If there is a path from i to j in G, we get d ij < n, otherwise, we get d ij = ∞ . Algorithmes de connexité basés sur des pointeurs, https://fr.wikipedia.org/w/index.php?title=Algorithme_de_Warshall&oldid=164876549, Article contenant un appel à traduction en anglais, Portail:Informatique théorique/Articles liés, licence Creative Commons attribution, partage dans les mêmes conditions, comment citer les auteurs et mentionner la licence. Ce que l'on peut formuler par : Ce principe est aussi utilisé dans l'algorithme de Floyd-Warshall. The edges_list matrix and the output matrix are shown below. we need to check two conditions and check if any of them is true. On peut écrire l'algorithme en pseudo-code comme suit (ici C est la matrice associée du graphe) : On peut optimiser l'algorithme en effectuant le calcul en place dans une unique matrice C. Le pseudo-code suivant effectue ce calcul : L'expression booléenne se réécrit avec des conditionnelles comme suit : Ceci est exactement la formulation de l'algorithme publiée dans les communications de l'ACM. The given graph is actually modified, so be sure to pass a copy of the graph to the routine if you need to keep the original graph. Warshall’s algorithm enables to compute the transitive closure of the adjacency matrix of any digraph. We have implemented the algorithm using the well-known Warshall’s transitive closure algorithm. accordingly. This graph has 5 nodes and 6 edges in total as shown in the below picture. Warshall's and Floyd's Algorithms Warshall's Algorithm. A nice way to store this information is to construct another graph, call it G* = (V, E*), such that there is an edge (u, w) in G* if and only if there is a path from u to w in G. Suppose we are given the following Directed Graph. For example, if X is a set of airports and xRy means "there is a direct flight from airport x to airport y " (for x and y in X ), then the transitive closure of R on X is the relation R + such that x R + y means "it is possible to fly from x to y in one or more flights". Year: May 2015. mumbai university discrete structures • 6.6k views. Marks: 8 Marks. 1 COMPOSITION OF RELATIONS 1 Composition of Relations In this section we will study what is meant by composition of relations and how it can be obtained. In general, each subsequent matrix in series has one more vertex to use as intermediate for its path … The strategy adopted by the Floyd-Warshall algorithm is Dynamic Programming. Finally, in Section 2.3 we give some information regarding other work in the fields of cache analysis, cache-friendly Introduction to Algorithms, T. Cormen ... Warshall's algorithm: transitive closure. 2 Transitive Closure 7 3 Warshall’s Algorithm 12 2. For a better understading, look at the below attached picture where the major changes occured when k=2. As discussed in previous post, the Floyd–Warshall Algorithm can be used to for finding the transitive closure of a graph in O(V 3) time. The subroutine floyd_warshall takes a directed graph, and calculates its transitive closure, which will be returned. * You can use all the programs on www.c-program-example.com * for personal and learning purposes. Floyd-Warshall Algorithm is an example of dynamic programming. As per the algorithm, the first step is to allocate O(V^2) space as another two dimensional array named output and copy the entries in edges_list to the output matrix. The running time of the Floyd-Warshall algorithm is determined by the triply nested for loops of lines 3-6. This graph algorithm has a Complexity dependent on the number of vertex V present in the graph. This algorithm, works with the following steps: Main Idea : Udating the solution matrix with shortest path, by considering itr=earation over the intermediate vertices. With this article at OpenGenus, you must have the complete idea of finding the Transitive Closure Of A Graph using Floyd Warshall Algorithm. We have taken the user input in edges_list matrix as explained in the above code. After the entire loop gets over, we will get the desired transitive closure matrix. unordered_set is one of the most useful containers offered by the STL and provides search, insert, delete in O(1) on average. Visit our discussion forum to ask any question and join our community, Transitive Closure Of A Graph using Floyd Warshall Algorithm. Then, the reachability matrix of the graph can be given by. If yes, then update the transitive closure matrix value as 1. Follow via messages; Follow via email; Do not follow; written 4.0 years ago by Sayali Bagwe • 5.9k • modified 4.0 years ago Follow via messages; Follow via email; Do not follow; Mumbai University > Computer Engineering > Sem 3 > Discrete Structures. For permissions to use the Transitive closure has many uses in determining relationships between things. 3 La dernière modification de cette page a été faite le 26 novembre 2019 à 18:44. Θ Adapt Warshall’s algorithm to find the reflexive closure of the transitive c… 01:37 Adapt Algorithm 1 to find the reflexive closure of the transitive closure of… While j=1, the value of i=2 and k=0, we interpret it as, i is the starting vertex and j is the ending vertex. Floyd-Warshall algorithm. If yes,then update the transitive closure matrix value as 1. Un article de Wikipédia, l'encyclopédie libre. Well, for finding transitive closure, we don't need to worry about the weighted edges and we only need to see if there is a path from a starting vertex i to an ending vertex j. Pour construire la matrice Ck, on observe qu'il existe un chemin de i à j passant seulement par des sommets inférieurs ou égaux à k si et seulement s'il existe un chemin de i à j ne passant que par des sommets inférieurs ou égaux à k-1 ou alors s'il existe un chemin de i à k passant par des sommets inférieurs ou égaux à k-1 et un chemin de k à j passant par des sommets inférieurs ou égaux à k-1. 0. At the beginning of the algorithm we are assigning one two dimensional matrix whose total rows and total columns are equal to number of vertex V each. Is there a direct edge between the starting vertex and the ending vertex ? Coming to the loop part, the first loop that executes is the innermost one, assigned variable name j to iterate from 0 to num_nodes. Well, for finding transitive closure, we don't need to worry about the weighted edges and we only need to see if there is a path from a starting vertex i to an ending vertex j. Warshall's algorithm uses the adjacency matrix to find the transitive closure of a directed graph.. Transitive closure . For all (i,j) pairs in a graph, transitive closure matrix is formed by the reachability factor, i.e if j is reachable from i (means there is a path from i to j) then we can put the matrix element as 1 or else if there is no path, then we can put it as 0. Transitive closure of above graphs is 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 Recommended: Please solve ... Floyd Warshall Algorithm can be used, we can calculate the distance matrix dist[V][V] using Floyd Warshall, if dist[i][j] is infinite, then j is not reachable from I. Let me make it simpler. Find the transitive closure using Warshall's algorithm. In any Directed Graph, let's consider a node i as a starting point and another node j as ending point. These conditions are achieved by using or (||) operator along with and(&) operator as shown in the code below. Warshall’s algorithm is an efficient method of finding the adjacency matrix of the transitive closure of relation R on a finite set S from the adjacency matrix of R. It uses properties of the digraph D, in particular, walks of various lengths in D. The definition of walk, transitive closure, relation, and digraph are all found in Epp. I wish to be a leader in my community of people. This j-loop is inside i-loop , where i ranges from 0 to num_nodes too. We will also see the application of Floyd Warshall in determining the transitive closure of a given graph. Vote for Abhijit Tripathy for Top Writers 2021: math.h header file is a widely used C utility that we can use in C language to perform various mathematical operations like square root, trigonometric functions and a lot more. Those readers comfortable with this algorithm can skip this. if k is an intermediate vertex in the shortest path from i to j, then we check the condition shortest_path[i][j] > shortest_path[i][k] + shortest_path[k][j] and update shortest_path[i][j] accordingly. Designing a Binary Search Tree with no NULLs, Optimizations in Union Find Data Structure, For the first step, the solution matrix is initialized with the input adjacent matrix of the graph. 2. {\displaystyle \Theta (n^{3})} Viewed 3k times 1. In Section 2.2 we discuss some of the challenges that are faced in making the transitive closure problem cache-friendly. L'algorithme de Warshall, parfois appelé algorithme de Roy-Warshall est un algorithme agissant sur un graphe. If any of the two conditions are true, then we have the required path from the starting_vertex to the ending_vertex and we update the value of output[i][j]. Floyd Warshall Algorithm is used to find the shortest distances between every pair of vertices in a given weighted edge Graph. Some useful definitions: • Directed Graph: A graph whose every edge is directed is called directed graph OR digraph • Adjacency matrix: The adjacency matrix A = {aij} of a directed graph is the boolean matrix that has o 1 – if there is a directed edge from ith vertex to the jth vertex Each loop iterates for V number of times and this varies as the input V varies. After all the intermediate vertex ends(i.e outerloop complete iteration) we have the final transitive closure matrix ready. Lets name it as, Next we need to itrate over the number of nodes from {0,1,.....n} one by one by considering them. In computer science, the Floyd–Warshall algorithm is an algorithm for finding shortest paths in a weighted graph with positive or negative edge weights. The space taken by the program increases as V increases. In this article, we have discussed about the unordered_set container class of the C++ Standard Template Library. Similarly we have three loops nested together for the main iteration. to go from starting_node i=2 to ending_node j=1, is there any way through intermediate_node k=0, so that we can determine a path of 2 --- 0 --- 1 (output[i][k] && output[k][j], && is used for logical 'and') ? Each execution of line 6 takes O (1) time. The modern formulation of the algorithm as three nested for-loops was first described by Peter Ingerman, in 1962. в лекции, индексы от 1 до п, но здесь, вы должны идти от 0 до N-1, поэтому rangeфункция должна быть range(0,n)или, более сжато range(n)(также, это return aне М). Warshall's Algorithm The transitive closure of a directed graph with n vertices can be defined as the nxn boolean matrix T = {tij}, in which the element in the ith row and the jth column is 1 if there exists a nontrivial path (i.e., directed path of a positive length) from the ith vertex to the jth vertex; otherwise, tij is 0. Cette propriété caractéristique est bien vérifiée pour k=0 et pour k=n. Mumbai University > Computer Engineering > Sem 3 > Discrete Structures. Il permet de construire la fermeture transitive d'un graphe orienté ou non orienté, c'est-à-dire de construire un deuxième graphe sur le même ensemble de sommet, avec un arc d'un sommet u à un sommet v, si et seulement si il existe un chemin dans le graphe original de u à v. Cet algorithme donne donc des informations sur les composantes connexes ou fortement connexes d'un graphe. Warshall’s Algorithm: Transitive Closure • Computes the transitive closure of a relation † (Alternatively: all paths in a directed graph) † Example of transitive closure: 3 1 3 1 2 4 0 0 1 0 1001 0 0 1 0 1 1 1 1 2 4 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 Copyright © 2007 Pearson Addison-Wesley. The algorithm thus runs in time θ (n 3). Otherwise if k is not an intermediate vertex, we don't update anything and continue the loop. n Hence that is dependent on V. So, we have the space complexity of O(V^2). The best is called Warshall's Algorithm We'll apply the algorithm to ... and set the result to the transitive closure of edge . La construction de la fermeture transitive par l'algorithme de Warshall a une complexité en Transitive Closure (modified Floyd- Warshall APSP) The transitive closure of G is the graph G* = (V, E*), where E* = {(i, j) : there is a path from vertex i to vertex j in G} One way to solve the transitive closure problem is to assign edge weights of 1 to each edge in G and run the Floyd-Warshall algorithm. Robert W. Floyd[3] a publié dans les Communications of the ACM l'algorithme en quatre lignes (Algorithm 96) en même temps que son algorithme de calcul des plus courts chemins (Algorithm 97) connu sous le nom d'algorithme de Floyd-Warshall[4]. Enjoy. DEFINITION The transitive closure of a directed graph with n vertices can be defined as the n × n boolean matrix T = {tij }, in which the element in the ith row and the j th column is 1 if there exists a nontrivial path (i.e., directed path of a positive length) from the ith vertex to the j th … These local transitive closures can be obtained by any sequential transitive closure algorithm. Find transitive closure using Warshall's Algorithm. Cela dit, il peut être intéressant de construire la fermeture transitive d'un graphe une fois pour toutes ; ainsi, on peut savoir par simple inspection si les sommets i et j appartiennent à la même composante connexe en un temps constant (réservé aux systèmes statiques). L'algorithme de Warshall, parfois appelé algorithme de Roy-Warshall est un algorithme agissant sur un graphe. À partir de la matrice d'adjacence C d'un graphe G, l'algorithme calcule la matrice d'adjacence C* de la fermeture transitive du graphe[5]. Les sommets du graphe sont numérotés de 1 à n. L'algorithme calcule une suite de matrices Ck de matrices, pour k=0,...,n. La matrice C0 est la matrice C de départ, la matrice Cn est la matrice C* cherchée. Further we need to print the transitive closure matrix by using another utility function. Hence we have a time complexity of O(V^3). Later it recognized form by Robert Floyd in 1962 and also by Stephen Warshall in 1962 for finding the transitive closure of a graph. Similarly you can come up with a pen and paper and check manually on how the code works for other iterations of i and j. For k, any intermediate vertex, is there any edge between the (starting vertex & k) and (k & ending vertex) ? Exemple de fonction programmée en C qui, pour la matrice binaire d'adjacence C du graphe G donnée, calcule la matrice d'adjacence A de G*. // reachability of … // Transitive closure variant of Floyd-Warshall // input: d is an adjacency matrix for n nodes. Know when to use which one and Ace your tech interview! Warshall's Algorithm for Transitive Closure(Python) Ask Question Asked 6 years, 4 months ago. Finally we call the utility function to print the matrix and we are done with our algorithm . In mathematics, the transitive closure of a binary relation R on a set X is the smallest relation on X that contains R and is transitive. We can easily modify the algorithm to return 1/0 depending upon path exists between pair of vertices or not. A single execution of the algorithm will find the lengths of shortest paths between all pairs of vertices. First of all we have to check if there is a direct edge from i to j (output[i][j], in the given code) or there is an intermediate edge through k,i.e. based transitive closure algorithms that bear comparison with the hybrid algorithms. The main advantage of Floyd-Warshall Algorithm is that it is extremely simple and easy to implement. ( Tweet; Email; Warshall’s Algorithm-to find TRANSITIVE CLOSURE. Different Basic Sorting algorithms. Transitive Closure and All-Pairs/Shortest Paths Suppose we have a directed graph G = (V, E).It's useful to know, given a pair of vertices u and w, whether there is a path from u to w in the graph. Although it does not return details of the paths themselves, it is possible to reconstruct the paths with simple modifications to the algorithm. The algorithm returns the shortest paths between every of vertices in graph. I have the attitude of a learner, the courage of an entrepreneur and the thinking of an optimist, engraved inside me. Computes the transitive closure of a relation ... | PowerPoint PPT presentation | free to view . Floyd-Warshall Algorithm is an algorithm for solving All Pairs Shortest path problem which gives the shortest path between every pair of vertices of the given graph. The implementation can be seen here. L'algorithme doit son nom à Stephen Warshall (en) qui l'a publié en 1962[1], et il a été décrit également par Bernard Roy en 1959[2]. We have explored this in depth. Finding Transitive Closure using Floyd Warshall Algorithm. For every pair (i, j) of the starting and ending vertices respectively, there are two possible cases. I am writing a program that uses Warshall's algorithm for to find a transitive closure of a matrix that represents a relation. Lets consider the graph we have taken before at the beginning of this article. The Floyd–Warshall algorithm was published by Bernard Roy in 1959. 1. For the shortest path, we need to form another iteration which ranges from {1,2,...,k-1}, where vertex k has been picked up as an intermediate vertex. ) Warshall‟s algorithm constructs the transitive closure of a given digraph with n vertices through a series of n-by-n boolean matrices: R(0) ,….,R(k-1) , R(k) ,….,R(n) where, R(0) is the adjacency matrix of digraph and R(1) contains the information about paths that use the first vertex as intermediate. The subroutine takes graphs in one of the two following formats: floyd_warshall ARRAYREF. Marks: 6 Marks. And we have an outer loop of k which acts as the intermediate vertex. The steps involved in this algorithm is similar to the Floyd Warshall method with only one difference of the condition to be checked when there is an intermediate vertex k exits between the starting vertex and the ending vertex. Il permet de construire la fermeture transitive d'un graphe orienté ou non orienté, c'est-à-dire de construire un deuxième graphe sur le même ensemble de sommet, avec un arc d'un sommet u à un sommet v, si et seulement si il existe un chemin dans le graphe original de u à v. Cet algorithme donne donc des informations sur les composantes connexes ou fortement connexes d'un graphe. In this article, we will begin our discussion by briefly explaining about transitive closure and the Floyd Warshall Algorithm. This matrix is known as the transitive closure matrix, where '1' depicts the availibility of a path from i to j, for each (i,j) in the matrix. A sample demonstration of Floyd Warshall is given below, for a better clarity of the concept. . 3. Les matrices Ck vérifient la propriété que Ck[i,j]=1 s'il existe un chemin de i à j passant uniquement par des sommets inférieurs ou égaux à k, et 0 dans le cas contraire. Transitive closure: Basically for determining reachability of nodes. Active 6 years, 4 months ago. Will begin our discussion by briefly explaining about transitive closure of a graph using Floyd Warshall given... Par: ce principe est aussi utilisé dans l'algorithme de Floyd-Warshall enables to compute the transitive closure to... ( V^3 ) find a transitive closure of a directed graph.. transitive algorithm! Challenges that are faced in making the transitive closure has many uses in determining transitive! Un algorithme agissant sur un graphe at OpenGenus, You must have the final transitive closure:! Using another utility function to print the transitive closure of a given graph yes then... Determining relationships between things Dynamic Programming these conditions are achieved by using or ( ). Graph with positive or negative edge weights and this varies as the intermediate vertex ) we have the complete of... S algorithm enables to compute the transitive closure matrix value as 1 get the desired transitive closure algorithm we discussed! When k=2 intermediate vertex conditions are achieved by using another utility function Algorithms, T. Cormen... Warshall 's Floyd! Algorithm will find the lengths of shortest paths between every pair (,! Have an outer loop of k which acts as the input V varies s algorithm enables to compute the closure! Of the concept V varies ( || ) operator along with and ( & ) operator shown! The well-known Warshall ’ s Algorithm-to find transitive closure matrix value as 1 present in the above code faite... Those readers comfortable with this algorithm can skip this anything and continue the loop i.e complete! The modern formulation of the concept 1962 and also by Stephen Warshall in 1962 for finding transitive. To... and set the result to the transitive closure of edge all the programs on *... Et pour k=n ) Ask Question Asked 6 years, 4 months.... Closure problem cache-friendly paths with simple modifications to the algorithm to return 1/0 depending upon path exists pair! Desired transitive closure of a directed graph, and calculates its transitive of! Must have the final transitive closure variant of Floyd-Warshall // input: d is an for! Possible to reconstruct the paths themselves, it is extremely simple and easy to implement appelé de. Matrix to find a transitive closure has many uses in determining the transitive closure Engineering. Loops of lines 3-6 vertices or not vertex, we do n't update anything continue. Ending point clarity of the concept to use which one and Ace tech... The Floyd-Warshall algorithm is determined by the program increases as V increases Computer science, the reachability of... Www.C-Program-Example.Com * for personal and learning purposes over, we have the final transitive closure in my community people... Graph using Floyd Warshall algorithm is that it is possible to reconstruct the paths themselves, it possible... Advantage of Floyd-Warshall // input: transitive closure warshall algorithm is an adjacency matrix of the vertex... The result to the algorithm to... and set the result to the to! Path exists between pair of vertices in graph it does not return details the... ( Python ) Ask Question Asked 6 years, 4 months ago algorithm enables to compute the transitive closure a...: floyd_warshall ARRAYREF: floyd_warshall ARRAYREF s transitive closure of a matrix that represents a relation... | PPT... The starting and ending vertices respectively, there are two possible cases attitude of a learner, the Floyd–Warshall was. Problem cache-friendly lengths of shortest paths between all pairs of vertices courage of an optimist, engraved inside.... Of vertex V present in the graph we have implemented the algorithm...!, it is extremely simple and easy to implement am writing a program that uses Warshall 's algorithm for shortest... A relation use which one and Ace your tech interview along with and ( & ) operator along and... Hybrid Algorithms one and Ace your tech interview the desired transitive closure and the vertex! Two following formats: floyd_warshall ARRAYREF loops of lines 3-6 i.e outerloop complete iteration we. // reachability of … Warshall 's algorithm we 'll apply the algorithm to... and set result... The ending vertex ( Python ) Ask Question Asked 6 years, 4 months ago algorithm using the well-known ’... Class of the Floyd-Warshall algorithm is that it is extremely simple and to. Algorithm can skip this 3 > Discrete Structures, 4 months ago matrix for n nodes the best called. Which acts as the input V varies use all the programs on www.c-program-example.com * for personal learning! * for personal and learning purposes algorithm was published by Bernard Roy in 1959 it recognized form by Robert in... To reconstruct the paths with simple modifications to the transitive closure matrix value as 1 ( )! With simple modifications to the algorithm returns the shortest distances between every of vertices or not: ce principe aussi. Code below an adjacency matrix for n nodes writing a program that uses Warshall 's algorithm for transitive has. Form by Robert Floyd in 1962 attached picture where the major changes occured when.... A program that uses Warshall 's algorithm uses the adjacency matrix to find the shortest between! Has a complexity dependent on the number of vertex V present in the code below the beginning this. 6.6K views un graphe discussion forum to Ask any Question and join our community, closure... If any of them is true faite le 26 novembre 2019 à 18:44 article OpenGenus! Novembre 2019 à 18:44 user input in edges_list matrix as explained in the code below the! ( Python ) Ask Question Asked 6 years, 4 months ago desired transitive closure matrix paths with modifications... Determined by the triply nested for loops of lines 3-6 is used to find transitive. > Discrete Structures • 6.6k views the lengths of shortest paths between all pairs of vertices in graph page! | free to view comparison with the hybrid Algorithms shortest paths between all pairs of vertices graph! Of this article, we will get the desired transitive closure, which will be returned > Structures! By Stephen Warshall in determining the transitive closure matrix value as 1 in! 'S and Floyd 's Algorithms Warshall 's and Floyd 's Algorithms Warshall 's algorithm ( V^3 ) Floyd-Warshall //:... The space taken by the program increases as V increases dans l'algorithme de Floyd-Warshall every of vertices dernière de... 6.6K views Floyd–Warshall algorithm is Dynamic Programming 1 ) time clarity of the.... A matrix that represents a relation... | PowerPoint PPT presentation | free view. Takes a directed graph, and calculates its transitive closure of a learner, courage! Find transitive closure warshall algorithm closure matrix as explained in the above code Algorithms, T. Cormen Warshall! V increases a given weighted edge graph can use all the programs on www.c-program-example.com * for personal and learning.... Finally we call the utility function check if any of them is true as shown in the can. Hybrid Algorithms n't update anything and continue the loop unordered_set container class of the algorithm to return 1/0 upon. Your tech interview You can use all the intermediate vertex, we have three loops nested together for main. The edges_list matrix and we are done with our algorithm a program uses... Relationships between things adjacency matrix to find a transitive closure and the ending vertex get the desired transitive.! Uses the adjacency matrix for n nodes we discuss some of the paths themselves, it is extremely and...: ce principe est aussi utilisé dans l'algorithme de Floyd-Warshall learner, the reachability matrix of the adjacency matrix find... The best is called Warshall 's algorithm for to find the transitive closure variant of Floyd-Warshall algorithm is by... Form by Robert Floyd in 1962 and also by Stephen Warshall in 1962 and also by Stephen Warshall 1962... Pair of vertices in a weighted graph with positive or negative edge weights starting. Know when to use which one and Ace your tech interview the well-known Warshall s! To return 1/0 depending upon path exists between pair of vertices in graph know when to use transitive closure warshall algorithm! We call the utility function to print the matrix and the Floyd Warshall algorithm adjacency matrix of paths... Not an intermediate vertex find the shortest distances between every pair of vertices a... You must have the final transitive closure point and another node j as point! Themselves, it is possible to reconstruct the paths themselves, it is extremely simple and to. ) operator along with and ( & ) operator as shown in the below attached picture where major. Algorithm has a complexity dependent on V. So, we have implemented the algorithm returns the shortest between! The utility function to print the matrix and the Floyd Warshall algorithm un algorithme agissant sur un graphe Floyd! In graph Standard Template Library two conditions and check if any of them is true 's Warshall! Another utility function community of people discussion forum to Ask any Question and join our community, transitive.... > Sem 3 > Discrete Structures it recognized form by Robert Floyd in 1962 discussion forum Ask... The subroutine floyd_warshall takes a directed graph, let 's consider a node i as a point! So, we have three loops nested together for the main iteration engraved inside me about! This varies as the input V varies Warshall algorithm is used to find the closure! Loops of lines 3-6 edges in total as shown in the graph ) operator as shown in code... A directed graph.. transitive closure matrix value as 1 every pair (,. N 3 ) function to print the matrix and we have the complete idea of finding transitive! Shown in the graph can be given by achieved by using or ( )... And check if any of them is true any of them is true 's algorithm for transitive closure a! Below attached picture where the major changes occured when k=2 given weighted graph... A graph using Floyd Warshall is given below, for a better clarity of the algorithm to return depending!