minimum-spanning-tree

I am trying to ensure a global correct orientation of my normals. I.e make sure they point outwards everywhere. This is my method: I have a coordinate list x y c n . At each point in coordinate list I create edges from the point to its 5 nearest neighbours. I Weight each edge with ( 1-n1.n2 ). Compute a minimal spanning tree. Traverse this tree in depth first order. Start traversing at the point with the greatest z value, force the normal of this point to be orientated towards the positive z axis. Assign orientation to subsequent points that are consistent with their preceeding point, for

Given a weighted, connected, simple undirected graph G with weights of only 1 and 2 on each edge, find the MST of G in O(V+E). Any ideas? Sorry for the phrasing of the question, I tried translating it as best as I could. In Prim's algorithm you need a way of storing active edges such that you can access and delete the edge with lowest weight. Normally there are a wide range of weights and some kind of heap data structure is used to store the edges. However, in this case, the weights are either 1 or 2, and so you can simply store the edges in 2 separate lists, one for edges with weight 1, and

I have a graph represented with adjacency lists and his MST represented by parent array. My problem is that I need to delete an edge from the graph and update parent array. I've already manage to do the cases when: the edge doesn't exist; the edge is in graph but not in MST (MST doesn't change); and the edge is the only path from two nodes(in this case I return null, because the graph is not connected). What can I do when the edge is in MST and the edge in graph is in a cycle? I need to do this in O(n+m) complexity . I write the costs of edges with red color. Just search the minimum-distance

A graph can have many different Minimum Spanning Trees (MSTs), but can different MSTs have different sets of edge weights? For example, if an MST uses edge weights {2,3,4,5}, must every other MST have edge weights {2,3,4,5}, or can some other MST use a different collection of weights? What gave me the idea is property that a graph has no unique MST only if its edge weights are different. The sets must have the same weight. Here's a simple proof: suppose they don't. Let's let T1 and T2 be MSTs for some graph G with different multisets of edge weights. Sort those edges into ascending order of