Minimum Lines to Represent a Line Chart

Minimum Lines to Represent a Line Chart

1. What is this problem about?

This problem gives you a list of coordinates [day, price] representing a line chart over time. You need to find the minimum number of straight line segments required to represent this chart.

A line segment can cover multiple points only if all those points are collinear—meaning they lie on the exact same line with the same slope. If the price changes direction or the rate of change (slope) varies between two consecutive pairs of points, a new line segment must start.

2. Why is this asked in interviews?

Google uses this problem to test a candidate's knowledge of Geometry, Sorting, and Precision. Key challenges:

• Collinearity Logic: How to check if three points (x1, y1), (x2, y2), (x3, y3) lie on the same line.
• Avoiding Precision Errors: Slopes can be floating-point numbers. In programming, checking slope1 == slope2 using division (y/x) is risky due to rounding errors. Using cross-multiplication is the "pro" way to solve this.
• Edge Cases: Handling the first line and arrays with very few points.

3. Algorithmic pattern used

The pattern is Sorting and Slope Comparison.

1. Sort the points by the day (x-coordinate) to process them chronologically.
2. Iterate through the sorted points from the third point onwards.
3. Compare the slope of the line formed by the previous two points with the slope formed by the current point and the previous point.
4. To check if (y2-y1)/(x2-x1) == (y3-y2)/(x3-x2), use the cross-multiplication formula: (y2-y1) * (x3-x2) == (y3-y2) * (x2-x1).
5. If the slopes are different, increment the line count.

4. Example explanation

Points: (1, 1), (2, 2), (3, 3), (4, 5) Sorted: (1, 1), (2, 2), (3, 3), (4, 5)

1. Points 1 and 2 form the first line. Count = 1.
2. Check Point 3: Slope of (1,1)-(2,2) is 1. Slope of (2,2)-(3,3) is 1. Same line. Count remains 1.
3. Check Point 4: Slope of (2,2)-(3,3) is 1. Slope of (3,3)-(4,5) is $(5-3)/(4-3) = 2$. $1 \neq 2$, so we need a new line. Count becomes 2. Total lines = 2.

5. Common mistakes candidates make

• Floating Point Division: Using (double)dy/dx which leads to precision errors with very large coordinates.
• Not Sorting: Assuming the points are given in chronological order.
• Slope between non-adjacent points: Trying to check the slope against the original starting point of the current line segment, which is much more complex than just checking consecutive triplets.

6. Interview preparation tip

Always remember the cross-multiplication trick for geometry problems. A/B = C/D is equivalent to A*D = B*C. This avoids division by zero and floating-point errors, making your code more robust and "interview-ready."