QuickHull_DivideConquer

# QuickHull_DivideConquer - Convex hulls Quickhull Quicksort...

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Convex hulls Quickhull Quicksort The Quickhull algorithm is based on the Quicksort algorithm. Recall how quicksort operates: at each level of recursion, an array of numbers to be sorted is partitioned into two subarrays, such that each term of the first (left) subarray is not larger than each term of the second (right) subarray. LEFT RIGHT Two pointers to the array cells (LEFT and RIGHT) initially point to the opposite extreme ends of the array. LEFT and RIGHT move towards each other, one cell at a time. At any given time, one pointer is moving and one is not. If the numbers pointed to by LEFT and RIGHT violate the desired sort order, they are swapped, then the moving pointer is halted and the halted pointer becomes the moving pointer. When the two pointers point to the same cell, the array is split at that cell and the process recurses on the subarrays. Quicksort runs in expected O ( N log N ) time, if the subarrays are well balanced, but can require as much as O ( N 2 ) time in the worst case.

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Convex hulls Quickhull Quickhull overview Quickhull operates in a similar manner. It recursively partitions the point set S, so as to find the convex hull for each subset. The hull at each level of the recursion is formed by concatenating the hulls found at the next level down . S
Convex hulls Quickhull Initial partition The initial partition of S is determined by a line L through the points l , r S with the smallest and largest absciss S (1) S is the subset of S on or above L . S (2) S is the subset of S on or below L . Note that { S (1) , S (2) } is not a strict partition of S , as S (1) S (2) { l , r }. This is not a difficulty. The idea now is to construct hulls H ( S (1) ) and H ( S (2) ), then concatenate them to get H ( S ). The process is the same for S (1) and S (2) , we consider S (1) . l r L S (1) S (2)

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Convex hulls Quickhull Finding the “apex” Find the point h S (1) such that (1) triangle hlr has the maximum area of all triangles { plr : p S (1) }, and if there are > 1 triangles with maximum area, (2) the one where angle hlr is maximum. This condition ensures that h H ( S ). Why? Construct a line parallel to line L through h , call it L . There will be no points of S (1) (or S ) above L , by condition (1). There may be other points on L , but h will be the leftmost, by condition (2), hence it is not a convex combination of any two points of S. h H ( S ). “Apex” h can be found in O ( N ) time by checking each point of S (1) .
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QuickHull_DivideConquer - Convex hulls Quickhull Quicksort...

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