Why CLRS example on residual networks does not follows its formula?
$begingroup$
I am learning algorithms to solve Maximum Flow problem by reading the CRLS book and confused by the following figure:
That is:
A flow in a residual network provides a roadmap for adding flow to the
original flow network. If $f$ is a flow in $G$ and $f'$ is a flow in
the corresponding residual network $G_f$, we define $f uparrow f'$,
the augmentation of flow $f$ by $f'$, to be a function from $V times V$ to
$R$, defined by
$$(f uparrow f')(u, v) = begin{cases} f(u,v) + f'(u, v) - f'(v, u) &
> text{if (u,v) $in$ E} \ 0 & text{otherwise} end{cases}$$
How the flow network in (c), for example $(s, v_2)$ got the flow 12 ?
If we follow the formula, it must have a flow 5:
$8 + 5 - 8 = 5$
algorithms network-flow
asked yesterday
maksadbekmaksadbek
1185
$endgroup$
add a comment |
$begingroup$
I am learning algorithms to solve Maximum Flow problem by reading the CRLS book and confused by the following figure:
That is:
A flow in a residual network provides a roadmap for adding flow to the
original flow network. If $f$ is a flow in $G$ and $f'$ is a flow in
the corresponding residual network $G_f$, we define $f uparrow f'$,
the augmentation of flow $f$ by $f'$, to be a function from $V times V$ to
$R$, defined by
$$(f uparrow f')(u, v) = begin{cases} f(u,v) + f'(u, v) - f'(v, u) &
> text{if (u,v) $in$ E} \ 0 & text{otherwise} end{cases}$$
How the flow network in (c), for example $(s, v_2)$ got the flow 12 ?
If we follow the formula, it must have a flow 5:
$8 + 5 - 8 = 5$
algorithms network-flow
asked yesterday
maksadbekmaksadbek
1185
$endgroup$
add a comment |
$begingroup$
I am learning algorithms to solve Maximum Flow problem by reading the CRLS book and confused by the following figure:
That is:
A flow in a residual network provides a roadmap for adding flow to the
original flow network. If $f$ is a flow in $G$ and $f'$ is a flow in
the corresponding residual network $G_f$, we define $f uparrow f'$,
the augmentation of flow $f$ by $f'$, to be a function from $V times V$ to
$R$, defined by
$$(f uparrow f')(u, v) = begin{cases} f(u,v) + f'(u, v) - f'(v, u) &
> text{if (u,v) $in$ E} \ 0 & text{otherwise} end{cases}$$
How the flow network in (c), for example $(s, v_2)$ got the flow 12 ?
If we follow the formula, it must have a flow 5:
$8 + 5 - 8 = 5$
algorithms network-flow
asked yesterday
maksadbekmaksadbek
1185
$endgroup$
I am learning algorithms to solve Maximum Flow problem by reading the CRLS book and confused by the following figure:
That is:
A flow in a residual network provides a roadmap for adding flow to the
original flow network. If $f$ is a flow in $G$ and $f'$ is a flow in
the corresponding residual network $G_f$, we define $f uparrow f'$,
the augmentation of flow $f$ by $f'$, to be a function from $V times V$ to
$R$, defined by
$$(f uparrow f')(u, v) = begin{cases} f(u,v) + f'(u, v) - f'(v, u) &
> text{if (u,v) $in$ E} \ 0 & text{otherwise} end{cases}$$
How the flow network in (c), for example $(s, v_2)$ got the flow 12 ?
If we follow the formula, it must have a flow 5:
$8 + 5 - 8 = 5$
algorithms network-flow
algorithms network-flow
asked yesterday
maksadbekmaksadbek
1185
asked yesterday
maksadbekmaksadbek
1185
asked yesterday
maksadbekmaksadbek
1185
asked yesterday
maksadbekmaksadbek
1185
asked yesterday
maksadbekmaksadbek
1185
1185
add a comment |
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2 Answers
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That's not what the formula gives you. As the caption says, the capacity of the augmenting path in the residual network in (b) is $4$. Therefore we send 4 units of flow along the augmenting path from $s$ to $t$, namely, the path $s to v_2 to v_3 to t$. In particular, $f(s,v_2)=8$, $f'(s,v_2)=4$, and $f'(v_2,s)=0$, so the updated flow is $8+4-0=12$.
answered yesterday
D.W.♦D.W.
103k12129294
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It is explained in part (b) of the caption of Figure 26.4.
The residual network $G_f$ with augmenting path $p$ shaded; its residual capacity is $c_f(p)=c_f(v_2,v_3)=4$.
Since the capacity of path $p$ is 4 (not 5), we find a flow $f'$ in the residual network $G_f$ that is defined by $f'(s,v_2)=f'(v_2,v_3)=f'(v_3,t)=4$. So for the network flow $fuparrow f'$ in (c), we have
$$ (fuparrow f')(v_2, v_3)=f(v_2,v_3)+f'(v_2,v_3) = 8+4=12.$$
answered yesterday
Apass.JackApass.Jack
14k1940
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2 Answers
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$begingroup$
That's not what the formula gives you. As the caption says, the capacity of the augmenting path in the residual network in (b) is $4$. Therefore we send 4 units of flow along the augmenting path from $s$ to $t$, namely, the path $s to v_2 to v_3 to t$. In particular, $f(s,v_2)=8$, $f'(s,v_2)=4$, and $f'(v_2,s)=0$, so the updated flow is $8+4-0=12$.
answered yesterday
D.W.♦D.W.
103k12129294
$endgroup$
add a comment |
$begingroup$
That's not what the formula gives you. As the caption says, the capacity of the augmenting path in the residual network in (b) is $4$. Therefore we send 4 units of flow along the augmenting path from $s$ to $t$, namely, the path $s to v_2 to v_3 to t$. In particular, $f(s,v_2)=8$, $f'(s,v_2)=4$, and $f'(v_2,s)=0$, so the updated flow is $8+4-0=12$.
answered yesterday
D.W.♦D.W.
103k12129294
$endgroup$
add a comment |
$begingroup$
That's not what the formula gives you. As the caption says, the capacity of the augmenting path in the residual network in (b) is $4$. Therefore we send 4 units of flow along the augmenting path from $s$ to $t$, namely, the path $s to v_2 to v_3 to t$. In particular, $f(s,v_2)=8$, $f'(s,v_2)=4$, and $f'(v_2,s)=0$, so the updated flow is $8+4-0=12$.
answered yesterday
D.W.♦D.W.
103k12129294
$endgroup$
That's not what the formula gives you. As the caption says, the capacity of the augmenting path in the residual network in (b) is $4$. Therefore we send 4 units of flow along the augmenting path from $s$ to $t$, namely, the path $s to v_2 to v_3 to t$. In particular, $f(s,v_2)=8$, $f'(s,v_2)=4$, and $f'(v_2,s)=0$, so the updated flow is $8+4-0=12$.
answered yesterday
D.W.♦D.W.
103k12129294
answered yesterday
D.W.♦D.W.
103k12129294
answered yesterday
D.W.♦D.W.
103k12129294
answered yesterday
D.W.♦D.W.
103k12129294
103k12129294
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$begingroup$
It is explained in part (b) of the caption of Figure 26.4.
The residual network $G_f$ with augmenting path $p$ shaded; its residual capacity is $c_f(p)=c_f(v_2,v_3)=4$.
Since the capacity of path $p$ is 4 (not 5), we find a flow $f'$ in the residual network $G_f$ that is defined by $f'(s,v_2)=f'(v_2,v_3)=f'(v_3,t)=4$. So for the network flow $fuparrow f'$ in (c), we have
$$ (fuparrow f')(v_2, v_3)=f(v_2,v_3)+f'(v_2,v_3) = 8+4=12.$$
answered yesterday
Apass.JackApass.Jack
14k1940
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add a comment |
$begingroup$
It is explained in part (b) of the caption of Figure 26.4.
The residual network $G_f$ with augmenting path $p$ shaded; its residual capacity is $c_f(p)=c_f(v_2,v_3)=4$.
Since the capacity of path $p$ is 4 (not 5), we find a flow $f'$ in the residual network $G_f$ that is defined by $f'(s,v_2)=f'(v_2,v_3)=f'(v_3,t)=4$. So for the network flow $fuparrow f'$ in (c), we have
$$ (fuparrow f')(v_2, v_3)=f(v_2,v_3)+f'(v_2,v_3) = 8+4=12.$$
answered yesterday
Apass.JackApass.Jack
14k1940
$endgroup$
add a comment |
$begingroup$
It is explained in part (b) of the caption of Figure 26.4.
The residual network $G_f$ with augmenting path $p$ shaded; its residual capacity is $c_f(p)=c_f(v_2,v_3)=4$.
Since the capacity of path $p$ is 4 (not 5), we find a flow $f'$ in the residual network $G_f$ that is defined by $f'(s,v_2)=f'(v_2,v_3)=f'(v_3,t)=4$. So for the network flow $fuparrow f'$ in (c), we have
$$ (fuparrow f')(v_2, v_3)=f(v_2,v_3)+f'(v_2,v_3) = 8+4=12.$$
answered yesterday
Apass.JackApass.Jack
14k1940
$endgroup$
It is explained in part (b) of the caption of Figure 26.4.
The residual network $G_f$ with augmenting path $p$ shaded; its residual capacity is $c_f(p)=c_f(v_2,v_3)=4$.
Since the capacity of path $p$ is 4 (not 5), we find a flow $f'$ in the residual network $G_f$ that is defined by $f'(s,v_2)=f'(v_2,v_3)=f'(v_3,t)=4$. So for the network flow $fuparrow f'$ in (c), we have
$$ (fuparrow f')(v_2, v_3)=f(v_2,v_3)+f'(v_2,v_3) = 8+4=12.$$
answered yesterday
Apass.JackApass.Jack
14k1940
edited yesterday
answered yesterday
Apass.JackApass.Jack
14k1940
answered yesterday
Apass.JackApass.Jack
14k1940
answered yesterday
Apass.JackApass.Jack
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