8/10/2019 chan alg

1/5

2. Chans Algorithm

2.1 Graham Scan (Successive Local Repair)

(p1, . . . , pn)

p10p4

p1

p3

p2

p5

p9

p7p6

p10p4p1p3p2p5p9p7p6p7p9p5p2p3p1p4p10

(p, q, r)

q

pr q

Theorem 2.1 P R2 n O(n

n)

Proof.

O(n

n)

2n 2

h

2n h 2

2n 2

4nh4

O(n n)

O(n

n)

2.2 Lower Bound

Theorem 2.2 (n n)

n

R2

Proof. (n n)

n

x1, . . . , xn

P = {pi| 1 i n} n R

2

pi = (xi, x2i

)

R2

x

7

8/10/2019 chan alg

2/5

P

xi o(n n)

o(n

n)

(n

n)

(n n)

n

2.3 Jarvis Wrap and Graham Scan in C++

Jarvis Wrap.

p[0..N)

p start x

q next

p[0..N)

int h = 0;Point_2 q_now = p_start;

do {

q[h] = q_now;

h = h + 1;

f o r ( i n t i = 0 ; i < N ; i = i + 1 )

if (rightturn_2(q_now, q_next, p[i]))

q_next = p[i];

q_now = q_next;

q_next = p_start;

} while (q_now != p_start);

q[0,h)

p[0..N)

Graham Scan.

p[0..N) N 2

q[0] = p[0];

int h = 0;

8

8/10/2019 chan alg

3/5

// Lower convex hull (left to right):

f o r ( i n t i = 1 ; i < N ; i = 1 + 1 ) {

while (h>0 && rightturn_2(q[h-1], q[h], p[i]))h = h - 1;

h = h + 1;

q[h] = p[i];

}

// Upper convex hull (right to left):

for (int i = N-2; i >= 0; i = i - 1) {

while (rightturn_2(q[h-1], q[h], p[i]))

h = h - 1;

h = h + 1;q[h] = p[i];

}

q[0,h)

p[0..N)

2.4 Chans Algorithm

O(nh) (n n) h= o( n)

Divide.

P

R2

n

H {1 , . . . , n}

P k= n/H P1, . . . , Pk |Pi| H

(

)

i 1

i

k

H ( )

O(n)

O(H H) Pi O(n H)

Conquer.

(

) i 1 i k

9

8/10/2019 chan alg

4/5

P

H

(

)

( )

(

)

(

) 1 i k

O(n)

H

k

H

O(Hk

H) =O(n

H)

O(n)

O(n H)

Searching for h.

H= h

h

H =

{22

t

, n}

t =

0 , . . . P

22s

H = 22

s1

22

s1

< h

2s1