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Each chapter presents an algorithm, a design technique, an application area, or a related topic. Algorithms are described in English and in a pseudocode designed to be readable by anyone who has done a little programming. D3d hacks blackshot hack. The book contains 244 figures—many with multiple parts—illustrating how the algorithms work. Since we emphasize efficiency as a design criterion, we include careful analyses of the running times of all our algorithms.

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The key to Strassen’s method is to make the recursion tree slightly less bushy. That is, instead of performing eight recursive multiplications of n=2 n=2 matrices, it performs only seven. The cost of eliminating one matrix multiplication will be several new additions of n=2 n=2 matrices, but still only a constant number of additions. As before, the constant number of matrix additions will be subsumed by ‚-notation when we set up the recurrence equation to characterize the running time.

Our first example is the birthday paradox. How many people must there be in a room before there is a 50% chance that two of them were born on the same day of the year?


We used some simplifying abstractions to ease our analysis of the INSERTION- SORT procedure. First, we ignored the actual cost of each statement, using the constants ci to represent these costs. Then, we observed that even these constants give us more detail than we really need: we expressed the worst-case running time as an2 C bn C c for some constants a, b, and c that depend on the statement costs ci. We thus ignored not only the actual statement costs, but also the abstract costs ci.

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The binary search al- gorithm repeats this procedure, halving the size of the remaining portion of the sequence each time. Write pseudocode, either iterative or recursive, for binary search.

We performed the same experiment as we did for mineral oil to obtain T1. Yamane, S: Perfect crystals of \(U_q(G^{(1)}_2)\). We obtained a value of T1 = 2.9 s for water by tting the data with Mathematica to Eq. 7. B. Water. Current Status: Released Game Version: XP/Gold/Drop rates: x1 Project Location: Central Europe Realmlist: set realmlist [HOST] Content release: Progressive (no partial/gated releases) Honor Distribution: Wednesday 6 AM Server Time (CET/UTC+1) How to connect Create Account. We write down and solve the Bethe equations for the generalized quantum model, and show that these take on a particularly simple form that allows for an exact solution: essentially, the quasiparticles behave like interacting hard rods.


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Let us further explore the distinction between probabilistic analysis and random- ized algorithms. In Section 5/2, we claimed that, assuming that the candidates ar- rive in a random order, the expected number of times we hire a new office assistant is about ln n. Note that the algorithm here is deterministic; for any particular input, the number of times a new office assistant is hired is always the same. Furthermore, the number of times we hire a new office assistant differs for different inputs, and it depends on the ranks of the various candidates.


The cost incurred at each of the two sub- nodes at the second level of recursion is cn=2. We continue expanding each node in the tree by breaking it into its constituent parts as determined by the recurrence, until the problem sizes get down to 1, each with a cost of c. Part (d) shows the resulting recursion tree.

In its i th iteration, it chooses the element AŒi� randomly from among elements AŒi� through AŒn�. Subsequent to the i th iteration, AŒi� is never altered.


One common method is to assign each element AŒi� of the array a random pri- ority P Œi�, and then sort the elements of A according to these priorities. For ex- ample, if our initial array is A D h1; 2; 3; 4i and we choose random priorities P D h36; 3; 62; 19i, we would produce an array B D h2; 4; 1; 3i, since the second priority is the smallest, followed by the fourth, then the first, and finally the third.

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That an efficient algorithm for one cannot exist. In other words, no one knows whether or not efficient algorithms exist for NP-complete problems. Second, the set of NP-complete problems has the remarkable property that if an efficient algo- rithm exists for any one of them, then efficient algorithms exist for all of them. This relationship among the NP-complete problems makes the lack of efficient solutions all the more tantalizing. Third, several NP-complete problems are similar, but not identical, to problems for which we do know of efficient algorithms. Computer scientists are intrigued by how a small change to the problem statement can cause a big change to the efficiency of the best known algorithm.


This section contains a proof of the master theorem (Theorem 4/1). You do not need to understand the proof in order to apply the master theorem.

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Your solution should be asymptotically tight. Do not worry about whether values are integral.

If k 0, and c > 1 are constants. Your answer should be in the form of the table with “yes” or “no” written in each box.


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We access a particular attribute using the syntax found in many object-oriented programming languages: the object name, followed by a dot, followed by the attribute name. For example, we treat an array as an object with the attribute length indicating how many elements it contains. To specify the number of elements in an array A, we write A: length.

Now that we have seen how recurrences characterize the running times of divide- and-conquer algorithms, we will learn how to solve recurrences. We start in this section with the “substitution” method.


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Give recurrences for the worst-case running times of binary search when arrays are passed using each of the three methods above, and give good upper bounds on the solutions of the recurrences. Let N be the size of the original problem and n be the size of a subproblem.


Generalizing your solution to part (b), suppose that there are k � 1 indices i such that AŒi� D x. What is the expected number of indices into A that we must pick before we find x and RANDOM-SEARCH terminates? Your answer should be a function of n and k.

Figure 4/1 Information about the price of stock in the Volatile Chemical Corporation after the close of trading over a period of 17 days. The horizontal axis of the chart indicates the day, and the vertical axis shows the price. The bottom row of the table gives the change in price from the previous day.


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What problem size n0 gives the crossover point at which the recursive algorithm beats the brute-force algorithm? Then, change the base case of the recursive algorithm to use the brute-force algorithm whenever the problem size is less than n0. Does that change the crossover point?


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Equation (3/20) is due to Robbins . Other properties of elementary math- ematical functions can be found in any good mathematical reference, such as Abramowitz and Stegun or Zwillinger , or in a calculus book, such as Apostol or Thomas et al. . Knuth and Graham, Knuth, and Patash- nik contain a wealth of material on discrete mathematics as used in computer science.

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What has changed between the second and third editions of this book? The mag- nitude of the changes is on a par with the changes between the first and second editions. As we said about the second-edition changes, depending on how you look at it, the book changed either not much or quite a bit.


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We use O-notation to give an upper bound on a function, to within a constant factor. Figure 3/1(b) shows the intuition behind O-notation.

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The time taken by the INSERTION-SORT procedure depends on the input: sorting a thousand numbers takes longer than sorting three numbers. Moreover, INSERTION- SORT can take different amounts of time to sort two input sequences of the same size depending on how nearly sorted they already are. In general, the time taken by an algorithm grows with the size of the input, so it is traditional to describe the running time of a program as a function of the size of its input. To do so, we need to define the terms “running time” and “size of input” more carefully.

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For a problem such as the hiring problem, in which it is helpful to assume that all permutations of the input are equally likely, a probabilistic analysis can guide the development of a randomized algorithm. Instead of assuming a distribution of inputs, we impose a distribution. In particular, before running the algorithm, we randomly permute the candidates in order to enforce the property that every permutation is equally likely. Although we have modified the algorithm, we still expect to hire a new office assistant approximately ln n times.

Chapter 27 presents a model for “multithreaded” algorithms, which take advantage of multiple cores. This model has advantages from a theoretical standpoint, and it forms the basis of several successful computer programs, including a championship chess program.


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Comparing Lemmas 5/2 and 5/3 highlights the difference between probabilistic analysis and randomized algorithms. In Lemma 5/2, we make an assumption about the input. In Lemma 5/3, we make no such assumption, although randomizing the input takes some additional time. To remain consistent with our terminology, we couched Lemma 5/2 in terms of the average-case hiring cost and Lemma 5/3 in terms of the expected hiring cost. In the remainder of this section, we discuss some issues involved in randomly permuting inputs.


Now consider a deterministic linear search algorithm, which we refer to as DETERMINISTIC-SEARCH. Specifically, the algorithm searches A for x in order, considering AŒ1�; AŒ2�; AŒ3�; AŒn� until either it finds AŒi� D x or it reaches the end of the array. Assume that all possible permutations of the input array are equally likely.

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In some particular cases, we shall be interested in the average-case running time of an algorithm; we shall see the technique of probabilistic analysis applied to various algorithms throughout this book. The scope of average-case analysis is limited, because it may not be apparent what constitutes an “average” input for a particular problem. Often, we shall assume that all inputs of a given size are equally likely. In practice, this assumption may be violated, but we can sometimes use a randomized algorithm, which makes random choices, to allow a probabilistic analysis and yield an expected running time. We explore randomized algorithms more in Chapter 5 and in several other subsequent chapters.

When the subproblems are large enough to solve recursively, we call that the recur- sive case. Once the subproblems become small enough that we no longer recurse, we say that the recursion “bottoms out” and that we have gotten down to the base case. Sometimes, in addition to subproblems that are smaller instances of the same problem, we have to solve subproblems that are not quite the same as the original problem. We consider solving such subproblems as part of the combine step.


That is, when we evaluate the expression “x and y” we first evaluate x. If x evaluates to FALSE, then the entire expression cannot evaluate to TRUE, and so we do not evaluate y. If, on the other hand, x evaluates to TRUE, we must evaluate y to determine the value of the entire expression. Similarly, in the expression “x or y” we eval- uate the expression y only if x evaluates to FALSE. Short-circuiting operators allow us to write boolean expressions such as “x ¤ NIL and x: f D y” without worrying about what happens when we try to evaluate x: f when x is NIL.

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Computational steps that we specify in English are often variants of a procedure that requires more than just a constant amount of time. For example, later in this book we might say “sort the points by x-coordinate,” which, as we shall see, takes more than a constant amount of time. Also, note that a statement that calls a subroutine takes constant time, though the subroutine, once invoked, may take more.


Because many programs use it as an intermediate step, sorting is a fundamental operation in computer science. As a result, we have a large number of good sorting algorithms at our disposal. Which algorithm is best for a given application depends on—among other factors—the number of items to be sorted, the extent to which the items are already somewhat sorted, possible restrictions on the item values, the architecture of the computer, and the kind of storage devices to be used: main memory, disks, or even tapes.

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We hope that this textbook provides you with an enjoyable introduction to the field of algorithms. We have attempted to make every algorithm accessible and interesting. To help you when you encounter unfamiliar or difficult algorithms, we describe each one in a step-by-step manner. We also provide careful explanations of the mathematics needed to understand the analysis of the algorithms. If you already have some familiarity with a topic, you will find the chapters organized so that you can skim introductory sections and proceed quickly to the more advanced material.


Chapter 5 introduces probabilistic analysis and randomized algorithms. We typ- ically use probabilistic analysis to determine the running time of an algorithm in cases in which, due to the presence of an inherent probability distribution, the running time may differ on different inputs of the same size. In some cases, we assume that the inputs conform to a known probability distribution, so that we are averaging the running time over all possible inputs. In other cases, the probability distribution comes not from the inputs but from random choices made during the course of the algorithm. An algorithm whose behavior is determined not only by its input but by the values produced by a random-number generator is a randomized algorithm. We can use randomized algorithms to enforce a probability distribution on the inputs—thereby ensuring that no particular input always causes poor perfor- mance—or even to bound the error rate of algorithms that are allowed to produce incorrect results on a limited basis.

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Write pseudocode for linear search, which scans through the sequence, looking for �. Using a loop invariant, prove that your algorithm is correct. Make sure that your loop invariant fulfills the three necessary properties.

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Intuitively, the lower-order terms of an asymptotically positive function can be ignored in determining asymptotically tight bounds because they are insignificant for large n. When n is large, even a tiny fraction of the highest-order term suf- fices to dominate the lower-order terms. Thus, setting c1 to a value that is slightly smaller than the coefficient of the highest-order term and setting c2 to a value that is slightly larger permits the inequalities in the definition of ‚-notation to be sat- isfied. The coefficient of the highest-order term can likewise be ignored, since it only changes c1 and c2 by a constant factor equal to the coefficient.


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Knuth traces the origin of the O-notation to a number-theory text by P. Bach- mann in 1892. The o-notation was invented by E. Landau in 1909 for his discussion of the distribution of prime numbers. The � and ‚ notations were advocated by Knuth to correct the popular, but technically sloppy, practice in the literature of using O-notation for both upper and lower bounds. Many people continue to use the O-notation where the ‚-notation is more technically precise. Further dis- cussion of the history and development of asymptotic notations appears in works by Knuth [209, 213] and Brassard and Bratley .

The merge sort algorithm closely follows the divide-and-conquer paradigm. In- tuitively, it operates as follows.


Bollobás , Hofri , and Spencer contain a wealth of advanced prob- abilistic techniques. The advantages of randomized algorithms are discussed and surveyed by Karp and Rabin . The textbook by Motwani and Raghavan gives an extensive treatment of randomized algorithms.

Because the factorial function grows faster than even an exponential function, we cannot feasibly generate each possible order and then verify that, within that order, each part appears before the parts using it (unless we have only a few parts). This problem is an instance of topological sorting, and we shall see in Chapter 22 how to solve this problem efficiently.


We have designed this book to be both versatile and complete. You should find it useful for a variety of courses, from an undergraduate course in data structures up through a graduate course in algorithms. Because we have provided considerably more material than can fit in a typical one-term course, you can consider this book to be a “buffet” or “smorgasbord” from which you can pick and choose the material that best supports the course you wish to teach.

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By equation (3/15), changing the base of a logarithm from one constant to an- other changes the value of the logarithm by only a constant factor, and so we shall often use the notation “lg n” when we don’t care about constant factors, such as in O-notation. Computer scientists find 2 to be the most natural base for logarithms because so many algorithms and data structures involve splitting a problem into two parts.


We understand that if you are using this book outside of a course, then you might be unable to check your solutions to problems and exercises against solutions provided by an instructor. Our Web site, links to solutions for some of the problems and exercises so that you can check your work. Please do not send your solutions to us.

A randomized algorithm is often the simplest and most efficient way to solve a problem. We shall use randomized algorithms occasionally throughout this book.


We now use “D” to indicate assignment and “==” to test for equality, just as C, C++, Java, and Python do. Likewise, we have eliminated the keywords do and then and adopted “//” as our comment-to-end-of-line symbol. We also now use dot-notation to indicate object attributes. Our pseudocode remains procedural, rather than object-oriented. In other words, rather than running methods on objects, we simply call procedures, passing objects as parameters.

We also thank the many readers of the first and second editions who reported errors or submitted suggestions for how to improve this book. We corrected all the bona fide errors that were reported, and we incorporated as many suggestions as we could. We rejoice that the number of such contributors has grown so great that we must regret that it has become impractical to list them all.


Conquer the subproblems by solving them recursively. If the subproblem sizes are small enough, however, just solve the subproblems in a straightforward manner.

In this chapter, we shall see more algorithms based on divide-and-conquer. The first one solves the maximum-subarray problem: it takes as input an array of num- bers, and it determines the contiguous subarray whose values have the greatest sum. Then we shall see two divide-and-conquer algorithms for multiplying n n matri- ces.


For example, we might extend the notation to the domain of real numbers or, alternatively, restrict it to a subset of the natural numbers. We should make sure, however, to understand the precise meaning of the notation so that when we abuse, we do not misuse it. This section defines the basic asymptotic notations and also introduces some common abuses.

We wish to determine, for each possible value of k, the probability that we hire the most qualified applicant. We then choose the best possible k, and implement the strategy with that value. For the moment, assume that k is fixed.


Which we can prove by induction (Exercise 3/2-7). Since ˇ̌y� ˇ̌ 0, be a degree-d polynomial in n, and let k be a constant. Use the definitions of the asymptotic notations to prove the following properties.

This section reviews some standard mathematical functions and notations and ex- plores the relationships among them. It also illustrates the use of the asymptotic notations.


In this book, the functions to which we apply asymptotic notation will usually characterize the running times of algorithms. But asymptotic notation can apply to functions that characterize some other aspect of algorithms (the amount of space they use, for example), or even to functions that have nothing whatsoever to do with algorithms.

Even when we use asymptotic notation to apply to the running time of an al- gorithm, we need to understand which running time we mean. Sometimes we are interested in the worst-case running time. Often, however, we wish to characterize the running time no matter what the input. In other words, we often wish to make a blanket statement that covers all inputs, not just the worst case. We shall see asymptotic notations that are well suited to characterizing running times no matter what the input.


The ‚-notation asymptotically bounds a function from above and below. When we have only an asymptotic upper bound, we use O-notation.

In the previous section, we showed how knowing a distribution on the inputs can help us to analyze the average-case behavior of an algorithm. Many times, we do not have such knowledge, thus precluding an average-case analysis. As mentioned in Section 5/1, we may be able to use a randomized algorithm.


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Termination: At termination, k D r C 1. By the loop invariant, the subarray AŒp: k � 1�, which is AŒp: r�, contains the k � p D r � p C 1 smallest elements of LŒ1: n1 C 1� and RŒ1: n2 C 1�, in sorted order. The arrays L and R together contain n1 C n2 C 2 D r � p C 3 elements. All but the two largest have been copied back into A, and these two largest elements are the sentinels.

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We shall see the details of steps 2–4 in a moment, but we already have enough information to set up a recurrence for the running time of Strassen’s method. Let us assume that once the matrix size n gets down to 1, we perform a simple scalar mul- tiplication, just as in line 4 of SQUARE-MATRIX-MULTIPLY-RECURSIVE.


The total number of levels of the recursion tree in Figure 2/5 is lg nC 1, where n is the number of leaves, corresponding to the input size. An informal inductive argument justifies this claim. The base case occurs when n D 1, in which case the tree has only one level. Since lg 1 D 0, we have that lg n C 1 gives the correct number of levels. Now assume as an inductive hypothesis that the number of levels of a recursion tree with 2i leaves is lg 2i C 1 D i C 1 (since for any value of i, we have that lg 2i D i). Because we are assuming that the input size is a power of 2, the next input size to consider is 2iC1.

Before we can analyze an algorithm, we must have a model of the implemen- tation technology that we will use, including a model for the resources of that technology and their costs. For most of this book, we shall assume a generic one- processor, random-access machine (RAM) model of computation as our imple- mentation technology and understand that our algorithms will be implemented as computer programs. In the RAM model, instructions are executed one after an- other, with no concurrent operations.


This method would make n calls to the RANDOM procedure. If n is much larger than m, we can create a random sample with fewer calls to RANDOM.

Moreover, as we go down from the root, more and more internal nodes are absent. Consequently, levels toward the bottom of the recursion tree contribute less than cn to the total cost. We could work out an accu- rate accounting of all costs, but remember that we are just trying to come up with a guess to use in the substitution method.


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This chapter gives several standard methods for simplifying the asymptotic anal- ysis of algorithms. The next section begins by defining several types of “asymp- totic notation,” of which we have already seen an example in ‚-notation. We then present several notational conventions used throughout this book, and finally we review the behavior of functions that commonly arise in the analysis of algorithms.

You might think that to prove that a permutation is a uniform random permuta- tion, it suffices to show that, for each element AŒi�, the probability that the element winds up in position j is 1=n. Exercise 5/3-4 shows that this weaker condition is, in fact, insufficient.

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Subproblems become sufficiently small. Consider a modification to merge sort in which n=k sublists of length k are sorted using insertion sort and then merged using the standard merging mechanism, where k is a value to be determined.

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Knowing which points are vertices of the convex hull is not quite enough, either, since we also need to know the order in which they appear. There are many choices, therefore, for the vertices of the convex hull. Chapter 33 gives two good methods for finding the convex hull.


An algorithm can be specified in English, as a computer program, or even as a hardware design. The only requirement is that the specification must provide a precise description of the computational procedure to be followed.

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We have been working with the MIT Press for over two decades now, and what a terrific relationship it has been! We thank Ellen Faran, Bob Prior, Ada Brunstein, and Mary Reilly for their help and support.


Similar to FIND-MAX-CROSSING-SUBARRAY, the recursive procedure FIND- MAXIMUM-SUBARRAY returns a tuple containing the indices that demarcate a maximum subarray, along with the sum of the values in a maximum subarray. Line 1 tests for the base case, where the subarray has just one element. A subar- ray with just one element has only one subarray—itself—and so line 2 returns a tuple with the starting and ending indices of just the one element, along with its value. Lines 3–11 handle the recursive case. Line 3 does the divide part, comput- ing the index mid of the midpoint. Let’s refer to the subarray AŒlow: mid� as the left subarray and to AŒmid C 1: high� as the right subarray. Because we know that the subarray AŒlow: high� contains at least two elements, each of the left and right subarrays must have at least one element. Lines 4 and 5 conquer by recur- sively finding maximum subarrays within the left and right subarrays, respectively. Lines 6–11 form the combine part.

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Many useful algorithms are recursive in structure: to solve a given problem, they call themselves recursively one or more times to deal with closely related sub- problems. These algorithms typically follow a divide-and-conquer approach: they break the problem into several subproblems that are similar to the original prob- lem but smaller in size, solve the subproblems recursively, and then combine these solutions to create a solution to the original problem.


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Unfortunately, there is no general way to guess the correct solutions to recurrences. Guessing a solution takes experience and, occasionally, creativity. Fortunately, though, you can use some heuristics to help you become a good guesser. You can also use recursion trees, which we shall see in Section 4/4, to generate good guesses.

A recursion tree is best used to generate a good guess, which you can then verify by the substitution method. When using a recursion tree to generate a good guess, you can often tolerate a small amount of “sloppiness,” since you will be verifying your guess later on. If you are very careful when drawing out a recursion tree and summing the costs, however, you can use a recursion tree as a direct proof of a solution to a recurrence. In this section, we will use recursion trees to generate good guesses, and in Section 4/6, we will use recursion trees directly to prove the theorem that forms the basis of the master method.