COMS 482: unofficial class blog

Some simple recurrences:

• T(n) <= 2*T(n/2) + cn. This is O(n log n), because it forms a call tree log n levels high with cn work per level.
• T(n) <= a*T(n/2) + cn. This runs in a^{log n}/2^{log n}*cn, because there are a copies on the first level, a² on the second, etc, for a total of log n levels. This is a generic recurrence you will see solved for various a.

Another technique for working with recurrences is called “summing over all levels of recursion.” Imagine a recurrence with log n levels of work, and a/2 work performed:

• T(n) <= Sum((a/2)ⁱ) from i = 0 to log n. Using the fact that this represents a geometric sequence, we can simplify this to T(n) <= cn*(1-(a/2)^{1+log n})/(1 - (a/2)).

We’ve covered the a=2 case already (n log n), so lets look at cases where a > 2 and a < 2:

• a < 2 –> T(n) <= cn*(1-(a/2)^{1+log n})/(1 - (a/2)) <= cn * 1/(1-a/2) = O(n)
• a > 2 –> T(n) <= cn*(1-(a/2)^{1+log n})/(1 - (a/2)) <= cn*((a/2)^{1+log n} -1)/(a/2-1) = O(n*(a/2)^{log n} = O(a^{log n}) = O(2^{log alog n}) = O(n^{log a}).

We also had some discussion of the “master theorem” for describing generic recurrences, but didn’t explain it in class.

Finding Closest Pair of Points:

The basic idea? Divide, find closest pair of points, and then merge:

We have to check pairs across the dividing line, or which there are O(n²). Let d be the min(closest point on left, closest point on right). Where can we find points on the right that might be closer than d from a point on the left? It turns out you can fit at most 4 points on a circle that satisfy this bound. But, to make the recurrences easier, we will look at a rectangle 2d tall and d wide, on which we can fit at most 6 points. So, sort the points in the y-band by y-coordinate, and compare the left and right sides 6 at a time.

T(n) <= 2*T(n/2) + 4*n*log*n <= 2*T(n/2) + c*n*log*n = O(n log² n)

If we sort by y coordinate first, and keep the points sorted, we can eliminate the extra n log n times, and get O(n log n) instead!

This entry was posted on Wednesday, February 16th, 2005 at 11:01 pm and is tagged with lt 2, master theorem, n levels, geometric sequence, closest point, gt 2, log2, dividing line, time t, recurrence, rectangle, a2, pairs, cn. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback.

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