1. Chapter 10 Search Structures
KAIST- CS206

2. 10.1 Optimal Binary Search Trees
Extended Two binary trees corresponding to search trees
for
for do
while do
return
return if
while if
An optimal binary search tree for a1, a2, … an (a1 < .. < an)
is one that minimize the quantity of I+E with
(1) pi = the probability of searching for each ai,
(2) qi = the probability of searching for id in between
ai-1 and ai.
Internal path length I = = 7
External path length E = = 17
E = I + 2n
KAIST- CS206

3. 10.1 Optimal Binary Search Tree: Example
If assume pi = qi = 1/7,
for all i and j 
If assume p1=0.5, p2=0.1,
p3=0.05, q0=0.15,
q1=0.1, q2=0.05,q3=0.05, 
while if if do
while do
(a) Cost = 15/7, 2.65
(b) Cost = 13/7, 1.9 do
while do if do
while
while if if
(c) Cost = 15/7, 1,5
(d) Cost = 15/7, 2.05
(e) Cost = 15/7, 1.6
KAIST- CS206

4. 10.1 Optimal Binary Search Tree: Algorithm
Ti,j : an optimal binary search tree for ai+1, …, aj , i<j.
ci,j : cost of Ti,j
ri,j : root of Ti,j
ci,j = min {pk + cost(L) + cost(R) + weight(L) + weight(R) | k = i+1, …, j}
= min{wi,j + ci,k-1 + ck, j | k = i+1, …, j}
= wi,j + min{wi,j + ci,k-1 + ck, j | k = i+1, …, j} ak L R
Example 10.2: (a1, a2, a3, a4)=(do, if, return, while),
(p1, p2, p3, p4) = (3, 3, 1, 1), (q0, q1, q2, q3, q4) = (2, 3, 1, 1, 1) 1 2 3 4
W00 = 2
C00 = 0
r00 = 0
W11 = 3
C11 = 0
r11 = 0
W22 = 1
C22 = 0
r22 = 0
W33 = 1
C33 = 0
r33 = 0
W44 = 1
C44 = 0
r44 = 0
W01 = 8
C01 = 8
r01 = 1
W12 = 7
C12 = 7
r12 = 2
W23 = 3
C23 = 3
r23 = 3
W34 = 3
C34 = 3
r34 = 4
W02 = 12
C02 = 19
r02 = 1
W13 = 9
C13 = 12
r13 = 2
W24 = 5
C24 = 8
r24 = 3
W03 = 14
C03 = 25
r03 = 2
W14 = 11
C14 = 19
r14 = 2
W04 = 16
C04 = 32
r04 = 2
for do
return
while

5. 10.1 Optimal Binary Search Tree: Algorithm(2)
template <class KeyType>
BST<KeyType>::obst(int * p, int* q, Element<KeyType>* a, int n)
// Given n distinct identifiers a1 < a2 < … < an, and probabilities pj, 1 <= j <= n, and
// qi, 0 <= i <= n this algorithm computes the cost cij of optimal binary search trees Tij
// for identifiers ai+1, …, aj. It also computes rij, the root of Tij. Wij is the weight of Tij {
for (int i=0;i<n;i++) {
w[i][i] = q[i]; r[i][i] = c[i][i] = 0; // initialize
w[i][i+1] = q[i] + q[i+1] + p[i+1]; // optimal trees with one node
r[i][i+1] = i+1;
c[i][i+1] = w[i][i+1]; }
w[n][n] = q[n]; r[n][n] = c[n] [n] =0;
for (int m=2; m <= n; m++) // find optimal trees with m nodes
for (int i=0; i<=n-m; i++) {
int j = i+m;
w[i][j] = w[i][j-1] + p[j] + q[j];
int k = KnuthMin(i, j);
// KnuthMin returns a value k in the range[r[i, j-1], r[i+1, j]]
// minimizing c[i, j-1[ + c[k, [
c[i][j] = w[i[[j] + c[i][k-1] + c[k][j];
r[i][j] = k;
} // end of obst
O(n3)
O(n2)

6. 10.2 AVL Trees : 동기
Objective : maintain binary search tree structure for O(log n) access time with dynamic changes of
identifiers (i.e., elements) in the tree.
APR
AUG
DEC
FEB
JAN
JULY
JUNE
MAR
MAY
NOV
OCT
SEPT
JULY
FEB
MAY
AUG
JAN
MAR
OCT
NOV
DEC
JUNE
NOV
SEPT
최상의 dynamic insertions
최악의 dynamic insertions
KAIST- CS206

7. 10.2 AVL Trees : 정의
Height-balanced: An empty tree is height-balanced. If T is a non-empty binary tree with TL and TR
as its left and right subtrees, respectively, then T is height-balanced iff (1) TL and TR are height-
balanced and (2) |hL-hR| < 2 where hL and hR are the heights of TL and TR, respectively.
Balance factor: The balance factor, BF(k), of a node k in a binary tree : hL – hR.
AVL is a binary search tree, satisfying for any node k in the tree, BF(k) = -1, 0, or 1.
KAIST- CS206

8. 10.2 AVL Trees : Insertion NOV MAR AUG MAY +1 (d) Insert AUGUST -1 MAR
(d) Insert AUGUST -1
MAR
MAR
MAY
(a) Insert MARCH
(b) Insert MAY
MAR
MAY
NOV -2 -1
MAY RR
MAR
NOV
(c) Insert NOV

9. 10.2 AVL Trees : Insertion (2)LL
(e) Insert APRIL
NOV
MAR
MAY +2
AUG
APR +1 +2
MAY
MAR -1 -1
AUG
NOV
AUG
MAY +1 LR
APR
MAR
APR
JAN
NOV
(f) Insert JANUARY
JAN

10. 10.2 AVL Trees : Insertion (3)LR
(g) Insert DECEMBER
MAY
AUG
MAR +1 -1
APR
JAN
DEC
NOV
JULY
(h) Insert JULY +2 +1
MAR
MAR -2 -1 -1
DEC
MAY
DEC
MAY +1 RL +1
APR
JAN
NOV
AUG
JAN
NOV -1
DEC
JULY
APR
FEB
JULY
FEB
(i) Insert FEBRUARY

11. 10.2 AVL Trees : Insertion (4)+2
MAR
JAN -1 -1 +1 LR
DEC
MAY
DEC
MAR +1 -1 +1 +1
AUG
JAN
NOV
AUG
FEB
AUG
NOV -1
APR
FEB
JULY
APR
JULY
JULY
JUNE
(j) Insert JUNE RR
(k) Insert OCTOBER
MAR
DEC
JAN -1 +1
AUG
FEB
MAY -2
APR
JUNE
JULY
NOV
OCT

12. 10.2 AVL Trees : Insertion (4)+2
MAR
JAN -1 -1 +1 LR
DEC
MAY
DEC
MAR +1 -1 +1 +1
AUG
JAN
NOV
AUG
FEB
AUG
NOV -1
APR
FEB
JULY
APR
JULY
JULY
JUNE
(j) Insert JUNE RR
(k) Insert OCTOBER
MAR
DEC
JAN -1 +1
AUG
FEB
MAY -2
APR
JUNE
JULY
NOV
OCT

13. 10.2 AVL Trees : Rebalancing rotations
Balanced subtree
Unbalanced following insertion
rotation
type
Rebalanced subtree +1 A +2 A B LL B AR +1 B AR BL A
h+2 h BL BR BL BR
h+2 BR AR
Height of BL increases to h+1
Height of subtrees
of B remain h+1
Rebalanced subtree
Height of BR increases to h+1 -1 A -2 A B
height+2 AL B AL -1 B A BR RR BL BR BL BR
height+2 AL BL

14. 10.2 AVL Trees : Rebalancing rotations(2)
Balanced subtree
Unbalanced following insertion
rotation
type
Rebalanced subtree +2 A C +1 A
LR(a) -1 B B A B C +1 A +2 A C
h+2
h+2 -1 B AR B -1 A B AR h h BL C BL +1 C
LR(b) BL CL CR BR CL CR CL CR
h-1 h

15. 10.2 AVL Trees : Rebalancing rotations(3)+2 A C
h+2 -1 B +1 B A AR
LR(c) BL -1 C BL CL CR BR CL CR h

16. 10.2 AVL Trees : Performance comparisons
Output in order
O(log n)
O(1)2
Insert x
O(k)
O(n-k)
Delete k-th item
O(1)1
Delete x
O(1)
Search for k-th item
Search for x
AVL tree
Linked List
Sequential list
Operation
Doubly linked list and position of x known
Position for insertion known
KAIST- CS206

17. 10.3 2-3 Trees a special case of B-trees Definition (2-3 tree) :
node degree is more than 2
a special case of B-trees
Definition (2-3 tree) :
(1) Each internal node is a 2-node or a 3-node.
(2) if e is a 2-node, key of every element in LeftChild(node e) < key of e
key of every element in MiddleChild(node e) > key of e
(3) if e is a 3-node, key of every element in LeftChild(node e) < keyL of e
KeyL of e < key of every element in MiddleChild(node e) > keyR of e
key of every element in RightChild(node e) > keyR of e
(4) All external nodes are at the same level. A 40 20 10 B 80 C

18. 10.3.3 Inserting into a 2-3 Tree G 40 20 A 70 F 60 80 10 30 B E C D
Searching : O(log n)
Inserting : O(log n)
(b) 60 inserted A 40 20 10 B 80 C A A 40 40 20 B C C B D 20 10 80 70 10 30 80 70
(a) 70 inserted
(b) 30 inserted

19. 10.3.4 Deletion from a 2-3 Tree 80 50 20 10 B 95 90 D 60 C A
70 deleted
Deleting : O(log n) A 80 50 20 10 B 95 90 D 70 60 C A 80 50 20 10 B 60 C 95 D
90 deleted

20. 10.3.4 Deletion from a 2-3 Tree (2)80 50 20 10 B 95 D 60 C B 80 20
10 deleted A 20 10 B 80 C
50 deleted A 80 20 10 B 95 D 50 C
60 deleted A 20 10 B 80 50 C
95 deleted

21. 10.3.4 Deletion from a 2-3 Tree : Rotations
(a) p is the left child r r ? x r ? y p q x p q z y z a b c d a b c d r ? z r ? y
(b) p is the middle child r y x q p x q p z a b c d a b c d
(c) p is the right child r r r z x y w a a q p q p z y x z b c d e b c d e

22. 10.3.4 Deletion from a 2-3 Tree : Procedure
Step 1: Modify node p as necessary to reflect its status after the desired element has
been deleted.
Step 2: for (; p has zero elements && p!=root; p = r) {
let r be the parent of p, and let q be the left or right sibling of p
(as appropriate);
if(q is a 3-node) perform a rotation
else perform a combine; }
Step 3: If p has zero elements, then p must be the root. The left child of p becomes
the new root, and node p is deleted.
KAIST- CS206

23. Figure 10.35: Example of a 3-way search tree that is not a 2-3 tree
10.6 B-Trees
Definition: An m-way search tree satisfies
(1) The root has at most m subtrees n, A0, (K1, A1), (K2, A2), …, (Kn, An).
(2) Ki < Ki+1, i = 1, …, n.
(3) Ki < All key values in subtree Ai < Ki+1, i = 1, …, n.
(4) Kn < All key values in subtree An, All key values in subtree A0 < K1.
(5) The subtrees Ai, i = 1, …, n, are also m-way search tree. a
20, 40 T
node
schematic format a
2, b, (20, c), (40, d) b
2, 0, (10, 0), (15, 0) c
2, 0, (25, e), (30, 0) d
2, 0, (45, 0), (50, 0) e
1, 0, (28, 0) b c d
10, 15
25, 30
45, 50 e 28
Figure 10.35: Example of a 3-way search tree that is not a 2-3 tree
Definition: A B-tree of order m is an m-way search tree, satisfying
(1) The root node has at least two children.
(2) All nodes other than root and failure nodes have at least m/2 children.
(3) All failure nodes are at the same level.

24. 10.6.3 B-Trees: Properties N: minimum number of keys in a B-tree
N+1 = the number of failure nodes
= the number of nodes at level l+1
> 2(m/2)l-1
 If there are N key values, the level of B-tree l is l < log m/2 {N+1)/2} +1
Total maximum search time
6.8
5.7 m 50
125
400
Choice of m
- depending on access time :
time for reading nodes from disk
+ time to search the nodes for x

25. 10.9 Tries blank a b c g o t w oriole h blank l u a h o u wren r d s b
oriole h
blank l u a h o u
wren r d s b
bluebird
bunting
gull a u
cardinal
chickadee
godwit
goshawk
thrasher
thrush
KAIST- CS206

26. 10.9 Tries : Searching and Sampling Strategies
1. Searching : O(l) where l is the number of level
2. How to reduce l  sampling strategy at the i-th level for key value x
Example: Sample(x, i) = xr(x,i) for r(x,i) a randomization function
blank a b c d e f g h i j k l m n o p q r s t u v w x y z b
bunting
goshawk
wren
godwit
bluebird
thrush
thrasher e l a h
A tri : sampling one character at a time,
from right to left
chickadee
oriole
cardinal
gull
blank a b c d e f g h i j k l m n o p q r s t u v w x y z b
thrasher
An optimal tri :
sampling on the first level
done by using the fourth
character
cardinal
goshawk
wren
bunting
godwit
chickadee
bluebird
gull
thrush
oriole

27. 10.9 Tries : Insertion and Deletionb
Shrink when deleting l o u σ u
bunting δ 1
Need a count
data member
in each branch
node
bobwhite e δ 2 b j δ 3
Section of tri showing changes resulting from
inserting bobwhile and bluejay ρ
bluebird
bluejay
Grow when inserting