Topological Sort Carola Wenk 11/1/07 CS 3343 Analysis of - - PowerPoint PPT Presentation

CS 3343 Analysis of Algorithms 1 11/1/07

CS 33433 – Fall 2007

Topological Sort

Carola Wenk

CS 3343 Analysis of Algorithms 2 11/1/07

Paths, Cycles, Connectivity

Let G=(V,E) be a directed (or undirected) graph

• A path from v1 to vk in G is a sequence of vertices v1, v2,…,vk such that

(vi,v{i+1})∈E (or {vi,v{i+1}} ∈E if G is undirected) for all i∈{1,…,k-1}.

• A path is simple if all vertices in the path are distinct.
• A path v1, v2,…,vk forms a cycle if v1=vk and k≥2.
• A graph with no cycles is acyclic.
• An undirected acyclic graph is called a tree. (Trees do not have to

have a root vertex specified.)

• A directed acyclic graph is a DAG. (A DAG can have undirected

cycles if the direction of the edges is not considered.)

• An undirected graph is connected if every pair of vertices is connected

by a path. A directed graph is strongly connected if for every pair u,v∈V there is a path from u to v and there is a path from v to u.

• The (strongly) connected components of a graph are the equivalence

classes of vertices under this reachability relation.

CS 3343 Analysis of Algorithms 3 11/1/07

Topological Sort

Topologically sort the vertices of a directed acyclic graph (DAG):

• Determine f : V → {1, 2, …, |V|} such that (u, v) ∈ E

⇒ f(u) < f(v).

3 3 5 5 6 6 4 4 2 2 7 7 9 9 8 8 1 1 3 3 5 5 6 6 4 4 2 2 7 7 9 9 8 8 1 1

CS 3343 Analysis of Algorithms 4 11/1/07

Topological Sort Algorithm

• Store vertices in a priority min-queue, with the

in-degree of the vertex as the key

• While queue is not empty
• Extract minimum vertex v, and give it next number
• Decrease keys of all adjacent vertices by 1

3 3 5 5 6 6 4 4 2 2 7 7 9 9 8 8 1 1

2 2 1 1 3 1 1

CS 3343 Analysis of Algorithms 5 11/1/07

Topological Sort Algorithm

• Store vertices in a priority min-queue, with the

in-degree of the vertex as the key

• While queue is not empty
• Extract minimum vertex v, and give it next number
• Decrease keys of all adjacent vertices by 1

3 3 5 5 6 6 4 4 2 2 7 7 9 9 8 8 1 1

2 2 1 1 3 1 1 1 2 1 1

CS 3343 Analysis of Algorithms 6 11/1/07

Topological Sort Runtime

Runtime:

• O(|V|) to build heap + O(|E|) DECREASE-KEY ops

⇒ O(|V| + |E| log |V|) with a binary heap ⇒ O(|V| + |E|) with a Fibonacci heap

CS 3343 Analysis of Algorithms 7 11/1/07

Depth-First Search revisited

DFS_rec(G, v) visit v; d[v]=++time; //discover time for each w adjacent to v do if w is unvisited Add edge (v,w) to tree T DFS_rec(G,w) f[v]=++time; //finish time DFS(G=(V,E)) Mark all vertices in G as “unvisited” time=0; for each vertex v ∈ V do if v is unvisited DFS_rec(G,v)

CS 3343 Analysis of Algorithms 8 11/1/07

DFS Edge Classification

• Edge (u,v) in depth-first forest:
• Tree edge: v was discovered by by exploring edge (u,v)
• Edge (u,v) not in depth-first forest:
• Back edge: v is ancestor of u in depth-first forest
• Forward edge: v is descendant of u in depth-first forest
• Cross edge: Any other edge

CS 3343 Analysis of Algorithms 9 11/1/07

DFS-Based Topological Sort Algorithm

• Call DFS on the directed acyclic graph G=(V,E)

⇒ Finish time for every vertex

• Reverse the finish times (highest finish time

becomes the lowest finish time,…) ⇒ Valid function f : V → {1, 2, …, | V |} such that (u, v) ∈ E ⇒ f (u) < f (v). Runtime: O(|V|+|E|)

CS 3343 Analysis of Algorithms 10 11/1/07

DFS-Based Topological Sort

• Run DFS:

1 2 3 4 /5 /6 7 8 /9 /10 /11 13 14 15/16 /17 /18 /12

• Reverse finish times:

9 8 6 7 5 3 2 1 4

CS 3343 Analysis of Algorithms 11 11/1/07

Topological Sort Runtime

Runtime:

• O(|V|) to build heap + O(|E|) DECREASE-KEY ops

⇒ O(|V| + |E| log |V|) with a binary heap ⇒ O(|V| + |E|) with a Fibonacci heap

• DFS-based algorithm: O(|V| + |E|)