SSSP in DAGs (directed acyclic graphs)

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

Topological Sort Topological Sort: Ordering of vertices in a DAG based on precedence : if path then v i before v j 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)

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

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

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

Theorem 1: A directed G has a TS  G is a DAG Proof: – Assume G has TS and a cycle C. Let V i  C be the vertex with smallest index assigned by TS. There is then an edge (V j,V i ), with V j  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) = true Assume P(m) is true, Let G have m+1 vertices => source V 0. G\{V 0 } has TS V 1,V 2,…, V m => G has TS V 0,V 1,…, V m

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

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

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?

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.