We will need to be able to grab the minimum value from our heap. My source node looks at all of its neighbors and updates their provisional distance from the source node to be the edge length from the source node to that particular neighbor (plus 0). Note that I am doing a little extra — since I wanted actual node objects to hold data for me I implemented an array of node objects in my Graphclass whose indices correspond to their row (column) number in the adjacency matrix. We can keep track of the lengths of the shortest paths from K to every other node in a set S, and if the length of S is equal to N, we know that the graph is connected (if not, return -1). Remember when we pop() a node from our heap, it gets removed from our heap and therefore is equivalent in logic to having been “seen”. I then make my greedy choice of what node should be evaluated next by choosing the one in the entire graph with the smallest provisional distance, and add E to my set of seen nodes so I don’t re-evaluate it. Destination node: j. The only idea I have come up with would consist on turning to infinity the last edge towards my destination vertex if the overall distance lies below N. However, this would make this edge no longer available for use for the other paths that would arrive to destination vertex. It fans away from the starting node by visiting the next node of the lowest weight and continues to … Now for our last method, we want to be able to update our heap’s values (lower them, since we are only ever updating our provisional distances to lower values) while maintaining the heap property! This is the best place to expand your knowledge and get prepared for your next interview. [ provisional_distance, [nodes, in, hop, path]] , our is_less_than lambda could have looked like this: lambda a,b: a[0] < b[0], and we could keep the second lambda at its default value and pass in the nested array ourselves into decrease_key. If the next node is a neighbor of E but not of A, then it will have been chosen because its provisional distance is still shorter than any other direct neighbor of A, so there is no possible other shortest path to it other than through E. If the next node chosen IS a direct neighbor of A, then there is a chance that this node provides a shorter path to some of E's neighbors than E itself does. We need our heap to be able to: To accomplish these, we will start with a building-block which will be instrumental to implement the first two functions. We want to find the shortest path in between a source node and all other nodes (or a destination node), but we don’t want to have to check EVERY single possible source-to-destination combination to do this, because that would take a really long time for a large graph, and we would be checking a lot of paths which we should know aren’t correct! Because each recursion of our method performs a fixed number of operations, i.e. # 2. I think you are right. Each element at location {row, column} represents an edge. P.S. Dijkstra's algorithm is an iterative algorithm that provides us with the shortest path from one particular starting node (a in our case) to all other nodes in the graph.To keep track of the total cost from the start node to each destination we will make use of the distance instance variable in the Vertex class. If we implemented a heap with an Adjacency Matrix representation, we would not be changing the asymptotic runtime of our algorithm by using a heap! 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