comp2123 Assignment 5 s1 2023
Problem 1. We want to design a divide and conquer algorithm for computing
the union of a collection of rectangles. The input rectangles are aligned with
the axes and they are all stabbed by the y-axis. Each rectangle is represented
by the coordinates of its top-left and bottom-right corners, and the union is
representation by a sequence of interior-disjoint rectangles listed from top to
bottom. We require that no two consecutive rectangles in the representation can
be merged into a single rectangle.
For example, the union of the three rectangles on the left leads to the union
represented by the four rectangles on the right:
y
union
y
In each case, the top-left and bottom-right corners of the rectangles in question have been highlighted for reference.
Here is another example where the union of two rectangles on the left leads
to the union represented by a single rectangle on the right1
:
y
union
y
The scaffold provided in Ed uses the following classes:
• Point: Represent a point on the plane with integer coordinates.
• Rectangle: Represents a single rectangle R. Supports membership queries
of the form does x ∈ R?
• Union: Represents the union U of a set of rectangles using a sequence of
disjoint rectangles in top-to-bottom order. Supports membership queries
of the form x ∈ U.
1Please note that in this second example, representing the union with two ore more rectangles
would be incorrect.
1
comp2123 Assignment 5 s1 2023
The scaffold provides the top level code for divided and conquer algorithms
to computing the union of a set of rectangles, but the merge step of each of these
algorithms is missing.
Using the scaffold provided in Ed, your task is to implement the missing
functions marked with a TODO note. Here is a non-exhaustive list of the main
functions:
• merge_unions(union_left, union_right): Compute the union of two Union
objects. You can assume the function will only be called by the divide and
conquer algorithm, so you only need to handle inputs that can arise from
the execution of the union divide and conquer algorithm. Your implementation should run in O(|union_le f t| + |union_right|) time.
• Rectangle.__init__( topleft_point , bottomright_point): Validate input and
throw ValueError if it doesn’t. Rectangle must intersect the y-axis and the
bottom-right corner must lie strictly to the right and strictly below the topleft corner.
• Rectangle.contains(point): Check if point lies on the rectangle.
• Union.contains(point): You can assume that the object is the result of some
union divide and conquer execution. Your implementation should run in
O(|union|) time.
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