问题
In my algorithms class I've been told that a draw back of Adjacency Lists for graph representation is the O(n) look up time for iterating through the array of adjacent nodes corresponding to each node. I implement my adjacency list by using a HashMap that maps nodes to a HashSet of their adjacent nodes, wouldn't that only take O(1) look up time? Is there something I'm missing?
回答1:
As you know look up for value using key in HashMap is O(1). However, in adjacency list the value of the HashMap is also a list of its adjacent nodes. The main purpose of the adjacency list is to iterate the adjacent nodes. For example: graph traversal algorithms like DFS and BFS. In your case HashSet. Suppose number of elements in HashSet is n. Then for iterating all the elements even in HashSet is O(n).
So, total complexity would be O(1)+O(n).
Where O(1)= look up in HashMap
O(n)= iterate all the elements
Generally, Adjacency List is preferable for sparse graph because it is the graph with only a few edges. It means the number of adjacent elements in each node(key of HashMap) is less. So the look up for a element wont cost more.
回答2:
I implement my adjacency list by using a HashMap that maps nodes to a HashSet of their adjacent nodes, wouldn't that only take O(1) look up time? [emphasis mine]
Right — but "adjacency list" normally implies a representation as an array or a linked-list rather than a HashSet: in other words, adjacency lists are optimized for iterating over a vertex's neighbors rather than for querying if two vertices are neighbors.
回答3:
It may be possible to produce more time-efficient graph representations than adjacency lists, particularly for graphs where vertices vertex often have many edges.
With a map of vertices where each vertex contains a map of neighbor vertices and/or edge objects, we can look if nodes are connected in O(1) time by indexing a vertex id and then indexing a neighbor. That's potentially a big savings over an adjacency list where we might have to loop over many edges to find specific neighbors. Furthermore, a map-of-maps data structure can allow us to store arbitrary data in edge objects. That's useful for weighted graphs and features of actions/edges
来源：https://stackoverflow.com/questions/43024638/adjacency-list-with-o1-look-up-time-using-hashset