Max Stream Learning Note (3)-Push sticker labeling algorithm One

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1 Pre-flow: The pre-flow is a function $f$ of $v\times V \rightarrow R, which satisfies the capacity constraints and the following properties for each node $u\in v-\{s\}$:

$\sum_{v\in v}f (v,u)-\sum_{v\in v}f (u,v) \geq 0$

Define $E (U) =\sum_{v\in v}f (v,u)-\sum_{v\in v}f (u,v) $ for excess flow into the node $u$. If $e (u) >0$, call $u$ overflow

2 definition of the height function: Set g is a flow network if the function $f:v\rightarrow n$ satisfies $h (s) =| V|,h (t) =0$, and on any side $ (u,v) \in e_{f}$, with $h (u) \leq h (v) +1$, then $h$ is a height function

3 for any two nodes $u,v\in v$, if $h (u) >h (v) +1$, then $ (u,v) \notin e_{f}$

4 The idea of a push-to-paste label algorithm is to maintain a pre-stream $f$ and height function $h$ at any time of the algorithm, and to update it using push operations and a label-labeling Operation .

5 Push operation: If $u$ is an overflow node, $c _{f} (u,v) >0$, and $h (U) =h (v) +1$, then the push Operation $push (U,V) $ acts on the node $u,v$. The excess flow of the set node $u$ is stored in the $u.e$ and is stored in the $u.h$ height. Here is the pseudo-code for the push operation:

Each time the traffic from the node $u$ to the node $v$ is $\delta _{f} (u,v) =min (U.e,c_{f} (U,V)) $, the $u.e$ is guaranteed not to turn negative and $c (U,V) $.

If, after a $push$ operation, $ (u,v) $ is saturated, that is, $c_{f} (u,v) =0$, the push operation is called saturation push , otherwise unsaturated push .

6 node $u$ will no longer overflow after an unsaturated push from node $u$ to node $v$.

7 Re-labeling operation: If the node $u$ overflow and for all sides $ (u,v) \in e_{f}$, there is $u.h\leq v.h$, then the label operation will be re-labeled $relabel (U) $.

When this operation is called, the $E _{f}$ contains at least one edge from $u$. Because $e (u) =\sum_{v\in v}f (v,u)-\sum_{v\in v}f (u,v) >0$, there is at least one $v$ that makes $ (v,u). f>0$, so at least one side $ (u,v) \in e_{f}$.

8 The overall flow of the algorithm:

(1) initialization

(2) Main loop

9 Set $f$ as a pre-stream for the stream network G, $h $ for the height function. If a node $u$ overflow, either the $u$ can be pushed or the $u$ can be re-labeled.

10 the height of each node is not reduced throughout the Generic-push-relabel process, and each re-labeling operation will increase the $u$ height by at least 1.

11 the whole process of generic-push-relabel will always be maintained $h$ is a height function

12 The path from the source point to the sink point does not exist throughout the Generic-push-relabel process.

13 If the Generic-push-relabel algorithm is able to terminate, the computed $f$ is a maximum flow.

The following is proof

Proof of 9

For any remaining edge $ (u,v) $ has $h (u) \leq h (v) +1$. If the $u$ cannot be pushed, then for any of the $ (u,v) \in e_{f}$ there is $h (u)

Proof of 11

(1) Initially, $h $ is a height function;

(2) Re-labeling operation after $u$, for all $ (u,v) $ still satisfies $h (u) \leq h (v) +1$. For edges entering $u$, such as $ (w,u) $, there is $w.h\leq u.h+1$ before the label is re-labeled, followed by $w.h< u.h+1$. So the re-labeling operation will keep the $h$ still as the height function.

(3) $PUSH (U,V) $ operation may add an edge $ (v,u) $, $v. h=u.h-1<u.h+1$, so $h$ is still a height function. It is also possible to delete the side $ (u,v) $, which has no effect.

Proof of 12

Suppose there is a $p=<v_{0},v_{1},..., v_{k}>$, where $s=v_{0},t=v_{k}.$ because $p$ passes through each node at most once, then there is $k<| v|$. Because of $h (t) =0$, and for any one edge $ (v_{i},v_{i+1}) $, the height of the $v _{i}$ is up to 1 higher than the $v_{i+1}$, so $h (s) \leq k<| v|$, this with $h (s) =| v|$ contradiction

Proof of 13

(1) First, during the entire process of the algorithm, $f $ is a pre-stream. Initially, the algorithm in each loop, the re-labeling does not affect the $f$, push operation after the end of $f$ is still a pre-flow;

(2) If the algorithm can terminate, then terminate, no node need push operation and re-labeling operation, by 9, no overflow node exists, so $f$ is a stream. By 12 It is known that no augmented path exists, so by the maximum flow minimum cut theorem, $f $ is the maximum flow

Max Stream Learning Note (3)-Push sticker labeling algorithm One