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CMSC 132:
Object-Oriented Programming II
Advanced Tree Structures
Department of Computer Science
University of Maryland, College Park
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Overview
Binary trees
Balance Rotation
Multi-way trees
Search Insert
Indexed tries
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Tree Balance
Degenerate
Worst case Search in O(n) time
Balanced
Average case Search in O( log(n) ) time
Degenerate binary tree
Balanced binary tree
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Tree Balance
Question
Can we keep tree (mostly) balanced?
Self-balancing binary search trees
AVL trees Red-black trees
Approach
Select invariant (that keeps tree balanced) Fix tree after each insertion / deletion Maintain invariant using rotations Provides operations with O( log(n) ) worst case
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AVL Trees
Properties
Binary search tree Heights of children for node differ by at most 1
Example
88
44
17 78
32 50
48 62
2
4
1
1
2
3
1
1
Heights of children shown in red
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AVL Trees
History
Discovered in 1962 by two Russian mathematicians, Adelson-Velskii & Landis
Algorithm
- Find / insert / delete as a binary search tree
- After each insertion / deletion a) If height of children differ by more than 1 b) Rotate children until subtrees are balanced c) Repeat check for parent (until root reached)
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Red-black Trees
Properties
Binary search tree Every node is red or black The root is black Every leaf is black All children of red nodes are black For each leaf, same # of black nodes on path to root
Characteristics
Properties ensures no leaf is twice as far from root as another leaf
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Red-black Trees
Example
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Red-black Trees
History
Discovered in 1972 by Rudolf Bayer
Algorithm
Insert / delete may require complicated bookkeeping & rotations
Java collections
TreeMap, TreeSet use red-black trees
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Tree Rotations
Changes shape of tree
Move nodes Change edges
Types
Single rotation Left Right Double rotation Left-right Right-left
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Tree Rotation Example
Single right rotation
1
2
3
1
2
3
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Tree Rotation Example
Single right rotation
1
2
3
5
6
4 1 6
2
3
5
4
Node 4 attached
to new parent
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Multi-way Search Trees
Insert Example (for 2-3 tree)
Insert( 4 )
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Multi-way Search Trees
Insert Example (for 2-3 tree)
Insert( 1 )
Split node Split parent
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B-Trees
Characteristics
Height of tree is O( logT (n) ) Reduces number of nodes accessed Wasted space for non-full nodes
Popular for large databases
1 node = 1 disk block Reduces number of disk blocks read
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Indexed Search Tree ( Trie)
Special case of tree
Applicable when
Key C can be decomposed into a sequence of subkeys C 1 , C 2 , … Cn Redundancy exists between subkeys
Approach
Store subkey at each node Path through trie yields full key
Example
Huffman tree
C 3
C 1
C 2
C 3 C 4
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Tries
Useful for searching strings
String decomposes into sequence of letters Example “ART” ⇒ “A” “R” “T”
Can be very fast
Less overhead than hashing
May reduce memory
Exploiting redundancy
May require more memory
Explicitly storing substrings
S
A
R
E T
“ART”
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Types of Tries
Standard
Single character per node
Compressed
Eliminating chains of nodes
Compact
Stores indices into original string(s)
Suffix
Stores all suffixes of string
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Standard Tries
Approach
Each node (except root) is labeled with a character Children of node are ordered (alphabetically) Paths from root to leaves yield all input strings
Trie for Morse Code 26
Standard Trie Example
For strings
{ a, an, and, any, at }
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Standard Trie Example
For strings
{ bear, bell, bid, bull, buy, sell, stock, stop }
a
e
b
r
l
l
s
u
l
l
y
e t
l
l
o
c
k
p
i
d
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Standard Tries
Node structure
Value between 1…m Reference to m children Array or linked list
Example
Class Node { Letter value; // Letter V = { V 1 , V 2 , … Vm } Node child[ m ]; }
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Standard Tries
Efficiency
Uses O(n) space Supports search / insert / delete in O(d×m) time For n total size of strings indexed by trie d length of the parameter string m size of the alphabet
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Word Matching Trie
Insert words into trie Each leaf stores occurrences of word in the text
s e e b e a r? s e l l s t o c k! s e e b u l l? b u y s t o c k! b i d s t o c k!
a a
h e t h e b e l l? s t o p!
b i d s t o c k!
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86
a r 87 88
a
e
b
l
s u l
e t e 0, 24
o c
i l r 6
l 78
d 47, 58 (^) l 30
y 36 l (^12) k 17, 40, 51, 62
p 84
h e
r 69
a
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Suffix Trie Example
e nimize
nimize ze
i mi ze
mize nimize ze
m i n i m i z e 0 1 2 3 4 5 6 7
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Tries and Web Search Engines
Search engine index
Collection of all searchable words Stored in compressed trie
Each leaf of trie
Associated with a word List of pages (URLs) containing that word Called occurrence list
Trie is kept in memory (fast)
Occurrence lists kept in external memory
Ranked by relevance
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Computational Biology
DNA
Sequence of 4 different nucleotides (ATCG) Portions of DNA sequence produce proteins (genes)
Genome
Master DNA sequence for organism For Human 46 chromosomes 3 billion nucleotides
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Tries and Computational Biology
ESTs
Fragments of expressed DNA Indicator for genes (& location) 5.5 million sequences at NIH
ESTmapper
Build suffix trie of genome 8 hours, 60 Gbytes Search for ESTs in suffix trie 11 hours w/ 8 processor Sun
Search genome w/ BLAST
5 +^ years (predicted)
Genome
ESTs
Suffix tree
Mapping
Gene
