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def count_crossings_and_nestings(arcs): n = len (arcs) // 2 crossings = 0 nestings = 0 for i in range(2 * n): for j in range(i + 1, 2 * n): if arcs [i] [0] < arcs [j] [0] < arcs [i] [1] < arcs [j] [1]: crossings += 1 elif arcs [i] [0] > arcs [j] [0] > arcs[i] [1] > arcs[j][1]: crossings += 1 elif arcs[i][0] < arcs [j] [0] < arcs [j] [1] < arcs [i] [1]: nestings += 1 elif arcs [i] [0] > arcs [j] [0] > arcs [i] [1] > arcs [j][1]: nestings += 1 return crossings, nestings #Test cases test_cases=[ [], [(0,1)], [(0,1),(2,3)], [(0,3), (1,2)], [(0,2), (1,3)] ] # Empty set of arcs # Single arc # Two non-overlapping arcs # Two overlapping arcs # Two nested arcs for i, arcs in enumerate(test_cases, 1): crossings, nestings = count_crossings_and_nestings(arcs) print (f"Test case {i}:") print (f"Number of arcs: {len(arcs)}") print (f"Number of crossings: {crossings}") print (f"Number of nestings: {nestings}") print() Explanation below: {} (empty set of arcs): The output can be defined as (0, 0). {(0, 1)} (single arc): The output can be defined as (0, 0) because there are no crossings or nestings. {(0, 1), (2, 3)} (two non-overlapping arcs): The output can be defined as (0, 0) because there are no crossings or nestings. {(0, 3), (1, 2)} (two overlapping arcs): The output can be defined as (1, 0) because there is one crossing but no nestings. {(0, 2), (1, 3)} (two nested arcs): The output can be defined as (0, 1) because there are no crossings but one nesting. The output matches the expectations for all the test cases. Therefore, the function is working.

def count_crossings_and_nestings(arcs): n = len (arcs) // 2 crossings = 0 nestings = 0 for i in range(2 * n): for j in range(i + 1, 2 * n): if arcs [i] [0] < arcs [j] [0] < arcs [i] [1] < arcs [j] [1]: crossings += 1 elif arcs [i] [0] > arcs [j] [0] > arcs[i] [1] > arcs[j][1]: crossings += 1 elif arcs[i][0] < arcs [j] [0] < arcs [j] [1] < arcs [i] [1]: nestings += 1 elif arcs [i] [0] > arcs [j] [0] > arcs [i] [1] > arcs [j][1]: nestings += 1 return crossings, nestings #Test cases test_cases=[ [], [(0,1)], [(0,1),(2,3)], [(0,3), (1,2)], [(0,2), (1,3)] ] # Empty set of arcs # Single arc # Two non-overlapping arcs # Two overlapping arcs # Two nested arcs for i, arcs in enumerate(test_cases, 1): crossings, nestings = count_crossings_and_nestings(arcs) print (f"Test case {i}:") print (f"Number of arcs: {len(arcs)}") print (f"Number of crossings: {crossings}") print (f"Number of nestings: {nestings}") print() Explanation below: {} (empty set of arcs): The output can be defined as (0, 0). {(0, 1)} (single arc): The output can be defined as (0, 0) because there are no crossings or nestings. {(0, 1), (2, 3)} (two non-overlapping arcs): The output can be defined as (0, 0) because there are no crossings or nestings. {(0, 3), (1, 2)} (two overlapping arcs): The output can be defined as (1, 0) because there is one crossing but no nestings. {(0, 2), (1, 3)} (two nested arcs): The output can be defined as (0, 1) because there are no crossings but one nesting. The output matches the expectations for all the test cases. Therefore, the function is working.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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When we want to write Python code to count the number of crossings and nestings of an arc diagram,an example is attached

Question: For diagrams with three arcs, there are 15 possibilities. Find one for which the function count_crossings_and_nestings provides a wrong result. Print the result and draw the corresponding diagram.

def count_crossings_and_nestings(arcs): n = len (arcs) // 2 crossings = 0 nestings = 0 for i in range(2 * n): for j in range(i + 1, 2 * n): if arcs [i] [0] < arcs [j] [0] < arcs [i] [1] < arcs [j] [1]: crossings += 1 elif arcs [i] [0] > arcs [j] [0] > arcs[i] [1] > arcs[j][1]: crossings += 1 elif arcs[i][0] < arcs [j] [0] < arcs [j] [1] < arcs [i] [1]: nestings += 1 elif arcs [i] [0] > arcs [j] [0] > arcs [i] [1] > arcs [j][1]: nestings += 1 return crossings, nestings #Test cases test_cases=[ [], [(0,1)], [(0,1),(2,3)], [(0,3), (1,2)], [(0,2), (1,3)] ] # Empty set of arcs # Single arc # Two non-overlapping arcs # Two overlapping arcs # Two nested arcs for i, arcs in enumerate(test_cases, 1): crossings, nestings = count_crossings_and_nestings(arcs) print (f"Test case {i}:") print (f"Number of arcs: {len(arcs)}") print (f"Number of crossings: {crossings}") print (f"Number of nestings: {nestings}") print() Explanation below: {} (empty set of arcs): The output can be defined as (0, 0). {(0, 1)} (single arc): The output can be defined as (0, 0) because there are no crossings or nestings. {(0, 1), (2, 3)} (two non-overlapping arcs): The output can be defined as (0, 0) because there are no crossings or nestings. {(0, 3), (1, 2)} (two overlapping arcs): The output can be defined as (1, 0) because there is one crossing but no nestings. {(0, 2), (1, 3)} (two nested arcs): The output can be defined as (0, 1) because there are no crossings but one nesting. The output matches the expectations for all the test cases. Therefore, the function is working.
Transcribed Image Text:def count_crossings_and_nestings(arcs): n = len (arcs) // 2 crossings = 0 nestings = 0 for i in range(2 * n): for j in range(i + 1, 2 * n): if arcs [i] [0] < arcs [j] [0] < arcs [i] [1] < arcs [j] [1]: crossings += 1 elif arcs [i] [0] > arcs [j] [0] > arcs[i] [1] > arcs[j][1]: crossings += 1 elif arcs[i][0] < arcs [j] [0] < arcs [j] [1] < arcs [i] [1]: nestings += 1 elif arcs [i] [0] > arcs [j] [0] > arcs [i] [1] > arcs [j][1]: nestings += 1 return crossings, nestings #Test cases test_cases=[ [], [(0,1)], [(0,1),(2,3)], [(0,3), (1,2)], [(0,2), (1,3)] ] # Empty set of arcs # Single arc # Two non-overlapping arcs # Two overlapping arcs # Two nested arcs for i, arcs in enumerate(test_cases, 1): crossings, nestings = count_crossings_and_nestings(arcs) print (f"Test case {i}:") print (f"Number of arcs: {len(arcs)}") print (f"Number of crossings: {crossings}") print (f"Number of nestings: {nestings}") print() Explanation below: {} (empty set of arcs): The output can be defined as (0, 0). {(0, 1)} (single arc): The output can be defined as (0, 0) because there are no crossings or nestings. {(0, 1), (2, 3)} (two non-overlapping arcs): The output can be defined as (0, 0) because there are no crossings or nestings. {(0, 3), (1, 2)} (two overlapping arcs): The output can be defined as (1, 0) because there is one crossing but no nestings. {(0, 2), (1, 3)} (two nested arcs): The output can be defined as (0, 1) because there are no crossings but one nesting. The output matches the expectations for all the test cases. Therefore, the function is working.
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