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  Theory of Searching and their types

Searching:

Searching is very significant in computer science. It is very closely associated to sorting. We generally search after sorting the data. Remember Binary Search? This is the best instance of an algorithm that employs both Sorting and Searching.

The search algorithm mainly depends on data structure employed. If we want to search a graph, then we have a set of well recognized algorithms like DFS, BFS and so on.

Binary Search:

The most general application of binary search is to find a particular value in a sorted list. The search starts by examining the value in the center of list; as the values are sorted, it then knows whether the value takes place before or subsequent to the center value, and searches via the correct half in similar way. Here is simple a pseudocode that determines the index of a given value in the sorted list between indices right and left:

function binarySearch(a, value, left, right)
    if right < left
        return not found
    mid := floor((left+right)/2)
    if a[mid] = value
        return mid
    if value < a[mid]
        binarySearch(a, value, left, mid-1)
    else
        binarySearch(a, value, mid+1, right)

In both situations, the algorithm terminates since on each recursive call or iteration, the range of indexes right minus left for all time gets smaller, and therefore must eventually become negative.

Binary search is a logarithmic algorithm and executes in O(log n) time. Specially, 1 + log2N iterations are required to return an answer. This is considerably faster than a linear search. This can be implemented by using recursion or iteration, though in many languages it is more sophisticatedly expressed recursively.

Binary Search Tree:

Binary Search Tree (or BST) allow you to search a collection of objects (each with a real or integer value) quickly to find out if a given value present in the collection.

Principally, a binary search tree is a node-weighted and rooted binary ordered tree. That collection of adjectives signifies that each node in the tree might contain no child, one left child, one right child, or both right and left child. Additionally, each node has an object related with it, and the weight of node is the value of object.

The binary search tree as well has the property that each node's left child and the descendants of its left child encompass a value less than that of the node, and each node's right child and its descendants encompass a value greater or equivalent to it. 

427_BST.jpg

Figure: Binary Search Tree

The nodes are usually symbolized as a structure with four fields, a pointer to node's left child, a pointer to node's right child, the weight of the object stored in this node, and a pointer to the object itself. At times, for easier access, people add up pointer to the parent too.

Why is Binary Search Tree very useful?

We are given a collection of n objects, a binary search tree takes just O(height) time to find an objects, supposing that the tree is not actually poor (or unbalanced), O(height) is O(log n). In addition, dissimilar just keeping a sorted array, inserting and deleting objects just takes O(log n) time also. You as well can get all the keys in Binary Search Tree in a sorted order by traversing it employing O(n) in order traversal.

Variations on Binary Trees:

There are some variants which make sure that the trees are not at all poor. Splay trees, B-trees, Red-black trees and AVL trees are some of the more general illustrations. They are all much more complex to code, and random trees are usually good, therefore it is usually not worth it.

Tips: If you are concerned that the tree you made might be bad (it is being formed by inserting elements from an input file, for illustration), then arbitrarily order the elements before insertion.

Dictionary:

A dictionary or hash table stores the data with a very fast manner to do lookups. Let us say that there is a collection of objects and a data structure should quickly answer the question: 'Is this object is the data structure?' (Example: is this word in the dictionary?). A hash table does this in less time than it takes to do the binary search.

The idea is this: determine a function which maps the elements of the collection to an integer between 1 and x (here x, in this explanation, is bigger than the number of elements in your collection). Keep an array indexed from 1 to x, and store each and every element at the position which the function computes the element as. Then, to find out if something is in your collection, just plug it into the function and observe whether or not that position is empty. When it is not check, then the element there to see if it is similar as something you are holding.

For illustration, assume that the function is defined over 3-character words, and is (initial letter + (second letter * 3) + (third letter * 7)) mod 11 (A=1, B=2, etc.), and the words are 'CAT', 'CAR', and 'COB'. Whenever employing ASCII, this function takes 'CAT' and maps it into 3, maps 'CAR' to 0, and maps 'COB' to 7, therefore the hash table would appear like this:

0: CAR
1
2
3: CAT
4
5
6
7: COB
8
9
10 

Now, to see when 'BAT' is in there; plug it into the hash function to obtain 2. This position in the hash table is empty; therefore it is not in the collection. 'ACT', on the other hand, returns the value 7, therefore the program should check to see if that entry, 'COB', is similar as 'ACT' (no, therefore 'ACT' is not in the dictionary either). In other hand, when the search input is 'CAR', 'CAT', 'COB', then the dictionary will return true.

Collision Handling:

This glossed over a slight trouble which arises. What can be done when two entries map to similar value (example: we wanted to add 'ACT' and 'COB')? This is termed as a collision. There is couple of ways to correct collisions; however this document will focus on one method, termed as chaining.

Rather than having one entry at each and every position, maintain a linked list of entries with similar hash value. Therefore, whenever an element is added, determine its position and add it to the starting (or tail) of that list. Therefore, to have both 'ACT' and 'COB' in the table, it would look something similar to this:

0: CAR
1
2
3: CAT
4
5
6
7: COB -> ACT
8
9
10 

Now, to check an entry, all the elements in linked list should be examined to determine the element is not in the collection. This, obviously, reduces the efficiency of employing the hash table; however it is frequently quite handy.

Hash Table variations:

It is frequently quite helpful to store more information that just the value. One illustration is when searching a small subset of a big subset, and employing the hash table to store locations visited, you might want the value for searching a position in the hash table with it.

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