Instructor’s Manual by Thomas H. Cormen to Accompany Introduction to Algorithms Third Edition by Thomas H. Cormen Charles E. Leiserson Ronald L. Rivest Clifford Stein
The MIT Press Cambridge, Massachusetts London, England Instructor’s Manual to Accompany Introduction to Algorithms, Third Edition by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein
Published by the MIT Press. Copyright c 2009 by The Massachusetts Institute of Technology. All rights reserved.
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Revision History R-1 Preface P-1 Chapter 2: Getting Started Lecture Notes 2-1 Solutions 2-17 Chapter 3: Growth of Functions Lecture Notes 3-1 Solutions 3-7 Chapter 4: Divide-and-Conquer Lecture Notes 4-1 Solutions 4-17 Chapter 5: Probabilistic Analysis and Randomized Algorithms Lecture Notes 5-1 Solutions 5-9 Chapter 6: Heapsort Lecture Notes 6-1 Solutions 6-10 Chapter 7: Quicksort Lecture Notes 7-1 Solutions 7-9 Chapter 8: Sorting in Linear Time Lecture Notes 8-1 Solutions 8-10 Chapter 9: Medians and Order Statistics Lecture Notes 9-1 Solutions 9-10 Chapter 11: Hash Tables Lecture Notes 11-1 Solutions 11-16 Chapter 12: Binary Search Trees Lecture Notes 12-1 Solutions 12-15 Chapter 13: Red-Black Trees Lecture Notes 13-1 Solutions 13-13 Chapter 14: Augmenting Data Structures Lecture Notes 14-1 Solutions 14-9 iv Contents
Chapter 15: Dynamic Programming Lecture Notes 15-1 Solutions 15-21 Chapter 16: Greedy Algorithms Lecture Notes 16-1 Solutions 16-9 Chapter 17: Amortized Analysis Lecture Notes 17-1 Solutions 17-14 Chapter 21: Data Structures for Disjoint Sets Lecture Notes 21-1 Solutions 21-6 Chapter 22: Elementary Graph Algorithms Lecture Notes 22-1 Solutions 22-13 Chapter 23: Minimum Spanning Trees Lecture Notes 23-1 Solutions 23-8 Chapter 24: Single-Source Shortest Paths Lecture Notes 24-1 Solutions 24-13 Chapter 25: All-Pairs Shortest Paths Lecture Notes 25-1 Solutions 25-9 Chapter 26: Maximum Flow Lecture Notes 26-1 Solutions 26-12 Chapter 27: Multithreaded Algorithms Solutions 27-1 Index I-1 Revision History
Revisions are listed by date rather than being numbered.
22 February 2014. Corrected an error in the solution to Exercise 4.3-7, courtesy of Dan Suthers. Corrected an error in the solution to Exercise 23.1-6, courtesy of Rachel Ginzberg. Updated the Preface. 3 January 2012. Added solutions to Chapter 27. Added an alternative solution to Exercise 2.3-7, courtesy of Viktor Korsun and Crystal Peng. Corrected a minor error in the Chapter 15 notes in the recurrence for T.n/ for the recursive CUT-ROD procedure. Updated the solution to Problem 24-3. Corrected an error in the proof about the Edmonds-Karp algorithm performing O.VE/ flow augmentations. The bodies of all pseudocode procedures are indented slightly. 28 January 2011. Corrected an error in the solution to Problem 2-4(c), and removed unnecessary code in the solution to Problem 2-4(d). Added a missing parameter to recursive calls of REC-MAT-MULT on page 4-7. Changed the pseudocode for HEAP-EXTRACT-MAX on page 6-8 and MAX-HEAP-INSERT on page 6-9 to assume that the parameter n is passed by reference. 7 May 2010. Changed the solutions to Exercises 22.2-3 and 22.3-4 because these exercises changed. 17 February 2010. Corrected a minor error in the solution to Exercise 4.3-7. 16 December 2009. Added an alternative solution to Exercise 6.3-3, courtesy of Eyal Mashiach. 7 December 2009. Added solutions to Exercises 16.3-1, 26.1-1, 26.1-3, 26.1-7, 26.2-1, 26.2-8, 26.2-9, 26.2-12, 26.2-13, and 26.4-1 and to Problem 26-3. Cor- rected spelling in the solution to Exercise 16.2-4. Several corrections to the solution to Exercise 16.4-3, courtesy of Zhixiang Zhu. Minor changes to the solutions to Exercises 24.3-3 and 24.4-7 and Problem 24-1. 7 August 2009. Initial release.
Preface
This document is an instructor’s manual to accompany Introduction to Algorithms, Third Edition, by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. It is intended for use in a course on algorithms. You might also find some of the material herein to be useful for a CS 2-style course in data structures. Unlike the instructor’s manual for the first edition of the text—which was organized around the undergraduate algorithms course taught by Charles Leiserson at MIT in Spring 1991—but like the instructor’s manual for the second edition, we have chosen to organize the manual for the third edition according to chapters of the text. That is, for most chapters we have provided a set of lecture notes and a set of exercise and problem solutions pertaining to the chapter. This organization allows you to decide how to best use the material in the manual in your own course. We have not included lecture notes and solutions for every chapter, nor have we included solutions for every exercise and problem within the chapters that we have selected. We felt that Chapter 1 is too nontechnical to include here, and Chap- ter 10 consists of background material that often falls outside algorithms and data- structures courses. We have also omitted the chapters that are not covered in the courses that we teach: Chapters 18–20 and 27–35 (though we do include some solutions for Chapter 27), as well as Appendices A–D; future editions of this man- ual may include some of these chapters. There are two reasons that we have not included solutions to all exercises and problems in the selected chapters. First, writing up all these solutions would take a long time, and we felt it more important to release this manual in as timely a fashion as possible. Second, if we were to include all solutions, this manual would be much longer than the text itself. We have numbered the pages in this manual using the format CC-PP, where CC is a chapter number of the text and PP is the page number within that chapter’s lecture notes and solutions. The PP numbers restart from 1 at the beginning of each chapter’s lecture notes. We chose this form of page numbering so that if we add or change solutions to exercises and problems, the only pages whose numbering is affected are those for the solutions for that chapter. Moreover, if we add material for currently uncovered chapters, the numbers of the existing pages will remain unchanged.
The lecture notes
The lecture notes are based on three sources: P-2 Preface
Some are from the first-edition manual; they correspond to Charles Leiserson’s lectures in MIT’s undergraduate algorithms course, 6.046. Some are from Tom Cormen’s lectures in Dartmouth College’s undergraduate algorithms course, CS 25. Some are written just for this manual. You will find that the lecture notes are more informal than the text, as is appro- priate for a lecture situation. In some places, we have simplified the material for lecture presentation or even omitted certain considerations. Some sections of the text—usually starred—are omitted from the lecture notes. (We have included lec- ture notes for one starred section: 12.4, on randomly built binary search trees, which we covered in an optional CS 25 lecture.) In several places in the lecture notes, we have included “asides” to the instruc- tor. The asides are typeset in a slanted font and are enclosed in square brack- ets. [Here is an aside.] Some of the asides suggest leaving certain material on the board, since you will be coming back to it later. If you are projecting a presenta- tion rather than writing on a blackboard or whiteboard, you might want to replicate slides containing this material so that you can easily reprise them later in the lec- ture. We have chosen not to indicate how long it takes to cover material, as the time nec- essary to cover a topic depends on the instructor, the students, the class schedule, and other variables. There are two differences in how we write pseudocode in the lecture notes and the text:
Lines are not numbered in the lecture notes. We find them inconvenient to number when writing pseudocode on the board. We avoid using the length attribute of an array. Instead, we pass the array length as a parameter to the procedure. This change makes the pseudocode more concise, as well as matching better with the description of what it does. We have also minimized the use of shading in figures within lecture notes, since drawing a figure with shading on a blackboard or whiteboard is difficult.
The solutions
The solutions are based on the same sources as the lecture notes. They are written a bit more formally than the lecture notes, though a bit less formally than the text. We do not number lines of pseudocode, but we do use the length attribute (on the assumption that you will want your students to write pseudocode as it appears in the text). As of the third edition, we have publicly posted a few solutions on the book’s web- site. These solutions also appear in this manual, with the notation “This solution is also posted publicly” after the exercise or problem number. The set of pub- licly posted solutions might increase over time, and so we encourage you to check whether a particular solution is posted on the website before you assign an exercise or problem to your students. Preface P-3
The index lists all the exercises and problems for which this manual provides solu- tions, along with the number of the page on which each solution starts. Asides appear in a handful of places throughout the solutions. Also, we are less reluctant to use shading in figures within solutions, since these figures are more likely to be reproduced than to be drawn on a board.
Source files
For several reasons, we are unable to publish or transmit source files for this man- ual. We apologize for this inconvenience. You can use the clrscode3e package for LATEX 2" to typeset pseudocode in the same way that we do. You can find this package at http://www.cs.dartmouth.edu/thc/ clrscode/. That site also includes documentation. Make sure to use the clrscode3e package, not the clrscode package; clrscode is for the second edition of the book.
Reporting errors and suggestions
Undoubtedly, instructors will find errors in this manual. Please report errors by sending email to [email protected]. If you have a suggestion for an improvement to this manual, please feel free to submit it via email to [email protected]. As usual, if you find an error in the text itself, please verify that it has not already been posted on the errata web page before you submit it. You can use the MIT Press web site for the text, http://mitpress.mit.edu/algorithms/, to locate the errata web page and to submit an error report. We thank you in advance for your assistance in correcting errors in both this manual and the text.
How we produced this manual
Like the third edition of Introduction to Algorithms, this manual was produced in LATEX 2". We used the Times font with mathematics typeset using the MathTime Pro 2 fonts. As in all three editions of the textbook, we compiled the index using Windex, a C program that we wrote. We drew the illustrations using MacDraw Pro,1 with some of the mathematical expressions in illustrations laid in with the psfrag package for LATEX 2". We created the PDF files for this manual on a MacBook Pro running OS 10.5.
Acknowledgments
This manual borrows heavily from the manuals for the first two editions. Julie Sussman, P.P.A., wrote the first-edition manual. Julie did such a superb job on the
1See our plea in the preface for the third edition to Apple, asking that they update MacDraw Pro for OS X. P-4 Preface
first-edition manual, finding numerous errors in the first-edition text in the process, that we were thrilled to have her serve as technical copyeditor for both the second and third editions of the book. Charles Leiserson also put in large amounts of time working with Julie on the first-edition manual. The manual for the second edition was written by Tom Cormen, Clara Lee, and Erica Lin. Clara and Erica were undergraduate computer science majors at Dart- mouth at the time, and they did a superb job. The other three Introduction to Algorithms authors—Charles Leiserson, Ron Rivest, and Cliff Stein—provided helpful comments and suggestions for solutions to exercises and problems. Some of the solutions are modifications of those written over the years by teaching assistants for algorithms courses at MIT and Dartmouth. At this point, we do not know which TAs wrote which solutions, and so we simply thank them collectively. Several of the solutions to new exercises and problems in the third edition were written by Sharath Gururaj of Columbia University; we thank Sharath for his fine work. The solutions for Chapter 27 were written by Priya Natarajan. We also thank the MIT Press and our editors—Ada Brunstein, Jim DeWolf, and Marie Lee— for moral and financial support. Tim Tregubov and Wayne Cripps provided computer support at Dartmouth.
THOMAS H.CORMEN Hanover, New Hampshire August 2009 Lecture Notes for Chapter 2: Getting Started
Chapter 2 overview
Goals Start using frameworks for describing and analyzing algorithms. Examine two algorithms for sorting: insertion sort and merge sort. See how to describe algorithms in pseudocode. Begin using asymptotic notation to express running-time analysis. Learn the technique of “divide and conquer” in the context of merge sort.
Insertion sort
The sorting problem
Input: A sequence of n numbers a ;a ;:::;a . h 1 2 ni Output: A permutation (reordering) a10 ;a20 ;:::;an0 of the input sequence such that a a a . h i 10 20 n0 The sequences are typically stored in arrays. We also refer to the numbers as keys. Along with each key may be additional information, known as satellite data. [You might want to clarify that “satellite data” does not necessarily come from a satellite.] We will see several ways to solve the sorting problem. Each way will be expressed as an algorithm: a well-defined computational procedure that takes some value, or set of values, as input and produces some value, or set of values, as output.
Expressing algorithms
We express algorithms in whatever way is the clearest and most concise. English is sometimes the best way. When issues of control need to be made perfectly clear, we often use pseudocode. 2-2 Lecture Notes for Chapter 2: Getting Started
Pseudocode is similar to C, C++, Pascal, and Java. If you know any of these languages, you should be able to understand pseudocode. Pseudocode is designed for expressing algorithms to humans. Software en- gineering issues of data abstraction, modularity, and error handling are often ignored. We sometimes embed English statements into pseudocode. Therefore, unlike for “real” programming languages, we cannot create a compiler that translates pseudocode to machine code.
Insertion sort
A good algorithm for sorting a small number of elements. It works the way you might sort a hand of playing cards:
Start with an empty left hand and the cards face down on the table. Then remove one card at a time from the table, and insert it into the correct position in the left hand. To find the correct position for a card, compare it with each of the cards already in the hand, from right to left. At all times, the cards held in the left hand are sorted, and these cards were originally the top cards of the pile on the table.
Pseudocode We use a procedure INSERTION-SORT.
Takes as parameters an array AŒ1::n and the length n of the array. As in Pascal, we use “::” to denote a range within an array. [We usually use 1-origin indexing, as we do here. There are a few places in later chapters where we use 0-origin indexing instead. If you are translating pseudocode to C, C++, or Java, which use 0-origin indexing, you need to be careful to get the indices right. One option is to adjust all index calculations in the C, C++, or Java code to compensate. An easier option is, when using an array AŒ1::n, to allocate the array to be one entry longer—AŒ0::n—and just don’t use the entry at index 0.] [In the lecture notes, we indicate array lengths by parameters rather than by using the length attribute that is used in the book. That saves us a line of pseu- docode each time. The solutions continue to use the length attribute.] The array A is sorted in place: the numbers are rearranged within the array, with at most a constant number outside the array at any time. Lecture Notes for Chapter 2: Getting Started 2-3
INSERTION-SORT.A;n/ cost times for j 2 to n c n D 1 key AŒj c2 n 1 // InsertD AŒj into the sorted sequence AŒ1::j 1. 0 n 1 i j 1 c4 n 1 D n while i > 0 and AŒi > key c5 j 2 tj nD AŒi 1 AŒi c6 j 2.tj 1/ C D PnD i i 1 c7 j 2.tj 1/ D P D AŒi 1 key c8 n 1 C D P [Leave this on the board, but show only the pseudocode for now. We’ll put in the “cost” and “times” columns later.]
Example j j j 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 5 2 4 6 1 3 2 5 4 6 1 3 2 4 5 6 1 3
j j 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 2 4 5 6 1 3 12 4 5 6 3 1 2 3 4 5 6
[Read this figure row by row. Each part shows what happens for a particular itera- tion with the value of j indicated. j indexes the “current card” being inserted into the hand. Elements to the left of AŒj that are greater than AŒj move one position to the right, and AŒj moves into the evacuated position. The heavy vertical lines separate the part of the array in which an iteration works—AŒ1::j—from the part of the array that is unaffected by this iteration—AŒj 1::n. The last part of the figure shows the final sorted array.] C
Correctness
We often use a loop invariant to help us understand why an algorithm gives the correct answer. Here’s the loop invariant for INSERTION-SORT:
Loop invariant: At the start of each iteration of the “outer” for loop—the loop indexed by j —the subarray AŒ1::j 1 consists of the elements orig- inally in AŒ1::j 1 but in sorted order. To use a loop invariant to prove correctness, we must show three things about it: Initialization: It is true prior to the first iteration of the loop. Maintenance: If it is true before an iteration of the loop, it remains true before the next iteration. Termination: When the loop terminates, the invariant—usually along with the reason that the loop terminated—gives us a useful property that helps show that the algorithm is correct. 2-4 Lecture Notes for Chapter 2: Getting Started
Using loop invariants is like mathematical induction: To prove that a property holds, you prove a base case and an inductive step. Showing that the invariant holds before the first iteration is like the base case. Showing that the invariant holds from iteration to iteration is like the inductive step. The termination part differs from the usual use of mathematical induction, in which the inductive step is used infinitely. We stop the “induction” when the loop terminates. We can show the three parts in any order.
For insertion sort Initialization: Just before the first iteration, j 2. The subarray AŒ1::j 1 is the single element AŒ1, which is the elementD originally in AŒ1, and it is trivially sorted. Maintenance: To be precise, we would need to state and prove a loop invariant for the “inner” while loop. Rather than getting bogged down in another loop invariant, we instead note that the body of the inner while loop works by moving AŒj 1, AŒj 2, AŒj 3, and so on, by one position to the right until the proper position for key (which has the value that started out in AŒj) is found. At that point, the value of key is placed into this position. Termination: The outer for loop ends when j >n, which occurs when j n 1. Therefore, j 1 n. Plugging n in for j 1 in the loop invariant,D C the subarray AŒ1::n consistsD of the elements originally in AŒ1::n but in sorted order. In other words, the entire array is sorted.
Pseudocode conventions
[Covering most, but not all, here. See book pages 20–22 for all conventions.] Indentation indicates block structure. Saves space and writing time. Looping constructs are like in C, C++, Pascal, and Java. We assume that the loop variable in a for loop is still defined when the loop exits (unlike in Pascal). // indicates that the remainder of the line is a comment. Variables are local, unless otherwise specified. We often use objects, which have attributes. For an attribute attr of object x, we write x:attr. (This notation matches x:attr in Java and is equivalent to x->attr in C++.) Attributes can cascade, so that if x:y is an object and this object has attribute attr, then x:y:attr indicates this object’s attribute. That is, x:y:attr is implicitly parenthesized as .x:y/:attr. Objects are treated as references, like in Java. If x and y denote objects, then the assignment y x makes x and y reference the same object. It does not cause attributes ofD one object to be copied to another. Parameters are passed by value, as in Java and C (and the default mechanism in Pascal and C++). When an object is passed by value, it is actually a reference (or pointer) that is passed; changes to the reference itself are not seen by the caller, but changes to the object’s attributes are. Lecture Notes for Chapter 2: Getting Started 2-5
The boolean operators “and” and “or” are short-circuiting: if after evaluating the left-hand operand, we know the result of the expression, then we don’t evaluate the right-hand operand. (If x is FALSE in “x and y” then we don’t evaluate y. If x is TRUE in “x or y” then we don’t evaluate y.)
Analyzing algorithms
We want to predict the resources that the algorithm requires. Usually, running time. In order to predict resource requirements, we need a computational model.
Random-access machine (RAM) model
Instructions are executed one after another. No concurrent operations. It’s too tedious to define each of the instructions and their associated time costs. Instead, we recognize that we’ll use instructions commonly found in real com- puters:
Arithmetic: add, subtract, multiply, divide, remainder, floor, ceiling). Also, shift left/shift right (good for multiplying/dividing by 2k ). Data movement: load, store, copy. Control: conditional/unconditional branch, subroutine call and return. Each of these instructions takes a constant amount of time.
The RAM model uses integer and floating-point types.
We don’t worry about precision, although it is crucial in certain numerical ap- plications. There is a limit on the word size: when working with inputs of size n, assume that integers are represented by c lg n bits for some constant c 1. (lg n is a very frequently used shorthand for log2 n.) c 1 we can hold the value of n we can index the individual elements. ) ) c is a constant the word size cannot grow arbitrarily. )
How do we analyze an algorithm’s running time?
The time taken by an algorithm depends on the input.
Sorting 1000 numbers takes longer than sorting 3 numbers. A given sorting algorithm may even take differing amounts of time on two inputs of the same size. For example, we’ll see that insertion sort takes less time to sort n elements when they are already sorted than when they are in reverse sorted order. 2-6 Lecture Notes for Chapter 2: Getting Started
Input size Depends on the problem being studied. Usually, the number of items in the input. Like the size n of the array being sorted. But could be something else. If multiplying two integers, could be the total number of bits in the two integers. Could be described by more than one number. For example, graph algorithm running times are usually expressed in terms of the number of vertices and the number of edges in the input graph.
Running time On a particular input, it is the number of primitive operations (steps) executed. Want to define steps to be machine-independent. Figure that each line of pseudocode requires a constant amount of time. One line may take a different amount of time than another, but each execution of line i takes the same amount of time ci . This is assuming that the line consists only of primitive operations. If the line is a subroutine call, then the actual call takes constant time, but the execution of the subroutine being called might not. If the line specifies operations other than primitive ones, then it might take more than constant time. Example: “sort the points by x-coordinate.”
Analysis of insertion sort
[Now add statement costs and number of times executed to INSERTION-SORT pseudocode.]
Assume that the ith line takes time ci , which is a constant. (Since the third line is a comment, it takes no time.)
For j 2;3;:::;n, let tj be the number of times that the while loop test is executedD for that value of j . Note that when a for or while loop exits in the usual way—due to the test in the loop header—the test is executed one time more than the loop body. The running time of the algorithm is .cost of statement/ .number of times statement is executed/: all statementsX Let T.n/ running time of INSERTION-SORT. D n n T.n/ c n c .n 1/ c .n 1/ c t c .t 1/ D 1 C 2 C 4 C 5 j C 6 j j 2 j 2 XD XD n c .t 1/ c .n 1/ : C 7 j C 8 j 2 XD The running time depends on the values of tj . These vary according to the input. Lecture Notes for Chapter 2: Getting Started 2-7
Best case The array is already sorted. Always find that AŒi key upon the first time the while loop test is run (when i j 1). D All tj are 1. Running time is T.n/ c n c .n 1/ c .n 1/ c .n 1/ c .n 1/ D 1 C 2 C 4 C 5 C 8 .c c c c c /n .c c c c /: D 1 C 2 C 4 C 5 C 8 2 C 4 C 5 C 8 Can express T.n/ as an b for constants a and b (that depend on the statement costs c ) T.n/ is a linearC function of n. i )
Worst case The array is in reverse sorted order. Always find that AŒi > key in while loop test. Have to compare key with all elements to the left of the j th position compare with j 1 elements. ) Since the while loop exits because i reaches 0, there’s one additional test after the j 1 tests tj j . n n ) nD n t j and .t 1/ .j 1/. j D j D j 2 j 2 j 2 j 2 D D D D Xn X X X j is known as an arithmetic series, and equation (A.1) shows that it equals j 1 Xn.nD 1/ C . 2 n n n.n 1/ Since j j 1, it equals C 1. D 2 j 2 j 1 ! [The parenthesesXD XD around the summation are not strictly necessary. They are there for clarity, but it might be a good idea to remind the students that the meaning of the expression would be the same even without the parentheses.] n n 1 n.n 1/ Letting k j 1, we see that .j 1/ k . D D D 2 j 2 k 1 XD XD Running time is n.n 1/ T.n/ c n c .n 1/ c .n 1/ c C 1 D 1 C 2 C 4 C 5 2 n.n 1/ n.n 1/ c c c .n 1/ C 6 2 C 7 2 C 8 c c c c c c 5 6 7 n2 c c c 5 6 7 c n D 2 C 2 C 2 C 1 C 2 C 4 C 2 2 2 C 8 .c c c c /: 2 C 4 C 5 C 8 Can express T.n/ as an2 bn c for constants a;b;c (that again depend on statement costs) T.n/ isC a quadraticC function of n. ) 2-8 Lecture Notes for Chapter 2: Getting Started
Worst-case and average-case analysis
We usually concentrate on finding the worst-case running time: the longest run- ning time for any input of size n.
Reasons The worst-case running time gives a guaranteed upper bound on the running time for any input. For some algorithms, the worst case occurs often. For example, when search- ing, the worst case often occurs when the item being searched for is not present, and searches for absent items may be frequent. Why not analyze the average case? Because it’s often about as bad as the worst case. Example: Suppose that we randomly choose n numbers as the input to inser- tion sort. On average, the key in AŒj is less than half the elements in AŒ1::j 1 and it’s greater than the other half. On average, the while loop has to look halfway through the sorted subarray AŒ1::j) 1 to decide where to drop key. t j=2. ) j Although the average-case running time is approximately half of the worst-case running time, it’s still a quadratic function of n.
Order of growth
Another abstraction to ease analysis and focus on the important features. Look only at the leading term of the formula for running time.
Drop lower-order terms. Ignore the constant coefficient in the leading term. Example: For insertion sort, we already abstracted away the actual statement costs to conclude that the worst-case running time is an2 bn c. Drop lower-order terms an2. C C Ignore constant coefficient) n2. ) But we cannot say that the worst-case running time T.n/ equals n2. It grows like n2. But it doesn’t equal n2. We say that the running time is ‚.n2/ to capture the notion that the order of growth is n2. We usually consider one algorithm to be more efficient than another if its worst- case running time has a smaller order of growth. Lecture Notes for Chapter 2: Getting Started 2-9
Designing algorithms
There are many ways to design algorithms. For example, insertion sort is incremental: having sorted AŒ1::j 1, place AŒj correctly, so that AŒ1::j is sorted.
Divide and conquer
Another common approach.
Divide the problem into a number of subproblems that are smaller instances of the same problem. Conquer the subproblems by solving them recursively. Base case: If the subproblems are small enough, just solve them by brute force. [It would be a good idea to make sure that your students are comfortable with recursion. If they are not, then they will have a hard time understanding divide and conquer.] Combine the subproblem solutions to give a solution to the original problem.
Merge sort
A sorting algorithm based on divide and conquer. Its worst-case running time has a lower order of growth than insertion sort. Because we are dealing with subproblems, we state each subproblem as sorting a subarray AŒp::r. Initially, p 1 and r n, but these values change as we recurse through subproblems. D D To sort AŒp::r: Divide by splitting into two subarrays AŒp::q and AŒq 1::r, where q is the halfway point of AŒp::r. C Conquer by recursively sorting the two subarrays AŒp::q and AŒq 1::r. C Combine by merging the two sorted subarrays AŒp::q and AŒq 1::r to pro- duce a single sorted subarray AŒp::r. To accomplish this step,C we’ll define a procedure MERGE.A;p;q;r/. The recursion bottoms out when the subarray has just 1 element, so that it’s trivially sorted.
MERGE-SORT.A;p;r/ if p Initial call: MERGE-SORT.A;1;n/ [It is astounding how often students forget how easy it is to compute the halfway point of p and r as their average .p r/=2. We of course have to take the floor to ensure that we get an integer indexCq. But it is common to see students perform calculations like p .r p/=2, or even more elaborate expressions, forgetting the easy way to computeC an average.] Example Bottom-up view for n 8: [Heavy lines demarcate subarrays used in subprob- lems.] D sorted array 1 2 3 4 5 6 7 8 12 2 3 4 5 6 7 merge 2 4 5 7 1 2 3 6 merge 2 5 4 7 1 3 2 6 merge 5 2 4 7 1 3 2 6 1 2 3 4 5 6 7 8 initial array [Examples when n is a power of 2 are most straightforward, but students might also want an example when n is not a power of 2.] Bottom-up view for n 11: D sorted array 1 2 3 4 5 6 7 8 9 10 11 1 2 2 3 4 4 5 6 6 7 7 merge 1 2 4 4 6 7 2 3 5 6 7 merge 2 4 7 1 4 6 3 5 7 2 6 merge 4 7 2 1 6 4 3 7 5 2 6 merge 4 7 2 6 1 4 7 3 5 2 6 1 2 3 4 5 6 7 8 9 10 11 initial array [Here, at the next-to-last level of recursion, some of the subproblems have only 1 element. The recursion bottoms out on these single-element subproblems.] Lecture Notes for Chapter 2: Getting Started 2-11 Merging What remains is the MERGE procedure. Input: Array A and indices p;q;r such that p q