dijkstra最短路径算法的时间复杂度是否取决于所使用的数据结构？

Question

One way to store the graph is to implement nodes as structures, like

struct node {
int vertex; node* next; 
};

where vertex stores the vertex number and next contains link to the other node.

Another way I can think of is to implement it as vectors, like

vector<vector< pair<int,int> > G;

Now, while applying Dijkstra's algorithm for shortest path, we need to build priority queue and other required data structures and so as in case 2 (vector implementation). Will there be any difference in complexity in above two different methods of applying graph? Which one is preferable?

EDIT: In first case, every node is associated with a linked list of nodes which are directly accessible from the given node. In second case, G.size() is the number of vertices in our graph G[i].size() is the number of vertices directly reachable from vertex with index i G[i][j].first is the index of j-th vertex reachable from vertex i G[i][j].second is the length of the edge heading from vertex i to vertex G[i][j].first

Answer 1

Both are adjacency list representations . If implemented correctly, that would be expected to result in the same time complexity. You'd get a different time complexity if you use an adjacency matrix representation .

In more detail - this comes down to the difference between an array ( vector ) and a linked-list . When all you're doing is iterating through the entire collection (ie the neighbours of a vertex), as you do in Dijkstra's algorithm, this takes linear time ( O(n) ) regardless of whether you're using an array or linked-list.

The resulting complexity for running Dijkstra's algorithm, as noted on Wikipedia , would be
O(|E| log |V|) with a binary heap in either case.