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algorithm

Algorithms have proven to be the most important domain over the last century which has reformed the way we do tasks. It is actively used to design industry systems and forms the building blocks of companies like Google. We cover all types of algorithms in depth

Find Mother Vertex in a Graph【O(V+E)】

mother vertex in a graph is a vertex from which we can reach all the nodes in the graph through directed path. If there exist mother vertex (or vertices), then one of the mother vertices is the last finished vertex in DFS. (Or a mother vertex has the maximum finish time in DFS traversal).

Kosaraju's Algorithm for Strongly Connected Components 【O(V+E)】

Kosaraju algorithm is a DFS based algorithm used to find Strongly Connected Components(SCC) in a graph. It is based on the idea that if one is able to reach a vertex v starting from vertex u, then one should be able to reach vertex u starting from vertex v

Count paths from Top Left to Bottom Right of a Matrix using Dynamic Programming【O(M*N)】

Count all the possible paths from top left to bottom right of a m x n matrix with the constraints that from each cell you can either move only to right or down. Brute force approach takes exponential time while dynamic programming takes O(M*N) time complexity

Using Farach Colton and Bender Algorithm to solve LCA【O(V) query】

The basic idea of Farach Colton and Bender Algorithm is to traverse all the nodes using the Depth First Search graph traversal and keep a record of all the visited nodes and there corresponding heights. This allows to answer LCA queries in O(V) time complexity

Finding the Lexicographical Next Permutation in O(N) time complexity

In Lexicographical Permutation Algorithm we will find the immediate next smallest Integer number or sequence permutation. Finding all permutations take O(N!) time complexity but we present an efficient algorithm which can solve this in O(N) time complexity

Heap’s algorithm for generating permutations

Heap's Algorithm is used to generate all the possible permutation of n-decimals of a number. It generates each permutation from the previous one by interchanging a single pair of elements; the other n−2 elements are not disturbed. For N numbers, it takes O(N!) time complexity

Algorithm to find cliques of a given size k【O(n^k) time complexity】

We can find all the 2-cliques by simply enumerating all the edges. To find k+1-cliques, we can use the previous results. Compare all the pairs of k-cliques. If the two subgraphs have k-1 vertices in common and graph contains the missing edge, we can form a k+1-clique.

Minimum Product Subset of an array

For a given array of elements, we have to find the non-empty subset having the minimum product. We will explore two techniques brute force O(2^N) and Greedy algorithm O(N)

Maximum Product Subset of an array

For a given array of elements, we have to find the non-empty subset having the maximum product. We will explore two techniques brute force O(2^N) and Greedy algorithm O(N)

Understanding Bit mask/ Bit map in depth

A bit mask is a data structure, usually an integer or array of bits, that represent a bit pattern which signifies a particular state in a particular computational problem. It takes advantage of bitwise operations.

Greedy approach to find a single maximal clique in O(V^2) time complexity

There can be more than one single maximal clique in a non-complete graph (since complete graph is a maximal clique itself). To find a single maximal clique in a graph we use a straightforward greedy algorithm in O(V^2) time complexity

Johnson Algorithm to find the shortest paths between all pair of vertices

Johnson Algorithm is used to find shortest paths between every pair of vertices in a given weighted directed graph and here weights may be negative. Johnson Algorithm uses both Dijkstra and Bellman-Ford algorithms as subroutines.

Divide numbers from 1 to n into two groups with minimum sum difference from O(2^N) to O(N)

For numbers from 1 to given n, we have to find the division of the elements into two groups having minimum absolute sum difference. We will explore two techniques Brute force which will take O(2^N) time complexity and Greedy algorithm which will take O(N) time complexity

Using Bron Kerbosch algorithm to find maximal cliques in O(3^(N/3))

Bron–Kerbosch algorithm is an enumeration algorithm for finding maximal cliques in an undirected graph. Any n-vertex graph has at most 3^n⁄3 maximal cliques, and the worst-case running time of the Bron–Kerbosch algorithm (with a pivot strategy) is O(3^n⁄3).

Centroid Decomposition of Tree

Centroid Decomposition is a divide and conquer technique which is used on trees. Given a tree with N nodes, a centroid is a node whose removal splits the given tree into a forest of trees, where each of the resulting tree contains no more than N/2 nodes.

Clique in Graphs

A clique is a subset of vertices of an undirected graph G such that every two distinct vertices in the clique are adjacent; that is, its induced subgraph is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs.

Solving Vertex Cover Problem from O(2^n) to O(n^2)

Vertex Cover Problem is a known NP Complete problem. We will see Naive approach and Dynamic programming approach to solve the vertex cover problem for a binary tree graph and reduce the complexity from O(2^n) to O(n^2).

Understanding pairing nodes in Graphs (Maximum Matching)

A maximal matching is a matching M of a graph G with the property that if any edge not in M is added to M, it is no longer a matching, that is, M is maximal if it is not a subset of any other matching in graph G. We cover Blossom, Hungarian and Hopcroft Karp algorithm

Blossom Maximum Matching Algorithm

The blossom algorithm, sometimes called the Edmonds' matching algorithm, can be used on any graph to construct a maximum matching. The blossom algorithm improves upon the Hungarian algorithm by shrinking cycles in the graph to reveal augmenting paths. The blossom algorithm will work on any graph.

Hungarian Maximum Matching Algorithm

The Hungarian maximum matching algorithm, also called the Kuhn-Munkres algorithm, is a O(V3) algorithm that can be used to find maximum-weight matchings in bipartite graphs, which is sometimes called the assignment problem. A bipartite graph can easily be represented by an adjacency matrix

Hopcroft Karp algorithm

The Hopcroft–Karp algorithm is an algorithm that takes as input a bipartite graph and produces as output a maximum cardinality matching, it runs in O(E√V) time in worst case.

Pigeonhole Sort

Pigeonhole sorting is a sorting algorithm that is suitable for sorting lists of elements where the number of elements and the number of possible key values are approximately the same.

Cocktail Shaker Sort / Bidirectional bubble sort

Cocktail Sort is a variation of Bubble sort. The Bubble sort algorithm always traverses elements from left and moves the largest element to its correct position in first iteration and second largest in second iteration and so on. It traverses through a given array in both directions alternatively.

Odd Even Sort / Brick Sort

Odd Even Sort uses parallel algorithm which is based on bubble sort technique. It is also known as Brick Sort. It has a best time complexity of O(N) whereas the worst case time complexity is O(N^2)

Number of unique partitions of an integer

Given a positive integer, the problem is to find all the Unique Partitions of the number. We can understand the above problem using some basic examples. Naive approach takes O((N^N)log N) time complexity while Dynamic programming takes O(2^N)

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