The single source shortest path algorithm (for arbitrary weight positive or negative) is also known Bellman-Ford algorithm is used to find minimum distance from source vertex to any other vertex. The main difference between this algorithm with Dijkstra’s algorithm is, in Dijkstra’s algorithm we cannot handle the negative weight, but here we can handle it easily.Bellman-Ford algorithm finds the distance in bottom up manner. At first it finds those distances which have only one edge in the path. After that increase the path length to find all possible solutions.Input − The cost matrix of the graph:0 6 ∞ 7 ∞ ∞ ... Read More

The single source shortest path algorithm (for non-negative weight) is also known Dijkstra algorithm. There is a given graph G(V, E) with its adjacency matrix representation, and a source vertex is also provided. Dijkstra’s algorithm to find the minimum shortest path between source vertex to any other vertex of the graph G.From starting node to any other node, find the smallest distances. In this problem the graph is represented using the adjacency matrix. (Cost matrix and adjacency matrix is similar for this purpose).Input − The adjacency matrix −0 3 6 ∞ ∞ ∞ ∞ 3 0 2 1 ∞ ∞ ... Read More

There is a connected graph G(V, E) and the weight or cost for every edge is given. Prim’s Algorithm will find the minimum spanning tree from the graph G.It is growing tree approach. This algorithm needs a seed value to start the tree. The seed vertex is grown to form the whole tree.The problem will be solved using two sets. One set holds the nodes that are already selected, and another set holds the item that are not considered yet. From the seed vertex, it takes adjacent vertices, based on minimum edge cost, thus it grows the tree by taking ... Read More

There is a connected graph G(V, E) and the weight or cost for every edge is given. Kruskal’s algorithm will find the minimum spanning tree using the graph and the cost.It is merge tree approach. Initially there are different trees, this algorithm will merge them by taking those edges whose cost is minimum, and form a single tree.In this problem, all of the edges are listed, and sorted based on their cost. From the list, edges with minimum costs are taken out and added in the tree, and every there is a check whether the edge forming cycle or not, ... Read More

The Breadth First Search (BFS) traversal is an algorithm, which is used to visit all of the nodes of a given graph. In this traversal algorithm one node is selected and then all of the adjacent nodes are visited one by one. After completing all of the adjacent vertices, it moves further to check another vertices and checks its adjacent vertices again.To implement this algorithm, we need to use the Queue data structure. All the adjacent vertices are added into the queue, when all adjacent vertices are completed, one item is removed from the queue and start traversing through that ... Read More

The Depth First Search (DFS) is a graph traversal algorithm. In this algorithm one starting vertex is given, and when an adjacent vertex is found, it moves to that adjacent vertex first and try to traverse in the same manner.It moves through the whole depth, as much as it can go, after that it backtracks to reach previous vertices to find new path.To implement DFS in iterative way, we need to use the stack data structure. If we want to do it recursively, external stacks are not needed, it can be done internal stacks for the recursion calls.Input: The Adjacency ... Read More

The graph is a non-linear data structures. This represents data using nodes, and their relations using edges. A graph G has two sections. The vertices, and edges. Vertices are represented using set V, and Edges are represented as set E. So the graph notation is G(V, E). Let us see one example to get the idea.In this graph, there are five vertices and five edges. The edges are directed. As an example, if we choose the edge connecting vertices B and D, the source vertex is B and destination is D. So we can move B to D but not ... Read More

Amortize AnalysisThis analysis is used when the occasional operation is very slow, but most of the operations which are executing very frequently are faster. In Data structures we need amortized analysis for Hash Tables, Disjoint Sets etc.In the Hash-table, the most of the time the searching time complexity is O(1), but sometimes it executes O(n) operations. When we want to search or insert an element in a hash table for most of the cases it is constant time taking the task, but when a collision occurs, it needs O(n) times operations for collision resolution.Aggregate MethodThe aggregate method is used to ... Read More

Little o NotationsThere are some other notations present except the Big-Oh, Big-Omega and Big-Theta notations. The little o notation is one of them.Little o notation is used to describe an upper bound that cannot be tight. In other words, loose upper bound of f(n).Let f(n) and g(n) are the functions that map positive real numbers. We can say that the function f(n) is o(g(n)) if for any real positive constant c, there exists an integer constant n0 ≤ 1 such that f(n) > 0.Mathematical Relation of Little o notationUsing mathematical relation, we can say that f(n) = o(g(n)) means, Example ... Read More

Asymptotic NotationsAsymptotic notations are used to represent the complexities of algorithms for asymptotic analysis. These notations are mathematical tools to represent the complexities. There are three notations that are commonly used.Big Omega NotationBig-Omega (Ω) notation gives a lower bound for a function f(n) to within a constant factor.We write f(n) = Ω(g(n)), If there are positive constants n0 and c such that, to the right of n0 the f(n) always lies on or above c*g(n).Ω(g(n)) = { f(n) : There exist positive constant c and n0 such that 0 ≤ c g(n) ≤ f(n), for all n ≤ n0}Big Theta ... Read More