SSSP in DAGs (directed acyclic graphs)

1. SSSP in DAGs (directed acyclic graphs)

2. DFS (depth first search) DFS(vertex v) { v.visited = TRUE; for each w adjacent to v do if(!w.visited) then dfs(w); } • If G=(V,E) not (strongly) connected then it may have DFS forest

3. Topological Sort Topological Sort: Ordering of vertices in a DAG based on precedence : if path then vi before vj in ordering • not unique: • V  V : indegree(v): #edges(u,v)  E • Can use DFS to obtain TS order • Given directed graph G, is it a DAG? • Can use TS to answer it ( cannot find vertex with indegree 0)

4. TS algorithm • Output (number) v V with indegree 0 • (remove v and its outgoing edges) • Repeat until no vertex left

5. TS algorithm • How to find v  V with indgree(v)=0 • Use queue: • Each time incoming edge at uV is removed, decrease indegree(u) • If indegree(u) is 0, place u in queue • O(|V|+|E|) time • Init: • Find v  V with indegree(v)=0 in O(|V|) time • Place in queue

6. DAG and TS • Lemma 1: A DAG has a sink(outdegree=0) and a source(indegree=0) • Proof: • Let P be the longest path in G, • Claim: u is a source • If not, (w,u). If w∉P => P’ = {(w,u)} U P is longer than P. If wp => cycle: • A similar argument shows v is a sink

7. Theorem 1: A directed G has a TS G is a DAG • Proof: • Assume G has TS and a cycle C. Let ViC be the vertex with smallest index assigned by TS. There is then an edge (Vj,Vi), with VjC and j>I => contradiction with TS assignment. Now Assume G acyclic. Use induction: • Define P(n): a DAG with n vertices has a TS • P(1) = trueAssume P(m) is true, Let G have m+1 vertices => source V0. G\{V0} has TS V1 ,V2 ,…, Vm => G has TS V0 ,V1 ,…, Vm

8. SSSP in DAG (cont.) • Use vertex selection rule: select in TS order • SP well defined, even if negative weight edges ( negative weight cycle cannot exist)DAG-SP(G,w,s) Topologically sort vertices of G Init-Single-Source ( G, s) For each u ∈V, in TS order do For each v ∈Adj[u] do Relax(u,v,w) • O(|V|+|E|) time

9. SSSP in DAG (cont.) • Critical path in a DAG: longest path through the DAG • Can find one by: • (1) negating edge weights and running DAG-SP, or • (2) run DAG-SP with modifications: • Replace “∞” by “- ∞” in init-single-source • Replace “>” by “<“ in RELAX • Note: Longest simple path in unweighted graph is NP-Complete

10. BFS( breadth first search) • Visit all nodes at same level before going to next level; use queue for this • Queue_init();id=0;visited[u]=0, uV; Bfs(vV){ enqueue(v); While(!queue_empty()){ u=dequeue();visited[u]=++id; (need 3 values) for each wadj[u] do if(visited[w]==0){enqueue(w);visited[w]=-1} } } How many levels in the queue at a time?

11. Application:Unweighted SPs • δ(s,v) : #edges on the path (all edges have weight 1) • Use BFS: need min # edges to reach each uV from s. • Process vertices by layers: • Process those that can be reached with1 edge • Process those that can be reached with2 edge, etc. • Use a queue: • Check d[v]: if d[v]= ∞ , place in queue.