Prove that the divide-and-conquer algorithm for the closest-pair problem examines, for every point p...

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Prove that the divide-and-conquer algorithm for the closest-pair problem examines, for every point p in the vertical strip (see Figures 5.7a and 5.7b), no more than seven other points that can be closer to p than d_min, the minimum distance between two points encountered by the algorithm up to that point.

Figures 5.7a and 5.7b

.

David · Accepted Answer

a. Given that 
  L R R L L R L R L L R RX Y X Y X X Y Y X Y X Y       
Let us suppose that X and Y are two values. If onebreaks them into halves, consisting 
most significant digits and least significant digits.
Then, X XL XR   
And, Y YL YR   
Multiply X *Y  – 
   L R L RXY X X Y Y    
 
Now using algebraic expression   a b c d ac ad bc bd      , multiply X and Y 
  L R L R L L L R R L R RX X Y Y X Y X Y X Y X Y       
  L R L R L L R R L R R LX X Y Y X Y X Y X Y X Y       
Or, 
        L R R L L R L R L L R RX Y X Y X X Y Y X Y X Y .
b. When the multiplication between any two numbers of N digit is computed with the 
help of Divide and Conquer multiplication algorithm then the time consumed by the 
algorithm is  1.59O N . 
 
Multiplication equation is  
 L L L R R L R RXY X Y X Y X Y X Y     
when one uses   L R L R L L R RX X Y Y X Y X Y    in the place of L R R LX Y X Y . 
 
Then,

Prove that the divide-and-conquer algorithm for the closest-pair problem examines, for every point p in the vertical strip (see Figures 5.7a and 5.7b), no more than seven other points that can be...

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