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# Trees and Searches

#### Depth-First Search (DFS)

Depth-first search (DFS) is an algorithm for traversing or searching tree or graph data structures. One starts at the root (selecting some arbitrary node as the root in the case of a graph) and explores as far as possible along each branch before backtracking.

Breadth-first search (BFS) is an algorithm for traversing or searching tree or graph data structures. It starts at the tree root (or some arbitrary node of a graph, sometimes referred to as a 'search key') and explores the neighbor nodes first, before moving to the next level neighbors.

#### Linear Search

In computer science, 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. Linear search runs in at worst linear time and makes at most n comparisons, where n is the length of the list.

Complexity
Time Complexity: O(n) - since in worst case we're checking each element exactly once.

#### Jump Search

Like Binary Search, Jump Search (or Block Search) is a searching algorithm for sorted arrays. The basic idea is to check fewer elements (than linear search) by jumping ahead by fixed steps or skipping some elements in place of searching all elements.

For example, suppose we have an array arr[] of size n and block (to be jumped) of size m. Then we search at the indexes arr, arr[m], arr[2 * m], ..., arr[k * m] and so on. Once we find the interval arr[k * m] < x < arr[(k+1) * m], we perform a linear search operation from the index k * m to find the element x.

What is the optimal block size to be skipped? In the worst case, we have to do n/m jumps and if the last checked value is greater than the element to be searched for, we perform m - 1 comparisons more for linear search. Therefore the total number of comparisons in the worst case will be ((n/m) + m - 1). The value of the function ((n/m) + m - 1) will be minimum when m = √n. Therefore, the best step size is m = √n.

Complexity
Time complexity: O(√n) - because we do search by blocks of size √n.

#### Binary Search

In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the middle element of the array; if they are unequal, the half in which the target cannot lie is eliminated and the search continues on the remaining half until it is successful. If the search ends with the remaining half being empty, the target is not in the array.

Complexity
Time Complexity: O(log(n)) - since we split search area by two for every next iteration.

#### Interpolation Search

Interpolation search is an algorithm for searching for a key in an array that has been ordered by numerical values assigned to the keys (key values).

For example we have a sorted array of n uniformly distributed values arr[], and we need to write a function to search for a particular element x in the array.

Linear Search finds the element in O(n) time, Jump Search takes O(√ n) time and Binary Search take O(Log n) time.

The Interpolation Search is an improvement over Binary Search for instances, where the values in a sorted array are uniformly distributed. Binary Search always goes to the middle element to check. On the other hand, interpolation search may go to different locations according to the value of the key being searched. For example, if the value of the key is closer to the last element, interpolation search is likely to start search toward the end side.

To find the position to be searched, it uses following formula:

``````// The idea of formula is to return higher value of pos
// when element to be searched is closer to arr[hi]. And
// smaller value when closer to arr[lo]
pos = lo + ((x - arr[lo]) * (hi - lo) / (arr[hi] - arr[Lo]))
arr[] - Array where elements need to be searched
x - Element to be searched
lo - Starting index in arr[]
hi - Ending index in arr[]
``````

Complexity
Time complexity: O(log(log(n))