diff --git a/Gruntfile.js b/Gruntfile.js
index 0e5ad77..9b72e8d 100644
--- a/Gruntfile.js
+++ b/Gruntfile.js
@@ -10,7 +10,7 @@ module.exports = function(grunt) {
options: {
port: port,
base: base,
- livereload: true,
+ livereload: false,
open: true
}
}
@@ -18,7 +18,7 @@ module.exports = function(grunt) {
watch: {
options: {
- livereload: true
+ livereload: false
},
// js: {
// files: [ 'Gruntfile.js', 'js/reveal.js' ],
diff --git a/documentation/contents/chapter3-background.tex b/documentation/contents/chapter3-background.tex
index d3cc76b..712ac33 100644
--- a/documentation/contents/chapter3-background.tex
+++ b/documentation/contents/chapter3-background.tex
@@ -69,6 +69,19 @@ \section{The \maxflow{}}
\section{\pushRelabel{}}
+The push-relabel algorithm operates locally and does not, in contrast to the less efficient globally operating maximum flow algorithms based on augmenting paths, maintain a feasible flow during the algorithm execution. Instead, it maintains a preflow. The capacity constraints of the edges are still maintainted, but the flow conservation at the nodes is not:
+
+\begin{definition}[preflow]
+ The preflow $\tilde f$ is a generalization of the flow $f$, the equality sign $=$ in the flow conservation is replaced with $\geq$. This means that more flow can enter an inner node than leaving it, e.g. we allow leaks in our pipe system.
+\end{definition}
+
+Another important concept in in the push-relabel algorithm is the height function, which is an approximation of the distance of a node towards the sink. The local \textit{push} operations try to move \textit{excess} at inner nodes 'downwards' towards the sink. If the current node is at a local minimum and still has excess, we \textit{relabel} the node by increasing its \textit{height} so that subsequent push operations can remove the excess.
+\begin{definition}[valid edge with respect to the height function]
+ An edge $e'=(v,w)$ of the \textbf{residual graph} is valid with respect to the current height function, if $h(v)==h(w)+1$, meaning that the current node is one level above the one to where we wish to push excess to.
+\end{definition}
+
+The important property of the push-relabel algorithm is that when the algorithm terminates, the computed preflow is actually a flow!
+
\input{pseudocode/pseudocode-maxflow}
@@ -106,10 +119,9 @@ \section{The \spprc{}}
\begin{example}[SPPTW]
An illustrative example is the two-resource SPPRC, the shortest path problem with time windows (SPPTW). In addition to the \textit{cost} $c$, each edge additionally bears the resource \textit{time} $t$. Thus each edge is associated with the two-dimensional resource vector $(t,c) \forall e \in E$. The secondary resource \textit{time} is constrained, while the primary resource \textit{cost} is unconstrained, but seeks to be minimized.
-The accumulated consumptions of the resource \textit{time} along a path are constrained at the intermediate vertices along that path by lower and upper limits, that is the earliest arrival time $t_a$ and the latest departure time $t_b$ and called the \textit{resource window}, denoted as a tuple $[t_a,t_b] \forall v \in V$.
+The accumulated consumptions of the resource \textit{time} along a path are constrained at the intermediate vertices along that path by lower and upper limits, that is the earliest arrival time $t_a$ and the latest departure time $t_b$ and called the \textit{resource window}, denoted as tuples $[t_a,t_b] \, \forall v \in V$.
\end{example}
-$[t_{arrival},t_{departure}] \forall v \in V$
\section{\labelSetting{}}
diff --git a/documentation/pseudocode/pseudocode-maxflow.tex b/documentation/pseudocode/pseudocode-maxflow.tex
index 16bcbbb..96e6a26 100644
--- a/documentation/pseudocode/pseudocode-maxflow.tex
+++ b/documentation/pseudocode/pseudocode-maxflow.tex
@@ -6,42 +6,39 @@
\begin{algorithm}[h!]
\caption{Goldberg-Tarjan Push-Relabel algorithm}
-\DontPrintSemicolon % Some LaTeX compilers require you to use \dontprintsemicolon instead
-\KwIn{directed Graph $G=(V,E)$ with
-\begin{itemize}
- \setlength\itemsep{0pt}
- \setlength{\parskip}{0pt}
- \item digraph $G=(V,E)$ with start $s$, sink $t$
- \item $\forall$ edges $e \in E$: (cap) capacity
-\end{itemize}
+\DontPrintSemicolon
+\KwIn{digraph $G=(V,E)$ with nodes $s,t \in V$ and edge capacities $c(e) \, \forall e \in E$
+%\begin{itemize}
+% \setlength\itemsep{0pt}
+% \setlength{\parskip}{0pt}
+% \item nodes start $s \in V$, sink $t \in V$
+% \item edge capacities $c(e) \, \forall e \in E$
+%\end{itemize}
}
\KwOut{A feasible maximum s-t flow f(e)}\vspace{0.2cm}
(* Initialize the preflow *)\;
\ForAll{$e=(u,w) \in E$}{
- \If{$u==s$}{
- $f(e) \gets c(e)$\;
- \If{$w \neq t$}{$Q$.add(w)}
- }
- \Else{$f(e) \gets 0$}
+ $f(e) \gets (u==s) \; ? \; c(e) : 0$\;
+ \If{$u==s$ AND $w \neq t$}{$Q$.add(w)}
}
((* Initialize the height function *))\;
-$h(s) = |V|$\;
-\ForAll{$v \in V$}{
- $h(v) \gets$ number of arcs on directed v-t path\;
+$h(s) \gets |V|$\;
+\ForAll{$v \in V \setminus \{s \}$}{
+ $h(v) \gets$ number of arcs on shortest v-t path\;
}
((* Main Loop *))\;
\While{$\setfont{Q} \neq \emptyset$}{
$v \gets \setfont{Q}$.pop()\;
- \While{$e(v)>0$ AND $\exists e'=(v,w) \in E' | h(v)==h(w)+1$}{
+ \While{$e(v)>0$ AND $\exists \, e'=(v,w) \in E' \, | \, h(v)==h(w)+1$}{
(* Push *)\;
push $\min(e(v),c'(e'))$ flow from v to w\;
\If{$w \neq s,t$ AND $w \notin \setfont{Q}$}{
$\setfont{Q}$.add($w$)\;
}
}
- \If{$e(v)>0$ AND $\not\exists e'=(v,w) \in E' | h(v)==h(w)+1$}{
+ \If{$e(v)>0$}{
(* Relabel *)\;
- $h(v) \gets 1 + \min(h(w) | e*=(v,w)\in E'$ \;
+ $h(v) \gets 1 + \min(\{h(w) | e^*=(v,w)\in E'\})$ \;
$\setfont{Q}$.add($v$) \;
}
}
diff --git a/documentation/pseudocode/pseudocode-spprc.tex b/documentation/pseudocode/pseudocode-spprc.tex
index a977f9f..147d393 100644
--- a/documentation/pseudocode/pseudocode-spprc.tex
+++ b/documentation/pseudocode/pseudocode-spprc.tex
@@ -6,34 +6,30 @@
\begin{algorithm}[h!]
\DontPrintSemicolon % Some LaTeX compilers require you to use \dontprintsemicolon instead
-\KwIn{directed Graph $G=(V,E)$ with
-\begin{itemize}
- \setlength\itemsep{0pt}
- \setlength{\parskip}{0pt}
- \item $\forall$ nodes $v \in V$: [arrive, depart] time window resource constraint
- \item $\forall$ edges $e \in E$: (time,cost) 2-dimensional resource consumptions
- \item start Node $s$, end Node $t$
-\end{itemize}}
-\KwOut{feasible, pareto-optimal $s \rightarrow t$ paths} \vspace{0.2cm}
+\KwIn{digraph $G=(V,E)$ with start node $s \in V$, target node $t \in V$, resource windows for all nodes %$[t_a, t_d] \, \forall v \in V$
+and resource vectors for all edges} %$(t,c) \, \forall e \in E$}
+\KwOut{feasible, pareto-optimal $s$-$t$ path $l^*$ with minimal cost} \vspace{0.2cm}
(* Initialize *) \;
-SET $\setfont{U} = \{(s)\}$ and $\setfont{P} = \emptyset$ \;
-\While{$\setfont{U} \neq \emptyset$}{
+$U \gets \{(\epsilon,s)\}$ and $P \gets \emptyset$ \;
+(* Main Loop *) \;
+\While{$\exists l=(\sim,v) \in U$}{
+ $U \gets U \setminus \{l\}$\;
(* Path extension step *)\;
- CHOOSE a path $\setfont{Q} \in \setfont{U}$ and REMOVE $\setfont{Q}$ from $\setfont{U}$\;
- \ForAll{arcs $(v(\setfont{Q}), w) \in A$ of the forward star of $v(\setfont{Q})$}{
- \If{$(\setfont{Q},w) \in \setfont{F}(s,w) \cap \setfont{G}$}{
- ADD $(\setfont{Q},w)$ to $\setfont{U}$
+ \ForAll{$e=(v,w) \in E$}{
+ $l'=(l,w) \gets$ EXTEND$(l,e)$\;
+ \If{$l' \in \mathrm{FEASIBLE}(w)$}{
+ $U \gets U \cup \{l'\}$\;
}
}
- ADD $\setfont{Q}$ to $\setfont{P}$\;
+ $P \gets P \cup \{l\}$\;
(* Dominance step *)\;
- \If{(* any condition *)}{
- APPLY dominance algorithm to paths from $\setfont{U} \cup \setfont{P}$ ending at some node $v$\;
+ \ForAll{$v \in V\setminus\{s\}$}{
+ $U,P \gets$ REMOVE-DOMINATED$(U,P)$\;
}
}
-(* Filtering step *)\;
-FILTER $\setfont{P}$, i.e., identify a solution $\setfont{S} \subseteq \setfont{P}$\;
-\caption{Generic Dynamic Programming SPPRC Algorithm}
+((* Filtering step *))\;
+$l^* \in P \; | \; cost(l^*) ==\min( \{ cost(l=(\sim,t)\in P) \} )$\;
+\caption{Generic Dynamic Programming SPPTW Label Setting Algorithm}
\end{algorithm}
diff --git a/implementation/library-d3-svg/js/AlgorithmTab.js b/implementation/library-d3-svg/js/AlgorithmTab.js
index e7510f1..4826926 100644
--- a/implementation/library-d3-svg/js/AlgorithmTab.js
+++ b/implementation/library-d3-svg/js/AlgorithmTab.js
@@ -88,7 +88,7 @@ function AlgorithmTab(algo,p_tab) {
return "pseudocode";
});
- d3.select("#tw_div_statusPseudocode").text(pseudocode.text())
+ d3.select("#tw_div_statusPseudocode").html(pseudocode.html())
Tab.prototype.init.call(this);
diff --git a/implementation/library-d3-svg/js/GraphDrawer.js b/implementation/library-d3-svg/js/GraphDrawer.js
index 04cf899..d08ddbf 100644
--- a/implementation/library-d3-svg/js/GraphDrawer.js
+++ b/implementation/library-d3-svg/js/GraphDrawer.js
@@ -101,7 +101,7 @@ GraphDrawer = function(svgOrigin,extraMargin,transTime){
left: global_KnotenRadius+wS +(extraMargin.left || 0)
}
- var width = xRange - margin.left - margin.right,
+ var width = xRange - margin.left - margin.right;
var height = yRange - margin.top - margin.bottom;
this.height = height;
diff --git a/implementation/maxflow-push-relabel/index_en.html b/implementation/maxflow-push-relabel/index_en.html
index 8780043..ff639e3 100644
--- a/implementation/maxflow-push-relabel/index_en.html
+++ b/implementation/maxflow-push-relabel/index_en.html
@@ -416,27 +416,25 @@ Finished
-(* Initializing the preflow *)
-
FORALL e=(u,w) ∈ E
-
IF u == s THEN f(e) ← c(e)
-
ELSE f(e) ← 0
-
IF u == s AND w ≠ t THEN Q.add(w)
(* Initializing the height function *)
-
h(s) = |V|
-
FORALL v ∈ V\{s}
-
h(v) ← #arcs on directed v-t path
(* Initialize the preflow *)
+
FORALL e=(u,w) ∈ E
+
f(e) ← (u == s) ? c(e) : 0
+
IF u == s AND w ≠ t THEN Q.add(w)
(* Initialize the height function *)
+
h(s) ← |V|
+
FORALL v ∈ V\{s}
+
h(v) ← #arcs on shortest v-t path
(* Main Loop *)
-
WHILE Q ≠ ∅
-
v ← Q.pop()
WHILE e(v)>0
-
AND ∃ e'=(v,w)∈E'|h(v)==h(w)+1
(* PUSH *)
-
push min{e(v),c'(e')} flow from v to w
-
IF w != s,t AND w ∉ Q THEN Q.add(w)
IF e(v)>0 AND ∄ e'=(v,w)∈E'|h(v)==h(w)+1
(* RELABEL *)
-
h(v) ← 1+min{h(w)|e*=(v,w)∈E'}
-
Q.add(v)
WHILE Q ≠ ∅
+ v ← Q.pop()
+ WHILE e(v)>0 AND ∃ e'=(v,w)∈E'|h(v)==h(w)+1
(* PUSH *)
+
push min(e(v),c'(e')) flow from v to w
+
IF w ≠ s,t AND w ∉ Q THEN Q.add(w)
(* RELABEL *)
+
h(v) ← 1+min({h(w)|e*=(v,w)∈E'})
+
Q.add(v)
What is the pseudocode of the algorithm?
The flow value of f(e) is maximized along all feasible flows
-
-
+
+
+
How fast is the algorithm?
diff --git a/implementation/spp-rc-label-setting/index_en.html b/implementation/spp-rc-label-setting/index_en.html
index 91b8034..ea3c55b 100644
--- a/implementation/spp-rc-label-setting/index_en.html
+++ b/implementation/spp-rc-label-setting/index_en.html
@@ -421,31 +421,31 @@
Filtering step
var STATUS_DOMINANCE = id++;
var STATUS_DOMINANCE_NODE = id++;
var STATUS_FINISHED = id;-->
-
-
-
-
(* Initialize *)
-
U ← {(ε,s)}, P ← ∅
-
(* Main Loop *)
-
WHILE ∃ l=(~,v) ∈ U
-
-
(* Path extension step *)
-
FORALL e=(v,w) ∈ E
-
l'=(l,w) ← EXTEND(l,e)
-
IF l' ∈ FEASIBLE(w)
-
U ← U ∪ {l'}
-
-
-
(* Dominance step *)
-
FORALL v ∈ V\{s}
-
U,P ← REMOVE-DOMINATED(U,P)
-
END
-
t ← pick(v)
-
(* Filtering step *)
-
solution ← l=(~,t) ∈ P :
-
cost(l) = min(cost(l)) ∀ l=(~,t) ∈ P
+
+
+
+
(* Initialize *)
+
U ← {(ε,s)} and P ← ∅
+
(* Main Loop *)
+
WHILE ∃ l=(~,v) ∈ U
+
U ← U \ {l}
+
(* Path extension step *)
+
FORALL e=(v,w) ∈ E
+
l'=(l,w) ← EXTEND(l,e)
+
IF l' ∈ FEASIBLE(w)
+
U ← U ∪ {l'}
+
+
+
(* Dominance step *)
+
FORALL v ∈ V\{s}
+
U,P ← REMOVE-DOMINATED(U,P)
+
END
+
(* Filtering step *)
+
t ← pick(v)
+
l* ∈ P | cost(l*) == min({cost(l=(~,t)∈P)})
Variable status
@@ -560,14 +560,17 @@
Negative cycles: Label-setting vs. Label-correcting
What is the pseudocode of the algorithm?
-Input: Weighted, undirected graph G=(V,E) with weight function l.
-Output: A list {d(v[j]) : j = 1,..,n} containing the distances dist(v[1],v[j]) = d(v[j]),
- if there are no negative circles reachable from v[1].
- The message "Negative Circle" is shown, if a negative circle can be reached from v[1].
+Input: directed graph G=(V,E) with
+ start node s and end node t
+ resource windows for all nodes
+ resource vectors for all edges
+Output: feasible, pareto-optimal s-t path l* with minimal cost
-
-
+
+
+
How fast is the algorithm?