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

Here we will see some sorting methods. There are 200+ sorting techniques. We will see few of them. Some sorting techniques are comparison based sort, some are non-comparison based sorting technique.Comparison Based Soring techniques are bubble sort, selection sort, insertion sort, Merge sort, quicksort, heap sort etc. These techniques are considered as comparison based sort because in these techniques the values are compared, and placed into sorted position in ifferent phases. Here we will see time complexity of these techniques.Analysis TypeBubble SortSelection SortInsertion SortMerge SortQuick SortHeap SortBest CaseO(n2)O(n2)O(n)O(log n)O(log n)O(logn)Average CaseO(n2)O(n2)O(n2)O(log n)O(log n)O(log n)Worst CaseO(n2)O(n2)O(n2)O(log n)O(n2)O(log n)some sorting algorithms are ... Read More

In different cases, we perform different searching schemes to find some keys. In this section we will see what are the basic differences between two searching techniques, the sequential search and binary search.Sequential SearchBinary SearchTime complexity is O(n)Time complexity is O(log n)Finds the key present at first position in constant timeFinds the key present at center position in constant timeSequence of elements in the container does not affect.The elements must be sorted in the containerArrays and linked lists can be used to implement thisIt cannot be implemented directly into the linked list. We need to change the basic rules of ... Read More

The abstract datatype is special kind of datatype, whose behavior is defined by a set of values and set of operations. The keyword “Abstract” is used as we can use these datatypes, we can perform different operations. But how those operations are working that is totally hidden from the user. The ADT is made of with primitive datatypes, but operation logics are hidden.Here we will see the stack ADT. These are few operations or functions of the Stack ADT.isFull(), This is used to check whether stack is full or notisEmpry(), This is used to check whether stack is empty or ... Read More

Here we will see one example on convex hull. Suppose we have a set of points. We have to make a polygon by taking less amount of points, that will cover all given points. In this section we will see the Jarvis March algorithm to get the convex hull.Jarvis March algorithm is used to detect the corner points of a convex hull from a given set of data points.Starting from left most point of the data set, we keep the points in the convex hull by anti-clockwise rotation. From a current point, we can choose the next point by checking ... Read More

A set of integers are given in the sorted order and another array freq to frequency count. Our task is to create a binary search tree with those data to find minimum cost for all searches.An auxiliary array cost[n, n] is created to solve and store the solution of sub problems. Cost matrix will hold the data to solve the problem in bottom up manner.Input − The key values as node and the frequency.Keys = {10, 12, 20} Frequency = {34, 8, 50}Output − The minimum cost is 142.These are possible BST from the given values.For case 1, the cost ... Read More

The Negative Binomial Distribution is a random number distribution that will produce integers according to a negative binomial discrete distribution. This is known as Pascal’s distribution So the negative binomial distribution can be written as$$P\lgroup i\arrowvert k,p\rgroup=\lgroup \frac{k+i-1}{i}\rgroup p^{k}\lgroup 1-p\rgroup^{i}$$Example Live Demo#include #include using namespace std; int main(){    const int nrolls = 10000; // number of rolls    const int nstars = 100; // maximum number of stars to distribute    default_random_engine generator;    negative_binomial_distribution distribution(3,0.5);    int p[10]={};    for (int i=0; i

The Geometric Distribution is a discrete probability distribution for n = 0, 1, 2, …. having the probability density function.$$P\lgroup n\rgroup=p\lgroup1-p\rgroup^{n}$$The distribution function is −$$D\lgroup n\rgroup=\displaystyle\sum\limits_{i=0}^n P\lgroup i \rgroup=1-q^{n+1}$$Example Live Demo#include #include using namespace std; int main(){    const int nrolls = 10000; // number of rolls    const int nstars = 100; // maximum number of stars to distribute    default_random_engine generator;    geometric_distribution distribution(0.3);    int p[10]={};    for (int i=0; i

The Binomial Distribution is a discrete probability distribution Pp(n | N) of obtaining n successes out of N Bernoulli trails (having two possible outcomes labeled by x = 0 and x = 1. The x = 1 is success, and x = 0 is failure. Success occurs with probability p, and failure occurs with probability q as q = 1 – p.) So the binomial distribution can be written as$$P_{p}\lgroup n\:\arrowvert\ N\rgroup=\left(\begin{array}{c}N\ n\end{array}\right) p^{n}\lgroup1-p\rgroup^{N-n}$$Example Live Demo#include #include using namespace std; int main(){    const int nrolls = 10000; // number of rolls    const int nstars = 100; // ... Read More

A spanning tree is a subset of an undirected Graph that has all the vertices connected by minimum number of edges.If all the vertices are connected in a graph, then there exists at least one spanning tree. In a graph, there may exist more than one spanning tree.Minimum Spanning TreeA Minimum Spanning Tree (MST) is a subset of edges of a connected weighted undirected graph that connects all the vertices together with the minimum possible total edge weight. To derive an MST, Prim’s algorithm or Kruskal’s algorithm can be used. Hence, we will discuss Prim’s algorithm in this chapter.As we ... Read More