Algorithm exercises with solutions

This website contains nearly complete solutions to the bible textbook - Introduction to Algorithms Third Editionpublished by Thomas H. CormenCharles E. LeisersonRonald L. Rivestand Clifford Stein. I hope to organize solutions to help people and myself study algorithms. By using Markdown. I build this website since I want to help everyone learn algorithms by providing something easy to read on mobile devices. Therefore, if any adjustment is needed or you have the same motivation to contribute to this work, please don't hesitate to give me your feedback.

You can press the "pencil icon" in the upper right corner to edit the content or open an issue in my repository. Your solution will be rebased to this repository after I review it and make some form modifications to your pull request. I use the static site generator MkDocs and the beautiful theme Material for MkDocs to build this website.

As for rendering math equations, I use KaTeXwhich is fast and beautiful. I also add overflow-x: auto to prevent the overflow issue on small screen devices so that you can scroll horizontally in the math display equations. For a clear commit history, I rebase my repository regularly.

Therefore, if you have forked the repository before, consider re-forking it again. For more information, please visit my GitHub. Skip to content. CLRS Solutions. More Information License. If I miss your name here, please tell me! Motivation I build this website since I want to help everyone learn algorithms by providing something easy to read on mobile devices. Thank you very much, and I hope that everyone will learn algorithms smoothly. How I Generate the Website?

More Information For a clear commit history, I rebase my repository regularly. Revised on July 21, GitHub is home to over 40 million developers working together to host and review code, manage projects, and build software together. If nothing happens, download GitHub Desktop and try again. If nothing happens, download Xcode and try again. If nothing happens, download the GitHub extension for Visual Studio and try again.

Solutions may not be optimal, but relied on the concepts taught in that particular chapter plus some extra coding techniques that I've veen playing around with. Skip to content. Dismiss Join GitHub today GitHub is home to over 40 million developers working together to host and review code, manage projects, and build software together. Sign up.

The Art of Computer Programming/Algorithms/Exercises

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algorithm exercises with solutions

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algorithm exercises with solutions

Jun 26, Chapter 1 Exercises. Chapter 2 Exercises. Moved dependencies to SampleFiles folder. Jun 25, Chapter 3 Exercises. Oct 20, Chapter 4 Exercises. Chapter 5 Exercises. Chapter 6 Exercises. Jul 6, This page is part of The Art of Computer Programming. Show how the values of four variables a, b, c, d can be rearranged to b, c, d, a by a sequence of replacements.

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In other words, the new value of a is the original value of b, etc. Try to use the minimum number of replacements. My solution: The first time this step occurs n and m are the input values, so there it is possible that m is greater than n. After the first time this step occurs, m will be the value of n, and n will be the value of r the remainder of m divided by n. Since the remainder of m divided by n will always be less than n, m which is the previous value of n is always greater than n which is the remainder r.

Write this new algorithm in the style of Algorithm E, and call it Algorithm F.

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My Solution: Here is my thinking on this one: In algorithm E, the value of m is not needed after finding the remainder of m divided by n. So instead of saving the remainder step 1 in variable r, it can overwrite the value of m.

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Step 2 then had to be changed since 1 the remainder is now stored in variable m, and 2 the answer if the remainder is zero, is in variable n. The reduction step E3 was replaced with steps F3 and F4 that reverse the division and save the remainder in variable n with the corresponding changes in checking if the remainder is zero. Then we can go back to step F1 since the remainder is now back in n.

algorithm exercises with solutions

Also mention some differences in format between it and Algorithm E. My Solution: It does not meet the guidelines for finiteness, definiteness, and effectiveness. Some of the differences between it and Algorithm E is that algorithm E terminates, it only used mathematical steps, and the result of performing the algorithm is clearly defined.

The algorithm is not finite since it never terminates so it does not satisfy the finiteness. The algorithm has steps that are not precisely defined. For example, how does one determine if the subject of the chapter is interesting step 5how tired do you have to be before you should sleep step Since it has zero inputs, it satisfies the input condition.

The output is specified in step 12 — work exercises so it satisfies the output condition. I am guessing that this algorithm is not effective — for example, step 9 asks if you are mathematically inclined, but there is no effective operation given to determine this. Analysis: I got the first part right at least. My solution: Using my solution to exercise 2 above, n will decrease by at least on each time through step E1, and the remainder of m divided by n will be at most 4.

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So the average is 2. Analysis: Note to self: Answers to questions that Knuth pose need to be preciously figured out, the answer is exact not just about. Oh well.

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Show that Um is well defined. Is Um in any way related to Tm? My Solution: No clue. Maybe this is a cop out, but I think I will see how this method goes.

Analysis: My first question is about the finite number. I did not see the difference right away between Um and Tm, but here is what I get: Um is the average number of times step E1 is performed when m is fixed and n varies. Tm, is the average number of times step E1 is performed when n is fixed and m varies.This is already in triangular form.

Substitute the z into Row 2 and solve for y. Then substitute the y and z into Row 1 to find the x. This is almost in triangular form. Try multiplying Row 2 by 2, then adding that to Row 3.

Get this system in triangular form.

algorithm exercises with solutions

Try multiplying Row 3 by 2, then add that to Row 2. Follow this by multiplying Row 3 by -7, then adding that to Row 1. Rewrite the original Row 1 and the two new equations. One more step to get it to triangular form. Can you do it?

Do you notice anything yet? If not, try multiplying Row 1 by -2, then add Row 1 and Row 2.


Log In. Systems of Equations. Example 1. Solve this system of equations using Gaussian Elimination. Gimme a Hint Hint This is already in triangular form.

Answer 2, 4, 6. Example 2. Gimme a Hint Hint This is almost in triangular form. Answer -1, -1, Example 3. Gimme a Hint Hint Get this system in triangular form. Answer -3, 3, 0. Example 4. Gimme a Hint Hint Do you notice anything yet?The Wiki is an experiment, a grass-roots effort to create an answer key to aid self-study with Steven Skiena's The Algorithm Design Manual. Please do not use this resource to cheat on your class homework.

Recognize that no authority certifies the correctness of these solutions; they could well have been submitted by the idiot who sits in the back row of your class. Also recognize that other students in your class have equal access to these solutions, and it is typically easy for professors to recognize when two students submit the same solution. I Steven Skiena would like to offer a hearty round of thanks to Gordon Rios for all his efforts in setting up this Wiki. It will be interesting to see how well distributed efforts come to populate it.

This is a community editable solutions manual for odd-numbered exercises -- please edit solutions for specific odd-numbered exercises only. The wiki markup is generated by scripts so please report bugs in the problem markup to Gordon. Jump to: navigationsearch. Navigation menu Personal tools Create account Log in.

Namespaces Page Discussion. Views Read View source View history. Navigation Main page Recent changes Random page Help. This page was last modified on 7 Decemberat This page has been accessedtimes.

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Writing code in comment? Please use ide. Find the path from root to the given nodes of a tree for multiple queries Reverse a singly Linked List in groups of given size Set 3 Shortest Path Faster Algorithm Number of subarrays with GCD equal to 1 Number of factors of very large number N modulo M where M is any prime number Maximum sum path in a Matrix Find last remaining element after reducing the Array Minimum increment or decrement required to sort the array Top-down Approach Find lexicographically smallest string in at most one swaps Minimize the maximum difference between adjacent elements in an array Number of pairs such that path between pairs has the two vertices A and B Minimum Cost Graph Traveling Salesman Problem using Genetic Algorithm Count of subarrays having exactly K distinct elements Minimum window size containing atleast P primes in every window of given range Maximum size of square such that all submatrices of that size have sum less than K Why Data Structures and Algorithms Are Important to Learn?

Count the numbers with N digits and whose suffix is divisible by K Huffman Coding using Priority Queue Count maximum occurrence of subsequence in string such that indices in subsequence is in A.

Load Comments.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. The dark mode beta is finally here. Change your preferences any time. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. The book is full of interesting questions, but since I am learning it myself, it would be a great help if I can find solutions to at least some of the questions.

The book's website, algorist. Learn more. Asked 9 years, 1 month ago. Active 2 years, 6 months ago. Viewed 80k times. Anyone knows anything about this? Steve Steve 4, 9 9 gold badges 50 50 silver badges 71 71 bronze badges. Link is working now: www2. The solutions are up once again!

Active Oldest Votes. Uwe Keim It did in February : They must have taken it off the mediawiki. The book's webpage still appears to point to sunysb's mediawiki, so they haven't got the memo yet! The Overflow Blog. Featured on Meta. Community and Moderator guidelines for escalating issues via new response…. Feedback on Q2 Community Roadmap. Technical site integration observational experiment live on Stack Overflow.

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