Chapter 4
Searching
Computer systems are often used to store large amounts of data from which individual records must be retrieved according to some search criterion. Thus the efficient storage of data to facilitate fast searching is an important issue.
Sequential Search
Linear search or sequential search is a method for finding a target value within a list. It sequentially checks each element of the list for the target value until a match is found or until all the elements have been searched
Number => 33
Figure 1 Step1
Checking on index 0, and value on index 0 is not matched move next
Figure 2 Step2
Checking on index 1, and value on index 1 is not matched move next
Figure 3 Step 3
Checking on index 2, and value on index 2 is not matched move next
Figure 4 Step 4
Checking on index 3, and value on index 3 is not matched move next
Figure 5 Step 5
Checking on index 4, and value on index 4 is not matched move next
Figure 6 Step 6
Checking on index 5, and value on index 5 is not matched move next
Figure 7 Step 6
Checking on index 6, and value on index 6 is not matched move next
Algorithm
Linear Search ( Array A, Value x)
Step 1: Set i to 0// from 0 index
Step 2: if i > n then go to step 7
Step 3: if A[i] = x then go to step 6
Step 4: Set i to i + 1
Step 5: Go to Step 2
Step 6: Print Element x Found at index i and go to step 8
Step 7: Print element not found
Step 8: Exit
Pseudocode
Procedure linear_search (list, value)
for each item in the list
if match item == value
return the item's location
end if
end for
end procedure
Binary Search
For a binary search to work, it is mandatory for the target array to be sorted. We shall learn the process of binary search with a pictorial example. The following is our sorted array and let us assume that we need to search the location of value 31 using binary search.
First, we shall determine half of the array by using this formula.
mid = low + ((high - low) / 2)
Here it is, 0 + (9 - 0) / 2 = 4 (integer value of 4.5). So, 4 is the mid of the array.
Now we compare the value stored at location 4, with the value being searched, i.e. 31. We find that the value at location 4 is 27, which is not a match. As the value is greater than 27 and we have a sorted array, so we also know that the target value must be in the upper portion of the array.
We change our low to mid + 1 and find the new mid value again.
low = mid + 1
mid = low + (high - low) / 2
Our new mid is 7 now. We compare the value stored at location 7 with our target value 31.
The value stored at location 7 is not a match, rather it is more than what we are looking for. So, the value must be in the lower part from this location.
Hence, we calculate the mid again. This time it is 5.
We compare the value stored at location 5 with our target value. We find that it is a match
We conclude that the target value 31 is stored at location 5.
Binary search halves the searchable items and thus reduces the count of comparisons to be made to very less numbers.
Pseudocode
Procedure binary_search
A ← sorted array
n ← size of array
x ← value to be searched
Set lowerBound = 1
Set upperBound = n
while x not found
if upperBound < lowerBound
EXIT: x does not exists.
set midPoint = lowerBound + ( upperBound - lowerBound ) / 2
if A[midPoint] < x
set lowerBound = midPoint + 1
if A[midPoint] > x
set upperBound = midPoint - 1
if A[midPoint] = x
EXIT: x found at location midPoint
end while
end procedure
Tree
When starting out programming, it is common to understand better the linear data structures than data structures like trees and graphs.
Trees are well-known as a non-linear data structure. They don’t store data in a linear way. They organize data hierarchically.
* Binary tree = a tree where each node has at most 2 children nodes
In computer science, a binary tree is a tree data structure in which each node has at the most two children, which are referred to as the left child and the right child
Terminology used in trees
Root is the topmost node of the tree
Node at the "top" of a tree - the one from which all operations on the tree commence. The root node may not exist (a NULL tree with no nodes in it) or have 0, 1 or 2 children in a binary tree.
Edge is the link between two nodes
Child is a node that has a parent node
Parent is a node that has an edge to a child node
Leaf is a node that does not have a child node in the tree
Node at the "bottom" of a tree - farthest from the root. Leaf nodes have no children.
Height is the length of the longest path to a leaf
Number of nodes which must be traversed from the root to reach a leaf of a tree.
Depth is the length of the path to its root
Figure 8 Example of Tree
Complete Tree
Definition:
This forms a complete tree, whose height is defined as the number of links from the root to the deepest leaf. Tree in which each leaf is at the same distance from the root.
First, we need to work out how many nodes, n, we have in such a tree of height, h.
Now,
Total number of Nodes.
n = 1 + 21 + 22 +.... + 2h
From which we have,
n = 2h+1 – 1
Detailed Explanation:
Figure 9 A tree where each node has at most 2 children nodes
Left and right child
Because each node has at most 2 children nodes, we can label the children distinctly as left and right:
Note
Some nodes (e.g. h) have only a left child node
Some nodes (e.g. b) have only a right child node
Perfect binary tree
A binary tree where each level contains the maximum number of nodes
I.e., every level is completely full of nodes
Some properties of the perfect binary tree
Depth of Tree
The number of nodes at depth d in a perfect binary tree = 2d
There is only 1 node (= the root node) at depth 0:
Figure 10, in a perfect binary tree, every node has 2 children nodes
So:
Figure 11 the number of nodes doubles every time the depth increases by 1!
Therefore:
# Nodes at depth d = 2d
Total Number of Node
The total number of nodes in a perfect binary tree of height h: has 2h+1 − 1 nodes
Figure 12 # nodes = 20 + 21 + ... 2h = 2h+1 − 1
Therefore
Height h=3;
By using Formula 23+1 – 1=24-1;
=16-1
=15 Nodes, all we have the Maximum number of Nodes in tree.
Total Number of Leaf nodes
All the leaf nodes in a perfect binary tree of height h has a depth equal to h:
Figure 13 # nodes at depth h in a perfect binary tree = 2h
Therefore
Number of leaf nodes in a perfect binary tree of height h = 2h
So the Height h =3;
By using Formula 2h=23;
Total number of Leafs=8 Ans
Number of internal nodes
Number of internal nodes in a perfect binary tree of height h = 2h − 1
# Nodes in a perfect binary tree of height h = 2h+1 − 1
# Leaf nodes in a perfect binary tree of height h = 2h
The other nodes are internal nodes (i.e., with at least 1 child node).
So:
# Internal nodes in a perfect binary tree of
Height h = (2h+1 − 1) − 2h = 2h − 1
Therefore Number of internal nodes
Height h=3;
The Formula is 2h-1
By putting values in formula
23-1=8-1;
=7 Ans
Minimum and maximum number of nodes in a binary tree of height h
The minimum number of nodes in a binary tree of height h = h + 1
The binary tree of height h with the minimum number of nodes is a tree where each node has one child
Because the height = h, they are h edges
h edges connects h+1 nodes
Therefore, the minimum number of nodes in a binary tree of height h = h + 1
The maximum number of nodes in a binary tree of height h = 2h+1 – 1
All possible node in a tree=2h+1 – 1.
General View
Figure 14 General View