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| The code in Part 1 counts the crossings and nestings for a given matching. We now want to use this to analyze the distribution of crossings and nestings for arc diagrams with 100 arcs. Using your function random_matching from Exercise [1.4], generate 104 such arc diagrams and create histograms for the numbers of crossings and nestings, respectively. Display them in a single plot. Also generate a two-dimensional histogram for the joint distribution of crossings and nestings. What do you observe? [1.4] The numpy module contains functions to generate random permutations. Given a non-negative integer n, every permutation of 2n numbers can be turned into a matching by taking the first half of the permutation entries (i.e., the first n numbers) and connecting each of them to the corresponding entry in the second half (i.e., the last n numbers). Use this method to write a function random_matching that takes n and returns a randomly selected matching of 2n numbers in the form of a list of n tuples, where each tuple contains two integers. Note: Ensure that the tuples are returned such that the first integer is less than the second integer. To demonstrate that your code works, draw the arc diagrams for five randomly chosen matchings of 20 numbers. import numpy as np def random_matching(n): # Generate a random permutation of 2n numbers permutation = np.random. permutation (2*n) # Create the matching from the permutation matching = [(min(permutation[i], permutation[i+n]), max(permutation[i], permutation [i+n])) for i in range(n)] return matching Now, let's use this function to generate five randomly chosen matchings of 20 numbers and draw their arc diagrams: import matplotlib.pyplot as plt def draw_arc_diagram (matching): for pair in matching: plt.plot([pair[0], pair[1]], [1, 1], 'k-') plt.plot(pair[0], 1, 'ro') plt.plot(pair[1], 1, 'ro') plt.yticks([]) plt.show() # Generate and draw five random matchings for i in range(5): matching = random_matching (10) # 10 for 20 numbers print("Matching", i+1, ":", matching) draw_arc_diagram (matching)

| The code in Part 1 counts the crossings and nestings for a given matching. We now want to use this to analyze the distribution of crossings and nestings for arc diagrams with 100 arcs. Using your function random_matching from Exercise [1.4], generate 104 such arc diagrams and create histograms for the numbers of crossings and nestings, respectively. Display them in a single plot. Also generate a two-dimensional histogram for the joint distribution of crossings and nestings. What do you observe? [1.4] The numpy module contains functions to generate random permutations. Given a non-negative integer n, every permutation of 2n numbers can be turned into a matching by taking the first half of the permutation entries (i.e., the first n numbers) and connecting each of them to the corresponding entry in the second half (i.e., the last n numbers). Use this method to write a function random_matching that takes n and returns a randomly selected matching of 2n numbers in the form of a list of n tuples, where each tuple contains two integers. Note: Ensure that the tuples are returned such that the first integer is less than the second integer. To demonstrate that your code works, draw the arc diagrams for five randomly chosen matchings of 20 numbers. import numpy as np def random_matching(n): # Generate a random permutation of 2n numbers permutation = np.random. permutation (2*n) # Create the matching from the permutation matching = [(min(permutation[i], permutation[i+n]), max(permutation[i], permutation [i+n])) for i in range(n)] return matching Now, let's use this function to generate five randomly chosen matchings of 20 numbers and draw their arc diagrams: import matplotlib.pyplot as plt def draw_arc_diagram (matching): for pair in matching: plt.plot([pair[0], pair[1]], [1, 1], 'k-') plt.plot(pair[0], 1, 'ro') plt.plot(pair[1], 1, 'ro') plt.yticks([]) plt.show() # Generate and draw five random matchings for i in range(5): matching = random_matching (10) # 10 for 20 numbers print("Matching", i+1, ":", matching) draw_arc_diagram (matching)

C++ for Engineers and Scientists
C++ for Engineers and Scientists
4th Edition
ISBN:9781133187844
Author:Bronson, Gary J.
Publisher:Bronson, Gary J.
Chapter13: Structures
Section: Chapter Questions
Problem 4PP
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| The code in Part 1 counts the crossings and nestings for a given matching. We now want to use this to analyze the distribution of crossings and nestings for arc diagrams with 100 arcs. Using your function random_matching from Exercise [1.4], generate 104 such arc diagrams and create histograms for the numbers of crossings and nestings, respectively. Display them in a single plot. Also generate a two-dimensional histogram for the joint distribution of crossings and nestings. What do you observe?
Transcribed Image Text:| The code in Part 1 counts the crossings and nestings for a given matching. We now want to use this to analyze the distribution of crossings and nestings for arc diagrams with 100 arcs. Using your function random_matching from Exercise [1.4], generate 104 such arc diagrams and create histograms for the numbers of crossings and nestings, respectively. Display them in a single plot. Also generate a two-dimensional histogram for the joint distribution of crossings and nestings. What do you observe?
[1.4] The numpy module contains functions to generate random permutations. Given a non-negative integer n, every permutation of 2n numbers can be turned into a matching by taking the first half of the permutation entries (i.e., the first n numbers) and connecting each of them to the corresponding entry in the second half (i.e., the last n numbers). Use this method to write a function random_matching that takes n and returns a randomly selected matching of 2n numbers in the form of a list of n tuples, where each tuple contains two integers. Note: Ensure that the tuples are returned such that the first integer is less than the second integer. To demonstrate that your code works, draw the arc diagrams for five randomly chosen matchings of 20 numbers. import numpy as np def random_matching(n): # Generate a random permutation of 2n numbers permutation = np.random. permutation (2*n) # Create the matching from the permutation matching = [(min(permutation[i], permutation[i+n]), max(permutation[i], permutation [i+n])) for i in range(n)] return matching Now, let's use this function to generate five randomly chosen matchings of 20 numbers and draw their arc diagrams: import matplotlib.pyplot as plt def draw_arc_diagram (matching): for pair in matching: plt.plot([pair[0], pair[1]], [1, 1], 'k-') plt.plot(pair[0], 1, 'ro') plt.plot(pair[1], 1, 'ro') plt.yticks([]) plt.show() # Generate and draw five random matchings for i in range(5): matching = random_matching (10) # 10 for 20 numbers print("Matching", i+1, ":", matching) draw_arc_diagram (matching)
Transcribed Image Text:[1.4] The numpy module contains functions to generate random permutations. Given a non-negative integer n, every permutation of 2n numbers can be turned into a matching by taking the first half of the permutation entries (i.e., the first n numbers) and connecting each of them to the corresponding entry in the second half (i.e., the last n numbers). Use this method to write a function random_matching that takes n and returns a randomly selected matching of 2n numbers in the form of a list of n tuples, where each tuple contains two integers. Note: Ensure that the tuples are returned such that the first integer is less than the second integer. To demonstrate that your code works, draw the arc diagrams for five randomly chosen matchings of 20 numbers. import numpy as np def random_matching(n): # Generate a random permutation of 2n numbers permutation = np.random. permutation (2*n) # Create the matching from the permutation matching = [(min(permutation[i], permutation[i+n]), max(permutation[i], permutation [i+n])) for i in range(n)] return matching Now, let's use this function to generate five randomly chosen matchings of 20 numbers and draw their arc diagrams: import matplotlib.pyplot as plt def draw_arc_diagram (matching): for pair in matching: plt.plot([pair[0], pair[1]], [1, 1], 'k-') plt.plot(pair[0], 1, 'ro') plt.plot(pair[1], 1, 'ro') plt.yticks([]) plt.show() # Generate and draw five random matchings for i in range(5): matching = random_matching (10) # 10 for 20 numbers print("Matching", i+1, ":", matching) draw_arc_diagram (matching)
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