Graph Algorithms Graph Algorithms Graph Algorithms

10 -1
Chapter 10
Amortized Analysis

10 -2
 A sequence of operations: OP1, OP2, … OPm
OPi : several pops (from the stack) and
one push (into the stack)
ti : time spent by OPi
the average time per operation:
An example– push and pop



m
i
i
t
m 1
ave
1
t

10 -3
 Example: a sequence of push and pop
p: pop , u: push
i 1 2 3 4 5 6 7 8
OPi 1u 1u 2p 1u 1u 1u 2p 1p
1u 1u 1u
ti 1 1 3 1 1 1 3 2
tave = (1+1+3+1+1+1+3+2)/8
= 13/8
= 1.625

10 -4
 Another example: a sequence of push
and pop
p: pop , u: push
tave = (1+2+1+1+1+1+6+1)/8
= 14/8
= 1.75
i 1 2 3 4 5 6 7 8
OPi 1u 1p 1u 1u 1u 1u 5p 1u
1u 1u
ti 1 2 1 1 1 1 6 1

10 -5
Amortized time and potential function
ai = ti +  
i i
 1
ai : amortized time of OPi
i : potential function of the stack after OPi
 
i i
 1: change of the potential
a t
i
i
m
i
i
m
i
i
m
i
  

     
1 1 1
1
( )
 
   

ti
i
m
m
1
0
 
If  
m  0  0, then ai
i
m


1
represents an upper
bound of ti
i
m


1

10 -6

 Suppose that before we execute Opi , there are k
elements in the stack and Opi consists of n pops and
1 push.
Amortized analysis of the
push-and-pop sequence
0
have
We
stack
the
in
elements
of
#
:
0 




m
i
2
)
1
(
)
1
(
1
Then,
1
1
1




















-k
k-n
n
t
a
n
k
n
t
k
i
i
i
i
i
i
i

10 -7

 By observation, at most m pops and m
pushes are executed in m operations. Thus,
2.
t
Then,
2
/
)
(
have
We
ave
1




m
i
i m
a
2.
tave 

10 -8
Skew heaps
1
50
13 20
10
16 25
5
12
30
19
40 14
1
50
13
20
10
16
25
5
12
30
19
40
14
Two skew heaps
Step 1: Merge the right paths.
5 right heavy nodes: yellow
 meld: merge + swapping

10 -9
1
5
13
20
10
16
25
50
12
30
19
40
14
Step 2: Swap the children along the right path.
No right heavy node

10 -10
Amortized analysis of skew heaps
 meld: merge + swapping
 operations on a skew heap:
 find-min(h): find the min of a skew heap h.
 insert(x, h): insert x into a skew heap h.
 delete-min(h): delete the min from a skew heap h.
 meld(h1, h2): meld two skew heaps h1 and h2.
The first three operations can be implemented
by melding.

10 -11
Potential function of skew heaps
 wt(x): # of descendants of node x, including
x.
 heavy node x: wt(x)  wt(p(x))/2, where
p(x) is the parent node of x.
 light node : not a heavy node
 potential function i: # of right heavy nodes
of the skew heap.

10 -12
light nodes  log2n
# heavy=k3 log2n
possible heavy nodes
# of nodes: n
 Any path in an n-node tree contains at most
log2n light nodes.
 The number of right heavy nodes attached to
the left path is at most log2n .

10 -13
Amortized time
heap: h1
# of nodes: n1
heap: h2
# of nodes: n2
# light  log2n1
# heavy = k1
# light  log2n2
# heavy = k2

10 -14
ai = ti + i  i-1
ti : time spent by OPi
ti  2+log2n1+k1+log2n2+k2
(“2” counts the roots of h1 and h2)
 2+2log2n+k1+k2
where n=n1+n2
i  i-1 = k3-(k1+k2)  log2nk1k2
ai = ti + i  i-1
2+2log2n+k1+k2+log2nk1k2
=2+3log2n
 ai = O(log2n)

10 -15
AVL-trees
height balance of node v:
hb(v)= (height of right subtree) – (height of left subtree)
 The hb(v) of every node never differ by more than 1.
M
I O
E K Q
N
G L
C J P R
B D F
-1
-1
-1
0
0
0 0 0
0
0
0
0
0
0
0
+1
Fig. An AVL-Tree with Height Balance Labeled

10 -16
 Add a new node A.
Before insertion, hb(B)=hb(C)=hb(E)=0
hb(I)0 the first nonzero from leaves.
M
I O
E K Q
N
G L
C J P R
B D F
-2
-2
-1
0 0
0
0
0
0
0
0
0
+1
A
0
-1
-1
-1
Critical
node
Critical
path

10 -17
Amortized analysis of AVL-trees
 Consider a sequence of m insertions on an
empty AVL-tree.
 T0: an empty AVL-tree.
 Ti: the tree after the ith insertion.
 Li: the length of the critical path involved in the ith
insertion.
 X1: total # of balance factor changing from 0 to +1
or -1 during these m insertions (the total cost for
rebalancing)
X1= Li
i
m


1
, we want to find X1.
Val(T): # of unbalanced node in T
(height balance  0)

10 -18
Case 1 : Absorption
newly
added
Ti-1 Ti
1
0
0
0
0
-1
-1
-1
Li
Val(Ti)=Val(Ti-1)+(Li1)
 The tree height is not increased, we need not
rebalance it.

10 -19
Case 2.1 single rotation
D
B
Li
E(n)
A(n)
C(n)
0
0
newly added
0
-1
E(n): height of tree E is n

10 -20
Case 2 : Rebalancing the tree
D
B F
E G
A C
D
B
F
E
G
A C
D
B
F
E G
A
C
left
rotation
right
rotation

10 -21
Case 2.1 single rotation
D
B
E(n)
A(n)
C(n)
0
-1
-1
0
 After a right rotation on the subtree rooted at
D:
Val(Ti)=Val(Ti-1)+(Li-2)

10 -22
Case 2.2 double rotation
F
B
E(n-1)
C(n-1)
0
-1
A(n)
G(n)
D
0
newly
added

10 -23
Case 2.2 double rotation
D
B
C(n-1)
0
1
A(n) G(n)
F
0
E(n-1)
Val(Ti)=Val(Ti-1)+(Li-2)
 After a left rotation on the subtree rooted at
B and a right rotation on the subtree rooted
at F:

10 -24
Case 3 : Height increase
Val(Ti)=Val(Ti-1)+Li
 Li is the height of the root.
Li
A(n) 0
0
newly
added
0
root -1
-1
-1
root

10 -25
Amortized analysis of X1
X2: # of absorptions in case 1
X3: # of single rotations in case 2
X4: # of double rotations in case 2
X5: # of height increases in case 3
Val(Tm) = Val(T0)+ Li
i
m


1
X22(X3+X4)
=0+X1X22(X3+X4)
Val(Tm)  0.618m (proved by Knuth)
 X1 = Val(Tm)+2(X2+X3+X4)X2
 0.618m+2m
= 2.618m

10 -26
A self-organizing sequential
search heuristics
 3 methods for enhancing the performance of
sequential search
(1)Transpose Heuristics:
Query Sequence
B B
D D B
A D A B
D D A B
D D A B
C D A C B
A A D C B

10 -27
(2)Move-to-the-Front Heuristics:
Query Sequence
B B
D D B
A A D B
D D A B
D D A B
C C D A B
A A C D B

10 -28
(3)Count Heuristics: (decreasing order by the count)
Query Sequence
B B
D B D
A B D A
D D B A
D D B A
A D A B
C D A B C
A D A B C

10 -29
Analysis of the move-to-the-
front heuristics
 interword comparison: unsuccessful
comparison
 intraword comparison: successful comparison
 pairwise independent property:
 For any sequence S and all pairs P and Q, # of
interword comparisons of P and Q is exactly # of
interword comparisons made for the subsequence
of S consisting of only P’s and Q’s.
(See the example on the next page.)

10 -30
e.g.
Query Sequence (A, B) comparison
C C
A A C
C C A
B B C A 
C C B A
A A C B 
# of comparisons made between A and B: 2
Pairwise independent
property in move-to-the-front

10 -31
Consider the subsequence consisting of A and B:
Query Sequence (A, B) comparison
A A
B B A 
A A B 
# of comparisons made between A and B: 2

10 -32
Query Sequence C A C B C A
(A, B) 0 1 1
(A, C) 0 1 1 0 1
(B, C) 0 0 1 1
0 1 1 2 1 2
There are 3 distinct interword comparisons:
(A, B), (A, C) and (B, C)
 We can consider them separately and then
add them up.
the total number of interword comparisons:
0+1+1+2+1+2 = 7

10 -33
Theorem for the move-to-the-
front heuristics
CM(S): # of comparisons of the move-to-the-
front heuristics
CO(S): # of comparisons of the optimal static
ordering
CM (S) 2CO(S)

10 -34
Proof
Proof:
 InterM(S): # of interword comparisons of the move to
the front heuristics
 InterO(S): # of interword comparisons of the optimal
static ordering
Let S consist of a A’s and b B’s, a  b.
The optimal static ordering: BA
InterO(S) = a
InterM(S)  2a
 InterM(S)  2InterO(S)

10 -35
 Consider any sequence consisting of more than two
items. Because of the pairwise independent property,
we have InterM(S)  2InterO(S)
 IntraM(S): # of intraword comparisons of the move-
to-the-front heuristics
 IntraO(S): # of intraword comparisons of the optimal
static ordering
 IntraM(S) = IntraO(S)
 InterM(S) + IntraM(S)  2InterO(S) + IntraO(S)
 CM(S)  2CO(S)
Proof (cont.)

10 -36
The count heuristics
 The count heuristics has a similar result:
CC(S)  2CO(S), where CC(S) is the cost
of the count heuristics

10 -37
The transposition heuristics
 The transposition heuristics does not possess the
pairwise independent property.
 We can not have a similar upper bound for the cost
of the transposition heuristics.
e.g.
Consider pairs of distinct items independently.
(A, B) 0 1 1
(A, C) 0 1 1 0 1
(B, C) 0 0 1 1
0 1 1 2 1 2
# of interword comparisons: 7 (not correct)

10 -38
the correct interword comparisons:
Data Ordering C AC CA CBA CBA CAB
Number of Interword
Comparisons
0 1 1 2 0 2

6

Graph Algorithms Graph Algorithms Graph Algorithms

More Related Content

Similar to Graph Algorithms Graph Algorithms Graph Algorithms

Recently uploaded

Graph Algorithms Graph Algorithms Graph Algorithms