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COMP 250
Lecture 29
graph traversal
- Nov. 15/16, 2017
Today
- Recursive graph traversal
- depth first
- Non-recursive graph traversal
- depth first
- breadth first
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graph traversal Nov. 15/16, 2017 1 Today Recursive graph - - PowerPoint PPT Presentation
graph traversal Nov. 15/16, 2017 1 Today Recursive graph - - PowerPoint PPT Presentation
COMP 250 Lecture 29 graph traversal Nov. 15/16, 2017 1 Today Recursive graph traversal depth first Non-recursive graph traversal depth first breadth first 2 Heads up! There were a few mistakes in the slides for Sec. 001
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Heads up! There were a few mistakes in the slides for Sec. 001 for today’s
- lecture. So if you are following
the lecture recordings and using these (corrected) slides, then you will notice some differences.
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depthfirst__Tree (root){ if (root is not empty){ root.visited = true // “preorder” for each child of root depthfirst__Tree( child ) } }
Recall: tree traversal (recursive)
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Graph traversal (recursive)
Need to specify a starting vertex. Visit all nodes that are “reachable” by a path from a starting vertex.
f g c d a e b h
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depthFirst_Graph(v){ v.visited = true for each w such that (v,w) is in E // w in v.adjList _______?___________ }
Graph traversal (recursive)
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// Here “visiting” just means “reaching”
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depthFirst_Graph(v){ v.visited = true for each w such that (v,w) is in E // w in v.adjList if ! (w.visited) // avoids cycles depthFirst_Graph(w) }
Graph traversal (recursive)
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// Here “visiting” just means “reaching”
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b e f f f f f c c c c c c c a f g c d a e b h
Call Stack for depthFirst(a)
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depthFirst_Graph(v){ v.visited = true for each w such that (v,w) is in E if ! (w.visited) depthFirst_Graph(w) }
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b e f f f f f c a a f g c d a e b h
Call Stack for depthFirst(a)
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depthFirst_Graph(v){ v.visited = true for each w such that (v,w) is in E if ! (w.visited) depthFirst_Graph(w) }
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b e f c c a a a f g c d a e b h
Call Stack for depthFirst(a)
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depthFirst_Graph(v){ v.visited = true for each w such that (v,w) is in E if ! (w.visited) depthFirst_Graph(w) }
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b e f f f f f c c c c c c c a a a a a a a a a f g c d a e b h
Call Stack for depthFirst(a)
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depthFirst_Graph(v){ v.visited = true for each w such that (v,w) is in E if ! (w.visited) depthFirst_Graph(w) }
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b e f f f f f c c c c c c c a a a a a a a a a f g c d a e b h
Call Stack for depthFirst(a)
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depthFirst_Graph(v){ v.visited = true for each w such that (v,w) is in E if ! (w.visited) depthFirst_Graph(w) }
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Call Tree
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f g c d a e b h
root
b e f f f f f c c c c c c c a a a a a a a a a
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Example 2
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g d a h e b i f c a - (b,d) b - (a,c,e) c - (b,f) d - (a,e,g) e - (b,d,f,h) f - (c,e,i) g - (d,h) h - (e,g,i) i - (f,h)
Adjacency List What is the call tree for depthFirst( a ) ?
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Example 2
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g d a h e b i f c g d a h e b i f c
call tree for depthFirst(a)
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Q: Non-recursive graph traversal ? A: Similar to tree traversal: Use a stack or a queue.
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Recall: depth first tree traversal
(with a slight variation)
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treeTraversalUsingStack(root){ initialize empty stack s visit root s.push(root) while s is not empty { cur = s.pop() for each child of cur{ visit child s.push(child) } } } Visit a node before pushing it onto the stack. Every node in the tree gets visited, pushed, and then popped.
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Generalize to graphs…
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graphTraversalUsingStack(v){ initialize empty stack s v.visited = true s.push(v) while (!s.empty) { u = s.pop() for each w in u.adjList{ if (!w.visited){ // the only new part w.visited = true s.push(w) } } } }
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Example: graphTraversalUsingStack(a)
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a g d a h e b i f c
a
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Example: graphTraversalUsingStack(a)
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d a b g d a h e b i f c
d a b ‘a’ is popped and both ‘b’ and ‘d’ are pushed. The traversal defines a tree, but it is not a “call tree”. Why not?
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Example: graphTraversalUsingStack(a)
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g d a e b g d a h e b i f c
g d e ‘d’ is popped and both ‘e’ and ‘g’ are pushed. a b b
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Example: graphTraversalUsingStack(a)
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g d a e b g d a h e b i f c
g h d e e ‘g’ is popped and ‘h’ is pushed. a b b b
h
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Example: graphTraversalUsingStack(a)
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g d a e b g d a h e b i f c
g h i d e e e ‘h’ is popped and ‘i’ is pushed. a b b b b
h i
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Example: graphTraversalUsingStack(a)
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g d a e b g d a h e b i f c
g h i f d e e e e ‘i’ is popped and ‘f’ is pushed. a b b b b b
h i f
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Example: graphTraversalUsingStack(a)
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g d a e b g d a h e b i f c
g h i f c d e e e e e ‘f’ is popped and ‘c’ is pushed. a b b b b b b
h i f c
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Example: graphTraversalUsingStack(a)
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g d a e b g d a h e b i f c
g h i f c d e e e e e e a b b b b b b b b
h i f c
Order of nodes visited: abdeghifc
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a
b c d i e f h g j k
treeTraversalUsingQueue(root){ initialize empty queue q q.enqueue(root) while q is not empty { cur = q.dequeue() visit cur for each child of cur q.enqueue(child) } }
Recall: breadth first tree traversal
(see lecture 20)
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for each level i visit all nodes at level i
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Breadth first graph traversal
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Given an input vertex, find all vertices that can be reached by paths of length 1, 2, 3, 4, ….
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Breadth first graph traversal
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graphTraversalUsingQueue(v){ initialize empty queue q v.visited = true q.enqueue(v) while (! q.empty) { u = q.dequeue() for each w in u.adjList{ if (!w.visited){ w.visited = true q.enqueue(w) } } } }
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Example
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f c d a e b
graphTraversalUsingQueue(c)
queue c
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Example
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f c d a e b
graphTraversalUsingQueue(c)
queue c f
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Example
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f c d a e b
graphTraversalUsingQueue(c)
queue c f be
Both ‘b’, ‘e’ are visited and enqueued before ‘b’ is dequeued.
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Example
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f c d a e b
graphTraversalUsingQueue(c)
queue c f be e
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f c d a e b
graphTraversalUsingQueue(c) It defines a tree whose root is the starting vertex. It finds the shortest path (number of vertices) to all vertices reachable from starting vertex.
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Example: graphTraversalUsingQueue(a)
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g d a h e b i f c a
1
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Example: graphTraversalUsingQueue(a)
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g d a h e b i f c a bd
1 2 3
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Example: graphTraversalUsingQueue(a)
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g d a h e b i f c a bd dce
1 2 4 3 5
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Example: graphTraversalUsingQueue(a)
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g d a h e b i f c a bd dce ceg
1 2 4 3 5 6
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Example: graphTraversalUsingQueue(a)
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g d a h e b i f c a bd dce ceg egf
1 2 4 3 5 7 6
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Example: graphTraversalUsingQueue(a)
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g d a h e b i f c a bd dce ceg egf gfh
1 2 4 3 5 7 6 8
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Example: graphTraversalUsingQueue(a)
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g d a h e b i f c a bd dce ceg egf gfh fh hi i
1 2 4 3 5 7 6 8
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Example: graphTraversalUsingQueue(a)
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g d a h e b i f c a bd dce ceg egf gfh fh hi
1 2 4 3 5 7 6 8 9
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Example: graphTraversalUsingQueue(a)
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g d a h e b i f c a bd dce ceg egf gfh fh hi i
1 2 4 3 5 7 6 8 9
Note order of nodes visited: paths of length 1,2, 3, 4
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Example: graphTraversalUsingQueue(a)
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g d a h e b i f c
The traversal defines a tree, but it is not a “call tree”. Why not?
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class Graph<T> { HashMap< String, Vertex<T> > vertexMap; class Vertex<T> { ArrayList<Edge> adjList; T element; boolean visited; } class Edge { Vertex endVertex; double weight; : } }
Recall: How to implement a Graph class in Java?
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HEADS UP ! Prior to traversal, …. for each w in V w.visited = false How to implement this ?
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class Graph<T> { HashMap< String, Vertex<T> > vertexMap; : public void resetVisited() { } }
HEADS UP ! Prior to traversal, …. for each w in V w.visited = false How to implement this ?
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class Graph<T> { HashMap< String, Vertex<T> > vertexMap; : public void resetVisited() { for( Vertex<T> v : vertexMap.values() ){ v.visited = false; } }
HEADS UP ! Prior to traversal, …. for each w in V w.visited = false How to implement this ?
[ASIDE: I did something unnecessarily complicated on the Sec.001 slides. What I have above is better. ]