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Search Algorithms

1

Chapter 8

11

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Chapter Objectives

Data Structures Using Java2

y Learn the various search algorithms

y Explore how to implement the sequential and binarysearch algorithms

y

Discover how the sequential and binary searchalgorithms perform

y Learn about hashing

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class ArrayListClass: Basic Operations


y isEmpty

y isFull

y listSize

y maxListSixey Print

y isItemAtEqual

y insertAt

y insertEnd

y removeAt

y retrieveAt

y replaceAt

y clearListy seqSearch

y insert

y remove

y copyList

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Search Algorithms


y Associated with each item in a data set is a specialmember (the key of the item) that uniquely identifies theitem in the data set

y

Keys are used in such operations as searching, sorting,insertion, and deletion

y Analysis of the algorithms enables programmers todecide which algorithm to use for a specific application

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Sequential Search


y Starts at the first element in the list

y Continues until either the item is found in the list or the

entire list is searched

y Works the same for both array-based and linked lists

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Sequential Search


public int seqSearch(DataElement searchItem){

int loc;boolean found = false;for(loc = 0; loc < length; loc++)

if(list[loc].equals(searchItem)){found = true;

break;}

if(found)

return loc;else

return -1;}//end seqSearch

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Search Algorithms


y Search item: target

y To determine the average number of comparisons in the

successful case of the sequential search algorithm:

y Consider all possible casesy Find the number of comparisons for each case

y Add the number of comparisons and divide by the number of

cases

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Sequential Search Analysis


Suppose that there are n elements in the list. The following

expression gives the average number of comparisons:

It is known that

Therefore, the following expression gives the average number ofcomparisons made by the sequential search in the successful case:

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Ordered Lists as Arrays


y List is ordered if its elements are ordered according to somecriteria

y Elements of a list usually in ascending order

y Define ordered list as an ADT

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Ordered Lists as Arrays


y Several operations can be performed on an ordered list;similar to the operations performed on an arbitrary listy Determining whether the list is empty or fully Determining the length of the listy Printing the listy Clearing the list

y Using inheritance, derive class to implement ordered lists

from class ArrayListClass

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Binary Search Algorithm


y Very fast

y Uses divide and conquer technique to search list

y First, search item compared with middle element of list

y If the search item is less than middle element of list,restrict the search to first half of list

y Otherwise, search second half of list

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Binary Search: middle element


first + last

2mid =

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Binary Search


public int binarySearch(DataElement item){

int first = 0;int last = length - 1;int mid = -1;

boolean found = false;while(first 0)

last = mid - 1;else

first = mid + 1;

}if(found)

return mid;else

return -1;}//end binarySearch

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Binary Search: Example


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Binary Search: Example


Unsuccessful search

Total number of comparisons is 6

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Performance of Binary Search


y Unsuccessful search

y for a list of length n, a binary search makes approximately

2*log2(n + 1) key comparisons

y Successful search

y for a list of length n, on average, a binary search makes 2*log2n

4 key comparisons

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Algorithm to Insert into an Ordered List


Use algorithm similar to binary search algorithm to findplace where item is to be inserted

if the item is already in this listoutput an appropriate message

elseuse the method insertAt to insert the item in

the list

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Search Algorithm Analysis Summary


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Hashing


y Main objectives to choosing hash methods:

y Choose a hash method that is easy to compute

y Minimize the number of collisions

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Commonly Used Hash Methods


y Mid-Square

y Hash method, h, computed by squaring the identifier

y Using appropriate number of bits from the middle of the square

to obtain the bucket address

y Middle bits of a square usually depend on all the characters, it is

expected that different keys will yield different hash addresses

with high probability, even if some of the characters are the

same

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y Folding

y Key Xis partitioned into parts such that all the parts, except

possibly the last parts, are of equal length

y Parts then added, in convenient way, to obtain hash address

y Division (Modular arithmetic)

y Key Xis converted into an integer iX

y This integer divided by size of hash table to get remainder,

giving address ofXin HT

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Suppose that each key is a string. The following Java

method uses the division method to compute the address

of the key:

int hashmethod(String insertKey)

{

int sum = 0;

for(int j = 0; j

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Collision Resolution


y Algorithms to handle collisions

y Two categories of collision resolution techniques

y Open addressing (closed hashing)

y

Chaining (open hashing)

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Open Addressing: Linear Probing


y Suppose that an item with key Xis to be inserted in HT

y Use hash function to compute index h(X) of item in HT

y Suppose h(X) = t. Then 0 = h(X) = HTSize 1

y IfHT[t] is empty, store item into array slot.y Suppose HT[t] already occupied by another item;

collision occurs

y Linear probing: starting at location t, search array

sequentially to find next available array slot

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Collision Resolution:

Open Addressing


Pseudocode implementing linear probing:

hIndex = hashmethod(insertKey);

found = false;

while(HT[hIndex] != emptyKey && !found)

if(HT[hIndex].key == key)

found = true;

else

hIndex = (hIndex + 1) % HTSize;

if(found)

System.out.println(Duplicate items not allowed);

else

HT[hIndex] = newItem;

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Linear Probing


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Random Probing


y Uses a random number generator to find the nextavailable slot

y ith slot in the probe sequence is: (h(X) + ri) % HTSizewhere ri is the ith value in a random permutation of thenumbers 1 to HTSize 1

y All insertions and searches use the same sequence ofrandom numbers

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Quadratic Probing


y Reduces primary clustering

y We do not know if it probes all the positions in the table

y When HTSize is prime, quadratic probing probes about

half the table before repeating the probe sequence

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Deletion: Open Addressing


y In open addressing, when an item is deleted, its position

in the array cannot be marked as empty

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Chapter Summary


y Search Algorithms

y Sequential

y Binary

y Algorithm Analysis

y Hashing

y Hash Table

y Hash method

y Collision Resolution

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