1. SPLAY TREE
By,
Ambooj Yadav, Anam Iqbal & Hina Firdaus
M.Tech CSE Ist year, Ist Semester
Advance Datastructure Algorithm and Analysis
Jamia Hamdard University, New Delhi-62
2. Introduction
Definition:-
• Splay trees another variation of the binary search tree, invented by Robert
Tarjan and Daniel Sleator.
• Splay means “to rearrange, so that the desired element is placed at the
root"
• Splay trees support all the operations of binary trees.
• A splay tree is a self-adjusting binary search tree
Novel characteristics
• It performs basic operations such as insertion, look-up and removal.
• Does not require any accounting information (color, level, height, etc.)
• Worst case for any single operation: O(N)
• Average case for all operation: O(log N)
• Frequently-accessed keys are moved up so that they are near the root.
• Easier to perform comparison with 2-3-4 trees.
3.
4. General question
1. What is difference between AVL trees and
splay trees?
2. On what basis do we select these tress?
3. What are positive's and negative's of these
trees?
4. What are the performances of these trees in
terms of big O notation?
5. 1. What is difference between AVL
trees and splay trees?
Splay trees always try to be
balanced after every operation.
Because of this rest operation
take less time to perform
They are cool…
6. 2. On what basis do we select these
tress?
• Splay trees are always better than binary search trees
when, your application deals with a lot of data in the
tree but, will need access to a subset of the data very
frequently than others.
• In this case the data you access frequently will come
near the root as a result of the splay.
• Because of this, any node can then be accessed with
less time than before.
7. 3. What are positive's and negative's of
these trees?
Positive Negative
• Positives for both is that you
get around log(n) in both these
data structures theoretically.
• Splay trees have average log(n)
over a number of operations.
• Getting n times complexity for
an operation in a set, but it can
be compensated while
accessing frequent items.
• In binary search tree, you
need to be lucky to have log(n)
always.
• If the keys are not random,
then the tree will reduce to a
list like form with only one
side.
8. 4. What are the performances of these
trees in terms of big O notation?
• AVL tree insertion, deletion, and lookups take
O(log n) time for each.
• Splay tree do take same time in amortized sense.
• Any long sequence of operations will take at
most O(n log n) time, but individual operations
might take as much as O(n) time in splay trees.
9. Splay tree
• Two varieties to approach splay trees are :-
▫ Bottom up : first search the tree and rotate at same
iteration.
▫ Top down: first search the tree and rotate in another
iteration
• There are three case :-
1) ZIG
2) ZIG-ZAG (Bottom-up approach)
3) ZIG-ZIG (Top-down approach)
10. Case 1: ZIG
• When p is the root.
• The tree is rotated on the edge between x and p.
• 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.
11. Case 2: Zig-Zag
• Left-right or right-left
• The order of rotations: bottom-up approach, which
guarantees that the tree height is reduced by one at each zig-zag
rotation.
• The tree is rotated on the edge between p and x, and then rotated
on the resulting edge between x and g.
12. Case 3: Zig-Zig
• Left–left or right–right
• The order of rotations: top-down, which guarantees that the
distance to every node encountered (except the root) is reduced to
half.
• When p is not the root and x and p are either both right children or
are both left children.
• The tree is rotated on the edge joining p with its parent g, then
rotated on the edge joining x with p.
13. Insertion
• To insert a value x into a splay tree:
▫ Insert x as with a normal binary search tree.
▫ when an item is inserted, a splay is performed.
▫ As a result, the newly inserted node x becomes the root of the tree.
• Algorithm:
insert(node) {
1. Insert the new node.
2. Let n be the length of the path traversed for the insertion. Then,
a. If n = 2m (i.e., even number) then perform m double rotations.
b. If n = 2m+1 (i.e., odd number) then perform one single rotation followed
by m double rotations.
}
15. Deletion
Delete x
Splay x to root and remove it. (note: the node does not have
to be a leaf or single child node like in BST delete.) Two
trees remain, right subtree and left subtree.
Splay the max in the left subtree to the root
Attach the right subtree to the new root of the left subtree.
Delete(x, T):
∗ Splay(x, T) and remove root → tree falls into T1 and T2.
∗ Splay(x, T1)
∗ Make T2 right son of new root of T1 after splay
21. SEARCH
• The search operation in splay trees does the standard BINARY
SEARCH TREE SEARCH.
• In addition to that it SPLAYS.
• Splay trees are very effective search trees relatively simple:
1. No extra fields required
2. Excellent locality properties:
• frequently accessed keys are cheap to find (near top of tree)
• infrequently accessed keys stay out of the way (near
bottom of tree)
22. Rules of splaying: Search
• If the search is successful, then the node that is found is splayed and becomes
the new root.
• If the search is unsuccessful, the last node accessed prior to reaching the
NULL pointer is splayed and becomes the new root.
Search (i, t)
If item i is in tree t, return a pointer to the node containing i; otherwise return a
pointer to the null node.
• Search down the root of t, looking for i.
• If the search is successful and we reach a node x containing i, we complete
the search by splaying at x and returning a pointer to x.
• If the search is unsuccessful, i.e., we reach the null node, we splay at the last
non-null node reached during the search and return a pointer to null.
• If the tree is empty, we omit any splaying operation.
30. AMORTIZED ANALYSIS
• Amortized analysis is a technique for analyzing an algorithm's running time.
• It is often appropriate when one is interested in understanding asymptotic behavior
over sequences of operations.
• For example, one might be interested in reasoning about the running time for an ar
bitrary operation to insert an item into a binary search tree structure. In cases such
as this, it might be straightforward to come up with an upper bound by say,
finding the worst-possible time required for any operation, them multiplying this by
the number of operations in the sequence.
• However, many real data structures, such as splay trees, have the property that it is
impossible for every operation in a sequence to take the worst case time, so
this approach can result in a horribly pessimistic bound!
31. • Amortized analysis allows a tighter bound that better reflects performance.
• It does not say anything about the cost of a specific operation in that sequence.
• Amortized analysis is concerned with the overall cost of arbitrary sequences.
• An amortized bound will hold regardless of the specific sequence; for example, if th
e amortized cost of insertion is O(log n), it is so regardless of whether you insert
the sequence '10,' '160,' '2' or the sequence '2', '160', '10,' '399', etc.
• An amortized bound says nothing about the cost of individual operations, it may
be possible that one operation in the sequence requires a huge cost.
• Practical systems in which it is important that all operations have low and/or
comparable costs may require an algorithm with a worse amortized cost but a better
worst case per operation bound.
32. • Amortized analysis can be understood to take advantage of the fact that some
expensive operations may “pay” for future operations by somehow limiting the
number or cost of expensive operations that can happen in the near future.
• If good amortized cost is a goal, an algorithm may be designed to explicitly perform
this “clean up” during expensive operations!
• The key to performing amortized analysis is picking a good “credit” or “potential
function” that captures how operations with different actual costs affect a data
structure and allows the desired bound to be shown.
33. AMORTISED ANALYSIS OF
SPLAY TREES
• Let S(x) denote subtree of S
rooted at x.
• |S| = number of nodes in tree S.
• µ(S) = rank = floor(log |S|).
• P(i) = ∑tree node x rank(x).
• P(i) is potential after i’th
operation.
• rank(x) is computed after i’th
operation.
• P(0) = 0
• Potential=10
2
1 8
4
3 6
5 7
10
9
|S| = 10
µ(2) = 3
µ(8) = 3
µ(4) = 2
µ(6) = 1
µ(5) = 0
34. CASE 1: If q = null or q is the root
do nothing (splay is over).
∆P = 0
amortized cost = actual cost + ∆P = 0.
CASE 2: If q is at level 2
do a one-level move and terminate the splay
operation.
r(x) = rank of x before splay step.
r'(x) = rank of x after splay step.
35. ∆P = r'(p) + r'(q) − r(p) − r(q) ≤ r'(q) − r(q)
amortized cost = actual cost + ∆P ≤ 1+ r'(q) − r(q)
p
q
b
a
c
q
c
b
a p
Potential
36. s
p
q
d
a b
c
q
a
s
p
c d
b
r'(q) = r(s)
r'(s) ≤ r’(q)
r’(p) ≤ r’(q)
r(q) ≤ r(p)
∆P = r’(s) +r’(p) +r’(q)−r(s)−r(p)−r(q) ≤ r'(q) + r'(q) − r(q) − r(q) =
2(r'(q)−r(q))
2(r'(q)−r(q)) ≤ 3(r'(q)−r(q))−1
amortized cost = actual cost + ∆P ≤ 1+3(r'(q)−r(q))−1 = 3(r'(q)−r(q))
CASE 3: If q is at level 3
37. Putting it together
We repeat the steps until X replaces the root R
• zig only happens once, so the 1 is only added once/
• Each time, the last Rf(X) is cancelled by the next –Ri(X)
The only terms left are: AT(total) <= 1 + 3*[Rroot(X) – Rinitial(X)]
Rinitial(X) could be as low as 0, Rroot(X) as high as log N •
Thus, total budget for whole sequence is O(log N)
OPERATION COST
AT(zig) < = 1+ ∆R(X) <= 1 + 3 [Rf(X) –
Ri(X)]
AT(zig-zag) <= 2 ∆R(X) <= 3 [Rf(X) –
Ri(X)]
AT(zig-zig) <= 3 ∆R(X) <= 3 [Rf(X) –
Ri(X)]
