Graph Traversals

Graph Problems

• Runtime $\Theta(V + E)$
  • Each vertex visited exactly once, each visit = constant time
  • Each edge considered once
• Space $\Theta(V)$
  • Recursive call stack depth $\leq V$
  • Space of recursive algorithm = depth of call stack

Graph Traversals

• Level-order/BFS for vertices at same distance from source can be in any permutation
• If $\exists$ multiple paths to a vertex, BFS always visits based on closest path
• Traversals & Graph Problems

Topological Sorting

• Indegree 0 vertices → $\Theta(E + V)$, have to look at $V$ lists w/ total length $E$
• Only works if graph is acyclic → $\varnothing$ if cyclic b/c can't have circular dependencies
  • No indegree 0 vertices → cyclic → $\varnothing$
• Topological ordering only possible iff graph is directed acyclic graph (no directed cycles)
• Linearizes graph
• Graph Problems

Implementation

public class DepthFirstOrder {

    private boolean[] marked;
    private Stack<Integer> reversePostorder; // using stack analogous to reversing list b/c LIFO

    public DepthFirstOrder(Digraph G) {
        reversePostorder = new Stack<>();
        marked = new boolean[G.V()];
        for (int v = 0; v < G.V(); v++) {
            if (!marked[v]) {
                dfs(G, v);
            }
        }
    }

    private void dfs(Digraph G, int v) {
        marked[v] = true;
        for (int w : G.adj(v)) {
            if (!marked[w]) {
                dfs(G, w);
            }
        }
        reversePostorder.push(v); // after each DFS is done, "visit" vertex by pushing on stack
    }
}

• Works even when starting from vertices not indegree 0

Breadth First Search

• Finding Level-Order
• DFS written recursively has implicit fringe → recursive call stack of vertices