Check If It Is a Straight Line

This problem is asking us to determine if a set of points in the 2D plane all fall on the same straight line.

A key insight to solving this problem is to understand that if points are on the same line, the slope between any two points on the line will be the same. The slope of a line through points (x1, y1) and (x2, y2) can be calculated using the formula:

slope = (y2 - y1) / (x2 - x1)

With this insight, we can solve this problem by calculating the slope between the first two points and then checking if the slope between every subsequent pair of points is the same.

One thing we need to be cautious about is division by zero when calculating the slope. This could happen if x2 and x1 are the same, which means the line is vertical. To avoid this, we could subtract the x’s and y’s separately and then compare the differences instead of calculating the slope directly.

Python solution:

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class Solution:
    def checkStraightLine(self, coordinates: List[List[int]]) -> bool:
        (x0, y0), (x1, y1) = coordinates[:2]
        for x, y in coordinates[2:]:
            if (x1 - x0) * (y - y1) != (y1 - y0) * (x - x1):
                return False
        return True

In the code, (x1 - x0) * (y - y1) != (x - x1) * (y0 - y1) is actually the cross product of the vector from point (x1, y1) to (x0, y0) and the vector from point (x1, y1) to (x, y). If the cross product is not zero, that means the three points are not collinear.

10 Prerequisite LeetCode Problems

Here are 10 problems related to similar concepts, which are about coordinate geometry, slopes, and line properties.

Two Sum (LeetCode 1): This is a simple problem that introduces you to the concept of using a hash map (dictionary in Python) to store and retrieve data based on certain conditions.
Valid Boomerang (LeetCode 1037): This problem will make you comfortable with using slopes to check the collinearity of points.
Valid Triangle Number (LeetCode 611): This problem allows you to understand the geometric constraints for the formation of a certain shape (triangle in this case).
Minimum Time Visiting All Points (LeetCode 1266): This problem introduces the concept of calculating distances in a grid and using it to solve problems.
Max Points on a Line (LeetCode 149): This problem is a more complex version of “Check If It Is a Straight Line” as it requires you to find the maximum number of points that can be located on the same line.
Robot Bounded In Circle (LeetCode 1041): This is a grid-based problem where you deal with movements based on instructions.
Perfect Rectangle (LeetCode 391): This problem focuses on verifying whether given points can form a perfect rectangle.
Rectangle Overlap (LeetCode 836): This problem involves understanding the necessary and sufficient conditions for rectangles to overlap.
Number of Islands (LeetCode 200): This problem helps you in understanding how to traverse grid-based structures and is fundamental for dealing with coordinate-based problems.
Island Perimeter (LeetCode 463): It gives you a different perspective on how to handle grid-based problems, where you need to count the perimeter of islands.

These cover how to manipulate and use coordinates and other geometric properties to your advantage.

Clarification Questions

What are the clarification questions we can ask about this problem?

Problem Analysis and Key Insights

What are the key insights from analyzing the problem statement?

Problem Boundary

What is the scope of this problem?

How to establish the boundary of this problem?

Problem Classification

Let’s break down the problem statement for the problem “Check If It Is a Straight Line”.

Problem Statement: You are given an array of coordinates, coordinates[i] = [x, y], where [x, y] represents the coordinate of a point. Check if these points make a straight line in the XY plane.

Domain Categorization: This problem falls under the domain of Geometry and Mathematics. Specifically, it’s about understanding properties of lines in a 2D Cartesian plane.

‘What’ Components:

Array of coordinates: The input is an array of points, each having an X and Y coordinate.
Straight Line: The goal is to determine whether the given points form a straight line in the XY plane.

Problem Classification: Based on the problem statement and its components, we can classify this problem as a “Decision Problem”. The reason for this is because our task is to make a binary decision: the points either form a straight line (true) or they don’t (false).

Explanation: When breaking down any problem, it’s crucial to identify the domain it belongs to, as that helps in framing the appropriate problem-solving strategy. This problem lies within the Geometry and Mathematics domain because it’s all about understanding the properties of lines and coordinates.

The ‘What’ components, array of coordinates and the concept of a straight line, help us in understanding the nature of the problem and the input-output relationship.

Finally, the classification as a ‘Decision Problem’ helps in establishing what kind of answer we’re looking for. Decision problems require a binary answer (Yes/No, True/False), which guides our problem-solving approach accordingly. For this problem, our task is to return a boolean value indicating whether or not the given points form a straight line.

Distilling the Problem to Its Core Elements

Fundamental Concept: The fundamental concept this problem is based upon is the property of a straight line in a 2D plane. Specifically, for a set of points to form a straight line, the slope between any two points in the set must be the same.

Simplest Description: Imagine you are given a list of locations on a map. Your task is to check if you can connect all these locations with just a single straight line.

Core Problem: The core problem we are trying to solve is identifying whether a set of 2D points share the same slope, and thus form a straight line.

Simplified Problem Statement: Given a set of points in a 2D space, do they all lie on the same straight line?

Key Components:

Set of 2D points: Each point is represented by its x and y coordinates.
Slope Calculation: For each pair of points, we need to calculate the slope between them.
Slope Comparison: Check if the calculated slopes are equal for all pairs of points.

Minimal Set of Operations:

Compute the slope between the first two points.
Iterate over the remaining points. For each point, compute the slope between this point and the first point.
If the slope is not the same as the one computed in the first step, return False (the points don’t form a straight line).
If we’ve gone through all points and the slopes were always equal, return True (the points form a straight line).

Visual Model of the Problem

Visualizing this problem involves picturing a two-dimensional (2D) Cartesian plane with x and y coordinates.

For example, let’s consider the problem’s first example: coordinates = [[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]]. Here we are given seven points, which we can plot on a 2D Cartesian plane:

7 -    .
6 -   .
5 -  .
4 - .
3 - .
2 -.
1 -
0 -------------------------------
   1 2 3 4 5 6 7 8 9 10 11 12 ...

From this visualization, it’s clear that these points fall along a straight line, meaning the slope between any two consecutive points is the same.

Now, consider the second example from the problem: coordinates = [[1,1],[2,2],[3,4],[4,5],[5,6],[7,7]]. Plotting these points:

7 -       .
6 -     .
5 -   .
4 - .
3 -
2 -.
1 -.
0 -------------------------------
   1 2 3 4 5 6 7 8 9 10 11 12 ...

In this visualization, it’s clear that the points do not fall on a straight line. The slope between the points changes, so we conclude that the points do not form a straight line.

Visualizing the problem in this way can be extremely useful in understanding the nature of the problem and finding a solution.

Problem Restatement

Certainly, I’ll paraphrase the problem statement:

We’re given an array of pairs of numbers. Each pair represents the x and y coordinates of a point in a 2D plane. Our task is to check whether these points form a straight line when plotted on the plane.

To simplify, imagine you’re given a few dots on a piece of graph paper, and you need to determine if you could connect all these dots with a single straight line. If yes, then return true. Otherwise, return false.

Here are the constraints:

The array will have at least two pairs of numbers and at most 1000.
Each pair will consist of two integers that can range from -10,000 to 10,000.
There will be no duplicate coordinates, meaning no two points are the same.

This is a simpler and more direct way to understand the problem. The goal is to determine if all the given points lie on a straight line, abiding by the constraints provided.

Abstract Representation of the Problem

You are given a set of N tuples, each containing two integer values. You can think of these tuples as representing points in a 2-dimensional space. Your task is to determine whether all of these points belong to a single line.

This essentially becomes a problem of checking the consistency of the difference (slope) between every two consecutive points. If the difference remains constant, then all points lie on a straight line.

To express this in more abstract terms:

You have a set of N tuples, where 2 ≤ N ≤ 1000.
Each tuple contains two integers, each of which can range from -10^4 to 10^4.
Each tuple in the set is unique.
Your task is to verify if a constant property (in this case, the slope) exists between each consecutive pair of tuples.

This abstract representation eliminates the specific context of “coordinates” and “straight lines”, focusing instead on the underlying mathematical concept and data structure manipulation.

Terminology

Understanding this problem does require knowledge of a few mathematical concepts:

Coordinates: A coordinate is a pair of values that specifies the position of a point in a space. In this case, we are dealing with 2-dimensional Cartesian coordinates, where each pair (x, y) defines a point in a 2D plane.
Straight Line: In 2D geometry, a straight line is the shortest distance between two points. It is uniquely defined by any two points on the line.
Slope: The slope of a line is a measure of the vertical change (delta y) for each unit of horizontal change (delta x). It is calculated by the formula (y2 - y1) / (x2 - x1) for any two points (x1, y1) and (x2, y2) on the line. For a straight line, the slope is constant for all pairs of points on the line.
Collinearity: Collinearity is the property where all points lie on a single straight line. To check for collinearity among three points, one usually checks whether the slopes between pairs of points are equal.

In the context of this problem:

We are given a set of coordinates.
We need to determine whether they all lie on a single straight line.
We do this by checking if the slope between each pair of consecutive points is constant.
If so, the points are collinear. If not, they are not.

Problem Simplification and Explanation

So, imagine you’re playing a game of “connect-the-dots” on a piece of graph paper. Each dot represents a point on the paper with a distinct x and y coordinate. Now, you have a list of dots you need to connect.

The key question you’re trying to answer is: “Can I draw one straight line that passes through all these dots?” If you can, then the dots make a straight line, if not, they do not.

To check this, you start by drawing a line from the first dot to the second. Then, you note the slope of this line - this is like measuring the “steepness” or “angle” of the line you’ve drawn.

Now, you compare this slope with the slopes of lines drawn between the rest of the consecutive dots. You continue connecting the dots in this manner, always checking the slope. If at any point, the slope is different from the initial one, it means you had to change your drawing angle, indicating that the dots don’t line up in a straight line.

But if all the slopes are the same, it means your pencil stayed at the same angle for the entire drawing, and you’ve successfully drawn a straight line through all the dots.

The key concept here is the mathematical principle that all points lie on a straight line if and only if the slopes between any two consecutive points are the same, or in other words, the dots are collinear.

Let’s put it into a real-life analogy: Consider these points as a set of cities you’re planning to visit on a road trip, and the straight line is the most efficient route. If all cities line up perfectly in a straight line, then you can drive in a straight line from the first city, through all the others, to the last one without turning your steering wheel (ignoring the practicality of real-world driving). But, if any city is off this line, you’d have to make a turn to visit it, meaning the cities don’t line up in a straight line.

Constraints

Let’s dissect the problem and its constraints to identify some useful characteristics that we could leverage for an efficient solution:

Points in 2D Space: The problem involves points in 2D space. This directly suggests that geometric properties, such as slope of a line, could be relevant in finding a solution. In fact, as discussed earlier, the concept of slope is crucial in determining whether points lie on the same straight line.
Size of the Array: The number of coordinates in the array is between 2 and 1000. Given that the size of the input array is fairly moderate, an algorithm with time complexity of O(n) or O(n log n) would be efficient. This allows us to consider linear scan or sort-based approaches.
Range of Coordinates: The coordinates range from -10^4 to 10^4. The relatively large range of values indicates that we should avoid solutions that require space proportional to the values of the coordinates. This rules out certain “counting” type solutions that may have otherwise been feasible with a smaller range.
Non-Duplicate Points: The problem statement specifies that the coordinates contain no duplicate point. This eliminates the need for handling special cases where multiple points have the same coordinates, simplifying our solution.
Two or More Points: There are always at least two points in the input. This is useful because a line can be defined by any two points. We can use the first two points to determine the desired slope and then verify that all other points fall on the line with this slope.
Same Difference: We are looking for points that share the same difference in x and y coordinates, i.e., the slope. This is crucial to understand as it forms the basis of the problem - finding a straight line through all points, or in other words, finding a common slope among all points.

Looking for such specific characteristics or conditions is an important step in devising efficient solutions, as it helps us determine what aspects of the problem can be leveraged and what algorithms or data structures might be appropriate.

Analyzing the constraints can often provide valuable insights that inform our approach to the problem. Here are some key insights we can gain from the constraints of this problem:

Fixed range of coordinate values: Each coordinate value is within the range of -10^4 to 10^4. This implies we need to handle relatively large integers, and this range eliminates the possibility of using solutions that rely on the coordinate values as indices in an array or similar data structure.
Moderate size of the coordinates array: The number of points (2 to 1000) is moderate. We can rule out concerns about handling large amounts of data, and it suggests that a solution with linear or near-linear time complexity should be feasible.
No Duplicate Points: Knowing that no two points share the same coordinates simplifies the problem. It removes the need to account for edge cases where points overlap.
Points Form a Straight Line: The nature of the problem (checking if points form a straight line) informs us about the most critical aspect - the slope. The fact that all points should lie on the same line implies that the difference in y and x coordinates (i.e., the slope) should be the same for all consecutive points.

These insights guide the solution towards a linear scan of the array, checking the slope between each pair of consecutive points. They also help to rule out certain strategies or data structures that might be inappropriate given the constraints. For example, the absence of duplicate points means we don’t need to consider data structures like sets or dictionaries to track unique points.

Case Analysis

Here are a few additional examples with different scenarios.

Normal Case Input: coordinates = [[1,1],[2,2],[3,3],[4,4],[5,5]] Output: true In this case, the coordinates form a straight line along the diagonal of the XY plane, so the function should return true.
Normal Case Input: coordinates = [[1,2],[2,4],[3,6],[4,8],[5,10]] Output: true Even though the points aren’t along the diagonal, they still form a straight line with a slope of 2. Therefore, the function should return true.
Edge Case - Two Points Only Input: coordinates = [[1,1],[2,2]] Output: true Any two points will always form a straight line. Therefore, the function should return true.
Edge Case - Not a Line Input: coordinates = [[1,1],[2,2],[3,4]] Output: false Here, the third point does not lie on the line formed by the first two points. The function should return false.
Edge Case - Vertical Line Input: coordinates = [[1,1],[1,2],[1,3],[1,4]] Output: true All points have the same x-coordinate, forming a vertical line on the XY plane. The function should return true.
Edge Case - Horizontal Line Input: coordinates = [[1,1],[2,1],[3,1],[4,1]] Output: true All points have the same y-coordinate, forming a horizontal line on the XY plane. The function should return true.
Edge Case - Large Input Input: coordinates = [[1,1],[10000,10000],[-10000,-10000]] Output: true Even with large inputs, as long as they lie on the same line, the function should return true.

Visualization of these examples would be to plot these points on the XY plane and visually confirm if they form a straight line or not.

Upon analyzing the different cases, we can gain several key insights:

The Two-Point Case: Any two points can always form a straight line. This is the simplest case and can be used as a base case in our solution.
The Line Case: If all points are on the same line, be it diagonal, horizontal, or vertical, the function should return true. This teaches us that orientation of the line doesn’t matter in the problem. What matters is if all points lie on the same line.
The Non-Line Case: If any point doesn’t lie on the line formed by any two points, the function should return false. This emphasizes the requirement that all points need to lie on the same line.
The Large Input Case: Even with large inputs, as long as they lie on the same line, the function should return true. This highlights that the size of the coordinates doesn’t affect whether they form a straight line or not.

These insights guide us to think about the slope when comparing three or more points. The slope between any two points in the line should be the same for all pairs of points. This is the core logic that will drive our solution.

Additionally, it’s worth noting that the coordinates could be in any order, so our solution needs to take that into account. It should not assume that the coordinates are presented in a specific sequence.

Identification of Applicable Theoretical Concepts

There are indeed a few mathematical and algorithmic concepts that can be applied to simplify the problem and make it more manageable:

Slope of a Line: In mathematics, the slope of a line is a measure of the direction and steepness of a line. The slope can be calculated by the formula (y2 - y1) / (x2 - x1) for any two points (x1, y1) and (x2, y2) on the line. For all points to be on a straight line, the slope between any two points must be the same.
Handling Division by Zero: If the line is vertical, then the calculation of the slope would cause division by zero. However, this can be handled by comparing the differences in the x-coordinates and the y-coordinates separately. If the difference in the x-coordinates (x2 - x1) is zero, then the difference in the y-coordinates (y2 - y1) must also be zero for the points to lie on the same line. This avoids the division by zero error and correctly handles vertical lines.
Sequential Checking: Since the order of the points in the input array does not matter, the algorithm can be designed to sequentially check each point with the first two points. If the slope between the first two points and the slope between the current point and the first point are the same, the algorithm can continue to the next point. If not, the algorithm can immediately return false without having to check the rest of the points.
Linear Time Complexity: The problem can be solved with a time complexity of O(n), where n is the number of points in the input array. This is because each point needs to be checked exactly once. The space complexity is O(1), since no additional space is needed.

By understanding and applying these mathematical and algorithmic concepts, the problem becomes more straightforward and manageable to solve.

Simple Explanation

Imagine you have a bunch of dots on a piece of graph paper. Now, you’re given a ruler. Your task is to find out if you can place the ruler in such a way that it passes through all the dots. This means all the dots should lie on a straight line.

The coordinates of the dots on the paper are the points given in the problem, and your job is to see if you can line up your ruler in such a way that all these dots touch the edge of the ruler. If they can, then you return ’true’, which means ‘yes, they do form a straight line’. If not, then you return ‘false’, which means ’no, they don’t form a straight line’.

To relate it to a daily life example, think of it like lining up a row of dominoes. If you can push the first domino and all the others fall in sequence, then they were lined up straight! But if you push the first one and any domino doesn’t fall, it means there was a domino not in the straight line. It’s the same principle here - we are just checking if all the points (like dominoes) are in a straight line or not.

Problem Breakdown and Solution Methodology

To solve this problem, I’d like you to think of it in terms of slopes. In geometry, the slope of a line is the measure of the vertical change (often called “rise”) for each unit of horizontal change (often called “run”). If all the points lie on a straight line, the slope between any two points on that line will be the same.

Here’s the step-by-step process:

Calculate the initial slope: Start by calculating the slope between the first two points in the given array. The formula for the slope between points (x1, y1) and (x2, y2) is (y2 - y1) / (x2 - x1). This will give us a baseline to compare against the slopes between other points.
Compare slopes: Now, for each subsequent point in the array, calculate the slope between the current point and the first point. If the slope isn’t the same as the initial slope calculated in step 1, return false because this indicates that the points don’t all lie on a single straight line.
Final result: If you’ve calculated the slope for all points and they’re all the same, return true, indicating that all the points lie on a single straight line.

To explain this using an analogy, think of this approach like checking if a group of people all have the same height. First, you measure the height of the first person (like calculating the initial slope). Then, you measure everyone else’s height and compare it to the first person’s height (like comparing slopes). If you find someone who doesn’t have the same height, you know not everyone in the group is the same height. If everyone does have the same height, then everyone in the group is the same height.

Now, if the points’ coordinates were to change or additional points were added, you would just apply the same process. Calculate the slope between the first two points, and then compare this slope with the slope between each subsequent point and the first point.

Let’s demonstrate this with an example:

Suppose the points are (1,2), (2,3), (3,4), (4,5), (5,6), (6,7).

Step 1: The slope between the first two points (1,2) and (2,3) is (3 - 2) / (2 - 1) = 1.

Step 2: Now we calculate the slope between each subsequent point and the first point (1,2):

For point (3,4), the slope is (4 - 2) / (3 - 1) = 1
For point (4,5), the slope is (5 - 2) / (4 - 1) = 1
For point (5,6), the slope is (6 - 2) / (5 - 1) = 1
For point (6,7), the slope is (7 - 2) / (6 - 1) = 1

All these slopes are the same as the initial slope, so we return true. Therefore, all the points lie on a straight line.

Inference of Problem-Solving Approach from the Problem Statement

Here are the key terms and concepts:

Point: A point in this problem is a pair of x and y coordinates, represented as an array of two integers. The concept of a point is fundamental in understanding this problem because we’re checking whether all given points lie on the same straight line.
Coordinates: These are pairs of numbers that indicate the position of a point in a plane. Understanding this term is crucial because it’s the main input of our problem.
Straight Line: This is a concept from geometry. A line is straight if it lies evenly with the points on itself. The problem is basically asking us to verify if all the points lie on the same straight line.
Slope: The slope of a line is a measure of how steep the line is, and it’s calculated as the vertical difference between two points (rise) divided by the horizontal difference (run). It is a key concept because it gives us a strategy for determining whether all points lie on the same line - if all points form the same slope with one fixed point, they are on the same line.
Iteration: Iteration is a programming concept where you repeat a certain set of operations for each item in a collection (in this case, the points in our list). In this problem, we iterate through the points to calculate and compare slopes.

In terms of visualizing these concepts, drawing a graph on a paper can be helpful.

Let’s say we have four points: A(1,2), B(2,3), C(3,4), and D(4,5). You can draw a Cartesian plane and plot these points on it. You’ll notice that these points form a straight line.

Next, understand that the slope between any two points in this line will be the same. For example, slope of AB will be equal to slope of AC or AD. This property of straight line gives us a way to determine if given points lie on the same straight line or not, which is the core of our solution to this problem.

The connection between the problem statement and the concept of slope comes from understanding the basic geometric concept of a straight line.

In a 2D space, a set of points lies on a straight line if and only if the slope between any two points in the set is the same. The slope, in simple terms, is a measure of the steepness of a line.

The problem statement asks us to check if given points form a straight line, which leads us to apply the geometric property of a straight line just described.

The slope between two points (x1, y1) and (x2, y2) is calculated as (y2 - y1) / (x2 - x1). If we choose one point as a reference, we can calculate the slope of this reference point with all other points. If these slopes are the same, it means all points lie on the same straight line. If any slope differs, it means that the points do not all lie on the same straight line.

So, by knowing the property of a straight line (the concept of a consistent slope), we infer from the problem statement that it can be solved using the concept of slope.

Simple Explanation of the Proof

Certainly, let’s break it down in a simplified manner. We’re checking whether a given set of points in a 2D space forms a straight line. We’re using the property of the slope of a line to determine this.

Here’s the fundamental geometric principle we’re leveraging: For a straight line in a two-dimensional space, the slope between any two points on the line is always the same.

The slope between two points A(x1, y1) and B(x2, y2) is calculated as: (y2 - y1) / (x2 - x1).

Here’s the simplified explanation of the proof:

We choose one point from the given set as a reference point.
We calculate the slope of the line between this reference point and every other point in the set.
We check if the slopes calculated in step 2 are all equal.
If all the slopes are equal, it means all the points lie on the same straight line. Hence, the given set of points forms a straight line.
If any slope is not equal to the others, it means that the corresponding point does not lie on the same straight line as the reference point and the other points. Hence, the given set of points does not form a straight line.

In this way, by using the property of the slope of a line, we can prove whether the given set of points forms a straight line or not.

Remember, this algorithm assumes that the coordinates are given in a two-dimensional space, and it uses the geometric properties of a line in such a space. The proof of this algorithm is therefore based on these geometric principles.

Let’s break down the approach into granular steps:

Input Interpretation: First, understand the given input. The input is a list of points in the 2D plane. Each point is represented by a list of two integers representing the x and y coordinates.
Determining the Base Slope: Pick the first two points in the list. Calculate the slope between these two points using the formula (y2 - y1) / (x2 - x1). This will be our reference slope.
Iteration Over Points: Iterate over the remaining points in the list. For each new point, calculate the slope it forms with the first point.
Slope Comparison: Compare this newly calculated slope with the reference slope. If the slopes are not equal, return False immediately because this indicates that the points do not form a straight line.
All Points Checked: If we’ve iterated over all points and all of them have the same slope with the first point, return True because this means all points are on the same line.

Now, answering your other questions:

Independent Parts of the Problem: The calculation of the slope between two points can be considered an independent problem. Given two points, you can calculate the slope regardless of other points in the array.

Repeatable Patterns in the Solution: The repeatable pattern in our solution is the calculation and comparison of the slopes. For each point in the array (after the first two), we calculate the slope it forms with the first point and then compare it with our reference slope. This pattern repeats until we’ve checked all points in the list.

Solution Approach and Analysis

Let’s use the metaphor of a ’trekking route’ to explain this problem in a simpler way.

Suppose you and your friends decided to go on a trekking adventure. You all agree to stick to the same straight route without diverting to make the journey straightforward and easy to follow. The route’s steepness remains consistent throughout the journey. In other words, the route forms a straight line. In this analogy:

Each friend is like a coordinate point. Their order represents their position on the path.
The steepness of the route is like the slope of the line.

With this analogy, our task is to ensure that all friends are on the same straight path or not. In technical terms, we have to check whether all points lie on the same straight line.

Here’s how we can break down this task:

Step 1: Understanding the route’s steepness: Determine the steepness of the route between the first two friends. Technically, we calculate the slope between the first two points using the formula (y2-y1) / (x2-x1). This slope will act as a reference slope.

Step 2: Checking if all friends are on the same route: Starting from the third friend, calculate the steepness of the route from the first friend to the current friend. If the calculated steepness is not equal to the reference steepness, that means the current friend is not on the same route, and we return false.

Step 3: Confirming all friends are on the same route: If we have gone through all the friends, and all have the same steepness with the first friend, we return true because all friends are on the same route, meaning all points are on the same straight line.

Let’s consider an example:

Suppose we have coordinates like [[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]], meaning we have six friends going on a trek. Here:

The steepness (slope) between the first two friends is (3-2) / (2-1) = 1. So, our reference slope is 1.
Now, we calculate the steepness from the first friend to each of the remaining friends:
- For the third friend at position [3,4], the steepness is (4-2) / (3-1) = 1, which matches our reference slope.
- We continue this process for all friends. If all of them have the same steepness as the reference slope, we can conclude all friends are on the same route, meaning all points lie on the same straight line.

This approach will change if the points are not ordered. In such a case, we would need to sort the points based on their x-coordinate before performing the steps above. However, for this problem, we can assume the points are already ordered, as mentioned in the constraints.

Identify Invariant

An invariant, in the context of algorithms, is a condition that remains unchanged throughout the execution of the algorithm. It’s used as a tool to reason about the correctness and the progress of an algorithm.

In the “Check If It Is a Straight Line” problem, the invariant is the slope between any two consecutive points. For the coordinates to form a straight line, the slope (i.e., the difference in the y-coordinates divided by the difference in the x-coordinates) between any two consecutive points must remain the same throughout the entire set of coordinates.

Here’s the invariant in a formal sense:

Let’s denote the coordinates as (x[i], y[i]) for i = 0, 1, …, n-1. The invariant here is:

For any i, 2 <= i < n, the slope between the point (x[i], y[i]) and the point (x[i-1], y[i-1]) is the same as the slope between the point (x[1], y[1]) and the point (x[0], y[0]).

We calculate the initial slope from the first two points (x[0], y[0]) and (x[1], y[1]). As we iterate through the rest of the points, we check that the slope between each consecutive pair of points is equal to this initial slope. If at any point we find a slope that differs, we know that the points do not form a straight line, and we can break out of our algorithm early. If we get through all the points without finding a differing slope, then we know that all points lie on the same straight line.

This invariant, that the slope remains constant, is the key to our approach and helps us determine whether the points can form a straight line.

Identify Loop Invariant

A loop invariant is a condition [among program variables] that is initially true and is maintained from iteration to iteration. At the end of the loop, this gives us a useful property that helps show that the algorithm is correct.

For the “Check If It Is a Straight Line” problem, the loop invariant would be the calculated slope between each pair of consecutive points in the coordinates array.

We initialize the loop invariant as the slope between the first two points. Inside the loop, we calculate the slope between the current point and the previous point. If at any iteration, the calculated slope does not equal the loop invariant (the initial slope), we break the loop and return False because the points do not form a straight line.

Here’s a formal definition of the loop invariant:

Let’s denote the coordinates as (x[i], y[i]) for i = 0, 1, …, n-1. If ’m’ is the slope between (x[0], y[0]) and (x[1], y[1]), then the loop invariant in the process of checking the straight line is:

For every i, 2 <= i < n, the slope between (x[i], y[i]) and (x[i-1], y[i-1]) equals ’m’.

This invariant is established before the loop starts (for i = 2), maintained from iteration to iteration (if for some i, it holds, then for i+1, it also holds), and when the loop terminates (i = n), this gives us a useful property which proves that the points form a straight line if it still holds.

Thus, the loop invariant in this problem is crucial to the solution’s correctness and performance. It allows us to confidently say that the points form a straight line if and only if the loop completes without breaking early.

For this problem, is invariant and loop invariant the same?

In the context of this problem, the concept of a loop invariant is particularly relevant. A loop invariant is a property or condition that holds true before and after each iteration of a loop. In this case, the loop invariant is the slope between any two consecutive points. This must remain constant for all points to lie on the same straight line.

However, the term “invariant” has a broader usage in computer science and can refer to a condition that remains true during the execution of an entire program or a specific part of a program, not just during a loop. It could be any property of a data structure or a set of variables that is maintained regardless of what else is happening in the algorithm.

So to answer your question, in the context of this specific problem, the invariant and the loop invariant refer to the same concept - the constant slope. However, it’s important to note that in other contexts, an invariant might refer to a broader range of conditions that aren’t necessarily tied to loops.

Thought Process

Let’s break down the thought process and steps involved in solving this problem:

Understand the Problem: The problem asks if a set of points in a 2D space forms a straight line. We’re given an array of coordinates, and we need to return True if the points are on a straight line and False if they are not.
Identify Key Concepts: The problem is essentially asking us to verify if a set of points all have the same slope, which is a fundamental property of straight lines. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by (y2 - y1) / (x2 - x1). If the slope between every two points is the same, then the points lie on a straight line.
Potential Pitfalls: A key concern is that the denominator in the slope calculation could be zero if two points have the same x-coordinate. In such a case, we need to handle this separately to avoid a division by zero error.
Outline the Approach: To solve this problem, we’ll first calculate the slope between the first two points. Then, we’ll iterate through the rest of the points, calculating the slope between each point and the first point. If we find any slope that doesn’t match the initial one, we’ll return False. If we complete the iteration without finding any mismatches, we’ll return True.
Code the Solution:

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def checkStraightLine(coordinates):
    (x0, y0), (x1, y1) = coordinates[:2]
    for x, y in coordinates:
        if (x1 - x0) * (y - y0) != (x - x0) * (y1 - y0):
            return False
    return True

In this code, (x0, y0) and (x1, y1) are the first two points. The expression (x1 - x0) * (y - y0) != (x - x0) * (y1 - y0) calculates the slope between (x0, y0) and (x1, y1) and between (x0, y0) and (x, y). By cross-multiplying the slopes, we avoid the division and the possibility of a division by zero.

Test the Solution: Run the function with various test cases, including edge cases, to ensure it works as expected.

Overall, the cue in this problem is the requirement that the points form a “straight line”. This immediately suggests that we should be thinking about the properties of straight lines, leading us to the concept of slope.

Establishing Preconditions and Postconditions

Parameters:
- The method receives one parameter: coordinates.
- This parameter is a list of lists (or a 2D array), where each sub-list is a pair of integers.
- In the context of the problem, coordinates represents the set of points on a 2D plane. Each sub-list represents the (x, y) coordinates of a point.
Preconditions:
- There are no specific conditions that must be true about the state of the program prior to calling this method.
- Constraints on the input parameters are as follows: 2 <= coordinates.length <= 1000, coordinates[i].length == 2, -10^4 <= coordinates[i][0], coordinates[i][1] <= 10^4. coordinates contains no duplicate point.
- There is no specific state that the program or part of it must be in.
Method Functionality:
- This method is expected to determine if all the given points lie on the same straight line in the XY plane.
- It does this by calculating the slope between pairs of points and checking whether this slope is the same for all pairs.
Postconditions:
- After the method has been called and returned, the state of the program or the values of the parameters remain unchanged. The method does not have any side effects.
- The return value indicates whether all the points lie on a straight line (True) or not (False).
- There are no side effects.
Error Handling:
- The method does not explicitly handle errors if the preconditions are not met. If the coordinates list is empty or only contains one point, the method would not throw an error but its result may not be meaningful. If the input is not a list or the points are not in the form of (x, y) pairs, Python will throw an error when the method attempts to unpack these values.
- It does not throw an exception or return a special value in these cases.

Problem Decomposition

Problem Understanding:
- The problem asks us to determine if a series of points in a 2D plane lie on the same straight line. The input is an array of (x,y) coordinates, and we are to return a boolean indicating whether or not all the points fall on a single straight line.
Initial Breakdown:
- Broadly, we can break this problem into a few key stages: a. Extracting the points from the array. b. Calculating the slope between each pair of points. c. Checking if all the slopes are the same.
Subproblem Refinement:
- Extracting points from the array is a relatively simple operation, as is checking if all slopes are the same (this can be done with a simple equality check in a loop). The calculation of the slope could potentially be broken down further, into steps for calculating the difference in x coordinates, the difference in y coordinates, and the division to get the slope.
Task Identification:
- The tasks that are repeated here include calculating the slope (which involves subtracting x and y coordinates and performing a division), and checking if two slopes are equal. These are not necessarily the same task, but they are similar in that they both involve simple arithmetic operations.
Task Abstraction:
- These tasks are already fairly abstract. The slope calculation task could be made more abstract by defining a general-purpose “calculate difference” function, but in this case, it’s probably clearer to keep it specific to the slope calculation.
Method Naming:
- We can call the task of extracting points “getPoints”, the task of calculating the slope “calculateSlope”, and the task of checking if all slopes are equal “checkEqualSlopes”.
Subproblem Interactions:
- These tasks need to be performed in the order laid out above: first get the points, then calculate the slopes, and finally check if all the slopes are equal. The calculation of each slope is dependent on having the points, and the check for equal slopes is dependent on having all the slopes.

From Brute Force to Optimal Solution

Let’s start with a brute force solution:

A brute force approach to this problem would be to calculate the slope between every pair of points and check if all the slopes are the same.

Here is what the brute force Python code might look like:

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def checkStraightLine(coordinates):
    slopes = []
    for i in range(len(coordinates) - 1):
        for j in range(i + 1, len(coordinates)):
            if (coordinates[j][0] - coordinates[i][0]) != 0: # prevent division by zero
                slope = (coordinates[j][1] - coordinates[i][1]) / (coordinates[j][0] - coordinates[i][0])
                if slope not in slopes:
                    slopes.append(slope)
    return len(slopes) == 1

This solution is inefficient because it unnecessarily computes the slope between each pair of points, leading to a time complexity of O(n^2), where n is the number of points.

To optimize this solution, we can calculate the slope between the first two points and then check if every subsequent point maintains this slope.

Here is the optimized Python code:

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def checkStraightLine(coordinates):
    if len(coordinates) == 2:
        return True

    x0, y0 = coordinates[0]
    x1, y1 = coordinates[1]
    if x1 - x0 != 0: # prevent division by zero
        slope = (y1 - y0) / (x1 - x0)
    else:
        slope = float('inf')

    for i in range(2, len(coordinates)):
        x, y = coordinates[i]
        if x - x0 != 0: # prevent division by zero
            new_slope = (y - y0) / (x - x0)
        else:
            new_slope = float('inf')

        if new_slope != slope:
            return False

    return True

This optimized solution only computes the slope between each point and the first point, reducing the time complexity to O(n). The space complexity is O(1), since we only store the initial slope and the new slope for comparison.

Please note: As we’re working with float numbers (division result), this may cause precision issues. But for this specific problem, it will work because of the constraints on the input size.

Code Explanation and Design Decisions

Initial Parameters:
- coordinates is a list of lists where each inner list contains two integers representing the x and y coordinates of a point in a 2D space. This is our primary data, and we are checking whether these points form a straight line.
Primary Loop or Iteration:
- The primary loop iterates over the coordinates list starting from the third point (index 2). Each iteration represents a check to see if the current point (x, y) maintains the same slope with the first point as the slope calculated between the first two points.
Conditions or Branches within the Loop:
- The condition if new_slope != slope checks if the slope between the current point and the first point deviates from the original slope (between the first two points). If the condition is true, it means the current point does not lie on the straight line formed by the first two points, so we return False.
Updates or Modifications to Parameters within the Loop:
- The new_slope variable is updated at each iteration. It represents the slope between the current point and the first point. This update is necessary as we need to compare this slope with the initial slope to verify whether all points lie on the same line.
Invariant:
- The invariant in this case is the slope between the first two points. This value does not change throughout the iterations. It’s used to compare with the new_slope of each subsequent point with the first point. The invariant helps to determine if all points fall on the same straight line by maintaining a constant comparison benchmark.
Final Output:
- The final output is a boolean value (True or False). If all points lie on the same straight line, we return True; otherwise, False. This output is significant as it directly answers the problem statement. It satisfies the problem’s requirements by determining if it’s possible to draw a single straight line through all the given points in the 2D space.

Coding Constructs

High-level problem-solving strategies or techniques: The code primarily uses the concept of calculating the slope between points in a 2D space. It uses a comparison-based approach, where it computes the slope between pairs of points and checks for consistency to determine if the points lie on the same straight line.
Explaining the purpose of this code to a non-programmer: Suppose you have several dots on a paper and want to determine if you can connect all of them using a straight ruler without lifting it. This code performs exactly that check. It tells us if it’s possible to draw a single straight line that passes through all the given dots.
Logical elements or constructs used in this code: The logical constructs used in this code include iteration (looping through a set of points), calculation (finding the slope between two points), comparison (checking if two slopes are equal), and conditional execution (performing actions based on the outcome of the comparison).
Algorithmic approach in plain English: The algorithm first calculates the slope between the first two points. Then, it iteratively calculates the slope between the first point and each subsequent point, comparing this to the initial slope. If it finds a slope that doesn’t match the initial slope, it concludes that all points don’t lie on the same line and stops. If all the slopes match, it determines that all points lie on the same line.
Key steps or operations on the input data: The key steps are calculating the slope between points and comparing these slopes. The slope calculation is necessary to quantify the ‘steepness’ of the line between two points, and the comparison is needed to verify the consistency of this steepness across all points, ensuring they all lie on the same line.
Algorithmic patterns or strategies: The algorithm uses a linear traversal pattern, iterating through the list of points. It applies the principle of invariance, where it expects the slope between any point and the first point to be invariant if all points are to lie on the same line. This is a form of consistency check or validation that helps determine the final result.

Language Agnostic Coding Drills

Dissect the code and identify distinct concepts:
- Basic Programming Syntax: This includes understanding variables, loops, conditionals, and functions.
- Arithmetic Operations: This is the ability to perform basic arithmetic calculations like subtraction and division.
- Data Structures: Particularly the usage of an array or list to hold the data points.
- Iteration over Collections: The code requires iterating over the list of points.
- Pairwise Comparisons: The concept of comparing pairs of elements from a list.
- Conditionals: Making decisions based on the outcome of the comparisons.
- Working with Coordinate Points: This involves understanding the 2D coordinate system and how to work with points within it.
- Calculating Slope in a 2D space: This requires understanding of basic geometric principles to calculate the slope between two points.
Coding drills listed in order of increasing difficulty:
- Basic Programming Syntax: This is the foundation of any coding problem.
- Arithmetic Operations: Building upon basic syntax, performing arithmetic operations is one of the first skills a novice programmer learns.
- Data Structures: Understanding how to use basic data structures like arrays or lists is slightly more complex but a core coding concept.
- Iteration over Collections: This builds upon the previous concepts and involves using loops to go through each element in a data structure.
- Conditionals: This introduces logic into your code, making decisions based on certain conditions.
- Pairwise Comparisons: This is more complex as it involves comparing elements in a data structure with each other, which requires nested iteration.
- Working with Coordinate Points: Requires some understanding of the coordinate geometry.
- Calculating Slope in a 2D space: The most advanced concept in this list, as it requires understanding of geometry and applying it to a coding problem.
Problem-solving approach from problem statement to final solution:
- Start with reading and understanding the problem statement. Identify the input, the required output and constraints.
- Next, understand that you need to calculate the slope between points, which implies a need for pairwise comparison. This forms the core of our problem-solving approach.
- Translate the problem into the language of code, starting by creating a function that accepts a list of points as input.
- Then, identify the first two points in the list and calculate the slope. This is your reference slope.
- Now, you need to iterate over the rest of the points in the list. For each point, calculate the slope it forms with the first point.
- Use a conditional statement to compare the slope of the current point with the reference slope.
- If the slopes are equal, continue to the next point. If they’re not, return a result indicating that not all points lie on the same line.
- If you finish the iteration without finding any discrepancies in the slopes, return a result indicating that all points lie on the same line.
- Throughout this process, every operation you perform is a drill you’ve learned, all working together to solve the problem.

Targeted Drills in Python

Python coding drills for each identified concept:

Basic Programming Syntax:

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def hello_world():
    print("Hello, World!")
hello_world()

Arithmetic Operations:

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def basic_arithmetic(a, b):
    print("Sum: ", a + b)
    print("Difference: ", a - b)
    print("Product: ", a * b)
    print("Division: ", a / b)
basic_arithmetic(10, 5)

Data Structures (list):

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def basic_list_operations():
    my_list = [1, 2, 3, 4, 5]
    print("Original list: ", my_list)
    my_list.append(6)  # add element
    print("List after adding element: ", my_list)
    my_list.remove(1)  # remove element
    print("List after removing element: ", my_list)
basic_list_operations()

Iteration over Collections:

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def iterate_over_list(my_list):
    for element in my_list:
        print(element)
iterate_over_list([1, 2, 3, 4, 5])

Conditionals:

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def basic_conditionals(a, b):
    if a > b:
        print(f"{a} is greater than {b}")
    elif a < b:
        print(f"{a} is less than {b}")
    else:
        print(f"{a} is equal to {b}")
basic_conditionals(5, 10)

Pairwise Comparisons:

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def pairwise_comparisons(my_list):
    for i in range(len(my_list)):
        for j in range(i + 1, len(my_list)):
            if my_list[i] > my_list[j]:
                print(f"{my_list[i]} is greater than {my_list[j]}")
            elif my_list[i] < my_list[j]:
                print(f"{my_list[i]} is less than {my_list[j]}")
            else:
                print(f"{my_list[i]} is equal to {my_list[j]}")
pairwise_comparisons([1, 2, 3])

Working with Coordinate Points:

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def working_with_coordinates(point1, point2):
    x1, y1 = point1
    x2, y2 = point2
    print(f"Point 1 is at ({x1}, {y1})")
    print(f"Point 2 is at ({x2}, {y2})")
working_with_coordinates((1, 2), (3, 4))

Calculating Slope in a 2D space:

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def calculate_slope(point1, point2):
    x1, y1 = point1
    x2, y2 = point2
    if x1 != x2:  # to avoid division by zero
        slope = (y2 - y1) / (x2 - x1)
        return slope
    else:
        return None
print(calculate_slope((1, 2), (3, 4)))

Integration of drills to solve the initial problem:
The problem requires us to determine whether given points are collinear. The steps to solve this are:
- Define a function that takes a list of points as an argument.
- Using the “Working with Coordinate Points” drill, extract the first two points from the list and calculate the slope between them using the “Calculating Slope in a 2D space” drill.
- Then, iterate over the remaining points in the list using the “Iteration over Collections” drill. For each of these points, again use the “Working with Coordinate Points” and “Calculating Slope in a 2D space” drills to calculate the slope this point forms with the first point.
- Compare this slope to the original slope using a conditional (the “Conditionals” drill). If the slopes are not equal, return False to indicate the points are not collinear.
- If you finish iterating through the list without finding differing slopes, return True to indicate the points are collinear.

These drills individually tackle specific aspects of the problem, but when integrated together following the right problem-solving strategy, they build up to form the final solution.