Binary Search Trees - Data Structures and Algorithm - Lecture Slides, Slides of Data Structures and Algorithms

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Data Structures & Algorithm

Analysis

How to Implement a Dictionary?

• Sequences

• ordered
• unordered

• Binary Search Trees

• Skip lists

• Hashtables

Basic Idea

• Use hash function to map keys into

positions in a hash table

Ideally

• If element e has key k and h is hash

function, then e is stored in position h(k) of

table

• To search for e , compute h(k) to locate

position. If no element, dictionary does not

contain e. Docsity.com

Example

• Dictionary Student Records
  • Keys are ID numbers (951000 - 952000), no more than 100 students
  • Hash function: h(k) = k-951000 maps ID into distinct table positions 0-
  • array table[1001]

...

0 1 2 3 1000

hash table

buckets

Ideal Case is Unrealistic

• Works for implementing dictionaries, but many

applications have key ranges that are too large to have 1- mapping between buckets and keys!

Example:

• Suppose key can take on values from 0 .. 65,535 (2 byte

unsigned int)

• Expect ≈ 1,000 records at any given time
• Impractical to use hash table with 65,536 slots!

Hash Functions

• If key range too large, use hash table with fewer

buckets and a hash function which maps

multiple keys to same bucket:

h(k 1 ) = β = h(k 2 ): k 1 and k 2 have collision at slot β

• Popular hash functions: hashing by division

h(k) = k%D, where D number of buckets in hash table

• Example: hash table with 11 buckets

h(k) = k%

80 → 3 (80%11= 3), 40 → 7, 65 → 10

58 → 3 collision! Docsity.com

Closed Hashing

• Associated with closed hashing is a rehash strategy :

“If we try to place x in bucket h(x) and find it occupied, find alternative location h 1 (x) , h 2 (x) , etc. Try each in order, if none empty table is full,”

• h(x) is called home bucket
• Simplest rehash strategy is called linear hashing

h (^) i (x) = (h(x) + i) % D

• In general, our collision resolution strategy is to generate a

sequence of hash table slots (probe sequence) that can hold the record; test each slot until find empty one (probing)

Example Linear (Closed) Hashing

• D=8, keys a,b,c,d have hash values h(a)=3, h(b)=0, h(c)=4,

h(d)=

0 2 3 4 5 6 7 1

b

a c

Where do we insert d? 3 already filled Probe sequence using linear hashing: h 1 (d) = (h(d)+1)%8 = 4%8 = 4 h 2 (d) = (h(d)+2)%8 = 5%8 = 5* h 3 (d) = (h(d)+3)%8 = 6%8 = 6 etc. 7, 0, 1, 2 Wraps around the beginning of the table!

d

Performance Analysis - Worst Case

• Initialization: O(b), b# of buckets

• Insert and search: O(n), n number of

elements in table; all n key values have

same home bucket

• No better than linear list for maintaining

dictionary!

Performance Analysis - Avg Case

• Distinguish between successful and

unsuccessful searches

• Delete = successful search for record to be deleted
• Insert = unsuccessful search along its probe sequence
• Expected cost of hashing is a function of how

full the table is: load factor α = n/b

• It has been shown that average costs under

linear hashing (probing) are:

• Insertion: 1/2(1 + 1/(1 - α)^2 )
• Deletion: 1/2(1 + 1/(1 - α))

Example

0 1 2 3 4 5 6 7 8 9

10

1001 9537 3016

9874 2009 9875

h(k) = k%

0 1 2 3 4 5 6 7 8 9

10

1001 9537 3016

9874 2009 9875

1. What if next element has home bucket 0? → go to bucket 3 Same for elements with home bucket 1 or 2! Only a record with home position 3 will stay. ⇒ p = 4/11 that next record will go to bucket 3
2. Similarly, records hashing to 7,8, will end up in 10
3. Only records hashing to 4 will end up in 4 (p=1/11); same for 5 and 6

I

II insert 1052 (h.b. 7)

1052

next element in bucket 3 with p = 8/11Docsity.com

Hash Functions - Numerical Value

• Consider: h(x) = x%
  • poor distribution, not very random
  • depends solely on least significant four bits of key
• Better, mid-square method
  • if keys are integers in range 0,1,…,K , pick integer C such that DC^2 about equal to K^2 , then h(x) = x 2 /C % D extracts middle r bits of x 2 , where 2

r =D (a base-D digit)

• better, because most or all of bits of key contribute to result

Hash Function –

Strings of Characters

• Much better: Cyclic Shift

static long hashCode(String key, int D) { int h=0; for (int i=0, i<key.length(); i++){ h = (h << 4) | ( h >> 27); h += (int) key.charAt(i); } return h%D; }

Open Hashing

• Each bucket in the hash table is the head of a

linked list

• All elements that hash to a particular bucket

are placed on that bucket’s linked list

• Records within a bucket can be ordered in

several ways

• by order of insertion, by key value order, or by

frequency of access order