The self-crossing problem in Python
The self-crossing problem is a fascinating challenge in computational geometry, where a sequence of line segments intersects itself. This phenomenon is significant in various applications, such as robotics, computer graphics, and pathfinding algorithms. For example, consider a path defined by the points A, B, C, and D. If segments AB and CD intersect, the path is deemed self-crossing, complicating calculations like area and distance. Understanding this problem is crucial for ensuring accuracy in algorithms that depend on well-defined, non-overlapping paths, thereby enhancing performance in real-world applications like navigation and design.
Problem statement
The self-crossing problem is a programming challenge in which we’re provided with a list of integers called distance on an X-Y plane. We need to determine if the path depicted by this list intersects itself. This distance represents the number of units we travel in our path.
In GPS navigation systems, detecting self-crossing paths can help identify incorrect routes. By applying the self-crossing algorithm, navigation apps can give users more accurate directions.
A path is considered self-crossing if it follows the steps below:
We begin at the coordinates
(0,0)and move counter-clockwise.Travel
distance[0]units to the north.Travel
distance[1]units to the west.Travel
distance[2]units to the south.Travel
distance[3]units to the east.After we’ve finished traveling, if the path crosses itself, it’s self-crossing.
In this problem, we return True if these steps are followed. The slides below illustrate an example of a self-crossing path. In the slides, our distance is (1, 2, 1, 3):
As we can see in the slides above, the path is self-crossing.
Knowledge test
In the context of the self-crossing problem, what does the term self-crossing refer to?
A path that never intersects itself.
A path that intersects itself at least once.
A perfectly straight path.
A path that is not defined by distances.
Algorithm
In this problem, we’ll define three helper functions to verify specific self-crossing situations. These helper functions check specific conditions for potential intersection points:
crossedThree(i): This function checks whether self-crossing has occurred involving three distances. If these conditions occur, its self-crossing:
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The current distance is greater than the distance two steps back.
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The distance one step back is less than or equal to the distance three steps back.
crossedFive(i): This function checks whether self-crossing has occurred involving five distances. If these conditions occur, its self-crossing:
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Verify if there are five distances in the list to evaluate.
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The distance two steps back is greater than or equal to the distance four steps back.
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The sum of the current distance and the distance four steps back is greater than or equal to two steps back.
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The distance one step back is less than three steps back.
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The sum of the distances one step back and five steps back is greater than or equal to the distances three steps back.
In summary, these functions check if, at any index i, the path is self-crossing by evaluating a specific amount of distances.
Code
The code for this problem can be seen below:
def isSelfCrossing(distance):def crossedThree(i):return distance[i] >= distance[i - 2] and distance[i - 1] <= distance[i - 3]def crossedFour(i):return i >= 4 and distance[i - 1] == distance[i - 3] and distance[i] + distance[i - 4] >= distance[i - 2]def crossedFive(i):return i >= 5 and distance[i - 2] >= distance[i - 4] and distance[i] + distance[i - 4] >= distance[i - 2] and distance[i - 1] <= distance[i - 3] and distance[i - 1] + distance[i - 5] >= distance[i - 3]if len(distance) < 4:return Falsefor i in range(3, len(distance)):if crossedThree(i) or crossedFour(i) or crossedFive(i):return Truereturn Falseprint(isSelfCrossing([2, 1, 1, 2]))
The Python code can be explained as follows:
Line 1: The definition of the
isSelfCrossingfunction takes distance (an array of integers) as a parameter.Lines 2–3: The
crossedThreehelper function, which verifies the following conditions:The current distance at
iis greater than or equal to the distance provided three steps back.The previous distance is less than or equal to the distance two steps back.
Lines 5–6: The
crossedFourhelper function, similar to the previous function:The current distance is greater than or equal to the distance provided three steps back.
The current distance plus the distance four steps back is greater than or equal to the distance two steps back.
Lines 8–9: The
crossedFivehelper function returnsTrueif the following conditions are met:The index is greater than or equal to
5.The distance two steps back is greater than or equal to the distance four steps back.
The sum of the current distance and the distance four steps back is greater than or equal to two steps back.
The distance one step back is less than or equal to the distance three steps back.
The sum of the distance one step back and five steps back is greater than or equal to the distance three steps back.
Lines 11–12: It returns
Falseif the list size is smaller than4, as a self-crossing pattern is not possible with less than four distances.Lines 14–18: Iterate from the fourth index and call the helper functions. If any of them return
True, it is self-crossing.Line 18: If we don’t encounter any self-crossing, return
False.Line 20: Calling the function with example values.
Time complexity
The time complexity of the isSelfCrossing function in all implementations is O(n), where n is the length of the distance array. This is because we are iterating through the array once, checking three conditions for each index starting from 3 to n-1.
Space complexity
The space complexity of the function is O(1), as we are using a constant amount of space for variables, regardless of the input size. We don't use any additional data structures that grow with the input; we're only storing a few integer variables and function references.
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