algorithmic problems.

The fact that closely related problems can vary greatly in complexity is an important theme of the book. A number of supplements are available in support of the book itself. For instructions on obtaining a professor. These files are available at wunv. A set of lecture slides. In our undergraduate class. Comments and reports of errors can be sent to us by e-mail. These subsections are highlighted in the text with an icon depicting a feather.

It is worth mentioning two points concerning the use of these problems as homework in a course. Chapter 3 covers the basic definitions and algorithmic primitives needed for working with graphs. The goal of this structure is to offer a relatively uniform style of presentation that moves from the initial discussion of a problem arising in a computing application through to the detailed analysis of a method to solve it.

Chapter 2 introduces the key mathematical definitions and notations used for analyzing algorithms. Our approach to data structures is to introduce them as they are needed for the implementation of the algorithms being developed in the book. To the extent that network flow is covered in algorithms courses. When we use the book at the undergraduate level. Our chapter on approximation algorithms discusses both the process of designing effective algorithms and the task of understanding the optimal solution well enough to obtain good bounds on it.

Next we develop dynamic programming by starting with the recursive intuition behind it. We also consider types of computational hardness beyond NPcompleteness. Chapters 8 and 9 cover computational intractability. This chapter concludes with some of the main applications of greedy algorithms.

This chapter concludes with extended discussions of the dynamic programming approach to two fundamental problems: Preface Preface Chapters 2 and 3 also present many of the basic data structures that will be used for implementing algorithms throughout the book. We devote most of our attention to NP-completeness.

Chapters 4 through 7 cover four major algorithm design techniques: As design techniques for approximation algorithms. Use of the Book The book is primarily designed for use in a first undergraduate course on algorithms. Our goal here is to provide a more compact introduction to some of the ways in which students can apply randomized techniques using the kind of background in probability one typically gains from an undergraduate discrete math course. This is a topic on which several nice graduate-level books have been written.

Our chapter on tractable special cases emphasizes that instances of NP-complete problems arising in practice may not be nearly as hard as worst-case instances. We build up to some fairly complex proofs of NPcompleteness. Chapter 13 covers the use of randomization in the design of algorithms.

This topic is often missing from undergraduate algorithms courses. Chapters 10 through 12 cover three maior techniques for dealing with computationally intractable problems: For divide and conquer. We illustrate how NP-complete problems are often efficiently solvable when restricted to tree-structured inputs. While this topic is more suitable for a graduate course than for an undergraduate one.

With greedy algorithms. Aravind Srinivasan. Leonid Meyerguz. Siddharth Alexander. Joe Polastre. Steve Baker.

Chapter Mike Priscott. We thank our undergraduate and graduate teaching assistants. We skip the starred sections. Shan-Leung Maverick Woo. Our own undergraduate course involves material from all these chapters.

Nadya Travinin. Tina Nolte. Sections Shanghua Teng. Justin Yang. Many of them have provided valuable insights. Matthew Wachs. Ara Hayrapetyan. John Bicket. Amit Kumar. The book also naturally supports an introductory graduate course on algorithms. We also tend to skip one or two other sections per chapter in the first half of the book for example.

Jon Peress. Sebastian Sllgardo. David Richardson. Bill McCloskey. Sasha Evfimievski. Dexter Kozen.! This last point is worth emphasizing: Mike Connor. Dieter van Melkebeek. Yeongwee Lee. John Hopcroft. Dexter Kozen. Ralph Benzinger. Tim Roughgarden. Preface Preface might be able to use particular algorithm design techniques in the context of their own work.

Lars Backstrom. Yuval Rabani. Although our focus in an introductory graduate course is on the more advanced sections. Monika Henzinger. For this type of course. Alexander Druyan. Allan Borodin. Paul Beame. Venu Ramasubramanian. Tom Wexler. Gene Kleinberg. Travis Ortogero.

Igor Kats. Matt Piotrowski. Brian Kulis. Chris Jeuell. Here we find the emphasis on formulating problems to be useful as well. Ashwin Machanavajjhala. Henry Lin. Rachit Siamwalla. Mike Molloy. Shaddin Doghmi. Chapters excluding 4.

Xin Qi. Rie Ando. We cover roughly half of each of Chapters The resulting syllabus looks roughly as follows: Chapter 1. Doug Burdick. A number of graduate students and colleagues have used portions of the book in this way. Ayan Mandal. Chaitanya Swamy. Kevin Wayne. Alex Slivkins. We also thank all the students in these classes who have provided comments and feedback on early drafts of the book over the years.

Vladimir Dizhoor. Chapter 9 briefly. Brian Sabino. Our view of such a course is that it should introduce students destined for research in all different areas to the important current themes in algorithm design.

Bowei Du. For the past several years. Devdatt Dubhashi. These courses have grown. David Kempe. Ronitt Rubinfeld. Elliot Anshelevich. Sergei Vassilvitskii. Niranjan Nagarajan. Evan Moran. Alexa Sharp. Alexei Kopylov.

More generally. Aditya Rao. Perry Tam. It was probably iust in our imaginations. Jon Kleinberg gva Tardos Ithaca. Many of these contributions have undoubtedly escaped our notice. Bart Selman. Duncan Watts. Lillian Lee. Dan Huttenlocher. Mark Newman. Baltimore County. Edgar Ramos University of Illinois. We thank Anselm Blumer Tufts University. It has been a pleasure working with Addison Wesley over the past year. Leon Reznik Rochester Institute of Technology. Maite Suarez-Rivas at Addison Wesley.

Dieter van Melkebeek University of Wisconsin. Stephan Olariu Old Dominion University. First and foremost. Sanjeev Khanna University of Pennsylvania. Sariel Har-Peled University of Illinois. Sanjay Ranka University of Florida. David Shmoys. Patty Mahtani. Ted Laux for the indexing. Marilyn Lloyd. Nancy Murphy of Dartmouth Publishing for her work on the figures. Prabhakar Raghavan.

We thank Matt and Susan.

This book was begun amid the irrational exuberance of the late nineties. We appreciate their support. We deeply appreciate their input and advice. Evie Kleinberg. Richard Chang University of Maryland. Philip Klein Brown University. We thank Joyce Wells for the cover design. We would like to additionally thank Kevin Wayne for producing supplementary material associated with the book.

Kevin Compton University of Michigan. And so to all students of the subject. Bobby Kleinberg. David Matthias Ohio State University. Ron Elber. Our early conversations about the book with Susan Hartman were extremely valuable as well.

In a number of other cases. David McAllester. Diane Cook University of Texas. We fln-ther thank Paul and Jacqui for their expert composition of the book.

Michael Mitzenmacher Harvard University. Olga Veksler. Each applicant has a preference ordering on companies. The crux of the application process is the interplay between two different types of parties: The problem itself--the Stable Matching Problem--has several origins.

The algorithm to solve the problem is very clean as well. Stable Matching As an opening topic. Could one design a college admissions process. What did they mean by this?

To set up the question. It is motivated by some very natural and practical concerns. Based on these preferences. In this case. If this holds. The world of companies and applicants contains some distracting asymmetries. CluNet offers a job to one of its wait-listed applicants. And from the point of view of this book. Before doing this. We might well prefer the following. Gale and Shapley proceeded to develop a striking algorithmic solution to this problem.

But in this case. It turns out that for a decade before the work of Gale and Shapley. Matching Gale and Shapley considered the sorts of things that could start going wrong with this process. Suddenly down one summer intern. Rai actually prefers WebExodus to CluNet--won over perhaps by the laid-back. Following Gale and Shapley. Consider another student. What has gone wrong? One basic problem is that the process is not self-enforcing--if people are allowed to act in their self-interest.

Things look just as bad. It is useful. A few days later. Situations like this can rapidly generate a lot of chaos. Formulating the Problem To get at the essence of this concept. A First Problem: We will see that doing this preserves the fundamental issues inherent in the problem.

This is motivated by related applications. So this is the question Gale and Shapley asked: Given a set of preferences among employers and applicants. Each applicant is looking for a single company. Some Representative Problems 1. Guided by our initial motivation in terms of employers and applicants. A matching S is a set of ordered pairs. We will refer to the ordered ranking of m as his preference list. On the other hand. Given a perfect matching S.

There is a unique stable matching here. Can we declare immediately that m. There are two pairs m. Some Examples To illustrate these definitions.

The preference lists are as follows: Both m and Iv would want to leave their respective partners and pair up.. Iv would form an instability with respect to this matching. The other perfect matching. We wil! Let us consider some of the basic ideas that.. Let M x W denote the set of all possible ordered pairs of the form m.

Iv wii1 be one of the pairs in our final stable matching? Not necessarily: Suppose an unmarried man m chooses the woman Iv who ranks highest on his preference list and proposes to her. Each woman.. Two questions spring immediately to mind: Does there exist a stable matching for every set of preference lists? Given a set of preference lists.

Now we can add the notion of preferences to this setting. If we think about this set of preference lists intuitively. Some Representative Problems!. Matchings and perfect matchings are objects that will recur frequently throughout the book. Figure 1. The matching consisting of the pairs m. Iv and m. In this second example. In the present situation. Our goal.. Here is a concrete description of the Gale-Shapley algorithm. For a while. He is free until he proposes to the highest-ranked woman on his list.

An arbitrary flee man m chooses the highest-ranked woman w to whom he has not yet proposed. As time goes on. In the case of the present algorithm. A useful strategy for upper-bounding the running time of an algorithm. If w is also free. Then a man m may propose to her. Now we show that the algorithm terminates. We proceed to prove this now. But there are only n2 possible pairs of men and women in total. So a natural idea would be to have the pair m.

So we discover the following. The view of a man m during the execution of the algorithm is rather different. It follows that there can be at most n2 iterations. The next step could look like this. Then by 1. Now we ask: At termination. Let us suppose that the algorithm terminates with a flee man m.!

But there are only n men total. So the main thing we need to show is the following. This example shows a certain "unfairness" in the G-S algorithm. And in larger examples. Why is this not immediately obvious? Suppose there comes a point when m is flee but has already proposed to every woman. If he did. We have already seen. We now consider some further questions about the behavior of the G-S algorithm and its relation to the properties of different stable matchings. Since the set of engaged pairs forms a matching.

It follows that S is a stable matching. Stable Matching 9 iteration. For example. As defined earlier. So this simple set of preference lists compactly summarizes a world in which someone is destined to end up unhappy: The set S is a stable matching.

The set of engaged pairs always forms a matching. To recap. Let us now establish that the set S returned at the termination of the algorithm is in fact a perfect matching.

But this contradicts 1. Let us suppose. Recall that this is true if all men prefer different women. Different choices specify different executions of the algprithm. Stable Matching 11 To begin With. We will say that iv is the best valid partner of m if iv is a valid parmer of m. For a woman w.

Despite all this. This statement is surprising at a number of levels. In the present context. Since iv is a valid parmer of m. But either way. We will prove the folloWing fact. We will use best m to denote the best valid partner of m. All Executions Yield the Same Matching There are a number of possible ways to prove a statement such as this.

So consider the first moment during the execution g in which some man. It turns out that the easiest and most informative approach for us will be to uniquely characterize the matching that is obtained and then show that al!

After all. Suppose there were a pair m. First of all. Do all executions of the G-S algorithm yield the same matching?

This is a genre of question that arises in many settings in computer science: We say that m is the ivorst valid partner of iv if m is a valid partner of w. What is the characterization? Since men propose in decreasing order of preference. Chapter 4 Greedy Algorithms. Chapter 5 Divide and Conquer. Chapter 6 Dynamic Programming.

Chapter 7 Network Flow. Chapter 8 NP and Computational Intractability. A Class of Problems beyond NP. Chapter 10 Extending the Limits of Tractability. Chapter 11 Approximation Algorithms. Chapter 12 Local Search. Uploaded by SlekPanther on January 1, Search the history of over billion web pages on the Internet.

There are n input wires and rt output wires. Some of your friends are working for CluNet.. The basic question is: Can a man or a woman end up better off by lying about his or her preferences? More concretely. Show that for any specified pattern in which the input wires and output wires meet each other each pair meeting exactly once. It does not matter. Here is the setup. Furthermore--and this is the trick3. For this problem. And similarly for the orders in which output wires meet input wires.

A valid solution is to switch the data stream of Input 1 onto Output 2. Each input wire meets each output. We can ask the same question for men. Suppose we have two ships and two ports. Input 2 has its junction with Output 1 upstream from its junction with Output 2.

Some Representative Problems Exercises 27 Example. Input 1 has its junction with Output 2 upstream from its junction with Output 1. If the stream of Input i is switched onto Output j. Now consider a woman w. Gusfield and Irving also provide a nice survey of the "paralle!

Stable matching has grown into an area of study in its own right. We will look for paradigmatic problems and approaches that illustrate.

We then develop running-time bounds for some basic algorithms. We will discuss the problems in these contexts later in the book. We begin this chapter by talking about how to put this notion on a concrete footing. Having done this. At this level of generality. As discussed in the chapter. Some Representative Problems b Give an example of a set of preference lists for which there is a switch that-Would improve the partner of a woman who switched preferences.

In some cases. At the same time. Before focusing on any specific consequences of this claim. Some Initial Attempts at Defining Efficiency The first major question we need to answer is the following: How should we turn the fuzzy notion of an "efficient" algorithm into something more concrete? A first attempt at a working definition of efficiency is the following.

The focus on worst-case performance initially seems quite draconian: This certainly is an issue in some cases. Even bad algorithms can run quickly when applied to small test cases on extremely fast processors. But there are some crucial things missing from this definition. The first is the omission of where. But what is a reasonable analytical benchmark that can tell us whether a running-time bound is impressive or weak?

A first simple guide. At a certain leve!. In considering the problem. Another property shared by many of the problems we study is their fundamentally discrete nature. So what we could ask for is a concrete definition of efficiency that is platform-independent. Proposed Definition of Efficiency 1: An algorithm is efficient if. As we seek to understand the general notion of computational efficiency.

But it is important that algorithms be efficient in their use of other resources as well. Average-case analysis--the obvious appealing alternative. A common situation is that two very different algorithms will perform comparably on inputs of size N is closely related to the other natural parameter in this problem: We can use the Stable Matching Problem as an example to guide us. The input has a natural "size" parameter N. As we observed earlier. And indeed.

Since there are 2n preference lists. In any case. Since d is a constant. Not only is this approach almost always too slow to be useful. Where our previous definition seemed overly vague. The answers are.

Suppose an algorithm has the following property: Even when the size of a Stable Matching input instance is relatively small. What do we mean by "qualitatively better performance? This will be a common theme in most of the problems we study: This was a conclusion we reached at an analytical level. There are certainly exceptions to this principle in both directions: Problems for which polynomial-time algorithms exist almost invariably turn out to have algorithms with running times proportional to very moderately growing polynomials like n.

Proposed Definition of Efficiency 2: An algorithm is efficient if it achieves qualitatively better worst-case performance. The surprising punchline. It really works. For now. This will turn out to be a very usefu! From this notion. Search spaces for natural combinatorial problems tend to grow exponentially in the size N of the input.

We did not implement the algorithm and try it out on sample preference lists. The natural "brute-force" algorithm for this problem would plow through all perfect matchings by enumeration.

If the input size increases from N to 2N. Proposed Definition of Efficiency 3 " An algorithm is efficient if it has a polynomial running time. But if there is a problem with our second working definition. For most of these problems. And so if there is a common thread in the algorithms we emphasize in this book. Algorithms that improve substantially on brute-force search nearly always contain a valuable heuristic idea that makes them work.

One further reason why the mathematical formalism and the empirical evidence seem to line up well in the case of polynomial-time solvability is that the gulf between the growth rates of polynomial and exponential functions is enormous.

At times we will need to become more formal. We will mainly express algorithms in the pseudo-code style that we used for the Gale-Shapley algorithm. When we provide a bound on the running time of an algorithm. When we seek to say something about the running time of an algorithm on inputs of size n. All this serves to reinforce the point that our emphasis on worst-case.

Each one of these steps will typically unfold into some fixed number of primitive steps when the program is compiled into which an algorithm with exponential worst-case behavior generally runs well on the kinds of instances that arise in practice.

As just discussed. The function f n then becomes a bound on the rtmning time of the algorithm. In Table 2. We now discuss a framework for talking about this concept. In contrast. There is a final. In a sense. But overwhelmingly. Our definition in terms of polynomial time is much more an absolute notion. It becomes possible to express the notion that there is no efficient algorithm for a particular problem. Note that O. As an example of how this definition lets us express upper bounds on running times.

There was nothing wrong with the first result. There is a notation to express this: Often when we analyze an algorithm--say we have just proven that its worst-case running time T n is O n2 --we want to show that this upper bound is the best one possible. More precisely. There are cases which meets what is required by the definition of f2. Asymptotic Lower Bounds There is a complementary notation for lower bounds. This inequality is exactly what the definition of O - requires: It is not hard to do this.

So the most we can safely say is that as we look at different levels of computational abstraction. To see why. This definition works just like 0. T n grows exactly like [ n to within a constant factor. It is important to note that this definition requires a constant c to exist that works for all n. The fact that a function can have many upper bounds is not just a trick of the notation. Given another function f n. By analogy with O - notation. We now discuss a precise way to do this.

Whereas establishing the upper bound involved "inflating" the terms in T n until it looked like a constant times n2. Asymptotic Upper Bounds Let T n be a function--say.

In this example. Just as we discussed the notion of "tighter" and "weaker" upper bounds. Asymptotically tight bounds on worst-case running times are nice things to find. Since the overall running time is a sum of two functions the running times of. Sometimes one can also obtain an asymptotically tight bound directly by computing a limit as n goes to infinity. Transitivity A first property is transitivity: Thus we have shown 2.

Sums of Functions It is also useful to have results that quantify the effect of adding two functions. The result can be stated precisely as follows. A similar property holds for lower bounds. There is also a consequence of 2. We write this more precisely as follows. For a. But this is a direct consequence of 2. Such algorithms are also polynomial time: To take another common kind of example.

We state this more formally in the following claim. One can directly translate between logarithms of different bases using the following fundamental identity: Since f is a sum of a constant number of functions. Thus each term in the polynomial is O na. The upper bound is a direct application of 2. This is not sloppy usage: So algorithms with running-time bounds like O n2 and O n3 are polynomial-time algorithms. Asymptotic Bounds for Some Common Functions There are a number of functions that come up repeatedly in the analysis of algorithms.

Since we are concerned here only with functions that take nonnegative values. One way to get an approximate sense of how fast logb n grows is to note that. So to complete the proof. So logarithms are very slowly growing functions. This is a good point at which to discuss the relationship between these types of asymptotic bounds and the notion of polynomial time. Using O - notation. This value d is called the degree of the polynomial. A basic fact about polynomials is that their asymptotic rate of growth is determined by their "high-order term"--the one that determines the degree.

In order to implement the algorithm. Unlike the liberal use of log n. One way to summarize the relationship between polynomials and exponentials is as follows. In order to asymptotically analyze the running time of. The very first question we need to discuss is how such a ranking wil! If we want to determine whether a particular element e belongs to the list i. In the Stable Matching Problem.

We can answer a query of the form "What is the ith element on the list? Maybe the simplest way to keep a list of rt elements is to use an array A of length n.

Logarithms grow more slowly than polynomials. The implementation of basic algorithms using data structures is something that you probably have had some experience with. Arrays and Lists To start our discussion we wi!

Such an array is simple to implement in essentially all standard programming languages. So asymptotically speaking. In this book. And as we saw in Table 2. Just as people write O log rt without specifying the base. To get such a bound for the Stable Matching algorithm. An important issue to note here is that the choice of data structure is up to the algorithm designer.

To get this process started. Taken together. This is not entirely fair. Given the relative advantages and disadvantages of arrays and lists. For simplicity. By starting at First and repeatedly following pointers to the next element until we reach null.

This operation is Before deleting e: Element e After deleting e: Element e Figure 2. We have already shown that the algorithm terminates in at most n2 iterations. An array is less good for dynamically maintaining a list of elements that changes over time. This allows us to freely choose the data structure that suits the algorithm better and not be constrained by the way the information is given as input.

It is generally cumbersome to frequently add or delete elements to a list that is maintained as an array. A doubly linked list can be modified as follows.

To delete the element e from a doubly linked list. As discussed earlier. Unlike arrays. To insert element e between elements d and f in a list. Prey to point to e. We also have a pointer First that points to the first element. While lists are good for maintaining a dynamically changing set. An alternate. A generic way to implement such a linked list. Inserting or deleting e at the beginning of the list involves updating the First pointer. Chapter 2 Basics of Algorithm Analysis 2.

We discuss how to do this now. The deletion operation is illustrated in Figure 2. We can create a doubly linked list. If the array elements are sorted in some clear way either numerically or alphabetically.

In a linked list. Next that contains a pointer to the next element in the list. To ensure this. A schematic illustration of part of such a list is shown in the first line of Figure 2. Prev that contains a pointer to the previous element in the list.

Implementing the Stable Matching Algorithm Next we will use arrays and linked lists to implement the Stable Matching algorithm from Chapter 1. We also include a pointer Last. Such a record would contain a field e. We set Current [w] to a special null symbol when we need to indicate that woman w is not currently engaged. Earlier we discussed the notion that most problems have a natural "search space"--the set of all possible solutions--and we noted that a unifying theme in algorithm design is the search for algorithms whose performance is more efficient than a brute-force enumeration of this search space.

Consider a step of the algorithm. We need. At the start of the algorithm. In approaching a new problem. Current[w] is initialized to this null symbol for all women w. To do this we will need to maintain an extra array Next that indicates for each man m the position of the next woman he wil! We need to be able to identify a free man.

We need to have a preference list for each man and for each woman. We can do this by maintaining an array Current of length n. When we need to select a flee man. We need to consider each step of the algorithm and understand what data structure allows us to implement it efficiently. Now assume man m proposes to woman w. We delete m from the list if he becomes engaged. To get things under way. Note that the amount of space needed to give the preferences for all 2rt individuals is O rt2.

If a man m needs to propose to a woman. While O rt is still polynomial. To do this we will haye two arrays. To sum up. Assume w is already engaged. The discussion of running times in this section will begin in rhany cases with an analysis of the brute-force algorithm. Learning to recognize these common styles of analysis is a long-term goal. Next [re]I. This allows us to execute step 4 in constant time. We need to identify the highest-ranked woman to whom he has not yet proposed.

We process the numbers an in order. Each time we encounter a number ai. Other algorithms achieve a linear time bound for more subtle reasons. One basic way to get an algorithm with this running time is to process the input in a single pass.

Sometimes the constraints of an application force this kind of one-pass algorithm on you--for example. Computing the Maxirrturrt Computing the maximum of n numbers. One way to think about designing a better algorithm is to imagine performing the merging of the two lists by hand: If you look at the top card on each stack.

But this clearly seems wasteful. A B Figure 2. To illustrate some of the ideas here. Suppose the numbers are provided as input in either a list or an array. We now describe an algorithm for merging two sorted lists that stretches the one-pass style of design just a little. Merging Two Sorted Lists Often. Linear Time An algorithm that runs in O n. Two different subareas of algorithms. But there are only 2n elements total. Quadratic time also arises naturally from a pair of nested loops: An algorithm consists of a!

O rt log n Time O n log n is also a very common running time.. An element can be charged only once. Suppose that n is an even number. The number b1 at the front of list B will sit at the front of the list for iterations while elements from A are repeatedly being selected. The brute-force algorithm for finding the closest pair of points can be written in an equivalent way with two nested loops: For each input point xi.

Sorting is perhaps the most well-known example of a problem that can be solved this way. Each iteration involves a constant amount of work.. One also frequently encounters O n log n as a running time simply because there are many algorithms whose most expensive step is to sort the input. We have just seen that the merging can be done in linear time..

Multiplying these two factors of n together gives the running time. The natural brute-force algorithm for this problem would.. This example illustrates a very common way in which a rtmning time of O n2 arises: Suppose we charge the cost of each iteration to the element that is selected and added to the output list.

What is the running time of this algorithm? The largest of these gaps is the desired subinterval While this merging algorithm iterated through its input lists in order.

This is a correct boflnd. More crudely. Note that this algorithm requires O rt log n time to sort the numbers. The better way to argue is to bound the number of iterations of the While loop by an "accounting" scheme.. In Chapter 3 we will see linear-time algorithms for graphs that have an even more complex flow of control: The distance between points xi. What is the running time needed to solve this problem?

Multiplying these three factors of n together. O nk Time In the same way that we obtained a running time of O n2 by performing bruteforce search over all pairs formed from a set of n items. In Chapter 5 we describe a very clever algorithm that finds the closest pair of points in the plane in only O n log n time.

Cubic Time More elaborate sets of nested loops often lead to algorithms that run in O n3 time. The natural brute-force aigorithm for this problem would enumerate all subsets of k nodes. Note how the "inner" loop. At first. The following is a direct way to approach the problem. Recall that a set of nodes is independent if no two are joined by an edge. Despite this enormous set of possible solutions.

Independent Set is a principal example of a problem believed to be computationally hard. Multiplying all these choices out. The function n! In Chapter 7. Hence this is a search over pairs. Since we are treating k as a constant here. Inside the loop. Thus see that 2n arises naturally as a running time for a search algorithm that must consider all subsets. Beyond Polynomial Time The previous example of the Independent Set Problem starts us rapidly down the path toward running times that grow faster than any polynomial.

The total number of subsets of an n-element set is 2n. This is a recurring kind of dichotomy in the study of algorithms: Inside this loop. A basic problem in this genre is the Traveling Salesman Problem: Search spaces of size n!

We assume that the salesman starts and ends at the first city. The definition of an independent set tells us that we need to check. To see this. Multiplying these two together. In the case of Independent Set.

But in the case of the Interval Scheduling Problem. Perhaps the best-known example of this is the binary search algorithm. Once one has an efficient algorithm to solve a problem. A priority queue is designed for applications in which elements have a priority value. Our implementation of priority queues will also support some additional operations that we summarize at the end of the section.

The point is that in each step. We could do this by reading the entire array. In general. Since it takes linear time just to read the input. So how large is the "active" region of A after k probes? It starts at size n. For our purposes here. Given this. A motivating application for priority queues. Priority Queues 57 implicit search over all orders of the remaining n.

Given a sorted array A of n numbers. In this section. In Chapter 8. Some complex data structures are essentially tailored for use in a single kind of algorithm. Priority Queues Our primary goal in this book was expressed at the outset of the chapter: So the running time of binary search is O log n.

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