BA 6 Karatsuba's Algorithm, Closest Pair in the Plane

T/F Karatsuba's Algorithm isn't divide and conquer

F

Karatsuba's algorithm

T(n) = 4T(n/2)+Θ(n)

Closest pair of points brute force

check every 2 pair (n 2) = n(n-1)/2 = Θ(n^2)

How to improve efficiency for divide and conquer algorithm in T(n)=a(T/b) + f(n)

decrease a and f(n), increase b

Optimization in closest pair problem

divide into sub-grid of δ/2, sort by y and check only 11 above the cell

Closest pair of points DC

divide to 2, solve subproblem for each half, combine by finding the closest across the boarder, output the min of the 3!

two n-bits integers, compute a x b

naive: Θ(2^n * n) - b * (n per addition) - b = O(2^n) elem algo: Θ(n^2)

For optimization in closest pair problem, we use what strategy to improve algorithm efficiency

reduce f(n), because we reduce the sorting in combine part of the DC problem from nlog^2n to nlogn

Improve Karatsuba's algorithm

reeduce a, achieve n^1.585

run time for cross-border closest pair optimization

sort once at the very beginning, then Θ(nlogn)! - Θ(nlogn) for sort y, Θ(nlog^2n) if not sort in the beginning;