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CSE 326: Data Structures
Lecture #13
Extendible Hashing and
Splay Trees
Alon Halevy
Spring Quarter 2001
Extendible Hashing
• Hashing technique for huge data sets
– optimizes to reduce disk accesses
– each hash bucket fits on one disk block
– better than B-Trees if order is not important
• Table contains
– buckets, each fitting in one disk block, with the data
– a directory that fits in one disk block used to hash to the
correct bucket
Extendible Hash Table
• Directory contains entries labeled by k bits plus a
pointer to the bucket with all keys starting with its bits
• Each block contains keys+data matching on the first
j  k bits
directory for k = 3
000
(2)
00001
00011
00100
00110
001
(2)
01001
01011
01100
010
011
(3)
10001
10011
100
101
(3)
10101
10110
10111
110
111
(2)
11001
11011
11100
11110
Inserting (easy case)
insert(11011)
000
(2)
00001
00011
00100
00110
010
(2)
01001
01011
01100
000
(2)
00001
00011
00100
00110
001
001
(2)
01001
01011
01100
011
100
(3)
10001
10011
010
011
(3)
10001
10011
101
110
(3)
10101
10110
10111
100
101
(3)
10101
10110
10111
111
(2)
11001
11100
11110
110
111
(2)
11001
11011
11100
11110
Splitting
insert(11000)
000
(2)
00001
00011
00100
00110
000
(2)
00001
00011
00100
00110
001
010
(2)
01001
01011
01100
001
(2)
01001
01011
01100
010
011
(3)
10001
10011
011
100
(3)
10001
10011
100
101
110
(3)
10101
10110
10111
101
(3)
10101
10110
10111
110
111
(2)
11001
11011
11100
11110
111
(3)
11000
11001
11011
(3)
11100
11110
Rehashing
insert(10010)
00
01
(2)
01101
000
001
010
10
(2)
10000
10001
10011
10111
011
100
No room to
insert and no
adoption!
11
(2)
11001
11110
101
110
Now, it’s just a normal split.
111
Expand
directory
Rehash of Hashing
• Hashing is a great data structure for storing unordered data
that supports insert, delete, and find
• Both separate chaining (open) and open addressing
(closed) hashing are useful
– separate chaining flexible
– closed hashing uses less storage, but performs badly with load
factors near 1
– extendible hashing for very large disk-based data
• Hashing pros and cons
+ very fast
+ simple to implement, supports insert, delete, find
- lazy deletion necessary in open addressing, can waste storage
- does not support operations dependent on order: min, max, range
Recall: AVL Tree
Dictionary Data Structure
• Binary search tree
properties
8
– binary tree property
– search tree property
5
11
• Balance property
– balance of every node is:
-1 b  1
– result:
• depth is (log n)
2
6
4
10
7
9
12
13 14
15
Splay Trees
“blind” rebalancing – no height info kept
• amortized time for all operations is O(log n)
• worst case time is O(n)
• insert/find always rotates node to the root!
– Good locality – most common keys move high in tree
Idea
10
You’re forced to make
a really deep access:
17
Since you’re down there anyway,
fix up a lot of deep nodes!
5
2
9
3
Splay Operations: Find
•
Find(x)
1. do a normal BST search to find n such that
n->key = x
2. move n to root by series of zig-zag and zig-zig
rotations, followed by a final zig if necessary
*
Zig-Zag
Helped
Unchanged
Hurt
g
n
p
X
g
p
n
W
X
Y
Y
Z
Z
*This
is just a double rotation
W
Zig-Zig
n
g
p
p
W
Z
n
g
X
Y
Y
Z
W
X
Zig
root
p
root
n
n
p
Z
X
Y
X
Y
Z
Why Splaying Helps
• Node n and its children are always helped (raised)
• Except for final zig, nodes that are hurt by a zigzag or zig-zig are later helped by a rotation higher
up the tree!
• Result:
– shallow (zig) nodes may increase depth by one or two
– helped nodes may decrease depth by a large amount
• If a node n on the access path is at depth d before
the splay, it’s at about depth d/2 after the splay
– Exceptions are the root, the child of the root, and the
node splayed
Locality
• Assume m  n access in a tree of size n
– Total amortized time O(m log n)
– O(log n) per access on average
• Gets better when you only access k distinct items
in the m accesses.
– Exercise.
Splaying Example
1
1
2
2
zig-zig
3
3
Find(6)
4
6
5
5
6
4
Still Splaying 6
1
1
2
6
zig-zig
3
3
6
5
4
2
5
4
Almost There, Stay on Target
1
6
6
1
zig
3
2
3
5
4
2
5
4
Splay Again
6
6
1
1
zig-zag
3
4
Find(4)
2
5
4
3
2
5
Example Splayed Out
6
4
1
1
6
zig-zag
3
4
3
2
5
2
5
Splay Tree Summary
• All operations are in amortized O(log n) time
• Splaying can be done top-down; better because:
– only one pass
– no recursion or parent pointers necessary
• Invented by Sleator and Tarjan (1985), now widely used in
place of AVL trees
• Splay trees are very effective search trees
– relatively simple
– no extra fields required
– excellent locality properties: frequently accessed keys are cheap to
find
Coming Up
• Project 3: implement a “smart” web server.
• Heaps!
• Disjoint Sets
• Graphs and more.