Pdf recursion

Download Free PDF. Jeff Erickson. A short summary of this paper. Download Download PDF. Translate PDF. Simplify, simplify. Reducing one problem X to another problem or set of problems Y means to write an algorithm for X , using an algorithm or Y as a subroutine or black box. Those data structure operations are black boxes; the apportionment algorithm does not depend on any specific implementation. Conversely, when we design a particular priority queue data structure, we typically neither know nor care how our data structure will be used.

Whether or not the Census Bureau plans to use our code to apportion Congress is completely irrelevant to our design choices. As a general rule, when we design algorithms, we may not know—and we should not care—how the basic building blocks we use are implemented, or how your algorithm might be used as a basic building block to solve a bigger problem.

Your only task is to simplify the original problem, or to solve it directly when simplification is either unnecessary or impossible. The Recursion Fairy will magically take care of the simpler subproblems. Eventually, the recursive reductions must stop with an elementary base case that can be solved by some other method; otherwise, the recursive algorithm will never terminate.

Free distribution is strongly encouraged; commercial distribution is expressly forbidden. The following year, Henri de Parville described the puzzle with the following remarkable story:3 In the great temple at Benares beneath the dome which marks the centre of the world, rests a brass plate in which are fixed three diamond needles, each a cubit high and as thick as the body of a bee.

On one of these needles, at the creation, God placed sixty-four discs of pure gold, the largest disc resting on the brass plate, and the others getting smaller and smaller up to the top one. This is the Tower of Bramah. Day and night unceasingly the priests transfer the discs from one diamond needle to another according to the fixed and immutable laws of Bramah, which require that the priest on duty must not move more than one disc at a time and that he must place this disc on a needle so that there is no smaller disc below it.

When the sixty-four discs shall have been thus transferred from the needle on which at the creation God placed them to one of the other needles, tower, temple, and Brahmins alike will crumble into dust, and with a thunderclap the world will vanish. Of course, being good computer scientists, we read this story and immediately substitute n for the hardwired constant sixty-four. How can we move a tower of n disks from one needle to another, using a third needles as an occasional placeholder, never placing any disk on top of a smaller disk?

The Tower of Hanoi puzzle The trick to solving this puzzle is to think recursively. So now all we have to figure out is how to. Our algorithm does make one subtle but important assumption: There is a largest disk. We must handle that base case directly. Fortunately, the monks at Benares, being good Buddhists, are quite adept at moving zero disks from one needle to another in no time at all.

In fact, for more complicated problems, unfolding the recursive calls is merely distracting. Our only task is to reduce the problem to one or more simpler instances, or to solve the problem directly if such a reduction is impossible. Rouse Ball and H. There is no spoon. Thus, even at the impressive rate of one move per second, the monks at Benares will be at work for approximately billion years before tower, temple, and Brahmins alike will crumble into dust, and with a thunderclap the world will vanish.

According to Donald Knuth, it was suggested by John von Neumann as early as Divide the array A[ Recursively mergesort the subarrays A[ Merge the newly-sorted subarrays A[ All the real work is done in the final step; the two sorted subarrays A[ The base case, where at least one subarray is empty, is straightforward; the algorithm just copies it into B.

Otherwise, the smallest remaining element is either A[i] or A[ j], since both subarrays are sorted, so B[k] is assigned correctly. Otherwise, by the inductive hypothesis, the two smaller subarrays A[ Aside: Domain Transformations. How can we just ignore the floors and ceilings? We can use similar domain transformations to remove floors, ceilings, and lower order terms from any recurrence. In this algorithm, the hard work is splitting the array into subsets so that merging the final result is trivial.

Choose a pivot element from the array. Split the array into three subarrays containing the items less than the pivot, the pivot itself, and the items bigger than the pivot. Recursively quicksort the first and last subarray.

In fact, as we will see later, it is possible to locate the median element in an unsorted array in linear time. However, the algorithm is fairly complicated, and the hidden constant in the O notation is quite large. So in practice, programmers settle for something simple, like choosing the first or last element of the array. This suggests that the average-case running time is O n log n.

Although this intuition is correct, we are still far from a proof that quicksort is usually efficient. We will formalize this intuition about average-case behavior in a later lecture.

Divide the problem into several smaller independent subproblems. Delegate each subproblem to the Recursion Fairy to get a sub-solution. Combine the sub-solutions together into the final solution. If the size of any subproblem falls below some constant threshold, the recursion bottoms out. Hopefully, at that point, the problem is trivial, but if not, we switch to a different algorithm instead.

Proving a divide-and-conquer algorithm correct usually involves strong induction. Analyzing the running time requires setting up and solving a recurrence, which often but unfortunately not always! Their algorithm actually solves the more general problem of selecting the kth largest element in an array, using the following recursive divide-and-conquer strategy. We find the median of each block by brute force and collect those medians into a new array. Finally, either we get lucky and the median-of-medians is the kth largest element of A, or we recursively search one of the two subarrays.

The key insight is that these two subarrays cannot be too large or too small. For purposes of illustration, imagine that we sort every column from top down, and then we sort the columns by their middle element.

In this arrangement, the median-of-medians is the element closest to the center of the grid. A symmetric argument applies when our target element is smaller than the median-of-medians. Finer analysis reveals that the hidden constants are quite large, even if we count only comparisons; this is not a practical algorithm for small inputs.

Similarly, multiplying an n-digit number by a one-digit number takes O n time, using essentially the same algorithm. What about multiplying two n-digit numbers? The algorithm runs in O n2 time—altogether, there are O n2 digits in the partial products, and for each digit, we spend constant time.

Each of the four sub-products e, f , g, h is computed recursively. The last line does not involve any multiplications, however; to multiply by a power of ten, we just shift the digits and fill in the right number of zeros.

We can take this idea even further, splitting the numbers into more pieces and combining them in more complicated ways, to get even faster multiplication algorithms. Ultimately, this idea leads to the development of the Fast Fourier transform, a more complicated divide-and-conquer algorithm that can be used to multiply two n-digit numbers in O n log n time. Notice that the input a could be an integer, or a rational, or a floating point number. For example, the same algorithm can be used to compute powers modulo some finite number an operation commonly used in cryptography algorithms or to compute powers of matrices an operation used to evaluate recurrences and to compute shortest paths in graphs.

All we really require is that a belong to a multiplicative group. The simplification presented here is due to Donald Knuth. The O n log n running time requires the standard assumption that O log n -bit integer arithmetic can be performed in constant time; the number of bit operations is O n log n log log n.

We do not guarantee that these techniques will work for you. Some of the techniques listed in Recursion may require a sound knowledge of Hypnosis, users are advised to either leave those sections or must have a basic understanding of the subject before practicing them.

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