In the merging step, the algorithm looks at the points that are inside a strip of width centered on line L. How many of the points can this strip contain?

Question

In the merging step, the algorithm looks at the points that are inside a strip of width  centered on line L. How many of the  points can this strip contain?

BlackTom AI · Accepted Answer

First, let's restate what the question is asking: in the merging step of the closest-pair algorithm, we consider a strip of width centered on the dividing line L, and we want to know how many points this strip can contain.

Option evaluation mindset:
- If we think about the strip in the worst case, nothing in the problem statement restricts how many points can lie within that strip besides the total number of points in the input. Therefore, in the absolute worst case, the strip could contain all n points, since they could all lie within distance d of the line L. This means there is no fixed constant upper bound smaller than n on the number of strip points in general.
- However, algorithmic analyses commonly use a packing argument to limit the number of comparisons needed when checking cross-strip pairs. When the points are sorted by y-coordinate within the strip, it is shown that for each point you need to compare it with only a constant number of subsequent points (often stated as up to 7 following points) to guarantee you don’t miss a closer pair across the split. This relies on a geometric packing bound within the d-by-2d (or similar) rectangle arguments and is what yields the O(n) total time for the merge step.
- Specifically, the commonly taught bound is that, within the strip, you can bound the number of points in any d-by-2d region to a constant (often cited as at most 6 or 7 points) when considering the cross-pairs. This is what underpins the constant-time check per point, not the total strip size.

Now, addressing the core question directly: how many of the points can this strip contain?
- If interpreted strictly as a worst-case count without additional assumptions, the strip can contain up to all the points in the set, i.e., O(n). There is no universal constant upper bound on the total number of points in the strip in the input as a whole.
- If interpreted in the context of the cross-pair checking step after sorting by y within the strip, the relevant practical bound is a constant number of nearby points to compare with for each point (often described as up to 7 following points). This does not say the strip contains only a constant number of points, but rather that the number of cross-strip comparisons per point is bounded by a constant due to packing.

Because the provided answer options are missing in this prompt, there is no single choice to evaluate here. The key takeaway is:
- The strip can contain as many as all input points in the worst case (O(n)).
- For efficient cross-strip checking, geometric packing arguments imply a constant bound on the number of nearby points to compare with for each point (often stated as 7), which is what drives the linear-time merge step.

CPSC_V 320 201/202/203 2024W2 Reading Quiz #5 (Divide and Conquer 2)

In the merging step, the algorithm looks at the points that are inside a strip of width centered on line L. How many of the points can this strip contain?

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