DSA1 | Week 8

Navigating Graphs using DFS

Try them all

Explore one thoroughly

What do you do when you have many different options?

Navigating Graphs

Navigating Graphs

Try them all

Explore one thoroughly

What do you do when you have many different options?

Phase 0.

Identify a

starting point \(s\).

Phase 0.

Identify a

starting point \(s\).

Phase 1.

Go to some friend of \(s\).

Phase 1.

Go to some friend of \(s\).

Phase 0.

Identify a

starting point \(s\).

Phase k.

Go to some friend of* friend of friend of friend of ... \(s\).

\(\cdots\)

What if the current person you are talking to has no new friends to introduce you to?

*whom you have

not met yet!

\(s\)

\(x\)

\(y\)

\(w\)

\(z\)

\(s\)

\(x\)

\(y\)

\(w\)

\(z\)

\(s\)

\(x\)

\(y\)

\(w\)

\(z\)

\(s\)

\(x\)

\(y\)

\(w\)

\(z\)

\(s\)

\(x\)

\(y\)

\(w\)

\(z\)

\(s\)

\(x\)

\(y\)

\(w\)

\(z\)

\(s\)

\(x\)

\(y\)

\(w\)

\(z\)

\(s\)

\(x\)

\(y\)

\(w\)

\(z\)

\(s\)

\(x\)

\(y\)

\(w\)

\(z\)

\(s\)

\(x\)

\(y\)

\(w\)

\(z\)

\(s\)

\(x\)

\(y\)

\(w\)

\(z\)

\(s\)

\(x\)

\(y\)

\(w\)

\(z\)

Implementation?

[Backtrack, rinse and repeat.]

Phase 1.

Go to some friend of \(s\).

Phase 0.

Identify a

starting point \(s\).

Phase k.

Go to some friend of* friend of friend of friend of ... \(s\).

\(\cdots\)

*whom you have

not met yet!

Maintain Visit Status

visited[s] = 1

//where s is the starting point

 

visited[u] = 0 

//for all other vertices u
find_unvisited_neighbor(v):

    if v has an unvisited neighbor:

        return w

    else:

        return -1

A Helper Function

current = s
while(true):    
x = find_unvisited_neighbor(current)
if(x > -1):
visited[current] = 1
current = x
?!?
else:
current = s
while(true):    
x = find_unvisited_neighbor(current)
visited[current] = 1
current = x
?!?
if(x > -1):
initalize stack T
visited[s] = 1; push(T,s)
while(T is nonempty):    
current = peek(T)
visited[x] = 1
push(T,x)
pop(T)
else:

1

2

1

2

1

3

2

1

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2

1

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1

5

1

1

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1

1

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1

1

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1

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1

x = find_unvisited_neighbor(current)
if(x > -1):
initalize stack T
visited[s] = 1; push(T,s)
while(T is nonempty):    
current = peek(T)
visited[x] = 1
pop(T)
else:
push(T,x)
; inc clock; pre[x] = clock
; inc clock; post[v] = clock
; clock = 1
v= 
; pre[s] = clock

Source: Algorithms, Jeff Erickson

new if v is not on the stack yet

active if it is on the stack

finished if it is removed from the stack

new if v is not on the stack yet, that is, if:

clock < pre[v]

active if it is on the stack, that is, if:

pre[v] \(\leq\) clock < post[v]

finished if it is removed from the stack, that is, if:

post[v] \(\leq\) clock

edge: u to v; status of v when DFS(u) is called.

(4) cross edges

already finished

(3) back edges

already active

(2) forward edges

new; DFS(v) not called by DFS(u)

(1) tree edges

new; DFS(v) called by DFS(u)

Source: Algorithms, Jeff Erickson

Fix an arbitrary depth-first traversal of any directed graph G.

The following statements are equivalent for all vertices u and v of G.

(a) u is an ancestor of v in the depth-first forest.

(b) u.pre \(\leq\) v.pre < v.post \(\leq\) u.post.

(c) Just after DFS(v) is called, u is active.

(d) Just before DFS(u) is called, there is a path from u to v in which every vertex (including u and v) is new.

Topological Sort

A topological ordering of a directed graph G is a total order on the vertices such that u appears before v for every edge (u,v).

Topological Sort

 A topological ordering
arranges the vertices along a horizontal line
so that all edges point from
left to right.

2023 DSA1 | Week 8

By Neeldhara Misra

2023 DSA1 | Week 8

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