data structures & algorithms radix search richard newman based on slides by s. sahni and book by...

Data Structures & Algorithms

Radix Search

Richard Newmanbased on slides by S. Sahniand book by R. Sedgewick

Radix-based Keys

• Key has multiple parts

• Each part is an element of some set

• Character

• Numeral

• Key parts can be accessed (e.g., string s[i])

• Size of set is radix

Advantages of Radix-based Search

• Good worst-case performance

• Simpler than balanced trees, etc.

• Fast access to data

• Easy way to handle variable-length keys

• Save space (part of key in structure)

Disadvantages of Radix-based Search

• May be space-inefficient

• Performance depends on access to bytes of keys

• Must have distinct keys, or other way to handle duplicate keys

Digital Search Trees

• Similar to binary search trees

• Difference is that we use bits of the key to determine subtree to search

• Path in tree = prefix of key


• Insert A-S-E-R-C-H-I-N-G

Key ReprA 00001S 10011E 00101R 10010C 00011H 01000I 01001N 01110G 00111

A

S1

E0

1010R

10C

10H

10

I10

N10

G10

Note that binary tree is not sorted in BST sense


Prop 15.1: A search or insertion into a DST takes about lg N comparisons on average, and about 2 lg N comparisons in the worst case, in a tree built from N keys. The number of comparisons is never more than the number of bits in the search key.

Tries

• Use bits of key to guide search like DST

• But keep keys in order like BST

• Allow recursive sort, etc.

• Pronounced “try-ee” or “try”

• Keys kept at leaves of a binary tree

Tries

• Defn. 15.1: A trie is a binary tree that has keys associated with each leaf, defined as follows:

a trie for an empty set is a null link

a trie for a single key is a leaf w/key

a trie for > 1 key is an internal node with left link referring to trie for keys that start with 0, right for keys 1xxx

Tries



A

S1

A0

Construct tree to point where prefixes match

Tries



A10

A E10

10

10

10

10R S

S10

A

Construct tree to point where prefixes match

Tries



10

A

10

10

10

10R SA

10C

E10

10H

Tries

• Insert A-S-E-R-C-H-I-N-GKey Repr

A 00001S 10011E 00101R 10010C 00011H 01000I 01001N 01110G 00111

10

10

10

10

10R SA

10C

E10

10H10

10

10H I

Tries

• Prop. 15.2: The structure of a trie is independent of key insertion order; there is one unique trie for any given set of distinct keys.

• Prop. 15.3: Insertion or search for a random key in a trie built from N random keys takes about lg N bit comparisons on average, in the worst case, bounded by bits in key

Tries

• Annoying feature of tries:

• One-way branching when keys have common prefix

• Prop. 15.4: A trie built from N random w-bit keys has about N/lg 2 nodes on the average (about 1.44 N)

Patricia Tries

• Annoying feature of tries:

• One-way branching when keys have common prefix

• Two different types of nodes in trie

• Patricia tries: fix both of these

• Practical Algorithm To Retrieve Information Coded In Alphanumeric

Patricia Tries

• Avoid one-way branching:

• Keep at each node the index of the next bit to test

• Skip over common prefix!

• Avoid two types of nodes:

• Store data in internal nodes

• Replace external links with back links

Patricia Tries

S

R4

H

0

1

E2

3

C4

A


Patricia Tries

• Prop 15.5: Insertion or search in a patricia trie built from N random bitstrings takes about lg N bit comparisons on average, and about 2 lg N in the worst case, but never more than the length of the key.

Map

• Radix search

• Digital Search Trees

• Tries

• Patricia Tries

• Multiway tries and TSTs

• Text string algorithms

Multiway Tries

• Like radix sort, can get benefit from comparing more than one bit at a time

• Compare r bits, speed up search by a factor of r

• What could possibly be bad?

• Number of links is now R=2r

• Can waste a lot of space!

Multiway Tries

• Structure is (almost) the same as binary tries

• Except there are R branches• Search: start at root, leftmost digit• Follow ith link if next R-ary digit is i• If null link, then miss• If reach leaf, it contains only key with

prefix matching path to it - compare

Existence Tries

• Only keys, no records• Insert/search• Defn. 15.2: The existence trie for a

set of keys is:• Empty set: null link• Non-empty set: internal node with

links for each possible digit to tries built with the leading digit omitted

Existence Tries

• Convenient to return null on miss, dummy record on hit

• Convenient to have no duplicate keys and no key a prefix of another key• Keys of fixed length, or• Use termination character with

value NULLdigit, only used as sentinel

Existence Tries

• No need to store any data• All keys captured in trie structure

• If reach NULLdigit at the same time we run out of key digits, search hit

• Otherwise, search miss• Insert: search until find null link, then

add nodes for each of the remaining digits in the key

Existence Triesnowisthetimefor

atn

h

e

i

i

m

e

s o

w

f

o

r

Multi-way Tries

• R-ary branching• Keys stored at leaves• Path to leaf defines prefix of key

stored at leaf• Only build tree downward until

prefixes become distinct

Multi-way Tries

• Defn. 15.3: The multiway trie for a set of keys associated with leaves is:• Set empty: null link• Singleton set: leaf with key• Larger set: internal node with links

for each possible digit to tries built with the leading digit omitted

Multi-way Tries

• Prop. 15.6: Search or insertion in a standard R-ary trie takes built from N random keys takes about logR N character comparisons, bounded by the length of the key; the number of links is about RN/ln R.

• Classic time-space tradeoff!• Larger R = faster but more space

Ternary Search Trie (TST)

• Each node has a character (digit) and three links

• Left link refers to subtrie with current key digit less than that of the node

• Middle link refers to subtrie with current key digit the same

• Right link refers to subtrie with current key digit greater than node’s


• TST equivalent to BST that used characters for non-null links as keys

• Like 3-way radix sorting• BSTs like QuickSort• M-ary tries like RadixSort


• Search: start at root• Recursively –

• Compare next character in key with character in node• If less, take left link• If greater, take right link• If equal, take middle and go to next

character in key• Miss if encounter null link or reach end of

key before NULLdigit


• Insert: start at root• Search –

• Find location where prefix diverges• Add new nodes for characters not

consumed by search

Existence TSTnowisthetimefor

n

h

e

i

i

m

e

s

o

wf

o

r

t


• Prop. 15.7: A search or insertion in a full TST requires time proportional to the key length. The number of links in a TST is at most three times the number of characters in all the keys.


• Can make more space efficient by • putting keys in leaves at point where

prefix is unique, and • eliminating one-way branching as we did

in Patricia Tries.• Can compromise speed and space by

having large branch at root (R or R2) and rest of trie is regular TST. • Works well if first char(s) well-distributed

Ternary Search Trie (TST)• Nice for practical use• Adapt to non-uniformity often seen• Though character set may be large, often

only a few are used, or are used after a particular prefix• Don’t make many links we don’t need

• Structured format keys• May have many symbols used• But only a few at each part of key

Ternary Search Trie (TST)• Nice for practical use• Search misses are really fast!• Can adapt for partial match searches

• “Don’t care” characters in search key• Can adapt for “almost match” searches

• All but (any) one character match• Access bytes or larger symbols rather

than bits (like Patricia tries), which are often better supported/efficient, or more natural to the keys

Text-String-Index• Recall String Index built with BST with

string pointers into a large text• Consider each position in text to be start

of a string key that runs to the end of the text

• Build a symbol table with these keys• Keys are all different (lengths alone

suffice)• Most are very long• Suffix Tree = search tree for this

Text-String-Index• BSTs are simple and work well for suffix

trees• Not likely to be a worst-case BST

• Patricia tries designed to do this!• Need to have bit-level access• Fast on misses

• TSTs• Simple, take advantage of byte ops• Can solve more complex problems• Can change == to mean “prefix”

Text-String-Index• If text is static, why not use Binary

Search?• Fast• No need to support insert/delete• Uses less memory (fewer links/pointers)

• But TSTs have some advantages• Never retrace steps• Support other operations

• Can also build FSM..• But better for linear search of new text

String Search• If problem is to look for a particular string s

in a large text t• Naïve method:

• Search t linearly for s[0]• When match found at t[i],

• Match s[j] with t[i+j] for j = 1 to |s|-1• If all |s| chars match, have a match!• Else go back to searching t at t[i+1]

• Time?• |s| times |t| - not good

FSM-based String Search• Fast way to look for a particular string s in

one or more (large) texts:• Build FSM for search string

• States represent prefix matched• Transition either extends match or• Fails to longest suffix of what has been

seen that is a prefix of s• Can also build for multiple search strings

Finite State Machinea.k.a. Finite State Automaton (FSA)

ca

d any

Set of States S – represented as nodes in graphSet of input symbols – labels on directed edgesTransition function – for state and input, next stateInitial state q0 – where to startFinal set of states F – subset of S for “accept”

Start state

F={q1,q2}

b

= {a,b,c,d}

a,b,d

a,b,d

c

c

q0 q3

q2

q1

q2

q1

Edge=transition

(q1,c)=q3

FSM-based String SearchSearch for abraca

aa

abb

abrr

abraa

abracc

abraca

a

Not a a

else

a

b

a

b

Build recognizer skeletonAdd suffix-is-prefix linksAdd failure links

aStart state

Final state

Is that all of them?

FSM-based String Search• Linear time in |s| to build FSM for s• Linear time (in |t|) to search large text t for all

instances of s• Can’t hope for better than that!• What about searching for more than one

string?• Build FSM for all the strings!• Linear time in sum of string lengths to build

FSM• Linear time in |t| to search all of t for all strings

Summary

• Radix search

• Digital Search Trees

• Tries

• Patricia Tries

• Multiway tries and TSTs

• Text string algorithms

• FSMs for fast string matching

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