Breadth-first search is just Dijkstra’s algorithm with all edge weights equal to 1. Dijkstra’s algorithm is conceptually breadth-first search that respects edge costs. The process for exploring the graph is structurally the same in both cases.
Finding all possible paths is a hard problem, since there are exponential number of simple paths. Even finding the kth shortest path [or longest path] are NP-Hard. One possible solution to find all paths [or all paths up to a certain length] from s to t is BFS, without keeping a visited set, or for … Read more
I hope this is helpful to anybody having trouble understanding computational time complexity for Breadth First Search a.k.a BFS. Queue graphTraversal.add(firstVertex); // This while loop will run V times, where V is total number of vertices in graph. while(graphTraversal.isEmpty == false) currentVertex = graphTraversal.getVertex(); // This while loop will run Eaj times, where Eaj is … Read more
Depth first search is more memory efficient than breadth first search as you can backtrack sooner. It is also easier to implement if you use the call stack but this relies on the longest path not overflowing the stack. Also if your graph is directed then you have to not just remember if you have … Read more
You should have look at http://en.wikipedia.org/wiki/Breadth-first_search first. Below is a quick implementation, in which I used a list of list to represent the queue of paths. # graph is in adjacent list representation graph = { ‘1’: [‘2’, ‘3’, ‘4’], ‘2’: [‘5’, ‘6’], ‘5’: [‘9′, ’10’], ‘4’: [‘7’, ‘8’], ‘7’: [’11’, ’12’] } def bfs(graph, … Read more
Dijkstra allows assigning distances other than 1 for each step. For example, in routing the distances (or weights) could be assigned by speed, cost, preference, etc. The algorithm then gives you the shortest path from your source to every node in the traversed graph. Meanwhile BFS basically just expands the search by one “step” (link, … Read more
Technically, Breadth-first search (BFS) by itself does not let you find the shortest path, simply because BFS is not looking for a shortest path: BFS describes a strategy for searching a graph, but it does not say that you must search for anything in particular. Dijkstra’s algorithm adapts BFS to let you find single-source shortest … Read more
Your sum v1 + (incident edges) + v2 + (incident edges) + …. + vn + (incident edges) can be rewritten as (v1 + v2 + … + vn) + [(incident_edges v1) + (incident_edges v2) + … + (incident_edges vn)] and the first group is O(N) while the other is O(E).
(I’m assuming that this is just some kind of thought exercise, or even a trick homework/interview question, but I suppose I could imagine some bizarre scenario where you’re not allowed any heap space for some reason [some really bad custom memory manager? some bizarre runtime/OS issues?] while you still have access to the stack…) Breadth-first … Read more
These two terms differentiate between two different ways of walking a tree. It is probably easiest just to exhibit the difference. Consider the tree: A / \ B C / / \ D E F A depth first traversal would visit the nodes in this order A, B, D, C, E, F Notice that you … Read more