Algorithms For Dummies. John Paul Mueller

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from itertools import permutations a = np.array([1,2,3]) for p in permutations(a): print(p)

      The output you see looks like this:

       (1, 2, 3)(1, 3, 2)(2, 1, 3)(2, 3, 1)(3, 1, 2)(3, 2, 1)

      If you want to use the list comprehension (https://www.w3schools.com/python/python_lists_comprehension.asp) approach, which is a shorter method of performing repetitive tasks, you can use [print(p) for p in permutations(a)] instead. You can read more about itertools at https://docs.python.org/3/library/itertools.html.

      Shuffling combinations

      In some cases, you don't need an entire dataset; all you really need are a few of the members in combinations of a specific length. For example, you might have a dataset containing four numbers and want only two-number combinations from it. (The ability to obtain parts of a dataset is a key function for generating a fully connected graph, which is described in Part 3 of the book.) The following code shows how to obtain such combinations:

       from itertools import combinations a = np.array([1,2,3,4]) for comb in combinations(a, 2): print(comb)

      which produces this output:

       (1, 2)(1, 3)(1, 4)(2, 3)(2, 4)(3, 4)

       import random pool = [] for comb in combinations(a, 2): pool.append(comb) print(random.sample(pool, 3))

      The precise combinations you see as output will vary, such as [(1, 2), (2, 3), (1, 4)]. However, the idea is that you've limited your dataset in two ways. First, you’re not using all the data elements all the time, and second, you’re not using all the possible combinations of those data elements. The effect is to create a relatively random-looking set of data elements that you can use as input to an algorithm. Python provides a whole host of randomizing methods that you can see at https://docs.python.org/3/library/random.html. Many of the later examples in this book also rely on randomization to help obtain the correct output from algorithms.

      Facing repetitions

      Repeated data can unfairly weight the output of an algorithm so that you get inaccurate results. Sometimes you need unique values to determine the outcome of a data manipulation. Fortunately, Python makes it easy to remove certain types of repeated data. Consider this example:

       a = np.array([1,2,3,4,5,6,6,7,7,1,2,3])b = np.array(list(set(a))) print(b)

      The output contains only the unique elements:

       [1, 2, 3, 4, 5, 6, 7]

      

In this case, a begins with an assortment of numbers in no particular order and with plenty of repetitions. In Python, a set never contains repeated data. Consequently, by converting the list in a to a set and then back to a list, and then placing that list in an array, you obtain a vector that has no repeats.

      Recursion, an elegant method of solving many computer problems, relies on the capability of a function to continue calling itself until it satisfies a particular condition. The term recursion actually comes from the Latin verb recurrere, which means to run back.

      When you use recursion, you solve a problem by calling the same function multiple times but modifying the terms under which you call it. The main reason for using recursion is that it provides an easier way to solve problems when working with some algorithms because it mimics the way a human would solve it. Unfortunately, recursion is not an easy tool because it requires some effort to understand how to build a recursive routine, and it can cause out-of-memory problems on your computer if you don't set some memory settings. The following sections detail how recursion works and give you an example of how recursion works in Python.

      Explaining recursion

      Notice the conditional in the center. To make recursion work, the function must have such a conditional or it could become an endless loop. The conditional determines one of two things:

       The conditions for ending recursion haven’t been met, so the function must call itself again.

       The conditions for ending recursion have been met, so the function returns a final value that is used to calculate the ending result.

Schematic illustration of the recursion process, a function continuously calls itself until it meets a condition.

      FIGURE 4-1: In the recursion process, a function continuously calls itself until it meets a condition.

When a function calls itself, it doesn’t use the same arguments that were passed to it. If it continuously used the same arguments, the condition would never change and the recursion would never end. Consequently, recursion requires that subsequent calls to the function must change the call arguments in order to bring the function closer to an ending solution.

      One of the most common examples of recursion for all programming languages is the calculation of a factorial. A factorial is the multiplication of a series of numbers between a starting point and an ending point in which each number in the series is one less than the number before it. For example, to calculate 5! (read as five factorial), you multiple 5 * 4 * 3 * 2 * 1. The calculation represents a perfect and simple example of recursion. Here’s the Python code you can use to perform the calculation.

       def factorial(n): print("factorial called with n = ", str(n)) if n

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