Splay Tree Algorithm
Mingda Zhao
CSC 252 Algorithms
Smith College
Fall, 2000
Contents
 What is Splay Tree?
 Splaying: zig-zig; zig-zag; zig.
 Rules: search; insertion; deletion.
 Complexity analysis.
 Implementation.
 Demos
 Conclusion
 References
What is Splay Tree?
 A balanced search tree data structure
 NIST Definition: A binary search tree in
which operations that access nodes
restructure the tree.
 Goodrich: A splay tree is a binary search
tree T. The only tool used to maintain
balance in T is the splaying step done after
every search, insertion, and deletion in T.
 Kingston: A binary tee with splaying.
Splaying
 Left and right rotation
 Move-to-root operation
 Zig-zig
 Zig-zag
 Zig
 Comparing move-to-root and splaying
 Example of splaying a node
Left and right rotation
Adel’son-Vel’skii and Landis (1962), Kingston
Left rotation:
Y
X
T1
X
T3
T2
T1
Y
T2
T3
Right Rotation:
X
Y
T1
Y
T3
T2
T1
X
T2
T3
Move-to-root operation (x=3)
Allen and Munro (1978), Bitner(1979), Kingston
4
2
1
6
3
5
7
Move-to-root operation (x=3)
4
3
6
2
5
1
7
Move-to-root operation (x=3)
3
4
2
6
1
5
7
Zig-zig splaying
 The node x and its parent y are both left
children or both right children.
 We replace z by x, making y a child of x
and z a child of y, while maintaining the
inorder relationships of the nodes in tree T.
 Example: x is 30, y is 20, z is 10
Before splaying:
10
20
T1
30
T2
T3
T4
After splaying:
30
20
T4
10
T3
T1
T2
Zig-zag splaying
 One of x and y is a left child and the other is
a right child.
 We replace z by x and make x have nodes y
and z as its children, while maintaining the
inorder relationships of the nodes in tree T.
 Example: z is 10, y is 30, x is 20
Before splaying:
10
30
T1
20
T4
T2
T3
After splaying:
20
10
T1
30
T2
T3
T4
Zig splaying
 X doesn’t have a grand parent(or the
grandparent is not considered)
 We rotate x over y, making x’s children be
the node y and one of x’s former children u,
so as to maintain the relative inorder
relationships of the nodes in tree T.
 Example: x is 20, y is 10, u is 30
Before splaying:
10
20
T1
30
T2
T3
T4
After splaying:
20
10
T1
30
T2
T3
T4
Move-to-root vs. splaying
 Move-to-root should improve the performance of
the BST when there is locality of references in the
operation sequence, but it is not ideal. The
example we use before, the result is not balanced
even the original tree was.
 Splaying also moves the subtrees of node x(which
is being splayed) up 1 level, but move-to-root
usually leaves one subtree at its original level.
 Example:
Example1:original tree
10
20
T1
30
T2
T3
T4
Example1:move-to-root
30
10
T4
20
T1
T2
T3
Example1:splaying
30
20
T4
10
T3
T1
T2
Example2:original tree
50
40
30
20
10
Example2:
move-to-root
10
50
40
30
20
Example2:splaying
10
40
50
20
30
Splaying a node: original tree
20
25
40
10
30
50
15
35
Splaying a node: cont.
20
25
35
10
30
40
15
50
Splaying a node: cont.
35
25
40
20
30
10
15
50
Rules of splaying: Search
 When search for key I, if I is found at node
x, we splay x.
 If not successful, we splay the parent of the
external node at which the search terminates
unsuccessfully.(null node)
 Example: see above slides, it can be for
successfully searched I=35 or
unsuccessfully searched say I=38.
Rules of splaying: insertion
 When inserting a key I, we splay the newly
created internal node where I was inserted.
 Example:
Original tree
10
After insert key 15
10
15
After splaying
15
10
After insert key 12
15
10
12
After splaying
12
10
15
Rules of splaying: deletion
 When deleting a key I, we splay the parent
of the node x that gets removed. x is either
the node storing I or one of its
descendents(root, etc.).
 If deleting from root, we move the key of
the right-most node in the left subtree of
root and delete that node and splay the
parent. Etc.
 Example:
Original tree
30
10
40
15
50
20
After delete key 30(root)
20
10
40
15
50
We are going to splay 15
20
10
40
15
50
After splaying
10
15
20
40
50
Complexity
 Worst case: O(n), all nodes are on one side
of the subtree. (In fact, it is (n).)
 Amortized Analysis:
We will only consider the splaying time,
since the time for perform search, insertion
or deletion is proportional to the time for the
splaying they are associated with.
Amortized analysis
 Let n(v) = the number of nodes in the subtree
rooted at v
 Let r(v) = log(n(v)), rank.
 Let r’(v) be the rank of node v after splaying.
 If a>0, b>0, and c>a+b, then
log a + log b <= 2 log c –2
 *Search the key takes d time, d = depth of the
node before splaying.
Before splaying:
Z
Y
T1
X
T2
T3
T4
After splaying:
X
Y
T4
Z
T3
T1
T2
Cont.
 Zig-zig:
variation of r(T) caused by a single splaying substep is:
r’(x)+r’(y)+r’(z)-r(x)-r(y)-r(z)
=r’(y)+r’(z)-r(x)-r(y)
(r’(x)=r(z))
<=r’(x)+r’(z)-2r(x)
(r’(y)<=r’(x) and r(y)>=r(x))
Also n(x)+n’(z)<=n’(x)
We have r(x) + r’(z)<=2r’(x)-2
(see previous slide)
 r’(z)<=2r’(x)-r(x)-2
so we have variation of r(T) by a single splaying step is:
<=3(r’(x)-r(x))-2
Since zig-zig takes 2 rotations, the amortized complexity will
be 3(r’(x)-r(x))
Before splaying:
Z
Y
T1
X
T4
T2
T3
After splaying:
X
Z
T1
Y
T2
T3
T4
Cont.
 Zig-zag:
variation of r(T) caused by a single splaying substep is:
r’(x)+r’(y)+r’(z)-r(x)-r(y)-r(z)
=r’(y)+r’(z)-r(x)-r(y)
(r’(x)=r(z))
<=r’(y)+r’(z)-2r(x)
(r(y)>=r(x))
Also n’(y)+n’(z)<=n’(x)
We have r’(y)+r’(z)<=2r’(x)-2
So we have variation of r(T0 by a single splaying substep is:
<=2(r’(x)-r(x))-2 <=3(r’(x)-r(x))-2
Since zig-zag takes 2 rotations, the amortized complexity will
be 3(r’(x)-r(x))
Before splaying:
Y
X
T1
Z
T2
T3
T4
After splaying:
X
Y
T1
Z
T2
T3
T4
Cont.
 Zig:
variation of r(T) caused by a single splaying
substep is:
r’(x)+r’(y)-r(x)-r(y)
<=r’(x)-r(x)
(r’(y)<=r(y) and
r’(x)>=r(x))
<=3(r’(x)-r(x))
Since zig only takes 1 rotation, so the amortized
complexity will be 3(r’(x)-r(x))+1
Cont.
 Splaying node x consists of d/2 splaying
substeps, recall d is the depth of x.
 The total amortized complexity will be:
<=(3(ri(x)-ri-1(x))) + 1 (1<=i<=d/2)
recall: the last step is a zig.
=3(rd/2(x)-r0(x)) +1
<=3(r(t)-r(x))+1
(r(t): rank of root)
Cont.
 So from before we have:
3(r(t)-r(x))+1
<=3r(t)+1
=3log n +1
thus, splaying takes O(log n).
 So for m operations of search, insertion and
deletion, we have O(mlog n)
 This is better than the O(mn) worst-case
complexity of BST
Implementation
 LEDA ?
 Java implementation:
SplayTree.java and etc from previous
files(?)
Modified printTree() to track level and child
identity, add a small testing part.
Demos
 A Demo written in Perl, it has some small
bugs http://bobo.link.cs.cmu.edu/cgibin/splay/splay-cgi.pl
 An animation java applet:
http://www.cs.technion.ac.il/~itai/ds2/frame
splay/splay.html
Conclusion
 A balanced binary search tree.
 Doesn’t need any extra information to be stored in
the node, ie color, level, etc.
 Balanced in an amortized sense.
 Running time is O(mlog n) for m operations
 Can be adapted to the ways in which items are
being accessed in a dictionary to achieve faster
running times for the frequently accessed
items.(O(1), AVL is about O(log n), etc.)
References
 Data Structure and Algorithms in Java, Michael T.





GoodRich.
Algorithms and Data Structures, Jeffrey H Kingston
http://www.cs.mcgill.ca/~rsinge/web251.html
http://hissa.nist.gov/dads/HTML/splaytree.html NIST
Data Strucrues and Problem Solving with C++, Frank M.
Carrano. Source code: http://www.aw.com/cseng/
http://www.labmed.umn.edu/~micha\el/Splay/readme.html