Computer Notes - Data Structures - 32

Class No.32

Data Structures

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Tables and Dictionaries


Tables: rows & columns of information

A table has several fields (types of information)• A telephone book may have fields name, address,

phone number• A user account table may have fields user id,

password, home folder

Name Address Phone

Sohail Aslam 50 Zahoor Elahi Rd, Gulberg-4, Lahore 576-3205

Imran Ahmad 30-T Phase-IV, LCCHS, Lahore 572-4409

Salman Akhtar 131-D Model Town, Lahore 784-3753



To find an entry in the table, you only need know the contents of one of the fields (not all of them).

This field is the key• In a telephone book, the key is usually “name”• In a user account table, the key is usually “user

id”



Ideally, a key uniquely identifies an entry• If the key is “name” and no two entries in the

telephone book have the same name, the key uniquely identifies the entries

Name Address Phone

Sohail Aslam 50 Zahoor Elahi Rd, Gulberg-4, Lahore 576-3205

Imran Ahmad 30-T Phase-IV, LCCHS, Lahore 572-4409

Salman Akhtar 131-D Model Town, Lahore 784-3753


The Table ADT: operations

insert: given a key and an entry, inserts the entry into the table

find: given a key, finds the entry associated with the key

remove: given a key, finds the entry associated with the key, and removes it


How should we implement a table?

How often are entries inserted and removed? How many of the possible key values are likely to

be used? What is the likely pattern of searching for keys?

E.g. Will most of the accesses be to just one or two key values?

Is the table small enough to fit into memory? How long will the table exist?

Our choice of representation for the Table ADT depends on the answers to the following


TableNode: a key and its entry

For searching purposes, it is best to store the key and the entry separately (even though the key’s value may be inside the entry)

“Saleem” “Saleem”, “124 Hawkers Lane”, “9675846”

“Yunus” “Yunus”, “1 Apple Crescent”, “0044 1970 622455”

key entry

TableNode


Implementation 1: unsorted sequential array

An array in which TableNodes are stored consecutively in any order

insert: add to back of array; (1)

find: search through the keys one at a time, potentially all of the keys; (n)

remove: find + replace removed node with last node; (n)

0

…

key entry

1

23

and so on


Implementation 2:sorted sequential array

An array in which TableNodes are stored consecutively, sorted by key

insert: add in sorted order; (n) find: binary search; (log n) remove: find, remove node

and shuffle down; (n)

0

…

key entry

1

23

We can use binary search because thearray elements are sorted

and so on


Searching an Array: Binary Search

Binary search is like looking up a phone number or a word in the dictionary• Start in middle of book• If name you're looking for comes before names on

page, look in first half• Otherwise, look in second half


Binary Search

If ( value == middle element ) value is found else if ( value < middle element )

search left-half of list with the same method else search right-half of list with the same method


Case 1: val == a[mid]val = 10low = 0, high = 8

5 7 9 10 13 17 191 271 2 3 4 5 6 70 8

a:

low high

Binary Search

mid

mid = (0 + 8) / 2 = 4

10


Case 2: val > a[mid]val = 19low = 0, high = 8mid = (0 + 8) / 2 = 4

Binary Search -- Example 2

5 7 9 101 13 17 19 271 2 3 4 5 6 70 8

a:

midlow highnew low

new low = mid+1 = 5

13 17 19 27


Case 3: val < a[mid]val = 7low = 0, high = 8mid = (0 + 8) / 2 = 4

Binary Search -- Example 3

10 13 17 195 7 91 271 2 3 4 5 6 70 8

a:

midlow highnew high

new high = mid-1 = 3

5 7 91


val = 7

Binary Search -- Example 3 (cont)

5 7 9 10 13 17 191 271 2 3 4 5 6 70 8

a:

5 7 9 10 13 17 191 271 2 3 4 5 6 70 8

a:

5 7 9 10 13 17 191 271 2 3 4 5 6 70 8

a:

Binary Search – C++ Code

int isPresent(int *arr, int val, int N){ int low = 0; int high = N - 1; int mid; while ( low <= high ){

mid = ( low + high )/2;if (arr[mid]== val) return 1; // found!else if (arr[mid] < val)

low = mid + 1;else high = mid - 1;

} return 0; // not found} http://ecomputernotes.com

Binary Search: binary tree

The search divides a list into two small sub-lists till a sub-list is no more divisible.

First half

First half

An entire sorted list

First half Second half

Second half


Binary Search Efficiency

After 1 bisection N/2 items After 2 bisections N/4 = N/22 items

. . . After i bisections N/2i =1 item

i = log2 N


Implementation 3: linked list

TableNodes are again stored consecutively (unsorted or sorted)

insert: add to front; (1or n for a sorted list)

find: search through potentially all the keys, one at a time; (n for unsorted or for a sorted list

remove: find, remove using pointer alterations; (n)

key entry

and so on


Implementation 4: Skip List

Overcome basic limitations of previous lists• Search and update require linear time

Fast Searching of Sorted Chain Provide alternative to BST (binary search

trees) and related tree structures. Balancing can be expensive.

Relatively recent data structure: Bill Pugh proposed it in 1990.


Skip List Representation

Can do better than n comparisons to find element in chain of length n

20 30 40 50 60

head tail


Skip List Representation

Example: n/2 + 1 if we keep pointer to middle element

20 30 40 50 60

head tail


Higher Level Chains

For general n, level 0 chain includes all elements level 1 every other element, level 2 chain every

fourth, etc. level i, every 2i th element

40 50 60

head tail

20 3026 57

level 1&2 chains


Higher Level Chains

Skip list contains a hierarchy of chains In general level i contains a subset of

elements in level i-1

40 50 60

head tail

20 3026 57

level 1&2 chains

Skip List: formally

A skip list for a set S of distinct (key, element) items is a series of lists S0, S1 , … , Sh such that

• Each list Si contains the special keys and

• List S0 contains the keys of S in nondecreasing order

• Each list is a subsequence of the previous one, i.e.,

S0 S1 … Sh

• List Sh contains only the two special keys

Lecture No.38

Data Structure

Dr. Sohail Aslam

Skip List: formally

56 64 78 31 34 44 12 23 26S0

64 31 34 23S1

31S2

S3

Skip List: Search

We search for a key x as follows:

• We start at the first position of the top list

• At the current position p, we compare x with y key(after(p))

• x y: we return element(after(p))

• x y: we “scan forward” • x y: we “drop down”

• If we try to drop down past the bottom list, we return NO_SUCH_KEY

Skip List: Search

Example: search for 78

S0

S1

S2

S3

31

64 31 34 23

56 64 78 31 34 44 12 23 26

To insert an item (x, o) into a skip list, we use a randomized algorithm:

• We repeatedly toss a coin until we get tails, and we denote with i the number of times the coin came up heads

• If i h, we add to the skip list new lists Sh1, … , Si 1, each containing only the two special keys

Skip List: Insertion

To insert an item (x, o) into a skip list, we use a randomized algorithm: (cont)

• We search for x in the skip list and find the positions p0, p1 , …, pi of the items with largest key less than x in each list S0, S1, … , Si

• For j 0, …, i, we insert item (x, o) into list Sj after position pj


Example: insert key 15, with i 2


10 36

23

23

S0

S1

S2

S0

S1

S2

S3

10 362315

15

2315p0

p1

p2

Randomized Algorithms

A randomized algorithm performs coin tosses (i.e., uses random bits) to control its execution

It contains statements of the typeb random()if b <= 0.5 // head

do A …else // tail

do B … Its running time depends on the outcomes of the

coin tosses, i.e, head or tail

Skip List: Deletion

To remove an item with key x from a skip list, we proceed as follows:

• We search for x in the skip list and find the positions p0, p1 , …, pi of the items with key x, where position pj is in list Sj

• We remove positions p0, p1 , …, pi from the lists S0, S1, … , Si

• We remove all but one list containing only the two special keys

Skip List: Deletion

Example: remove key 34

4512

23

23

S0

S1

S2

S0

S1

S2

S3

4512 23 34

34

23 34p0

p1

p2

Skip List: Implementation

S0

S1

S2

S3

4512 23 34

34

23 34

Implementation: TowerNode

TowerNode will have array of next pointers. Actual number of next pointers will be

decided by the random procedure. Define MAXLEVEL as an upper limit on

number of levels in a node.

40 50 60

head tail

20 3026 57

Tower Node

Implementation: QuadNode

A quad-node stores:• item• link to the node before• link to the node after• link to the node below• link to the node above

This will require copying the key (jitem) at different levels

x

quad-node

Skip Lists with Quad Nodes

56 64 78 31 34 44 12 23 26

31

64 31 34 23

S0

S1

S2

S3

Performance of Skip Lists

In a skip list with n items

• The expected space used is proportional to n.

• The expected search, insertion and deletion time is proportional to log n.

Skip lists are fast and simple to implement in practice

Implementation 5: AVL tree

An AVL tree, ordered by key insert: a standard insert; (log n) find: a standard find (without

removing, of course); (log n) remove: a standard remove;

(log n)

key entry

key entry key entry

key entry

and so on

Anything better?

So far we have find, remove and insert where time varies between constant logn.

It would be nice to have all three as constant time operations!

An array in which TableNodes are not stored consecutively

Their place of storage is calculated using the key and a hash function

Keys and entries are scattered throughout the array.

Implementation 6: Hashing

key entry

Key hash function

array index

4

10

123

insert: calculate place of storage, insert TableNode; (1)

find: calculate place of storage, retrieve entry; (1)

remove: calculate place of storage, set it to null; (1)

Hashing

key entry

4

10

123

All are constant time (1) !

Hashing

We use an array of some fixed size T to hold the data. T is typically prime.

Each key is mapped into some number in the range 0 to T-1 using a hash function, which ideally should be efficient to compute.

Example: fruits

Suppose our hash function gave us the following values: hashCode("apple") = 5

hashCode("watermelon") = 3hashCode("grapes") = 8hashCode("cantaloupe") = 7hashCode("kiwi") = 0hashCode("strawberry") = 9hashCode("mango") = 6hashCode("banana") = 2

kiwi

bananawatermelon

applemango

cantaloupegrapes

strawberry

0

1

2

3

4

5

6

7

8

9

Example

Store data in a table array: table[5] = "apple"

table[3] = "watermelon" table[8] = "grapes" table[7] = "cantaloupe" table[0] = "kiwi" table[9] = "strawberry" table[6] = "mango" table[2] = "banana"

kiwi

bananawatermelon

applemango

cantaloupegrapes

strawberry

0

1

2

3

4

5

6

7

8

9

Example

Associative array: table["apple"]

table["watermelon"] table["grapes"] table["cantaloupe"] table["kiwi"] table["strawberry"] table["mango"] table["banana"]

kiwi

bananawatermelon

applemango

cantaloupegrapes

strawberry

0

1

2

3

4

5

6

7

8

9

Example Hash Functions

If the keys are strings the hash function is some function of the characters in the strings.

One possibility is to simply add the ASCII values of the characters:

TableSizeABChExample

TableSizeistrstrhlength

i

)%676665()(:

%][)(1

0

Finding the hash function

int hashCode( char* s ){

int i, sum;sum = 0;for(i=0; i < strlen(s); i++ ) sum = sum + s[i]; // ascii value

return sum % TABLESIZE;

}


Another possibility is to convert the string into some number in some arbitrary base b (b also might be a prime number):

TbbbABChExample

Tbistrstrhlength

i

i

)%676665()(:

%][)(

210

1

0


If the keys are integers then key%T is generally a good hash function, unless the data has some undesirable features.

For example, if T = 10 and all keys end in zeros, then key%T = 0 for all keys.

In general, to avoid situations like this, T should be a prime number.

Collision

Suppose our hash function gave us the following values:

• hash("apple") = 5hash("watermelon") = 3hash("grapes") = 8hash("cantaloupe") = 7hash("kiwi") = 0hash("strawberry") = 9hash("mango") = 6hash("banana") = 2

kiwi

bananawatermelon

applemango

cantaloupegrapes

strawberry

0

1

2

3

4

5

6

7

8

9• Now what?

hash("honeydew") = 6

Collision

When two values hash to the same array location, this is called a collision

Collisions are normally treated as “first come, first served”—the first value that hashes to the location gets it

We have to find something to do with the second and subsequent values that hash to this same location.

Solution for Handling collisions

Solution #1: Search from there for an empty location

• Can stop searching when we find the value or an empty location.

• Search must be wrap-around at the end.


Solution #2: Use a second hash function

• ...and a third, and a fourth, and a fifth, ...


Solution #3: Use the array location as the header of a linked list of values that hash to this location

Solution 1: Open Addressing

This approach of handling collisions is called open addressing; it is also known as closed hashing.

More formally, cells at h0(x), h1(x), h2(x), … are tried in succession where

hi(x) = (hash(x) + f(i)) mod TableSize,

with f(0) = 0. The function, f, is the collision resolution

strategy.

Linear Probing

We use f(i) = i, i.e., f is a linear function of i. Thus

location(x) = (hash(x) + i) mod TableSize

The collision resolution strategy is called linear probing because it scans the array sequentially (with wrap around) in search of an empty cell.

Linear Probing: insert

Suppose we want to add seagull to this hash table

Also suppose:• hashCode(“seagull”) = 143

• table[143] is not empty• table[143] != seagull

• table[144] is not empty• table[144] != seagull

• table[145] is empty

Therefore, put seagull at location 145

robin

sparrow

hawk

bluejay

owl

. . .

141

142

143

144

145

146

147

148

. . .

seagull


Suppose you want to add hawk to this hash table

Also suppose• hashCode(“hawk”) = 143

• table[143] is not empty• table[143] != hawk

• table[144] is not empty• table[144] == hawk

hawk is already in the table, so do nothing.

robin

sparrow

hawk

seagull

bluejay

owl

. . .

141

142

143

144

145

146

147

148

. . .


Suppose:• You want to add cardinal to

this hash table• hashCode(“cardinal”) = 147

• The last location is 148• 147 and 148 are occupied

Solution:• Treat the table as circular;

after 148 comes 0• Hence, cardinal goes in

location 0 (or 1, or 2, or ...)

robin

sparrow

hawk

seagull

bluejay

owl

. . .

141

142

143

144

145

146

147

148

Linear Probing: find

Suppose we want to find hawk in this hash table

We proceed as follows:• hashCode(“hawk”) = 143• table[143] is not empty• table[143] != hawk• table[144] is not empty• table[144] == hawk (found!)

We use the same procedure for looking things up in the table as we do for inserting them

robin

sparrow

hawk

seagull

bluejay

owl

. . .

141

142

143

144

145

146

147

148

. . .

Linear Probing and Deletion

If an item is placed in array[hash(key)+4], then the item just before it is deleted

How will probe determine that the “hole” does not indicate the item is not in the array?

Have three states for each location• Occupied• Empty (never used)• Deleted (previously used)

Clustering

One problem with linear probing technique is the tendency to form “clusters”.

A cluster is a group of items not containing any open slots

The bigger a cluster gets, the more likely it is that new values will hash into the cluster, and make it ever bigger.

Clusters cause efficiency to degrade.

Quadratic Probing

Quadratic probing uses different formula:• Use F(i) = i2 to resolve collisions• If hash function resolves to H and a search in cell

H is inconclusive, try H + 12, H + 22, H + 32, …

Probe array[hash(key)+12], thenarray[hash(key)+22], thenarray[hash(key)+32], and so on

• Virtually eliminates primary clusters

Collision resolution: chaining

Each table position is a linked list

Add the keys and entries anywhere in the list (front easiest)

4

10

123

key entry key entry

key entry key entry

key entry

No need to change position!

Collision resolution: chaining

Advantages over open addressing:• Simpler insertion and

removal• Array size is not a

limitation Disadvantage

• Memory overhead is large if entries are small.

4

10

123

key entry key entry

key entry key entry

key entry

Applications of Hashing

Compilers use hash tables to keep track of declared variables (symbol table).

A hash table can be used for on-line spelling checkers — if misspelling detection (rather than correction) is important, an entire dictionary can be hashed and words checked in constant time.

Applications of Hashing

Game playing programs use hash tables to store seen positions, thereby saving computation time if the position is encountered again.

Hash functions can be used to quickly check for inequality — if two elements hash to different values they must be different.

When is hashing suitable?

Hash tables are very good if there is a need for many searches in a reasonably stable table.

Hash tables are not so good if there are many insertions and deletions, or if table traversals are needed — in this case, AVL trees are better.

Also, hashing is very slow for any operations which require the entries to be sorted• e.g. Find the minimum key

Computer Notes - Data Structures - 32

Documents

key skip list

binary search binary

key insert

key remove

search example

largest key

half http

lowhigh binary search