Shortest Path
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Outline and Reading (§6.4) Reachability (§6.4.1)
Directed Graphs
BOS
ORD
Directed DFS Strong connectivity
JFK
Transitive closure (§6.4.2)
SFO
LAX
DFW
The Floyd-Warshall Algorithm
MIA
Directed Acyclic Graphs (DAG’s) (§6.4.4) Directed Graphs
Topological Sorting
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Directed Graphs
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Digraphs
Digraph Properties C
A digraph is a graph whose edges are all directed
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Directed Graphs
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We can specialize the traversal algorithms (DFS and BFS) to digraphs by traversing edges only along their direction In the directed DFS algorithm, we have four types of edges
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Directed Graphs
discovery edges back edges forward edges cross edges
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A directed DFS starting at a vertex s determines the vertices reachable from s
The good life
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Directed Graphs
Directed DFS
Scheduling: edge (a,b) means task a must be completed before b can be started ics22
A
If G is simple, m < n(n-1). If we keep in-edges and out-edges in separate adjacency lists, we can perform listing of of the sets of in-edges and out-edges in time proportional to their size.
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Digraph Application
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Each edge goes in one direction: Edge (a,b) goes from a to b, but not b to a.
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one-way streets flights task scheduling
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A graph G=(V,E) such that
Short for “directed graph”
Applications
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Directed Graphs
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Strong Connectivity
Reachability
Each vertex can reach all other vertices
DFS tree rooted at v: vertices reachable from v via directed paths E
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Directed Graphs
Strong Connectivity Algorithm a
G:
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If there’s a w not visited, print “no”. d
Let G’ be G with edges reversed. Perform a DFS from v in G’.
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G’:
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Running time: O(n+m).
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Directed Graphs
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B C A
We can perform DFS starting at each vertex
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{f,d,e,b}
Directed Graphs
Computing the Transitive Closure
Transitive Closure
{a,c,g}
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Directed Graphs
Maximal subgraphs such that each vertex can reach all other vertices in the subgraph Can also be done in O(n+m) time using DFS, but is more complicated (similar to biconnectivity).
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If there’s a w not visited, print “no”. Else, print “yes”.
Given a digraph G, the transitive closure of G is the digraph G* such that G* has the same vertices as G if G has a directed path from u to v (u ≠ v), G* has a directed edge from u to v The transitive closure provides reachability information about a digraph
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Strongly Connected Components
Pick a vertex v in G. Perform a DFS from v in G.
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Directed Graphs
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C A
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If there's a way to get from A to B and from B to C, then there's a way to get from A to C.
O(n(n+m))
Alternatively ... Use dynamic programming: the Floyd-Warshall Algorithm
G* 11
Directed Graphs
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Floyd-Warshall Transitive Closure
Floyd-Warshall’s Algorithm
Idea #1: Number the vertices 1, 2, …, n. Idea #2: Consider paths that use only vertices numbered 1, 2, …, k, as intermediate vertices: Uses only vertices numbered 1,…,k (add this edge if it’s not already in)
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Uses only vertices numbered 1,…,k-1 k
Uses only vertices numbered 1,…,k-1
Directed Graphs
Algorithm FloydWarshall(G) Input digraph G Output transitive closure G* of G i←1 for all v ∈ G.vertices() G0=G denote v as vi Gk has a directed edge (vi, vj) i←i+1 if G has a directed path from G0 ← G vi to vj with intermediate for k ← 1 to n do vertices in the set {v1 , …, vk} Gk ← Gk − 1 We have that Gn = G* for i ← 1 to n (i ≠ k) do for j ← 1 to n (j ≠ i, k) do In phase k, digraph Gk is computed from Gk − 1 if Gk − 1.areAdjacent(vi, vk) ∧ Gk − 1.areAdjacent(vk, vj) Running time: O(n3), if ¬Gk.areAdjacent(vi, vj) assuming areAdjacent is O(1) Gk.insertDirectedEdge(vi, vj , k) (e.g., adjacency matrix) return Gn
Floyd-Warshall’s algorithm numbers the vertices of G as v1 , …, vn and computes a series of digraphs G0, …, Gn
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Floyd-Warshall Example
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Directed Graphs
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Floyd-Warshall, Iteration 1
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Directed Graphs
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Floyd-Warshall, Iteration 2
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Directed Graphs
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Floyd-Warshall, Iteration 3
BOS ORD
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ORD
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v4 JFK
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SFO
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v7 BOS
JFK
LAX
v7 BOS
v6
SFO
DFW
LAX
v3
v1
MIA
DFW
v3 MIA
v5 Directed Graphs
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Directed Graphs
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Floyd-Warshall, Iteration 4
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Floyd-Warshall, Iteration 5
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ORD
JFK
SFO
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JFK
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DFW
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Directed Graphs
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Floyd-Warshall, Iteration 6
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Directed Graphs
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Floyd-Warshall, Conclusion
BOS
JFK
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SFO
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ORD
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LAX
v7 BOS
BOS
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Directed Graphs
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DAGs and Topological Ordering A directed acyclic graph (DAG) is a digraph that has no directed cycles A topological ordering of a digraph is a numbering v1 , …, vn of the vertices such that for every edge (vi , vj), we have i < j Example: in a task scheduling digraph, a topological ordering a task sequence that satisfies the v2 precedence constraints Theorem A digraph admits a topological v1 ordering if and only if it is a DAG Directed Graphs
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Topological Sorting wake up
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A typical student day
2 study computer sci.
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DAG G D
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Number vertices, so that (u,v) in E implies u < v
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Directed Graphs
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8 write c.s. program
9 make cookies for professors
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Directed Graphs
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Algorithm for Topological Sorting
Simulate the algorithm by using depth-first search
Note: This algorithm is different than the one in Goodrich-Tamassia
Algorithm topologicalDFS(G, v) Input graph G and a start vertex v of G Output labeling of the vertices of G in the connected component of v setLabel(v, VISITED) for all e ∈ G.incidentEdges(v) if getLabel(e) = UNEXPLORED w ← opposite(v,e) if getLabel(w) = UNEXPLORED setLabel(e, DISCOVERY) topologicalDFS(G, w) else {e is a forward or cross edge} Label v with topological number n n←n-1
Algorithm topologicalDFS(G) Input dag G Output topological ordering of G n ← G.numVertices() for all u ∈ G.vertices() setLabel(u, UNEXPLORED) for all e ∈ G.edges() setLabel(e, UNEXPLORED) for all v ∈ G.vertices() if getLabel(v) = UNEXPLORED topologicalDFS(G, v)
Method TopologicalSort(G) H←G // Temporary copy of G n ← G.numVertices() while H is not empty do Let v be a vertex with no outgoing edges Label v ← n n←n-1 Remove v from H
Running time: O(n + m). How…? Directed Graphs
Topological Sorting Algorithm using DFS
O(n+m) time. 25
Topological Sorting Example
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Topological Sorting Example
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Topological Sorting Example
Directed Graphs
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Topological Sorting Example
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Directed Graphs
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Topological Sorting Example
Topological Sorting Example
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Topological Sorting Example
Directed Graphs
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Topological Sorting Example 3
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Topological Sorting Example
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Topological Sorting Example
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