1. Suggested Solutions to Part of Homework 1
22.3-6: Eliminate recursion of DFS
22.4-2: Compute the number of st-paths in DAGs in linear time
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2. 22.3-6
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3. Eliminate recursion of DFS (1)
DFS(G)
Input: A graph G represented by adjacent list.
Output: Vertex list of G in DFS order.
1 for each vertex v  V(G) do
2 color[v]  WHITE
/* WHITE means that the corresponding vertex has not been visited. */
3 endfor
4 Stack ST  
5 for each vertex v  V(G) do
6 if color[v] = BLACK then loop
/* BLACK means the corresponding vertex has been visited */
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4. Eliminate recursion of DFS (1)
7 Push(ST, v)
8 while ST   do
u  Pop(ST)
if color[u] = BLACK then loop
Visit(u) /* to say, print u */
color[u]  BLACK
for each vertex a  Adj[u] do
if color[a] = WHITE then Push(ST, a)
endfor
endwhile
17 endfor
18 return
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5. Eliminate recursion of DFS (2)
To faithfully implement the DFS algorithm in the textbook without recursion.
Vertex color: WHITE – not visited; GRAY – visiting; BLACK – visited.
Time: d[u] – the beginning time of visiting vertex u f[u] – the end time of visiting vertex u.
Predecessor: [u] – the predecessor of vertex u in the DFS forest.
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6. Eliminate recursion of DFS (2)
DFS(G)
Input: A graph G represented by adjacent list.
Output: Visit beginning time d[u], visit end time f[u], and a DFS forest .
1 for each vertex v  V(G) do
2 color[v]  WHITE
3 [v]  NIL
4 endfor
5 time  0
6 Stack ST  
7 for each vertex v  V(G) do
8 if color[v]  WHITE then loop
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7. Eliminate recursion of DFS (2)
9 Push(ST, (v, HEAD))
/* Each unit of stack ST contains a pair (v, a), which means the vertex v and its an adjacent vertex a The adjacent vertex is gotten from the linked list of vertex v in the adjacent list of G. The adjacent vertex being HEAD means that we are currently at the dummy head node of the linked list (of vertex v). */
while ST   do
(u, a)  Pop(ST)
if a = HEAD then
color[u]  GRAY
d[u]  time  time + 1
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8. Eliminate recursion of DFS (2)
endif
a  NextAdjVertex(u, a)
/* NextAdjVertex(u, a) – getting the adjacent vertex of u which is after vertex a from u’s linked list If a is the last adjacent vertex in the linked list, the function would return NIL. */
if a  NIL then
Push(ST, (u, a))
if color[a] = WHITE then
[a]  u
Push(ST, (a, HEAD))
loop /* control returns back to while */
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9. Eliminate recursion of DFS (2)
endif
else
/* a = NIL. We have considered all the adjacent vertices of u. */
color[u]  BLACK
f[u]  time  time + 1
endif
endwhile
29 endfor
30 return
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10. 22.4-2
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11. Compute number of st-paths in DAG (1)
Transmit-Alg(G)
1 Compute a topological order L of V(G)
2 for each vertex v from s to t in order of L do
3 count[v]  0
4 endfor
5 count[s]  1
6 for each vertex u from s to the vertex just preceding t do
7 for each vertex v  Adj[u] do
count[v]  count[v] + count[u]
9 endfor
10 endfor
11 return count[t]
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12. Compute number of st-paths in DAG (2)
Transmit(G)
1 for each vertex v  V(G) do
2 color[v]  WHITE
3 count[v]  0
4 endfor
5 count[t]  1
6 DFS-Visit(s)
7 return count[s]
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13. Compute number of st-paths in DAG (2)
DFS-Visit(u)
1 if u = t then color[t]  BLACK and return
2 color[u]  GRAY
3 for each vertex v  Adj[u] do
4 if color[v] = WHITE then DFS-Visit(v)
5 count[u]  count[u] + count[v]
6 endfor
7 color[u]  BLACK
8 return
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14. Thanks for attention! (c) 2011 SDU