elementary data structures

Elementary Data Structures

Stacks, Queues, Lists, Vectors, Sequences, Trees, Priority Queues, Heaps, Dictionaries & Hash Tables

Elementary Data Structures 2

The Stack ADT (§2.1.1)The Stack ADT stores arbitrary objectsInsertions and deletions follow the last-in first-out schemeThink of a spring-loaded plate dispenserMain stack operations:

push(object): inserts an element

object pop(): removes and returns the last inserted element

Auxiliary stack operations:

object top(): returns the last inserted element without removing it

integer size(): returns the number of elements stored

boolean isEmpty(): indicates whether no elements are stored


Applications of StacksDirect applications Page-visited history in a Web browser Undo sequence in a text editor Chain of method calls in the Java

Virtual Machine or C++ runtime environment

Indirect applications Auxiliary data structure for algorithms Component of other data structures


The Queue ADT (§2.1.2)The Queue ADT stores arbitrary objectsInsertions and deletions follow the first-in first-out schemeInsertions are at the rear of the queue and removals are at the front of the queueMain queue operations:

enqueue(object): inserts an element at the end of the queue

object dequeue(): removes and returns the element at the front of the queue

Auxiliary queue operations:

object front(): returns the element at the front without removing it

integer size(): returns the number of elements stored

boolean isEmpty(): indicates whether no elements are stored

Exceptions Attempting the execution

of dequeue or front on an empty queue throws an EmptyQueueException


Applications of QueuesDirect applications Waiting lines Access to shared resources (e.g.,

printer) MultiprogrammingIndirect applications Auxiliary data structure for algorithms Component of other data structures


Position ADTThe Position ADT models the notion of place within a data structure where a single object is storedIt gives a unified view of diverse ways of storing data, such as a cell of an array a node of a linked list

Just one method: object element(): returns the element

stored at the position


List ADT (§2.2.2)

The List ADT models a sequence of positions storing arbitrary objectsIt allows for insertion and removal in the “middle” Query methods:

isFirst(p), isLast(p)

Accessor methods: first(), last() before(p), after(p)

Update methods: replaceElement(p, o),

swapElements(p, q) insertBefore(p, o),

insertAfter(p, o), insertFirst(o),

insertLast(o) remove(p)


Singly Linked ListA singly linked list is a concrete data structure consisting of a sequence of nodesEach node stores

element link to the next node

next

elem node

A B C D


Doubly Linked ListA doubly linked list provides a natural implementation of the List ADTNodes implement Position and store:

element link to the previous node link to the next node

Special trailer and header nodes

prev next

elem

trailerheader nodes/positions

elements

node


The Vector ADTThe Vector ADT extends the notion of array by storing a sequence of arbitrary objectsAn element can be accessed, inserted or removed by specifying its rank (number of elements preceding it)An exception is thrown if an incorrect rank is specified (e.g., a negative rank)

Main vector operations: object elemAtRank(integer r):

returns the element at rank r without removing it

object replaceAtRank(integer r, object o): replace the element at rank with o and return the old element

insertAtRank(integer r, object o): insert a new element o to have rank r

object removeAtRank(integer r): removes and returns the element at rank r

Additional operations size() and isEmpty()


Applications of VectorsDirect applications Sorted collection of objects

(elementary database)Indirect applications Auxiliary data structure for algorithms Component of other data structures


Sequence ADTThe Sequence ADT is the union of the Vector and List ADTsElements accessed by

Rank, or Position

Generic methods: size(), isEmpty()

Vector-based methods: elemAtRank(r),

replaceAtRank(r, o), insertAtRank(r, o), removeAtRank(r)

List-based methods: first(), last(),

before(p), after(p), replaceElement(p, o), swapElements(p, q), insertBefore(p, o), insertAfter(p, o), insertFirst(o), insertLast(o), remove(p)

Bridge methods: atRank(r), rankOf(p)


Applications of SequencesThe Sequence ADT is a basic, general-purpose, data structure for storing an ordered collection of elementsDirect applications: Generic replacement for stack, queue, vector,

or list small database (e.g., address book)

Indirect applications: Building block of more complex data structures


Trees (§2.3)In computer science, a tree is an abstract model of a hierarchical structureA tree consists of nodes with a parent-child relationApplications:

Organization charts File systems Programming

environments

Computers”R”Us

Sales R&DManufacturing

Laptops DesktopsUS International

Europe Asia Canada

Elementary Data Structures 15subtree

Tree TerminologyRoot: node without parent (A)Internal node: node with at least one child (A, B, C, F)External node (a.k.a. leaf ): node without children (E, I, J, K, G, H, D)Ancestors of a node: parent, grandparent, grand-grandparent, etc.Depth of a node: number of ancestorsHeight of a tree: maximum depth of any node (3)Descendant of a node: child, grandchild, grand-grandchild, etc.

A

B DC

G HE F

I J K

Subtree: tree consisting of a node and its descendants


Tree ADT (§2.3.1)We use positions to abstract nodesGeneric methods:

integer size() boolean isEmpty() objectIterator elements() positionIterator

positions()Accessor methods:

position root() position parent(p) positionIterator

children(p)

Query methods: boolean isInternal(p) boolean isExternal(p) boolean isRoot(p)

Update methods: swapElements(p, q) object replaceElement(p,

o)Additional update methods may be defined by data structures implementing the Tree ADT


Preorder Traversal (§2.3.2)A traversal visits the nodes of a tree in a systematic mannerIn a preorder traversal, a node is visited before its descendants Application: print a structured document

Make Money Fast!

1. Motivations References2. Methods

2.1 StockFraud

2.2 PonziScheme1.1 Greed 1.2 Avidity 2.3 Bank

Robbery

1

2

3

5

4 6 7 8

9

Algorithm preOrder(v)visit(v)for each child w of v

preorder (w)


Postorder Traversal (§2.3.2)

In a postorder traversal, a node is visited after its descendantsApplication: compute space used by files in a directory and its subdirectories

Algorithm postOrder(v)for each child w of v

postOrder (w)visit(v)

cs16/

homeworks/ todo.txt1Kprograms/

DDR.java10K

Stocks.java25K

h1c.doc3K

h1nc.doc2K

Robot.java20K

9

3

1

7

2 4 5 6

8


Inorder TraversalIn an inorder traversal a node is visited after its left subtree and before its right subtreeApplication: draw a binary tree

x(v) = inorder rank of v y(v) = depth of v

Algorithm inOrder(v)if isInternal (v)

inOrder (leftChild (v))visit(v)if isInternal (v)

inOrder (rightChild (v))

3

1

2

5

6

7 9

8

4


Binary Trees (§2.3.3)A binary tree is a tree with the following properties:

Each internal node has two children

The children of a node are an ordered pair

We call the children of an internal node left child and right childAlternative recursive definition: a binary tree is either

a tree consisting of a single node, or

a tree whose root has an ordered pair of children, each of which is a binary tree

Applications: arithmetic

expressions decision processes searching

A

B C

F GD E

H I


Properties of Binary TreesNotationn number of nodese number of

external nodesi number of

internal nodesh height

Properties: e i 1 n 2e 1 h i h (n 1)2 e 2h

h log2 e h log2 (n 1) 1


Array-Based Representation of Binary Trees

nodes are stored in an array

…

let rank(node) be defined as follows: rank(root) = 1 if node is the left child of parent(node),

rank(node) = 2*rank(parent(node)) if node is the right child of parent(node),

rank(node) = 2*rank(parent(node))+1

1

2 3

6 74 5

10 11

A

HG

FE

D

C

B

J


Priority Queue ADTA priority queue stores a collection of itemsAn item is a pair(key, element)Main methods of the Priority Queue ADT

insertItem(k, o)inserts an item with key k and element o

removeMin()removes the item with smallest key and returns its element

Additional methods minKey(k, o)

returns, but does not remove, the smallest key of an item

minElement()returns, but does not remove, the element of an item with smallest key

size(), isEmpty()Applications:

Standby flyers Auctions Stock market


What is a heap (§2.4.3)A heap is a binary tree storing keys at its internal nodes and satisfying the following properties:

Heap-Order: for every internal node v other than the root,key(v) key(parent(v))

Complete Binary Tree: let h be the height of the heap

for i 0, … , h 1, there are 2i nodes of depth i

at depth h 1, the internal nodes are to the left of the external nodes

2

65

79

The last node of a heap is the rightmost internal node of depth h 1

last node


Height of a Heap (§2.4.3)Theorem: A heap storing n keys has height O(log n)Proof: (we apply the complete binary tree property)

Let h be the height of a heap storing n keys Since there are 2i keys at depth i 0, … , h 2 and at least

one key at depth h 1, we have n 1 2 4 … 2h2 1

Thus, n 2h1 , i.e., h log n 1

1

2

2h2

1

keys0

1

h2

h1

depth


Dictionary ADTThe dictionary ADT models a searchable collection of key-element itemsThe main operations of a dictionary are searching, inserting, and deleting itemsMultiple items with the same key are allowedApplications:

address book credit card authorization mapping host names (e.g.,

cs16.net) to internet addresses (e.g., 128.148.34.101)

Dictionary ADT methods: findElement(k): if the

dictionary has an item with key k, returns its element, else, returns the special element NO_SUCH_KEY

insertItem(k, o): inserts item (k, o) into the dictionary

removeElement(k): if the dictionary has an item with key k, removes it from the dictionary and returns its element, else returns the special element NO_SUCH_KEY

size(), isEmpty() keys(), Elements()


Binary SearchBinary search performs operation findElement(k) on a dictionary implemented by means of an array-based sequence, sorted by key

similar to the high-low game at each step, the number of candidate items is halved terminates after a logarithmic number of steps

Example: findElement(7)

1 3 4 5 7 8 9 11 14 16 18 19

1 3 4 5 7 8 9 11 14 16 18 19

1 3 4 5 7 8 9 11 14 16 18 19

1 3 4 5 7 8 9 11 14 16 18 19

0

0

0

0

ml h

ml h

ml h

lm h


Lookup TableA lookup table is a dictionary implemented by means of a sorted sequence

We store the items of the dictionary in an array-based sequence, sorted by key

We use an external comparator for the keysPerformance:

findElement takes O(log n) time, using binary search insertItem takes O(n) time since in the worst case we have to

shift n2 items to make room for the new item removeElement take O(n) time since in the worst case we

have to shift n2 items to compact the items after the removalThe lookup table is effective only for dictionaries of small size or for dictionaries on which searches are the most common operations, while insertions and removals are rarely performed (e.g., credit card authorizations)


Binary Search TreeA binary search tree is a binary tree storing keys (or key-element pairs) at its internal nodes and satisfying the following property:

Let u, v, and w be three nodes such that u is in the left subtree of v and w is in the right subtree of v. We have key(u) key(v) key(w)

External nodes do not store items

An inorder traversal of a binary search trees visits the keys in increasing order

6

92

41 8


Hash Functions and Hash Tables (§2.5.2)

A hash function h maps keys of a given type to integers in a fixed interval [0, N1]Example:

h(x) x mod Nis a hash function for integer keysThe integer h(x) is called the hash value of key xA hash table for a given key type consists of Hash function h Array (called table) of size N

When implementing a dictionary with a hash table, the goal is to store item (k, o) at index i h(k)


ExampleWe design a hash table for a dictionary storing items (SSN, Name), where SSN (social security number) is a nine-digit positive integerOur hash table uses an array of size N10,000 and the hash functionh(x)last four digits of x

01234

999799989999

…451-229-0004

981-101-0002

200-751-9998

025-612-0001


Hash Functions (§ 2.5.3)A hash function is usually specified as the composition of two functions:Hash code map: h1: keys integers

Compression map: h2: integers [0, N1]

The hash code map is applied first, and the compression map is applied next on the result, i.e.,

h(x) = h2(h1(x))

The goal of the hash function is to “disperse” the keys in an apparently random way


Hash Code Maps (§2.5.3)

Memory address: We reinterpret the memory

address of the key object as an integer (default hash code of all Java objects)

Good in general, except for numeric and string keys

Integer cast: We reinterpret the bits of

the key as an integer Suitable for keys of length

less than or equal to the number of bits of the integer type (e.g., byte, short, int and float in Java)

Component sum: We partition the bits of

the key into components of fixed length (e.g., 16 or 32 bits) and we sum the components (ignoring overflows)

Suitable for numeric keys of fixed length greater than or equal to the number of bits of the integer type (e.g., long and double in Java)


Hash Code Maps (cont.)Polynomial accumulation:

We partition the bits of the key into a sequence of components of fixed length (e.g., 8, 16 or 32 bits) a0 a1 … an1

We evaluate the polynomialp(z) a0 a1 z a2 z2 … … an1zn1

at a fixed value z, ignoring overflows

Especially suitable for strings (e.g., the choice z 33 gives at most 6 collisions on a set of 50,000 English words)

Polynomial p(z) can be evaluated in O(n) time using Horner’s rule:

The following polynomials are successively computed, each from the previous one in O(1) timep0(z) an1

pi (z) ani1 zpi1(z) (i 1, 2, …, n 1)

We have p(z) pn1(z)


Compression Maps (§2.5.4)

Division: h2 (y) y mod N The size N of the

hash table is usually chosen to be a prime

The reason has to do with number theory and is beyond the scope of this course

Multiply, Add and Divide (MAD): h2 (y) (ay b) mod N a and b are

nonnegative integers such that

a mod N 0 Otherwise, every

integer would map to the same value b


Collision Handling (§ 2.5.5)

Collisions occur when different elements are mapped to the same cellChaining: let each cell in the table point to a linked list of elements that map there

Chaining is simple, but requires additional memory outside the table

01234 451-229-0004 981-101-0004

025-612-0001


Linear Probing (§2.5.5)Open addressing: the colliding item is placed in a different cell of the tableLinear probing handles collisions by placing the colliding item in the next (circularly) available table cellEach table cell inspected is referred to as a “probe”Colliding items lump together, causing future collisions to cause a longer sequence of probes

Example: h(x) x mod 13 Insert keys 18, 41,

22, 44, 59, 32, 31, 73, in this order

0 1 2 3 4 5 6 7 8 9 10 11 12

41 18445932223173 0 1 2 3 4 5 6 7 8 9 10 11 12


Search with Linear ProbingConsider a hash table A that uses linear probingfindElement(k)

We start at cell h(k) We probe consecutive

locations until one of the following occurs

An item with key k is found, or

An empty cell is found, or

N cells have been unsuccessfully probed

Algorithm findElement(k)i h(k)p 0repeat

c A[i]if c

return NO_SUCH_KEY else if c.key () k

return c.element()else

i (i 1) mod Np p 1

until p Nreturn NO_SUCH_KEY


Updates with Linear Probing

To handle insertions and deletions, we introduce a special object, called AVAILABLE, which replaces deleted elementsremoveElement(k)

We search for an item with key k

If such an item (k, o) is found, we replace it with the special item AVAILABLE and we return element o

Else, we return NO_SUCH_KEY

insert Item(k, o) We throw an exception

if the table is full We start at cell h(k) We probe consecutive

cells until one of the following occurs

A cell i is found that is either empty or stores AVAILABLE, or

N cells have been unsuccessfully probed

We store item (k, o) in cell i


Double HashingDouble hashing uses a secondary hash function d(k) and handles collisions by placing an item in the first available cell of the series

(i jd(k)) mod N for j 0, 1, … , N 1The secondary hash function d(k) cannot have zero valuesThe table size N must be a prime to allow probing of all the cells

Common choice of compression map for the secondary hash function: d2(k) q k mod qwhere

q N q is a prime

The possible values for d2(k) are

1, 2, … , q


Consider a hash table storing integer keys that handles collision with double hashing

N13 h(k) k mod 13 d(k) 7 k mod 7

Insert keys 18, 41, 22, 44, 59, 32, 31, 73, in this order

Example of Double Hashing

0 1 2 3 4 5 6 7 8 9 10 11 12

31 41 183259732244 0 1 2 3 4 5 6 7 8 9 10 11 12

k h (k ) d (k ) Probes18 5 3 541 2 1 222 9 6 944 5 5 5 1059 7 4 732 6 3 631 5 4 5 9 073 8 4 8


Performance of Hashing

In the worst case, searches, insertions and removals on a hash table take O(n) timeThe worst case occurs when all the keys inserted into the dictionary collideThe load factor nN affects the performance of a hash tableAssuming that the hash values are like random numbers, it can be shown that the expected number of probes for an insertion with open addressing is

1 (1 )

The expected running time of all the dictionary ADT operations in a hash table is O(1) In practice, hashing is very fast provided the load factor is not close to 100%Applications of hash tables:

small databases compilers browser caches


Universal Hashing (§ 2.5.6)

A family of hash functions is universal if, for any 0<i,j<M-1, Pr(h(j)=h(k)) < 1/N.Choose p as a prime between M and 2M.Randomly select 0<a<p and 0<b<p, and define h(k)=(ak+b mod p) mod N

Theorem: The set of all functions, h, as defined here, is universal.


Proof of Universality (Part 1)

Let f(k) = ak+b mod pLet g(k) = k mod NSo h(k) = g(f(k)).f causes no collisions: Let f(k) = f(j). Suppose k<j. Then

pp

bakbakpp

bajbaj

pp

bakp

bajkja

)(

So a(j-k) is a multiple of pBut both are less than pSo a(j-k) = 0. I.e., j=k. (contradiction)Thus, f causes no collisions.


Proof of Universality (Part 2)If f causes no collisions, only g can make h cause collisions. Fix a number x. Of the p integers y=f(k), different from x, the number such that g(y)=g(x) is at most Since there are p choices for x, the number of h’s that will cause a collision between j and k is at most

There are p(p-1) functions h. So probability of collision is at most

Therefore, the set of possible h functions is universal.

1/ Np

NppNpp )1(1/

NppNpp 1

)1(/)1(

elementary data structures

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