\n Coding tests are a great way to evaluate programming skills. Today, we will look at how to solve the Minimum Spanning Tree (MST)<\/strong> problem. \n Let’s solve the problem of finding a minimum spanning tree given the edges that represent the graph below. \n A graph consists of vertices (nodes) and edges. \n Kruskal’s Algorithm first sorts the edges based on their weights. \n To check whether a cycle occurs when adding edges, we use the Union-Find<\/strong> data structure. \n If no cycle occurs, we select the edge; otherwise, we ignore the edge. \n Now we will combine the implementations of all steps to ultimately find the MST. \n The results obtained through the implemented algorithm are as follows.
\n This problem is applicable in various fields, especially in problems related to computer networks.
\n There are several algorithms to construct a minimum spanning tree, but particularly Kruskal’s Algorithm<\/strong> and Prim’s Algorithm<\/strong> are widely used.\n<\/p>\nProblem Description<\/h2>\n
\n Each edge is assigned a weight, and we need to find a connected graph where the total weight is minimized.\n<\/p>\nProblem Definition<\/h3>\n
\n Input:\n [(1, 2, 4), (1, 3, 1), (2, 3, 2), (2, 4, 5), (3, 4, 8), (1, 4, 3)]\n Output:\n List of edges in the minimum spanning tree: [(1, 3, 1), (2, 3, 2), (1, 4, 3), (2, 4, 5)]\n Sum of minimum weights: 11\n<\/pre>\n
Problem Solving Process<\/h3>\n
Step 1: Understanding Graph Data Structure<\/h4>\n
\n Here, edges are provided in the form of tuples as `(start vertex, end vertex, weight)`.
\n We need to construct the graph using this data.\n<\/p>\nStep 2: Sorting Edges<\/h4>\n
\n We sort the given list of edges in ascending order to be able to select the edges with the minimum weights.\n<\/p>\nStep 3: Cycle Detection Using Union-Find Structure<\/h4>\n
\n This data structure has two main functions:\n<\/p>\n\n
\n We define the basic Union-Find class and its methods required to implement the algorithm.\n<\/p>\nStep 4: Implementing MST<\/h4>\n
\n Below is an example of the Kruskal’s algorithm implemented in JavaScript.\n<\/p>\n\n class UnionFind {\n constructor(size) {\n this.parent = Array.from({length: size}, (_, index) => index);\n this.rank = Array(size).fill(1);\n }\n\n find(node) {\n if (this.parent[node] !== node) {\n this.parent[node] = this.find(this.parent[node]);\n }\n return this.parent[node];\n }\n\n union(node1, node2) {\n const root1 = this.find(node1);\n const root2 = this.find(node2);\n if (root1 !== root2) {\n if (this.rank[root1] > this.rank[root2]) {\n this.parent[root2] = root1;\n } else if (this.rank[root1] < this.rank[root2]) {\n this.parent[root1] = root2;\n } else {\n this.parent[root2] = root1;\n this.rank[root1]++;\n }\n }\n }\n\n connected(node1, node2) {\n return this.find(node1) === this.find(node2);\n }\n }\n\n function kruskal(edges, numVertices) {\n edges.sort((a, b) => a[2] - b[2]); \/\/ Sort edges by weight\n const uf = new UnionFind(numVertices);\n const mst = [];\n let totalWeight = 0;\n\n for (const [u, v, weight] of edges) {\n if (!uf.connected(u - 1, v - 1)) {\n uf.union(u - 1, v - 1);\n mst.push([u, v, weight]);\n totalWeight += weight;\n }\n }\n\n return { mst, totalWeight };\n }\n\n const edges = [\n [1, 2, 4],\n [1, 3, 1],\n [2, 3, 2],\n [2, 4, 5],\n [3, 4, 8],\n [1, 4, 3]\n ];\n const numVertices = 4;\n\n const { mst, totalWeight } = kruskal(edges, numVertices);\n console.log(\"List of edges in the minimum spanning tree:\", mst);\n console.log(\"Sum of minimum weights:\", totalWeight);\n<\/pre>\nStep 5: Analyzing Results<\/h4>\n
\n We analyze whether the algorithm works correctly by checking the list of edges in the minimum spanning tree and the total weight.
\n As a result, we successfully formed the minimum spanning tree based on the given edges.\n<\/p>\n