October 26 2005 copyright 2001 5 by erik d demaine

October 26, 2005 Copyright © 2001-5 by Erik D. Demaine and Charles E. Leiserson L11.26
Proof of theorem
T
HEOREM
: With high probability, every search
in an n -element skip list costs O (lg n )
C
OOL
I
DEA
: Analyze search backwards—leaf to root
• Search starts [ends] at leaf (node in bottom level)
• At each node visited:
– If node wasn’t promoted higher (got
TAILS
here),
then we go [came from] left
– If node was promoted higher (got
HEADS
here),
then we go [came from] up
• Search stops [starts] at the root (or −∞ )

October 26, 2005 Copyright © 2001-5 by Erik D. Demaine and Charles E. Leiserson L11.27
Proof of theorem
T
HEOREM
: With high probability, every search
in an n -element skip list costs O (lg n )
C
OOL
I
DEA
: Analyze search backwards—leaf to root
P
ROOF
:
• Search makes “up” and “left” moves
until it reaches the root (or −∞ )
• Number of “up” moves < number of levels
≤ c lg n w.h.p. (Lemma)
• ⇒ w.h.p., number of moves is at most the number
of times we need to flip a coin to get c lg n
HEAD
s

October 26, 2005 Copyright © 2001-5 by Erik D. Demaine and Charles E. Leiserson L11.28
Coin flipping analysis
C
LAIM
: Number of coin flips until c lg n
HEAD
s
= Θ (lg n ) with high probability
P
ROOF
:
Obviously Ω (lg n ) : at least c lg n
Prove O (lg n ) “by example”:
• Say we make 10 c lg n flips
• When are there at least c lg n
HEAD
s?
(Later generalize to arbitrary values of 10 )

October 26, 2005 Copyright © 2001-5 by Erik D. Demaine and Charles E. Leiserson L11.29
Coin flipping analysis
C
LAIM
: Number of coin flips until c lg n
HEAD
s
= Θ (lg n ) with high probability
P
ROOF
:
• Pr{ exactly c lg n
HEAD
s } =
• Pr{ at most c lg n
HEAD
s } ≤
n c n c
n c
n c
lg 9 lg
2
1
2
1
lg
lg 10






⋅






⋅








orders
HEAD
s
TAIL
s
n c
n c
n c
lg 9
2
1
lg
lg 10






⋅








overestimate
on orders
TAIL
s

October 26, 2005 Copyright © 2001-5 by Erik D. Demaine and Charles E. Leiserson L11.30
Coin flipping analysis (cont’d)
• Recall bounds on :
• Pr{ at most c lg n
HEAD
s }
n c
n c
n c
lg 9
2
1
lg
lg 10






⋅








≤








x
y
x x
x
y
e
x
y
x
y






≤








≤






n c
n c
n c
n c
e
lg 9
lg
2
1
lg
lg 10






⋅








≤
( )
n c
n c
e
lg 9
lg
2 10
−
=
n c n c e lg 9 lg ) 10 lg(
2 2
− ⋅
=
n c e lg ] 9 ) 10 [lg( 2 ⋅ − =
α n / 1 =
for
[ ]
c e ⋅ − = ) 10 lg( 9 α

October 26, 2005 Copyright © 2001-5 by Erik D. Demaine and Charles E. Leiserson L11.31
Coin flipping analysis (cont’d)
• Pr{ at most c lg n
HEAD
s } ≤ 1/ n
α
for α = [9−lg(10 e )]c
• K
EY
P
ROPERTY
: α → ∞ as 10 → ∞ , for any c
• So set 10 , i.e., constant in O (lg n ) bound,
large enough to meet desired α
This completes the proof of the coin-flipping claim
and the proof of the theorem.

October 31, 2005 Copyright © 2001-5 by Erik D. Demaine and Charles E. Leiserson L13.1
Prof. Charles E. Leiserson
L
ECTURE
13
Amortized Analysis
• Dynamic tables
• Aggregate method
• Accounting method
• Potential method
Introduction to Algorithms
6.046J/18.401J

October 31, 2005 Copyright © 2001-5 by Erik D. Demaine and Charles E. Leiserson L13.2
How large should a hash
table be?
Problem: What if we don’t know the proper size
in advance?
Goal: Make the table as small as possible, but
large enough so that it won’t overflow (or
otherwise become inefficient).