Changeset 406:a578265aa8a6 in lemon-main

Timestamp:

11/30/08 21:53:24 (16 years ago)

Author:

Peter Kovacs <kpeter@…>

Branch:

default

Phase:

public

Message:

Improvements in groups.dox (#188)

• Unify the notations used for formulas.
• Add 'namespace lemon {...}' to simplify the references.
• Improved doc for algorithm groups.
• Extend the doc of the "shortest path" and "minimum cost flow" modules.

File:

• doc/groups.dox (modified) (13 diffs)

doc/groups.dox

@@ -17 +17 @@
  */
 
+namespace lemon {
+
 /**
 @defgroup datas Data Structures
@@ -89 +91 @@
 
 This group describes maps that are specifically designed to assign
-values to the nodes and arcs of graphs.
+values to the nodes and arcs/edges of graphs.
+
+If you are looking for the standard graph maps (\c NodeMap, \c ArcMap,
+\c EdgeMap), see the \ref graph_concepts "Graph Structure Concepts".
 */
 
@@ -100 +105 @@
 maps from other maps.
 
-Most of them are \ref lemon::concepts::ReadMap "read-only maps".
+Most of them are \ref concepts::ReadMap "read-only maps".
 They can make arithmetic and logical operations between one or two maps
 (negation, shifting, addition, multiplication, logical 'and', 'or',
@@ -202 +207 @@
 \brief Common graph search algorithms.
 
-This group describes the common graph search algorithms like
-Breadth-First Search (BFS) and Depth-First Search (DFS).
+This group describes the common graph search algorithms, namely
+\e breadth-first \e search (BFS) and \e depth-first \e search (DFS).
 */
 
@@ -211 +216 @@
 \brief Algorithms for finding shortest paths.
 
-This group describes the algorithms for finding shortest paths in graphs.
+This group describes the algorithms for finding shortest paths in digraphs.
+
+ - \ref Dijkstra algorithm for finding shortest paths from a source node
+   when all arc lengths are non-negative.
+ - \ref BellmanFord "Bellman-Ford" algorithm for finding shortest paths
+   from a source node when arc lenghts can be either positive or negative,
+   but the digraph should not contain directed cycles with negative total
+   length.
+ - \ref FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms
+   for solving the \e all-pairs \e shortest \e paths \e problem when arc
+   lenghts can be either positive or negative, but the digraph should
+   not contain directed cycles with negative total length.
+ - \ref Suurballe A successive shortest path algorithm for finding
+   arc-disjoint paths between two nodes having minimum total length.
 */
 
@@ -222 +240 @@
 feasible circulations.
 
-The maximum flow problem is to find a flow between a single source and
-a single target that is maximum. Formally, there is a \f$G=(V,A)\f$
-directed graph, an \f$c_a:A\rightarrow\mathbf{R}^+_0\f$ capacity
-function and given \f$s, t \in V\f$ source and target node. The
-maximum flow is the \f$f_a\f$ solution of the next optimization problem:
-
-\f[ 0 \le f_a \le c_a \f]
-\f[ \sum_{v\in\delta^{-}(u)}f_{vu}=\sum_{v\in\delta^{+}(u)}f_{uv}
-\qquad \forall u \in V \setminus \{s,t\}\f]
-\f[ \max \sum_{v\in\delta^{+}(s)}f_{uv} - \sum_{v\in\delta^{-}(s)}f_{vu}\f]
+The \e maximum \e flow \e problem is to find a flow of maximum value between
+a single source and a single target. Formally, there is a \f$G=(V,A)\f$
+digraph, a \f$cap:A\rightarrow\mathbf{R}^+_0\f$ capacity function and
+\f$s, t \in V\f$ source and target nodes.
+A maximum flow is an \f$f:A\rightarrow\mathbf{R}^+_0\f$ solution of the
+following optimization problem.
+
+\f[ \max\sum_{a\in\delta_{out}(s)}f(a) - \sum_{a\in\delta_{in}(s)}f(a) \f]
+\f[ \sum_{a\in\delta_{out}(v)} f(a) = \sum_{a\in\delta_{in}(v)} f(a)
+    \qquad \forall v\in V\setminus\{s,t\} \f]
+\f[ 0 \leq f(a) \leq cap(a) \qquad \forall a\in A \f]
 
 LEMON contains several algorithms for solving maximum flow problems:
-- \ref lemon::EdmondsKarp "Edmonds-Karp"
-- \ref lemon::Preflow "Goldberg's Preflow algorithm"
-- \ref lemon::DinitzSleatorTarjan "Dinitz's blocking flow algorithm with dynamic trees"
-- \ref lemon::GoldbergTarjan "Preflow algorithm with dynamic trees"
-
-In most cases the \ref lemon::Preflow "Preflow" algorithm provides the
-fastest method to compute the maximum flow. All impelementations
-provides functions to query the minimum cut, which is the dual linear
-programming problem of the maximum flow.
+- \ref EdmondsKarp Edmonds-Karp algorithm.
+- \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm.
+- \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees.
+- \ref GoldbergTarjan Preflow push-relabel algorithm with dynamic trees.
+
+In most cases the \ref Preflow "Preflow" algorithm provides the
+fastest method for computing a maximum flow. All implementations
+provides functions to also query the minimum cut, which is the dual
+problem of the maximum flow.
 */
 
@@ -253 +272 @@
 This group describes the algorithms for finding minimum cost flows and
 circulations.
+
+The \e minimum \e cost \e flow \e problem is to find a feasible flow of
+minimum total cost from a set of supply nodes to a set of demand nodes
+in a network with capacity constraints and arc costs.
+Formally, let \f$G=(V,A)\f$ be a digraph,
+\f$lower, upper: A\rightarrow\mathbf{Z}^+_0\f$ denote the lower and
+upper bounds for the flow values on the arcs,
+\f$cost: A\rightarrow\mathbf{Z}^+_0\f$ denotes the cost per unit flow
+on the arcs, and
+\f$supply: V\rightarrow\mathbf{Z}\f$ denotes the supply/demand values
+of the nodes.
+A minimum cost flow is an \f$f:A\rightarrow\mathbf{R}^+_0\f$ solution of
+the following optimization problem.
+
+\f[ \min\sum_{a\in A} f(a) cost(a) \f]
+\f[ \sum_{a\in\delta_{out}(v)} f(a) - \sum_{a\in\delta_{in}(v)} f(a) =
+    supply(v) \qquad \forall v\in V \f]
+\f[ lower(a) \leq f(a) \leq upper(a) \qquad \forall a\in A \f]
+
+LEMON contains several algorithms for solving minimum cost flow problems:
+ - \ref CycleCanceling Cycle-canceling algorithms.
+ - \ref CapacityScaling Successive shortest path algorithm with optional
+   capacity scaling.
+ - \ref CostScaling Push-relabel and augment-relabel algorithms based on
+   cost scaling.
+ - \ref NetworkSimplex Primal network simplex algorithm with various
+   pivot strategies.
 */
 
@@ -263 +309 @@
 This group describes the algorithms for finding minimum cut in graphs.
 
-The minimum cut problem is to find a non-empty and non-complete
-\f$X\f$ subset of the vertices with minimum overall capacity on
-outgoing arcs. Formally, there is \f$G=(V,A)\f$ directed graph, an
-\f$c_a:A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum
+The \e minimum \e cut \e problem is to find a non-empty and non-complete
+\f$X\f$ subset of the nodes with minimum overall capacity on
+outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a
+\f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum
 cut is the \f$X\f$ solution of the next optimization problem:
 
 \f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}
-\sum_{uv\in A, u\in X, v\not\in X}c_{uv}\f]
+    \sum_{uv\in A, u\in X, v\not\in X}cap(uv) \f]
 
 LEMON contains several algorithms related to minimum cut problems:
 
-- \ref lemon::HaoOrlin "Hao-Orlin algorithm" to calculate minimum cut
-  in directed graphs
-- \ref lemon::NagamochiIbaraki "Nagamochi-Ibaraki algorithm" to
-  calculate minimum cut in undirected graphs
-- \ref lemon::GomoryHuTree "Gomory-Hu tree computation" to calculate all
-  pairs minimum cut in undirected graphs
+- \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut
+  in directed graphs.
+- \ref NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for
+  calculating minimum cut in undirected graphs.
+- \ref GomoryHuTree "Gomory-Hu tree computation" for calculating
+  all-pairs minimum cut in undirected graphs.
 
 If you want to find minimum cut just between two distinict nodes,
-please see the \ref max_flow "Maximum Flow page".
+see the \ref max_flow "maximum flow problem".
 */
 
@@ -321 +367 @@
 graphs.  The matching problems in bipartite graphs are generally
 easier than in general graphs. The goal of the matching optimization
-can be the finding maximum cardinality, maximum weight or minimum cost
+can be finding maximum cardinality, maximum weight or minimum cost
 matching. The search can be constrained to find perfect or
 maximum cardinality matching.
 
-LEMON contains the next algorithms:
-- \ref lemon::MaxBipartiteMatching "MaxBipartiteMatching" Hopcroft-Karp
-  augmenting path algorithm for calculate maximum cardinality matching in
-  bipartite graphs
-- \ref lemon::PrBipartiteMatching "PrBipartiteMatching" Push-Relabel
-  algorithm for calculate maximum cardinality matching in bipartite graphs
-- \ref lemon::MaxWeightedBipartiteMatching "MaxWeightedBipartiteMatching"
-  Successive shortest path algorithm for calculate maximum weighted matching
-  and maximum weighted bipartite matching in bipartite graph
-- \ref lemon::MinCostMaxBipartiteMatching "MinCostMaxBipartiteMatching"
-  Successive shortest path algorithm for calculate minimum cost maximum
-  matching in bipartite graph
-- \ref lemon::MaxMatching "MaxMatching" Edmond's blossom shrinking algorithm
-  for calculate maximum cardinality matching in general graph
-- \ref lemon::MaxWeightedMatching "MaxWeightedMatching" Edmond's blossom
-  shrinking algorithm for calculate maximum weighted matching in general
-  graph
-- \ref lemon::MaxWeightedPerfectMatching "MaxWeightedPerfectMatching"
-  Edmond's blossom shrinking algorithm for calculate maximum weighted
-  perfect matching in general graph
+The matching algorithms implemented in LEMON:
+- \ref MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm
+  for calculating maximum cardinality matching in bipartite graphs.
+- \ref PrBipartiteMatching Push-relabel algorithm
+  for calculating maximum cardinality matching in bipartite graphs.
+- \ref MaxWeightedBipartiteMatching
+  Successive shortest path algorithm for calculating maximum weighted
+  matching and maximum weighted bipartite matching in bipartite graphs.
+- \ref MinCostMaxBipartiteMatching
+  Successive shortest path algorithm for calculating minimum cost maximum
+  matching in bipartite graphs.
+- \ref MaxMatching Edmond's blossom shrinking algorithm for calculating
+  maximum cardinality matching in general graphs.
+- \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating
+  maximum weighted matching in general graphs.
+- \ref MaxWeightedPerfectMatching
+  Edmond's blossom shrinking algorithm for calculating maximum weighted
+  perfect matching in general graphs.
 
 \image html bipartite_matching.png
@@ -356 +400 @@
 
 This group describes the algorithms for finding a minimum cost spanning
-tree in a graph
+tree in a graph.
 */
 
@@ -547 +591 @@
 \anchor demoprograms
 
-@defgroup demos Demo programs
+@defgroup demos Demo Programs
 
 Some demo programs are listed here. Their full source codes can be found in
@@ -557 +601 @@
 
 /**
-@defgroup tools Standalone utility applications
+@defgroup tools Standalone Utility Applications
 
 Some utility applications are listed here.
@@ -565 +609 @@
 */
 
+}