A Collection of Graph Programming Interview Questions Solved by Dr Antonio Gulli

By Dr Antonio Gulli

A suite of Graph Programming Interview Questions Solved in C++

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If the graph has weights associated to the edges, then the minimum spanning tree is a spanning tree with minimal sum of the edges’ weight. Solution One way to compute the MST is to adopt a classical greedy algorithm due to Prim[1]. At the beginning the tree contains a single vertex, chosen arbitrarily from the graph. Then the minimum weight edge is selected among those connecting a vertex in the tree with a vertex not being in the tree. This edge is added to the tree and the previous step is repeated until all vertices are in the tree.

Solution The algorithm proposed is due to Hopcroft and Tarjan[5]. The idea is to run a DFS while maintaining: the depth of each vertex in the DFS tree For each node , the lowest depth of neighbours of all descendants of in the DFS tree, called the lowpoint. The lowpoint of can be computed after visiting all descendants of as the minimum of the depth of , the depth of all neighbors of (excluding the parent of in the DFS tree) and the lowpoint of all children of in the DFS tree. A non-root vertex is an articulation point, if and only if there is a child of such that This property can be tested, once the depth-first search is returned from every child of v.

Then the minimum weight edge is selected among those connecting a vertex in the tree with a vertex not being in the tree. This edge is added to the tree and the previous step is repeated until all vertices are in the tree. priority << std::endl; } } } std::cout << "MST" << std::endl; for (MatrixGraph::NodeID i = 1; i < numNodes; ++i) { std::cout << parent[i] << '-' << i << " weight=" << g(parent[i], i) << std::endl; } } \ This particular implementation uses an adjacency matrix and therefore its complexity is .

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