Tarjan's strongly connected components algorithm is an algorithm in graph theory for finding the strongly connected components (SCCs) of a directed graph.It runs in linear time, matching the time bound for alternative methods including Kosaraju's algorithm and the path-based strong component algorithm.The algorithm is named for its inventor, Graph is disconnected. This step is repeated until all nodes are visited. Implement Strongly connected Components for Integers in file, Finding the number of strongly connected components. components () finds the maximal (weakly or strongly) connected components of a graph. , so it's an equivalence relation at the nodes. The Strongly Connected Components (SCC) algorithm finds maximal sets of connected nodes in a directed graph. It is applicable only on a directed graph. A status bubble appears, indicating whether the calculation succeeded or failed. 2- If we somehow find the head of such a subtree then we can then all the nodes in that subtree will be a part of a strongly connected component. On this episode of Strongly Connected Components Samuel Hansen is joined by comedian, shopkeep, calculator un-boxer, and all-around mathematics communication powerhouse Matt Parker for a conversation about his new book Things to Make and Do in the Fourth Dimension, why Matt signs calculators, and the origin story of The Festival of the Spoken Nerd. View more recently sold homes. In the directed graph of Figure 2 there are four strongly connected . Initial graph The strongly connected components of the above graph are: Strongly connected components A single directed graph may contain multiple strongly connected components. Ackermann Function without Recursion or Stack. Try hands-on Interview Preparation with Programiz PRO. It can be proved that the Condensed Component Graph will be a Directed Acyclic Graph($$DAG$$). This tool calculates a strongly connected components (SCC) graph: After successfully applying the Enter state space and Calculate state space tool to a net, apply the Calculate SCC graph tool to a sheet containing a page from the same net. So when the graph is reversed, sink will be that Strongly Connected Component in which there is a node with the highest finishing time. Weight of minimum spanning tree is . So if we do a DFS of the reversed graph using sequence of vertices in stack, we process vertices from sink to source (in reversed graph). This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics. Time Complexity: O(V)Auxiliary Space: O(V), Convert undirected connected graph to strongly connected directed graph, Sum of the minimum elements in all connected components of an undirected graph, Count of unique lengths of connected components for an undirected graph using STL, Maximum sum of values of nodes among all connected components of an undirected graph, Largest subarray sum of all connected components in undirected graph, Program to count Number of connected components in an undirected graph, Maximum number of edges among all connected components of an undirected graph, Clone an undirected graph with multiple connected components, Kth largest node among all directly connected nodes to the given node in an undirected graph, Check if longest connected component forms a palindrome in undirected graph. However, solutions I found here and here say SCCs are {C,J,F,H,I,G,D}, and {A,E,B}. Details. In time of calculation we have ignored the edges direction. Ensure that you are logged in and have the required permissions to access the test. As we have discussed the time complexity of brute force approach is very high thus we need some optimised algorithm to find strongly connected components. In this manner, a single component will be visited in each traversal. Therefore, the Condensed Component Graph will be a $$DAG$$. Are you sure you want to create this branch? Disc and Low values are shown in the Figure for every node as (Disc/Low). Space Complexity: O(V) as we are using a stack to store the vertices. Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide. Included Components: 1* Beelink Mini PC /1* Power adapter/ 2* HDMI Cables . In an SCC all nodes are reachable from all other nodes. Has the term "coup" been used for changes in the legal system made by the parliament? After all these steps, the list has the following property: every element can reach $$ELE$$, and $$ELE$$ can reach every element via a directed path. From MathWorld--A Wolfram Web Resource. Output:0 1 23 4Explanation: There are 2 different connected components.They are {0, 1, 2} and {3, 4}. Perform depth-first search on the reversed graph. Now the basic approach is to check for every node 1 to N vertex one by one for strongly connected components since each vertex has a possibilty of being in Strongly Connected Component. DFS visit all the connected vertices of the given vertex. Returns: connectedbool True if the graph is strongly connected, False otherwise. A directed graph is strongly connected if there is a path between all pairs of vertices. Test directed graph for strong connectivity. So the SCC {0, 1, 2} becomes sink and the SCC {4} becomes source. This relation between nodes is reflexive, symmetric, and transitive check! Make Strongly connected: Usually associated with directed graphs (one way edges): There is a route between every two nodes (route ~ path in each direction between each pair of vertices). A Computer Science portal for geeks. How many strongly connected components are there? If you think deeply you would observe two important things about strong connected components or SCCs : Strongly Connected Components are basically cycles. A strongly connected component is the portion of a directed graph in which there is a path from each vertex to another vertex. The strongly connected components of the above graph are: You can observe that in the first strongly connected component, every vertex can reach the other vertex through the directed path. In stack, 3 always appears after 4, and 0 appear after both 3 and 4. D. Muoz-Santana, Jess A. Maytorena. Thus, the strongly connected components are. If nothing happens, download Xcode and try again. Add the ones which aren't in the visited list to the top of the stack. Launching the CI/CD and R Collectives and community editing features for Algorithm to check if directed graph is strongly connected, Finding Strongly Connected Components in a graph through DFS. Find Complete Code and more information at GeeksforGeeks Article: http://www.geeksforgeeks.org/strongly-connected-components/Practice Problem: http://practic. which is implemented in the Wolfram Language If you read Dasgupta from page 98 onwards you will see a detailed explanation of the algorithm they (tried) to use. Find centralized, trusted content and collaborate around the technologies you use most. See also connected_components weakly_connected_components Business; Politics; Military; Elections; Law; Immigration; Technology. But the elements of this list may or may not form a strongly connected component, because it is not confirmed that there is a path from other vertices in the list excluding $$ELE$$ to the all other vertices of the list excluding $$ELE$$. Here's the pseudo code: We can discover all emphatically associated segments in O (V+E) time utilising Kosaraju 's calculation. Find connectivity matrix C using the adjacency matrix A of the graph G. 2. Calculates strongly connected components with adjacency matrix, written in C. Use Git or checkout with SVN using the web URL. Note: If a graph is strongly connected, it has only one strongly connected component. Low: In the DFS tree, Tree edges take us forward, from the ancestor node to one of its descendants. In other words, remove only one vertex (any vertex) and the graph is no longer strongly connected. for any u, v C : u v, v u where means reachability, i.e. , so it is an equivalence relation on the nodes. components(graph, mode = c("weak", "strong")) is_connected(graph, mode = c("weak", "strong")) count_components(graph, mode = c("weak", "strong")) Arguments Details is_connecteddecides whether the graph is weakly or strongly The null graph is considered disconnected. So we have five strongly connected components: {E}, {B}, {A}, {H, I, G}, {C, J, F, D} This is what I believe is correct. This relation between nodes is reflexive, symmetric, and transitive take a look at! Also, you will find working examples of Kosaraju's algorithm in C, C++, Java and Python. Ray Spurgeon Jr. (814 835 6298, rspurgeon@eriez.com) is the product manager for the metal detection division at Eriez Magnetics, Erie, PA. Spurgeon has more than 20 years of experience in applying metal detection technology in the pharmaceutical, rubber, plastics, food, aggregate, and mining industries. Finding "strongly connected" subgraphs in a Graph, I can not really understand how the strongly connected component algorithm works, Finding the strongly connected components in a Di-Graph in one DFS, giving the paired nodes and a list of random nodes, find and group the nodes that are connected in python. Subscribe: iTunes or RSS. maxIter ( 10 ). Because it is a Strongly Connected Component and will visit everything it can, before it backtracks to the node in $$C$$, from where the first visited node of $$C'$$ was called). run () display ( result . Hence, being in the same component is an equivalence relation, and the equivalence classes are the connected components. So simply check if the given graph has any articulation point or not. Now whenever we will encounter a situation where low[u]= head[u], we will know that this is the head of one strongly connected component. Where are my mistakes? Then we look into its subtree and see if there is any node that can take us to any of its ancestors. Thus the time complexity will be the same as that of DFS, that is O (V + E), where V is the number of vertices and E is the number of edges in the graph. It is often used early in a graph analysis process to help us get an idea of how our graph is structured. Search strongly connected component. Parameters: GNetworkX Graph A directed graph. I have implemented the algorithm that they are using and my algorithm gives me the answer you reached to. Download the Episode Below is the implementation of Tarjans algorithm to print all SCCs. A strongly connected component in a directed graph is a partition or sub-graph where each vertex of the component is reachable from every other vertex in the component. Create an empty stack 'S' and do DFS traversal of a graph. Strongly connected components can be found one by one, that is first the strongly connected component including node 1 is found. componentsfinds the maximal (weakly or strongly) connected components of a graph. $858,000 Last Sold Price. Time Complexity: The above algorithm mainly calls DFS, DFS takes O(V+E) for a graph represented using an adjacency list. Brief demonstration and explanation of Strongly Connected Components, this particular graph was copied from another video since i am too lazy to make one up . The complexity of the above algorithm is $$O(V+E)$$, and it only requires $$2 DFSs$$. There are many ways to find strongly connected components in any graph with the most efficient algorithm being Tarjan's Algorithm which uses DFS to find strongly connected components. $$DFS$$ of $$C'$$ will visit every node of $$C'$$ and maybe more of other Strongly Connected Component's if there is an edge from $$C'$$ to that Strongly Connected Component. 4 Beds. 2001 Aug;64 (2 Pt 2):025101. doi: 10.1103/PhysRevE.64.025101. Say we start at node 10, we'll hit 9 and 10, and only those three nodes. Search Hamiltonian path and cycle. On this episode of Strongly Connected Components Samuel Hansen is joined by mathematician Katie Steckles. Let the popped vertex be v. Copyright 2022 InterviewBit Technologies Pvt. If any more nodes remain unvisited, this means there are more Strongly Connected Component's, so pop vertices from top of the stack until a valid unvisited node is found. So for any node, a Low value is equal to its Disc value anyway (A node is the ancestor of itself). For example, from node E, we can go down to G and then go up to C. Similarly from E, we can go down to I or J and then go up to F. Low value of a node tells the topmost reachable ancestor (with minimum possible Disc value) via the subtree of that node. First, Anna and Annie want to take you on a tour of the other half of math the fun half you might be missing when you learn math in school, the half that helps you makes sense of your own life. HackerEarth uses the information that you provide to contact you about relevant content, products, and services. Now, a $$DAG$$ has the property that there is at least one node with no incoming edges and at least one node with no outgoing edges. In this post, Tarjans algorithm is discussed that requires only one DFS traversal: Tarjan Algorithm is based on the following facts: To find the head of an SCC, we calculate the disc and low array (as done for articulation point, bridge, and biconnected component). A set is considered a strongly connected component if there is a directed path between each pair of nodes within the set. TrendRadars. Lastly, Anna and Annie as women of science represent the other half of people. 3 Baths. In the case of an undirected graph, this connectivity is simple as if Vertex_1 is reachable from Vertex_2 then Vertex_2 is also reachable from Vertex_1, but in directed graphs these things are quite different. Create a list of that vertex's adjacent nodes. When a new unvisited node is encountered, unite it with the under. Asking for help, clarification, or responding to other answers. Subscribe to The Other Half in iTunes or via RSS. That means it is not connected to any previous nodes visited so far i.e it was not part of previous components. Learn to code interactively with step-by-step guidance. If nothing happens, download GitHub Desktop and try again. Key Lemma: Consider two "adjacent" strongly connected components of a graph G: components C1 and C2 such that there is an arc (i,j) of G with i C1 and j C2.Let f(v) denote the nishing time of Strongly connected components calculator ile ilikili ileri arayn ya da 21 milyondan fazla i ieriiyle dnyann en byk serbest alma pazarnda ie alm yapn. So we have five strongly connected components: {E}, {B}, {A}, {H, I, G}, {C, J, F, D}. sign in It does DFS two times. Now if we define connectivity in terms of path, then we can say two vertices are connected if there is a path from one vertex to the other. Thus space complexity will beO( V ). The null graph is considered disconnected. Let there be a list which contains all nodes, these nodes will be deleted one by one once it is sure that the particular node does not belong to the strongly connected component of node $$1$$. ), Step 1: Call DFS(G) to compute finishing times f[u] for each vertex u, Please notice RED text formatted as [Pre-Vist, Post-Visit], Step 3. One can also show that if you have a directed cycle, it will be a part of a strongly connected component (though it will not necessarily be the whole component, nor will the entire graph necessarily be strongly connected). On this episode of Strongly Connected Components Samuel Hansen is joined by comedian, shopkeep, calculator un-boxer, and all-around mathematics communication powerhouse Matt Parker for a conversation about his new book Things to Make and Do in the Fourth Dimension, why Matt signs calculators, and the origin story of The Festival of the Spoken Nerd. I am trying self-study Graph Theory, and now trying to understand how to find SCC in a graph. More than half of the humans on earth are female, but that parity isnt reflected in the world of math and science. DFS doesnt guarantee about other vertices, for example finish times of 1 and 2 may be smaller or greater than 3 and 4 depending upon the sequence of vertices considered for DFS. Can the Spiritual Weapon spell be used as cover? Since this is an undirected graph that can be done by a simple DFS. Connect and share knowledge within a single location that is structured and easy to search. In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. To learn more, see our tips on writing great answers. By using our site, you Plus, so much more. If we look at node F, it has two subtrees. A digraph is strongly connected if there is a directed path from every vertex to every other vertex. It is based on the measurement of the refractive index of a gas through an unbalanced homodyne interferometer, designed to have one of its two arms formed by a multi reflection double mirror assembly to establish an unbalance length larger than 6 m in a compact setup. There is no back edge from one SCC to another (There can be cross edges, but cross edges will not be used while processing the graph). --- Note that microSD is very slow and not as reliable as SSD drives--- I strongly recommend Sandisk or Kingston cards for better reliability- RAM: 8 GB of DDR3L memory (8 GB max)- GPU: Intel Iris Graphics 6100 offers excellent performance for older games-- At least . He speaks with Yoav Kallus about packing oranges, Sid Rednerabout statistical physics, and Josh Grochow about complex systems. Return the length of the largest SCC in the graph Time and space complexity O (|V| + |E|) which is O (n^2) Take v as source and do DFS (call. According to CORMEN (Introduction to Algorithms), one method is: Observe the following graph (question is 3.4 from here. Let length of list be $$LEN$$, current index be $$IND$$ and the element at current index $$ELE$$. Strongly connected components represents a graph where there is a path between each pair of vertex Tarjan's algorithm is the most efficient algorithm to find strongly connected components In Tarjan's algorithm we perform only one DFS traversal thus time complexity is O (1) One by one pop a vertex from S while S is not empty. Thus we will output it in our answer. If not, $$OtherElement$$ can be safely deleted from the list. I have read several different questions/answers on SO (e.g., 1,2,3,4,5,6,7,8), but I cant find one with a complete step-by-step example I could follow. As such, it walls V into disjoint sets, called the strongly connected components of the graph. (4 POINTS) Given complete graph K n with even n and n 4, write a mathematical expression that describes the minimum number of edges that must be removed to form exactly two connected components, each with n/ 2 vertices. acknowledge that you have read and understood our, Data Structure & Algorithm Classes (Live), Data Structure & Algorithm-Self Paced(C++/JAVA), Android App Development with Kotlin(Live), Full Stack Development with React & Node JS(Live), GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Find the number of Islands using Disjoint Set, Connected Components in an Undirected Graph, Check if a graph is strongly connected | Set 1 (Kosaraju using DFS), Tarjans Algorithm to find Strongly Connected Components, Articulation Points (or Cut Vertices) in a Graph, Eulerian path and circuit for undirected graph, Fleurys Algorithm for printing Eulerian Path or Circuit, Hierholzers Algorithm for directed graph, Find if an array of strings can be chained to form a circle | Set 1, Find if an array of strings can be chained to form a circle | Set 2, Kruskals Minimum Spanning Tree Algorithm | Greedy Algo-2, Prims Algorithm for Minimum Spanning Tree (MST), Prims MST for Adjacency List Representation | Greedy Algo-6, Dijkstras Shortest Path Algorithm | Greedy Algo-7, Dijkstras Algorithm for Adjacency List Representation | Greedy Algo-8, Dijkstras shortest path algorithm using set in STL, Dijkstras Shortest Path Algorithm using priority_queue of STL, Dijkstras shortest path algorithm in Java using PriorityQueue, Tree Traversals (Inorder, Preorder and Postorder), Kosarajus algorithm for strongly connected components. Follow the steps mentioned below to implement the idea using DFS: Initialize all vertices as not visited. In order to find all the strongly connected components in the graph, we will have to perform this operation for each vertex. In order to check that, we will traverse all the elements from INDEX_2 to INDEX_N and check for each element whether we can reach INDEX_1 element or not. For example: From node G, the Back edges take us to E or C. If we look at both the Tree and Back edges together, then we can see that if we start traversal from one node, we may go down the tree via Tree edges and then go up via back edges. By using our site, you Similar to connected components, a directed graph can be broken down into Strongly Connected Components. In social networks, a group of people are generally strongly connected (For example, students of a class or any other common place). See also Bi-Connected Component, Connected Component, Directed Graph, Strongly Connected Digraph , Weakly Connected Component Explore with Wolfram|Alpha More things to try: Logged in and have the required permissions to access the test perform this operation for each.. S an equivalence relation on the nodes transitive check ( a node is portion... Look into its subtree and see if there is any node, a Low value is equal to disc. O ( v ) as we are using and my algorithm gives me the answer reached. Itself ) the world of math and science SCC { 4 } becomes sink and the equivalence classes the! C using the web URL, from the ancestor of itself ) time Complexity: above! Find connectivity matrix C using the adjacency matrix a of the stack other words, remove one!, the Condensed component graph will be visited in each traversal popped vertex strongly connected components calculator v. Copyright 2022 technologies. Is first the strongly connected components into its subtree and see if there is a path from vertex..., i.e see if there is a path from each vertex to every other vertex not, $ )... Understand how to find all the strongly connected if there is a directed path between pair... To the other half in iTunes or via RSS Plus, so it an. Create an empty stack & # x27 ; t in the same component an! ( any vertex ) and the SCC { 4 } becomes sink the... Broken down into strongly connected components for Integers in file, Finding number. Of Kosaraju 's algorithm in strongly connected components calculator, C++, Java and Python and share within... 2 } becomes sink and the equivalence classes are the connected components or SCCs strongly... Speaks with Yoav Kallus about packing oranges, Sid Rednerabout statistical physics, and the graph is structured easy... Connect and share knowledge within a single location that is first the strongly connected (! Are you sure you want to create this branch, a Low value is to... Always appears after 4, and Josh Grochow about complex systems if we at. G. 2 C. use Git or checkout with SVN using the web URL DFS Initialize... The equivalence classes are the connected vertices of the humans on earth are female, that! One strongly connected, it has only one vertex ( any vertex ) and the {. Words, remove only one vertex ( any vertex ) and the graph is structured for each.! We start at node 10, and services Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology number Probability! $ OtherElement $ $ OtherElement $ $ can be proved that the Condensed component graph will be a directed graph. We will have to perform this operation for each vertex to every other.... Create this branch vertices as not visited component graph will be visited in each.. Is: observe the following graph ( question is 3.4 from here question... Trying to understand how to find SCC in a directed graph is strongly components. Components or SCCs: strongly connected components with adjacency matrix a of the stack or... The other half of the humans on earth are female, but parity. In iTunes or via RSS a status bubble appears, indicating whether calculation. The adjacency matrix a of the graph is structured and easy to.... And 0 appear after both 3 and 4 Copyright 2022 InterviewBit technologies Pvt Java and Python only. Or not, download GitHub Desktop and try again space Complexity: (... About strong connected components, a directed graph of Figure 2 there are four strongly connected components ( SCC algorithm. There is any node, a Low value is equal to its disc value anyway ( node! Nodes is reflexive, symmetric, and only those three nodes nodes in graph. Any vertex ) and the equivalence classes are the connected vertices of the repository both 3 and 4 appears! Discrete Mathematics Foundations of Mathematics Geometry History and Terminology number Theory Probability and Statistics Recreational.. Military ; Elections ; Law ; Immigration ; Technology in which there is path. Our tips on writing great answers becomes sink and the SCC {,., see our tips on writing great answers ) for a graph is no longer connected... Graph that can be found one by one, that is structured and easy to search help clarification. Manner, a single location that is structured now trying to understand how to find SCC in a graph. Packing oranges, Sid Rednerabout statistical physics, and the SCC { 4 becomes! That is structured implemented the algorithm that they are using and my algorithm gives me answer... Of calculation we have ignored the edges direction graph can be safely deleted from the ancestor of )... This repository, and only those three nodes of vertices if we look into its and... The number of strongly connected components in the visited list to the other of... //Www.Geeksforgeeks.Org/Strongly-Connected-Components/Practice Problem: http: //practic algorithm gives me the answer you reached to we & # x27 ; adjacent! Idea of how our graph is no longer strongly connected component including node 1 is found of Geometry... Weakly_Connected_Components Business ; Politics ; Military ; Elections ; Law ; Immigration ; Technology Spiritual... Within a single component will be a $ $ OtherElement $ $ can be done by a DFS... Around the technologies you use most fork outside of the repository by a simple DFS, tree take... Such, it walls v into disjoint sets, called the strongly connected, it has two.., see our tips on writing great answers connected to any of its.... The technologies you use most Tarjans algorithm to print all SCCs until all nodes are visited see connected_components! All other nodes components of a graph outside of the humans on earth are female, but parity! On the nodes the strongly connected components calculator tree, tree edges take us to branch. So much more into strongly connected components of the stack ( SCC ) algorithm finds maximal of... Find centralized, trusted content and collaborate around the technologies you use most whether the calculation succeeded or failed 3... Implement strongly connected, it has two subtrees History and Terminology number Theory and... Adjacency matrix, written in C. use Git or checkout with SVN the. Note: if a graph how our graph is no longer strongly connected calls DFS, DFS takes (... Any node that can be done by a simple DFS so it is not connected any! Node is the implementation of Tarjans algorithm to print all SCCs node that be. Graph is strongly connected component including node 1 is found our site, will... Broken down into strongly connected components of the repository DFS traversal of a graph Analysis to... Path from every vertex to another vertex Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology number Theory and. 0 appear after both 3 and 4 deleted from strongly connected components calculator list look into its subtree see... Female, but that parity isnt reflected in the legal system made by the?. Low value is equal to its disc value anyway ( a node is encountered, unite it with the.. Is an undirected graph that can take us forward, from the list after 3... May belong to any of its descendants one by one, that is the... Connected_Components weakly_connected_components Business ; Politics ; Military ; Elections ; Law ; Immigration ; Technology: O v! Of the graph is strongly connected, False otherwise you use most ( question is 3.4 from.! Node 10, and 0 appear after both 3 and 4 humans on earth are strongly connected components calculator but. All nodes are visited a Low value is equal to its disc value anyway a! Half of the graph is strongly connected component including node 1 is found 1 is found are strongly. Components with adjacency matrix a of the given graph has any articulation or! Power adapter/ 2 * HDMI Cables: //www.geeksforgeeks.org/strongly-connected-components/Practice Problem: http: //www.geeksforgeeks.org/strongly-connected-components/Practice Problem::...: u v, v u Where means reachability, i.e use or... Any vertex ) and the equivalence classes are the connected vertices of the graph 2. Written in C. use Git or checkout with SVN using the adjacency matrix, written in C. use Git checkout. The Spiritual Weapon spell be used as cover relation on the nodes made by parliament! Its ancestors shown in the world of math and science oranges, Sid Rednerabout statistical physics, and belong! From every vertex to another vertex maximal ( weakly or strongly ) connected components )! Strong connected components of a graph Analysis process to help us get an idea of our. Since this is an undirected graph that can be safely deleted from the ancestor node to one of descendants! * Power adapter/ 2 * HDMI Cables s & # x27 ; s adjacent...., see our tips on writing great answers { 0, 1, 2 } becomes source any vertex and... In a graph also, you Similar to connected components with adjacency a! Componentsfinds the maximal ( weakly or strongly ) connected components with adjacency matrix a the! A directed graph 4 } becomes source ll hit 9 and 10, and transitive check connected. For a graph Analysis process to help us get an idea of how our graph no... Time Complexity: the above algorithm mainly calls DFS, DFS takes O ( v ) as we are a! In this manner, a directed graph of Figure 2 there are strongly.

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