2.5 bfs & dfs 02
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Graph Traversals Depth-First Search • Think Stack Breadth-First Search • Think Queue
Transcript of 2.5 bfs & dfs 02
Graph Traversals
Depth-First Search• Think Stack
Breadth-First Search• Think Queue
Overview
Goal To systematically visit the nodes of a graph
A tree is a directed, acyclic, graph (DAG) If the graph is a tree,
DFS is exhibited by preorder, postorder, and (for binary trees) inorder traversals
BFS is exhibited by level-order traversal
Depth-First Search
// recursive, preorder, depth-first searchvoid dfs (Node v) { if (v == null) return;
if (v not yet visited) visit&mark(v); // visit node before adjacent nodes
for (each w adjacent to v) if (w has not yet been visited) dfs(w);
} // dfs
Depth-First Search
// recursive, postorder, depth-first searchvoid dfs (Node v) { if (v == null) return;
tag(v); // mark v as having been considered for (each w adjacent to v) if (w has not yet been tagged) dfs(w);
visit(v); // postorder traversal: visit node after // adjacent nodes
} // dfs
Depth-First Search
// non-recursive, preorder, depth-first searchvoid dfs (Node v) { if (v == null) return;
push(v); while (stack is not empty) { pop(v); if (v has not yet been visited) mark&visit(v);
for (each w adjacent to v) if (w has not yet been visited) push(w); } // while
} // dfs
Depth-First Search
// non-recursive, postorder, depth-first searchvoid dfs (Node v) {
// Lex 20 (Do not need to submit)
} // dfs
Example
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Policy: Visit adjacent nodes in increasing index order
Preorder DFS: Start with Node 5
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Preorder DFS: Start with Node 5
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Push 5
Preorder DFS: Start with Node 5
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Pop/Visit/Mark 5
Preorder DFS: Start with Node 5
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Push 2, Push 1
Preorder DFS: Start with Node 5
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Pop/Visit/Mark 1
Preorder DFS: Start with Node 5
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Push 4, Push 2, Push 0
Preorder DFS: Start with Node 5
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Pop/Visit/Mark 0
Preorder DFS: Start with Node 5
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Push 7, Push 3
Preorder DFS: Start with Node 5
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Pop/Visit/Mark 3
Preorder DFS: Start with Node 5
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Push 2
Preorder DFS: Start with Node 5
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Pop/Mark/Visit 2
Preorder DFS: Start with Node 5
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Pop/Mark/Visit 7
Preorder DFS: Start with Node 5
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Push 6
Preorder DFS: Start with Node 5
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Pop/Mark/Visit 6
Preorder DFS: Start with Node 5
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Pop (don’t visit) 2
Preorder DFS: Start with Node 5
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Pop/Mark/Visit 4
Preorder DFS: Start with Node 5
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Pop (don’t visit) 2
Preorder DFS: Start with Node 5
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Done
Preorder DFS: Start with Node 5Note: edge (0,3) removed
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Depth-First SearchPolicy: Don’t push nodes twice
// non-recursive, preorder, depth-first searchvoid dfs (Node v) { if (v == null) return;
push(v); while (stack is not empty) { pop(v); if (v has not yet been visited) mark&visit(v);
for (each w adjacent to v) if (w has not yet been visited && not yet stacked) push(w); } // while
} // dfs
Preorder DFS (Don’t push nodes twice). Start with Node 5
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Postorder DFS: Start with Node 5
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Breadth-first Search
Ripples in a pond Visit designated node Then visited unvisited nodes a distance i away, where i = 1,
2, 3, etc. For nodes the same distance away, visit nodes in systematic
manner (eg. increasing index order)
Breadth-First Search
// non-recursive, preorder, breadth-first searchvoid bfs (Node v) { if (v == null) return;
enqueue(v); while (queue is not empty) { dequeue(v); if (v has not yet been visited) mark&visit(v);
for (each w adjacent to v) if (w has not yet been visited && has not been queued) enqueue(w); } // while
} // bfs
BFS: Start with Node 5
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BFS: Start with Node 5
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BFS: Node one-away
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BFS: Visit 1 and 2
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BFS: Nodes two-away
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BFS: Visit 0 and 4
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BFS: Nodes three-away
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BFS: Visit nodes 3 and 7
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BFS: Node four-away
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BFS: Visit 6
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BiconnectivityA connected undirected graph is biconnected if there are no vertices whose removal disconnects the rest of the graph.
Example :
If the nodes are computers and the edges are links, then if any computer goes down, network mail is unaffected, except, of course, at the down computer.
Biconnectivity If a graph is not biconnected, the vertices whose removal
would disconnect the graph are known as articulation points. These nodes are critical in many applications.
BiconnectivityThe graph below is not biconnected: C and D are articulation points. The removal of C would disconnect G, and the removal of D would disconnect E and F, from the rest of the graph
Biconnectivity• Depth-first search provides a linear-time algorithm to find all
articulation points in a connected graph.
1.Starting at any vertex, perform a depth-first search and number the nodes as they are visited: num(v).
2.For every vertex v in the depth-first spanning tree, compute the lowest-numbered vertex called low(v), that is reachable from v by taking zero or more tree edges and then possibly one back edge (in that order).
Biconnectivity The depth-first spanning tree shows the preorder number
first, and then the lowest-numbered vertex reachable
(For each node num/low is given in the tree)
Biconnectivity• The lowest-numbered vertex , low(v), can be efficiently
computed by performing a postorder traversal of the
depth-first spanning tree, and it is the minimum of
1)num(v)
2)the lowest num(w) among all back edges (v, w)
3)the lowest low(w) among all tree edges (v, w)
Biconnectivity: How to decide Articulation Points
The root is an articulation point if and only if it has more than one child (because if it has two children, removing the root disconnects nodes in different subtrees, and if it has only one child, removing the root merely disconnects the root)
Any other vertex v is an articulation point if and only if v has some child w such that low(w) ≥ num(v).Example:Vertex D is an articulation point
low(E)= 4 ≥ num(D)=4Vertex C is an articulation point
low(G)= 7 ≥ num(C)=3
How many squares can you create in this figure by connecting any 4 dots (the corners of a square must lie upon a grid dot?
TRIANGLES:
How many triangles are located in the image below?
There are 11 squares total; 5 small, 4 medium, and 2 large.
27 triangles. There are 16 one-cell triangles, 7 four-cell triangles, 3 nine-cell triangles, and 1 sixteen-cell triangle.
GUIDED READING
ASSESSMENT
Do the DFS and BFS for this graph
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DIFFERENT TRAVERSALS STARTING FROM VERTEX 1
PRE ORDER TRAVERSAL (ALSO CALLED DEPTH FIRST SEARCH)
1 2 3 4 5 6 7 POST ORDER TRAVERSAL
6 5 4 7 3 2 1 BREADTH FIRST SEARCH
1 2 4 3 5 6 7
DEPTH FIRST SPANNING TREE
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DEPTH FIRST SPANNING TREE
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DEPTH FIRST SPANNING TREE
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DEPTH FIRST SPANNING TREE
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DEPTH FIRST SPANNING TREE
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DEPTH FIRST SPANNING TREE
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DEPTH FIRST SPANNING TREE
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DEPTH FIRST SPANNING TREE
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