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Bottom-up Mergesort

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Algoritmos, Parte I

Universidade de Princeton

4.9 (5,880 classificações) | 460K alunos inscritos

This course covers the essential information that every serious programmer needs to know about algorithms and data structures, with emphasis on applications and scientific performance analysis of Java implementations. Part I covers elementary data structures, sorting, and searching algorithms. Part II focuses on graph- and string-processing algorithms. All the features of this course are available for free. It does not offer a certificate upon completion.

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Habilidades que você aprenderá

Data Structure, Algorithms, Java Programming

Avaliações

4.9 (5,880 classificações)

5 stars
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BJ

Jun 03, 2018

Good contents and the logic of the whole course structure is very clear for a novice like me. The weekly homework is also awesome. Would recommend to anyone who wants to learn about computer science.

KR

Dec 02, 2016

An amazing course which enables one to appreciate and analyze various algorithms used in various applications. Must take for anyone who wants to build a career in computer science or related fields.

Na lição

Mergesort

We study the mergesort algorithm and show that it guarantees to sort any array of n items with at most n lg n compares. We also consider a nonrecursive, bottom-up version. We prove that any compare-based sorting algorithm must make at least n lg n compares in the worst case. We discuss using different orderings for the objects that we are sorting and the related concept of stability.

Bottom-up Mergesort3:20

Sorting Complexity9:05

Comparators6:43

Ministrado por

Kevin Wayne
Phillip Y. Goldman '86 Senior Lecturer
Robert Sedgewick
William O. Baker *39 Professor of Computer Science

Transcrição

Next, we're going to look at a bottom-up version of Mergesort. Well, Mergesort is easy to understand as a recursive program. This bottom-up version that has no recursion, it's also quite simple to understand and to code up. The basic idea is to think of the array as being a little at the begining a set of little sorted sub arrays of size one. And then what this method will do is go through and merge those little subarrays of size one together in pairs to get subarrays of size two. Then, the whole array consists of sorted subarrays to size two, and then we make another pass through to get size four, and then size eight, and so forth. So, as you can see in this example we start out by merging the first two sub arrays of size one to make a array of size two - E, M - that's sorted, and then do the same thing for the next two elements and the next two and so forth until eventually instead of sixteen individual elements we have eight sorted subarrays of size two. Then on another pass through, we can take the E, M and the G, R and merge them together to make EGMR, and the E, S and the O, R merge those together to make EORS, and so forth. And we have four subarrays of size four. One more pass makes two subarrays of size eight, and the last pass is just a sorted array. The bottom line in this is sequence of passes through the whole array and there's no recursion needed at all. It's extremely easy to code up as you can see from this code. We use the same merge code as before and we take a nested for loop. The first one is the size of the subarray and this loop gets executed on a log N times because each time we double the size of the subarray until we get to N. And then we pass through picking out from low to low+size-1, and then the next part is low+size+size-1 until we run to the end of the array where we might not have a full subarray of size sz. That is a fully complete industrial strength code for sorting. The only downsize as would regular Mergesort is that it uses extra space proportional to the size of the array. But otherwise, that's a fine method for merging. That's a bottom-up Mergesort. If you look at this visual trace you can see how it works. The thing is totally unsorted, then it gets sorted until subarrays to size four, then eight, sixteen, and 32. Now in this case the second subarray to be sorted is smaller but the merge routine doesn't really care about that so much. You can merge things that are not equal in size. And then we get a final sorted array. Whatever the size, bottom of Mergesort gets the job done in log N passes. Each pass using about N compares for a total cost of about N log N.

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