Design and Analysis of Algorithms (Divide & Conquer: Convex Hull, Median Finding)

Paradigm

Given a problem of size n divide it into subproblems of size n,  a ≥ 1, b > 1. Solve each subproblem recursively. Combine solutions of subproblems to get the overall solution.

T(n) = aT(n/b ) + [work for merge]

Convex Hull

Given n points in plane

S = {(xi, yi) | i = 1, 2,...,n}

assume no two have the same x coodinates, no two have sa`me y coordinates, no three in a line.

Convex Hull: smallest polygon contain all points in the plane. 

CH(S) represented by the sequence of points on the boundary in clockwise order as a doubly linked list.

Brute force for Convex Hull

Test each line segment to see if it makes up an edge of the convex hull

• If the rest of the points are on one side of the segment, the segment is on the convex hull.
• else the segment is not.
• O(n2) edges, O(n) tests ⇒ O(n3) complexity

Divide and Conquer Convex Hull

Sort points by x coordinates

1. divide into left half A and right half B by x coords
2. compute CH(A) and CH(B)
3. combine CH's of two halves (merge step)

How to merge?

• Find the upper tangent (ai, bj).
• Find the lower tangent (ak, bm).
• Cut and paste in time Θ(n).

Firstly,  link ai to bj, go down b list till you see bm and link bm to ak, continue along the a list until you return to ai.

Find Tangents

Assume ai maximizes x within CH(A) (a1, a2,...,ap). b1 minimizes x within CH(B) (b1, b2,..., bq)

L is the vertical line separating A and B. Define y(i, j) as y-coordinate of intersection between L and segment (ai, bj ).

Claim: (ai, bj ) is upper tangent if it maximizes y(i, j)

If y(i, j) is not maximum, there will be points on both sides of (ai, bj ) and it cannot be a tangent.

Algorithm: Obvious O(n2) algorithm looks at all ai, bj pairs. T(n)=2T(n/2)+ Θ(n^2) = Θ(n^2).

i = 1
j = 1
while(y(i, j+1) > y(i, j) or y(i − 1, j) > y(i, j))
    if (y(i, j + 1) > y(i, j)) -> move right finger clockwise
        j = j + 1( mod q)
    else
        i = i - 1( mod p) -> move left finger anti-cloclwise
    return (ai, bj ) as upper tangent

T(n)=2T(n/2) + Θ(n) = Θ(n log n) 

Median Finding

Given a set of n numbers, define rank(x) as the number of numbers in the set that are ≤ x. Find an element of rank (lower median) and  (upper median).

Select(S, i)
    Pick x in S
    Compute k = rank(x)
    B = {y ∈ S|yx}
    if k = i
        return x
    else if k > i
        return Select(B, i)
    else if k < i
        return Select(C, i - k)
        

Picking x Cleverly

Need to pick x so rank(x) is not extreme.

• Arrange S into columns of size 5 (n/5 cols)
• Sort each column (bigger elements on top) (linear time)
• Find “median of medians” as x