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Data structures
Skip Lists - Idea
Sorted linked list:
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Insertion and Deletion is fast (requires just constant time);
however Find operation requires (n) time.
Can we do better? (E.g. somehow do binary search on
linked list?)
Data structures
Skip Lists - Idea
We can try something like this:
[Adapted from J.Erickson]
Data structures
Skip Lists - "perfect"
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
Data structures
Skip Lists - real examples
[Adapted from B.Weems]
Data structures
Skip Lists - LookUp
[Adapted from J.Erickson]
Data structures
Skip Lists - LookUp
[Adapted from J.Erickson]
Data structures
Skip Lists - LookUp
procedure SkipListLookUp(int K, SkipList L):
P  Header(L)
for i from Level(L) downto 0 do
while Key(Forward(P)[i]) < K do P  Forward(P)[i]
P  Forward(P)[0]
if Key(P) = K then return Data(P)
else return fail
Data structures
Skip Lists - Insert
procedure SkipListInsert(int K, int I, SkipList L):
P  Header(L)
for i from Level(L) downto 0 do
while Key(Forward(P)[i]) < K do
P  Forward(P)[i]
Update[i]  P
P  Forward(P)[0]
if Key(P) = K then Data(P)  I
Data structures
Skip Lists - Insert (cont.)
else
NewLevel  RandomLevel()
if NewLevel > Level(L) then
for i from Level(L) + 1 to NewLevel do
Update[i]  Header(L)
Level(L)  NewLevel
P  NewCell(NewLevel)
Key(P)  K
Data(P)  I
for i from 0 to NewLevel do
Forward(P)[i]  Forward(Update[i])[i]
Forward(Update[i])[i]  P
Data structures
Skip Lists - Random Level
procedure RandomLevel():
v0
while Random() < 1/2 and v < MaxLevel do
vv+1
return v
Data structures
Skip Lists - Complexity
Let T be a path from the header (starting at level Level(L))
of skip list L to node P (at level i).
If we trace the path backwards (from P to header) what will
be the expected number of segments C(k) after which the
path will rise k levels?
Data structures
Skip Lists - Complexity
If we are at node P then there are 2 possibilities:
• P is a node of level i and previous node is of level i
(from which path has to rise k levels)
• P is a node of level i and previous node is of level i – 1
(from which path has to rise k – 1 levels)
C(k) = 1/2 (1 + C(k)) + 1/2(1 + C(k – 1))
C(k) = 2 + C(k – 1) = (k)
Data structures
Skip Lists - Complexity
Number of traversed segments from P (at level 0) to header
(at level log n) will not exceed
S(n) = C(log n) + L(log n) = (log n) + L(log n),
where L(log n) is expected number of nodes at level log n or
higher.
L(log n) = n (1/2) log n = const
S(n) = O(log n) + const = O(log n)
Data structures
Skip Lists - Summary
LookUp
(log n)
Insert
(log n)
Delete
(log n)
MakeEmpty
(1)
IsEmpty
(1)
Data structures
Skip Lists - Summary
[Adapted from B.Weems]
Data structures
Binary Search Trees
T is a binary search tree, if
• it is a binary tree (with a key associated with each node)
• for each node x in T the keys at all nodes of left subtree
of x are not larger than key at node x and keys at all nodes
of right subtree of x are not smaller than key at node x
Data structures
BST - Examples
This is BST
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Data structures
BST - Examples
This is BST
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Data structures
BST - Examples
This is not BST
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Data structures
BST - Examples
This is not BST
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Data structures
Tree traversal algorithms
Inorder - left subtree, root, right subtree
Postorder - left subtree, right subtree, root
Preorder - root, left subtree, right subtree
Data structures
Tree traversal algorithms - Preorder
procedure Preorder(Tree T):
if T   then
write Key(T)
Preorder(LC(T))
Preorder(RC(T))
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Data structures
Tree traversal algorithms - Inorder
procedure Inorder(Tree T):
if T   then
Inorder(LC(T))
write Key(T)
Inorder(RC(T))
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Data structures
Tree traversal algorithms - Postorder
procedure Postorder(Tree T):
if T   then
Postorder(LC(T))
Postorder(RC(T))
write Key(T)
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Data structures
BST - LookUp
procedure BSTLookUp(int K, BST T):
while T   do
if K = Key(T) then return Data(K)
else if K < Key(T) then T  LC(T)
else T  RC(T)
return fail
Data structures
BST - Insert
procedure BSTInsert(int K, int I, BST T):
while T   do
if Key(T) = K then
Data(T)  I
return
else if K < Key(T) then
T  LC(T)
else
T  RC(T)
Q  NewCell()
Key(Q)  K; Data(Q)  I
LC(Q)  ; RC(Q)  
T Q
Data structures
BST - Insert - Example
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Data structures
BST - Insert - Example 2
[Adapted from N.Dale]
Data structures
BST - Insert - Example 2
[Adapted from N.Dale]
Data structures
BST - Delete
procedure BSTDelete(int K, BST T):
while T   and Key(T)  K do
if K < Key(T) then T  LC(T)
else T  RC(T)
if T =  then return
if RC(T) =  then T  LC(T)
else
Q  RC(T)
while LC(Q)   do Q  LC(Q)
TQ
Q  RC(Q)
LC(Q)  LC(T)
RC(Q)  RC(T)
Data structures
BST - Delete - Example 1
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Data structures
BST - Delete - Example 1
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Data structures
BST - Delete - Example 2
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Data structures
BST - Delete - Example 2
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Data structures
BST - Delete - Example 3
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Data structures
BST - Delete - Example 3
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Data structures
BST - Delete - Example 4
[Adapted from N.Dale]
Data structures
BST - Delete - Example 4
[Adapted from N.Dale]
Data structures
BST - Complexity
LookUp
(h)
Insert
(h)
Delete
(h)
MakeEmpty
(1)
IsEmpty
(1)
Data structures
BST - Insertion order defines shape
[Adapted from N.Dale]
Data structures
BST - Insertion order defines shape
[Adapted from N.Dale]
Data structures
BST - Average Height
Theorem
After inserting of n keys in initially empty binary search tree
T, the average height of T is (log n).
Theorem
Let S(n) be the number of comparisons in a successful
search of randomly constructed n node BST and let U(n) be
the number of comparisons in a unsuccessful search of
randomly constructed n node BST. Then
• S(n)  1.39 log n  S(n) = (log n)
• U(n)  1.39 log (n + 1)  U(n) = (log n)
Data structures
External Binary Trees
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Data structures
External Binary Trees
Proposition
For BST with n nodes the corresponding external binary tree
has 2n + 1 node.
Proposition
Let internal path length I(n) be the sum of the depths of
internal nodes and let external path length E(n) be the
sum of the depths of all external nodes.
Then E(n) = I(n) + 2 n.
Data structures
BST - Average Height
• we need one more comparison to find a key in successful
search than we needed to insert the key in the unsuccessful
search that preceded its insertion
S(n) = 1/n (U(0) + 1 + U(1) + 1 +  +U(n – 1) + 1)
• the expected number of comparisons in successful search is
1 more than average internal path length and the expected
number of comparisons in an unsuccessful search is the
average external path length
S(n) = 1 + (I(n)/n)
U(n) = E(n)/(n + 1)
Data structures
BST - Average Height
[Adapted from H.Lewis, L.Denenberg]
Data structures
BST - Average Height
[Adapted from H.Lewis, L.Denenberg]
Data structures
BST - Average Height
[Adapted from H.Lewis, L.Denenberg]
Data structures
AVL Trees
T is an AVL tree, if
• it is a binary search tree
• for each node x in T we have
Height(LC(x)) – Height(RC(x))  {– 1, 0, 1}
Data structures
AVL Trees - Height
Theorem
Any AVL tree with n nodes has height less than 1.44 log n.
We will prove a slightly weaker statement that h = (log n)
Data structures
AVL Trees - “Worst” examples
Data structures
AVL Trees - Height
w(0) = 1, w(1) = 2, w(3) = 4, w(4) = 7, etc.
w(h) = 1 + w (h – 1) + w(h – 2)
F(i) = F(i – 1) + F(i – 2), F(0) = 0, F(1) = 1
w(h) = F(h + 3) – 1
Data structures
AVL Trees - Height
w(h) < 2h  h > log n  h = (log n)
w(h)  21/2 h  h  2 log n  h = O(log n)
Hence h = (log n)
Data structures
AVL Trees – Insertion – Case 1
+1 h + 2
h
0 h+2
h+1
h +1
h+1
Data structures
AVL Trees – Insertion – Case 2
0 h+1
h
-1 h + 2
h
h
h +1
Data structures
AVL Trees – Insertion – Case 3
–1 h + 1
h
–2 h + 2
h–1
h–1
h +1
Data structures
AVL Trees – Insertion – Rotation 1
A +2 h + 3
A +1 h + 2
C +1
C 0
h
h
T1
T1
h
T2
h
h
T3
T2
h+ 1
T3
Data structures
AVL Trees – Insertion – Rotation 1
A +2 h + 3
C 0
C +1
A
h+2
0
h+ 1
h
T1
h
T2
h+ 1
T3
T3
h
T1
T2
h
Data structures
AVL Trees – Insertion – Rotation 2
A +1 h + 2
0
h
0
T1
A +2 h + 3
C
-1 C
h
B
h
h–1
T2
T3
T4
+1 B
T1
h
h–1
T2
T3
T4
Data structures
AVL Trees – Insertion – Rotation 2
A +2 h + 3
B
T1
h–1
0
C
T2
h–1
h
T1
h
h+2
A -1
+2 B
h
0
T3
T4
0
T2
C
T3
T4
h
Data structures
AVL Trees – Insertion – Example 1
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Data structures
AVL Trees – Insertion – Example 1
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Data structures
AVL Trees – Insertion – Example 2
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Data structures
AVL Trees – Insertion – Example 2
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Data structures
AVL Trees – Insertion – Rotation 1
A +2 h + 3
A +1 h + 2
C +1
C 0
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h
T1
T1
h
T2
h
h
T3
T2
h+ 1
T3
Data structures
AVL Trees – Insertion – Rotation 1
A +2 h + 3
C 0
C +1
A
h+2
0
h+ 1
h
T1
h
T2
h+ 1
T3
T3
h
T1
T2
h
Data structures
AVL Trees – Insertion – Example 2
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A 12
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C 20
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Data structures
AVL Trees – Insertion – Example 2
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Data structures
AVL Trees – Insertion – Example 3
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Data structures
AVL Trees – Insertion – Example 3
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Data structures
AVL Trees – Insertion – Rotation 2
A +1 h + 2
0
h
0
T1
A +2 h + 3
C
-1 C
h
B
h
h–1
T2
T3
T4
+1 B
T1
h
h–1
T2
T3
T4
Data structures
AVL Trees – Insertion – Rotation 2
A +2 h + 3
B
T1
h–1
0
C
T2
h–1
h
T1
h
h+2
A -1
+2 B
h
0
T3
T4
0
T2
C
T3
T4
h
Data structures
AVL Trees – Insertion – Example 3
7
A 10
3
2
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4
C 20
8
9
B 12
11
24
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Data structures
AVL Trees – Insertion – Example 3
7
A 10
3
2
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B 12
8
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C 20
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Data structures
AVL Trees – Insertion – Example 3
7
B 12
3
2
A 10
5
4
8
C 20
11
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Data structures
AVL Trees – Rebalancing
[Adapted from P.Flocchini]
Data structures
AVL Trees – Insertion
[Adapted from P.Flocchini]
Data structures
AVL Trees – Deletion – Example 1
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h=3
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h=4
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Data structures
AVL Trees – Deletion – Example 1
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h=3
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h=4
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Data structures
AVL Trees – Deletion – Example 1
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h=3
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h=4
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Data structures
AVL Trees – Deletion – Example 1
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90
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h=3
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h=4
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Data structures
AVL Trees – Deletion – Example 1
90
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h=3
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h=4
47
Data structures
AVL Trees – Deletion – Example 2
[Adapted from P.Flocchini]
Data structures
AVL Trees – Deletion – Example 2
[Adapted from P.Flocchini]
Data structures
AVL Trees - Complexity
LookUp
(log n)
Insert
(log n)
Delete
(log n)
MakeEmpty
(1)
IsEmpty
(1)
Data structures
2 – 3 Trees
•
All leaves are at the same depth and contain 1 or 2 keys
•
An interior node either
- contains one key and has 2 children ( 2- node), or
- contains two keys and has three children ( 3- node)
•
A key in an interior node is between the keys in the subtrees
of its adjacent children. (For 2- node it is just BST property,
for 3- node the two keys split the keys in the subtrees in 3
groups: less than smaller key, between two keys, greater
than the larger key)
Data structures
2 – 3 Trees - Example
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Data structures
2 – 3 Trees - Insertion
1. Search for leave where key belongs. Remember path.
2. If there is one key in the leaf, add key and stop.
3. Split node in two (first and third keys), pass the middle key
to the parent.
4. If parent is a 2-node, then stop. Else return to step 3.
5. The process is repeated until there is a room for a key, or the
root is being split - then a new root is created and the height
increases by 1.
Data structures
2 – 3 Trees - Insertion
13-41
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Data structures
2 – 3 Trees - Insertion
13-33-41
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Data structures
2 – 3 Trees - Deletion
1. If key is in leaf, then remove it. Otherwise replace it by the
Inorder successor and remove the successor from the leaf.
2. A key has been removed from node N. If N still has a key
then stop. Otherwise:
a) If N is the root, delete it. If N has no child the tree becomes
empty; otherwise the child becomes the root.
b) If N has a sibling N’ immediately to its left or right that has
two keys, let S be a key in the parent that separates them.
Move S to N and replace it in the parent by the key of N’
that is adjacent to N. If N and N’ are interior nodes, then
move one child of N’ to be a child of N. (N and N’ now
have 1 key each, instead of 0 and 2). Stop.
Data structures
2 – 3 Trees - Deletion
c) N’ - a sibling of N with only one key. Let P be a parent
of N and N’ and let S be a key in P that separates them.
Consolidate S and the one key in N’ into a new 3-node,
which replaces both N and N’. This reduces by 1 the
number of keys in P and the number of children of P.
Let N  P and repeat step 2.
Data structures
2 – 3 Trees – Deletion 1
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13-41
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17-38
11 16
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Data structures
2 – 3 Trees – Deletion 2
13-41
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2
17-33
11 16
21
50
38-40
44
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13-41
17-33
2-9
16
21
50
38-40
44
99
Data structures
2 – 3 Trees – Deletion 2
13-41
17-33
2-9
16
21
50
38-40
44
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17-41
13
2-9
33
16
21
50
38-40
44
99
Data structures
Red-Black trees
A
A
B
C
B
A-B
C
A
B
or
C
D
E
C
B
D
A
E
C
E
D
Data structures
(a,b) Trees
We assume a  2 and b  2a – 1.
•
All leaves are at the same depth
•
All interior nodes has c children, where a  c  b (root may
have from 2 to b children).
Data structures
B-trees - motivation
So far we have assumed that we can store an entire
data structure in main memory
What if we have so much data that it won’t fit?
We will have to use disk storage but when this happens
our time complexity fails
The problem is that Big-Oh analysis assumes that all
operations take roughly equal time
This is not the case when disk access is involved
[Adapted from S.Garretti]
Data structures
B-trees - motivation
Assume that a disk spins at 3600 RPM
In 1 minute it makes 3600 revolutions, hence one
revolution occurs in 1/60 of a second, or 16.7ms
On average what we want is half way round this disk – it
will take 8ms
This sounds good until you realize that we get 120 disk
accesses a second – the same time as 25 million
instructions
In other words, one disk access takes about the same
time as 200,000 instructions
It is worth executing lots of instructions to avoid a disk
access
[Adapted from S.Garretti]
Data structures
B-trees - motivation
Assume that we use an AVL tree to store all the car
driver details in UK (about 20 million records)
We still end up with a very deep tree with lots of
different disk accesses; log2 20,000,000 is about 24, so
this takes about 0.2 seconds (if there is only one user of
the program)
We know we can’t improve on the log n for a binary tree
But, the solution is to use more branches and thus less
height!
As branching increases, depth decreases
[Adapted from S.Garretti]
Data structures
B-trees - definition
A B-tree of order m is an m-way tree (i.e., a tree where
each node may have up to m children) in which:
1. the number of keys in each non-leaf node is one less than the
number of its children and these keys partition the keys in the
children in the fashion of a search tree
2. all leaves are on the same level
3. all non-leaf nodes except the root have at least m / 2 children
4. the root is either a leaf node, or it has from two to m children
5. a leaf node contains no more than m – 1 keys
The number m should always be odd
[Adapted from S.Garretti]
Data structures
B-trees - example
26
6
A B-tree of order 5
containing 26 items
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1
2
4
27
7
29
8
13
45
46
15
48
Note that all the leaves are at the same level
[Adapted from S.Garretti]
18
51
62
25
53
55
60
64
70
90
Data structures
B*-trees - example
K
E
C-D
P-R
I
L-O
Q
T-X
|B| |D| |E| |H| |J| |L| |N| |P| |Q| |R| |T| |U| |Z|
Data structures
Self adjusting (Splay) BST
[Adapted from B.Weems]
Data structures
Self adjusting (Splay) BST
Basic operations:
LookUp(K,T)
Insert(K,I,T)
Delete(K,T)
Additional technical operations:
Splay(K,T)
Concat(T1,T2)
Data structures
Self adjusting BST - Splay operation
Splay(K,T)
Modifies T so that it remains BST on the same keys.
But the new tree has K at the root, if K is in tree;
otherwise the root contains the next largest or
smallest key compared to K.
Data structures
Self adjusting BST - Splay operation
K
Splay(K,T)
B
K
J
X
P
X
B
P
J
Data structures
Self adjusting BST - Splay operation
J
Splay(K,T)
B
X
J
P
B
X
P
Data structures
Self adjusting BST - LookUp
LookUp(K,T)
Execute Splay(K,T); examine the root to see if it
contains K.
?
Splay(K,T)
Data structures
Self adjusting BST - Insert
Insert(K,I,T)
Execute Splay(K,T); if K is in root, install I in this node.
Otherwise create a new node containing K and I and
break one link to make this node a new root.
Data structures
Self adjusting BST - Insert
J
K
J
Splay(K,T)
B
X
J
P
B
X
P
B
X
P
Data structures
Self adjusting BST - Concat
Concat(T1,T2)
T1, T2 - BST such that every key in T1 is less than
every key in T2. Concat(T1,T2) creates a BST
containing all keys from T1 and T2.
Execute Splay(+,T1). Now T1 has no right subtree,
attach T2 as the right child of T1.
Data structures
Self adjusting BST - Concat
Splay(+,T)
T2
T1
+
+
T2
T1
T2
T1
Data structures
Self adjusting BST - Delete
Delete(K,T)
Execute Splay(K,T1). If the root does not contain K,
there is nothing to do. Otherwise apply Concat to the
two subtrees of the root.
Data structures
Self adjusting BST - Delete
Splay(K,T)
K
Concat(T1,T2)
T2
T1
T2
T1
Data structures
Self adjusting BST - Splay
Splay(K,T)
Search for node K, remembering the search path by
stacking it. Let P be the last node inspected. If K is in
tree, then it is P; otherwise P has an empty child
where the search for K terminated. Return along the
path for P to root carrying out the following rotations,
which move P up the tree.
Data structures
Self adjusting BST - Splay - Case 1
Splay(K,T)
Case 1. P has no grandparent, i.e. Parent(P) is the
root. Perform the single rotation around the parent
of P.
Data structures
Self adjusting BST - Splay - Case 1
Q
P
P
Q
C
A
B
A
B
C
Data structures
Self adjusting BST - Splay - Case 2
Splay(K,T)
Case 2. P and Parent(P) are both left children, or both
right children. Perform two single rotations in the
same direction, first around the grandparent of P and
then around the parent of P.
Data structures
Self adjusting BST - Splay - Case 2
Q
R
P
Q
P
R
D
A
B
C
A
B
P
Q
R
A
B
C
D
C
D
Data structures
Self adjusting BST - Splay - Case 3
Splay(K,T)
Case 3. One of P and Parent(P) is a left child and the
other is a right child. perform single rotations in
opposite directions, firts around the parent of P and
then around its grandparent.
Data structures
Self adjusting BST - Splay - Case 3
P
R
R
Q
A
Q
P
A
D
B
C
B
C
D
Data structures
Self adjusting BST - Splay
[Adapted from B.Weems]
Data structures
SA BST - Worst case complexity
LookUp
(n)
Insert
(n)
Delete
(n)
MakeEmpty
(1)
IsEmpty
(1)
Data structures
SA BST - Amortised complexity
Theorem
Any sequence of m dictionary operations on a self-adjusting
tree that is initially empty and never has more than n nodes
uses O(m log n) time.
Data structures
SA BST - Amortised complexity - Tree
Potential
Node rank
r(N) = log w(N),
where w(N) - number of descendants of
N (including N itself)
Tree potential
P(Tree) = sum of all r(N), where N is a node in Tree
Amortised cost
TA(op) = T(op) + P
Data structures
SA BST - Amortised complexity
[Adapted from B.Weems]
Data structures
SA BST - Amortised complexity - Example
[Adapted from B.Weems]
Data structures
SA BST - Amortised complexity - Example
[Adapted from B.Weems]
Data structures
SA BST - Amortised complexity - Example
[Adapted from B.Weems]
Data structures
SA BST - Amortised complexity
Theorem
The amortised complexity of Splay operation on tree with
n nodes is at most 3 log n + 1.
LookUp
- TA(Splay) + const = O(log n)
Insert
- TA(Splay) + r(root) + const = O(log n)
Concat
- TA(Splay) + (part of) r(root) + const = O(log n)
Delete
- TA(Splay) + TA(Concat) + const = O(log n)
Data structures
SA BST - Amortised complexity
Lemma
The amortised cost of Splay step involving node P, the
parent of P and (possibly) the grandparent of P is at most
3 (r’(P) – r(P)), where r(P) and r’(P) are rank of P before and
after rotation (at most 3 (r’(P) – r(P)) + 1 for the last step
in Splay).
Data structures
SA BST - Amortised complexity
TA(Splay step)  3 (r’(P) – r(P))
TA(Splay) 
( +1)
3 (r’(P) – r(P)) +
3 (r(2)(P) – r’(P)) +
3 (r(3)(P) – r(2) (P)) +
...
3 (r(k)(P) – r(k–1) (P)) + 1 = 3 (r(k)(P) – r(P)) + 1
r(k)(P) - the rank of original root  log n
TA(Splay)  3 (log n – r(P)) + 1  3 log n + 1
Data structures
SA BST - Amortised complexity
Lemma (Rank Rule)
If a node has two children of equal rank, then its rank is greater
than that of each child.
w(P)2r(P)
w(Q)2r(Q)
R
P
Q
If r(Q) = r(P) then w(R) = w(P)+W(Q)+1>2r(P)+1
Thus r(R)= log w(r)  r(P)+1
Data structures
SA BST - Amortised complexity - Case 1
This must be the last step!
Q
P
P
Q
C
A
B
TA(Splay step) 
A
B
r’(P) + r’(Q) – r(P) – r(Q) + T 
3 (r’(P) – r(P)) + 1
C
Data structures
SA BST - Amortised complexity - Case 1
[Adapted from B.Weems]
Data structures
SA BST - Amortised complexity - Case 2
Q
R
P
Q
P
R
D
A
B
C
A
C
P
B
Q
TA(Splay step) 
r’(P) + r’(Q) + r’(R) –
r(P) – r(Q) – r(R) + T 
3 (r’(P) – r(P))
D
R
A
B
C
D
Data structures
SA BST - Amortised complexity - Case 2
[Adapted from B.Weems]
Data structures
SA BST - Amortised complexity - Case 2
[Adapted from B.Weems]
Data structures
SA BST - Amortised complexity - Case 3
P
R
R
Q
A
Q
P
A
B
C
D
B
C
TA(Splay step) 
r’(P) + r’(Q) + r’(R) –
r(P) – r(Q) – r(R) + T 
3 (r’(P) – r(P))
D
Data structures
SA BST - Amortised complexity - Case 3
[Adapted from B.Weems]
Data structures
SA BST - Amortised complexity - Case 3
[Adapted from B.Weems]