Breadth-first search for unweighted shortest path: basic idea

• By distance between two nodes u,v we mean the number of edges on the shortest path between u and v . Now:
  
• Start at the start vertex s . It is at distance 0 from itself, and there are no other nodes at distance 0
  
• Consider all the nodes adjacent to s . These all are at distance at most 1 from s (maybe less than 1, if s has an edge to itself; but then we would have found a shorter path already) and there are no other nodes at distance 1
  
• Consider all the nodes adjacent to the nodes adjacent to s . These are all at distance at most 2 from s (maybe less than 2; but then we would have found a shorter path already) and there are no other nodes at distance 2
  
• ... and so on. In this breadth-first search, as soon as we visit a node in the graph, we know the shortest path from s to it; and so by the time we have visited all the nodes in the graph, we know the shortest path from s to each of them