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=== Finite Search Spaces ===
 
=== Finite Search Spaces ===
  
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The finite search space is characterized by its finite amount of nodes and edges. You can use algorithms or heuristics to find the best solution, but if you have a finite but larger search space it is often more efficient to use a heuristic due to the shorter runtime.
+
The finite search space is characterized by its finite amount of nodes and edges. You can use algorithms or heuristics to find the best solution, but if you have a finite but larger search space it is often more efficient to use heuristic due to the shorter runtime.
  
 
=== Infinite Search Spaces ===  
 
=== Infinite Search Spaces ===  
  
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The infinite search space is characterized by his infinite amount of nodes and edges. You can only use heuristics to find the optimal solution. If we would use algorithms in infinite search spaces, they would not terminate, as there are infinite possibilites to form a solution. Heuristics have the advantage that they do not compute every last possibility but rather trie to narrow down the good solutions, so use heuristics to handle with infinite search spaces.
+
The infinite search space is characterized by its infinite amount of nodes and edges. You can only use heuristics to find the optimal solution. If one would use algorithms in infinite search spaces, it would not terminate, as there are infinite solution possibilites. Heuristics have the advantage of not computing every last possibility, but rather to narrow down to the best solutions. So heuristics should be used for infinite search spaces.
  
 
=== Solution Spaces ===
 
=== Solution Spaces ===
  
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In the solution space, the objective function defines a preference order over the solutions shown.
+
In the solution space, the objective function defines an order of preference among all the possible solutions.
−
There are already potential solutions ( e.g. valid tours of a TSP) which are resembled by the nodes of the graph.
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There are given potential solutions (e.g. valid tours of a TSP) which are represented by the nodes of the graph.
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The edges between the nodes visualize the transition from a given solution to a new (neighboring) solution.
+
The edges between the nodes illustrate the transition from a given solution to a new (neighboring) solution.
  
−
As you can see below, the problem space can be portrayed as a non-directed graph.
+
As you can see below, the problem space can be portrayed as a undirected graph.
  
 
[[Datei:BSP_Solutionspace.jpg ]]
 
[[Datei:BSP_Solutionspace.jpg ]]
−
 
  
 
=== Problem Spaces ===
 
=== Problem Spaces ===
  
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In the problem space the nodes resemble the problems and sub-problems.
+
In the problem space the nodes depict the problems and sub-problems.
 
The root of the tree is the initial problem while the interim nodes are the subproblems.
 
The root of the tree is the initial problem while the interim nodes are the subproblems.
 
The leaves represent the solutions. The edges are the decomposition of a problem into a simpler sub-problem.
 
The leaves represent the solutions. The edges are the decomposition of a problem into a simpler sub-problem.
Zeile 49: Zeile 48:
  
 
[[Datei:BSP_Problemspace.jpg ]]
 
[[Datei:BSP_Problemspace.jpg ]]
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−
 
  
 
== Neighborhoods in General (Mathematically) ==
 
== Neighborhoods in General (Mathematically) ==
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=== In directed Graphs ===
 
=== In directed Graphs ===
 
Let <math>G= \left ( V, R, \alpha, \omega \right )</math> be the directed graph  
 
Let <math>G= \left ( V, R, \alpha, \omega \right )</math> be the directed graph  
−
with the functions <math> \alpha: R \rightarrow V</math> and <math>\omega: R \rightarrow V</math> which define starting and end node
+
with the functions <math> \alpha: R \rightarrow V</math> and <math>\omega: R \rightarrow V</math> which define starting and end node.
 
Let <math>v \in V</math> be a node of <math>G</math>, then you could define:
 
Let <math>v \in V</math> be a node of <math>G</math>, then you could define:
  
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* <math>N_{G}^{-}(v) = \left \{ \alpha(r)\mid r \in \delta_{G}^{-}(v) \right \}</math> set of predecessors of <math>v</math>
 
* <math>N_{G}^{-}(v) = \left \{ \alpha(r)\mid r \in \delta_{G}^{-}(v) \right \}</math> set of predecessors of <math>v</math>
  
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Two nodes <math>u,v \in V</math> are called adjacent also known as neighbored if there exists an arc with <math>\alpha(r)=u</math> and <math>\omega(r)=v</math> or <math>\alpha(r)=v</math> and <math>\omega(r)=u</math>.
+
Two nodes <math>u,v \in V</math> are called adjacent, also known as neighbored, if an arc exists with <math>\alpha(r)=u</math> and <math>\omega(r)=v</math> or <math>\alpha(r)=v</math> and <math>\omega(r)=u</math>.
 
Therefore you could declare the Neighbordhood of <math>v</math> as <math>N(v) = N_{G}^{+}(v) + N_{G}^{-}(v)</math>
 
Therefore you could declare the Neighbordhood of <math>v</math> as <math>N(v) = N_{G}^{+}(v) + N_{G}^{-}(v)</math>
  
 
=== In undirected Graphs ===
 
=== In undirected Graphs ===
−
Let <math> G=(V,E,\gamma)</math> bet the undirected graph  
+
Let <math> G=(V,E,\gamma)</math> be the undirected graph  
−
with the function <math> \gamma : E \rightarrow V * V</math> which defines the two end nodes of the edge
+
with the function <math> \gamma : E \rightarrow V * V</math> which defines the two end nodes of the edge.
−
Two nodes <math>u,v \in V</math> are called adjacent also known as neighbored if there exists an edge with <math>\gamma(e) = \left \{ u, v \right \}</math>.
+
Two nodes <math>u,v \in V</math> are called adjacent, also known as neighbored, if an edge exists with <math>\gamma(e) = \left \{ u, v \right \}</math>.
 
Therefore you could declare the Neighbordhood of <math>v</math> as <math>N(v) = \{ u \in V \mid \gamma(e) = \{ u, v \}\}</math>.
 
Therefore you could declare the Neighbordhood of <math>v</math> as <math>N(v) = \{ u \in V \mid \gamma(e) = \{ u, v \}\}</math>.
  
 
=== Conclusion ===
 
=== Conclusion ===
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Two nodes are neighbored if there is an arc respectively an edge which connects both nodes.
+
Two nodes are neighbored if an arc or an edge connects both nodes.
 +
 
 +
If you want the neighborhood of a set of nodes, you take a look at the neighborhoods of single nodes and sum them up.
 +
 
  
−
If you want the neighborhood of a set of nodes you take a look at the neighborhoods of single nodes and sum them up.
 
  
 
== Heuristics in reference to Search Space and Neighborhoods ==
 
== Heuristics in reference to Search Space and Neighborhoods ==
Zeile 85: Zeile 84:
  
  
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Opening procedures construct a feasible solution with the help of algorithms.
+
The opening procedures construct a feasible solution with the help of algorithms.
  
−
A few examples for opening procedures:
+
A few examples for opening procedures are:
 
::* Nearest Neighbor
 
::* Nearest Neighbor
 
::* Double-sided nearest Neighbor
 
::* Double-sided nearest Neighbor
−
::* Nearest Neighbor
 
 
::* Nearest Addition
 
::* Nearest Addition
 
::* Nearest Insert
 
::* Nearest Insert
Zeile 99: Zeile 97:
  
  
−
Improving procedures take these feasible solutions and try to further improve them.
+
The improving procedures take these feasible solutions and try to further improve them.
  
−
A few examples for improving procedures:
+
A few examples for improving procedures are:
 
::* 2-Opt
 
::* 2-Opt
 
::* 3-Opt
 
::* 3-Opt
Zeile 114: Zeile 112:
  
  
−
Let <math>G=(V,E,\gamma)</math> be the undirected and weigthed graph and <math>T=\{ marked</math> <math>nodes \}</math>
+
Let <math>G=(V,E,\gamma)</math> be the undirected and weighted graph and <math>T=\{ marked</math> <math>nodes \}</math>
 
# Set <math>T=\{ starting </math> <math> node\}</math>
 
# Set <math>T=\{ starting </math> <math> node\}</math>
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# Regarding the Neighborhood of the nodes of T: Chose the edge <math>e=\{u,v\}</math> with a minimum of weight from <math>v \in V \setminus T</math> to <math>u \in T</math> the node you have chosen least
+
# Choose the edge <math>e=\{u,v\}</math> with the least weight from <math>v \in V \setminus T</math> to the node you chose previously <math>u \in T</math>
 
# Add <math>v</math> to <math>T</math>
 
# Add <math>v</math> to <math>T</math>
 
# Repeat step (2) and (3) until <math>T=\{all</math> <math>nodes\}</math>
 
# Repeat step (2) and (3) until <math>T=\{all</math> <math>nodes\}</math>
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# Take the edge which connects the starting node with the last node you have chosen
+
# Take the edge which connects the starting node with the previously chosen node
  
 
=== Example ===
 
=== Example ===
Zeile 134: Zeile 132:
 
<br />Take the nearest neighbor of the current node, which has not been visited yet. Otherwise if all nodes have been visited, go back to the starting node E.
 
<br />Take the nearest neighbor of the current node, which has not been visited yet. Otherwise if all nodes have been visited, go back to the starting node E.
 
<br />E
 
<br />E
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<br />You have a round trip {E,C,A,B,D,F,E} through all nodes.
+
<br />You have a cycle {E,C,A,B,D,F,E} through all nodes.
 
|
 
|
 
|}
 
|}
  
 
==== Initial state ====
 
==== Initial state ====
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Here we have a classic (strongly simplified) Traveling Salesman Problem which we want to solve by using the nearest neighbor heuristic. As the nearest neighbor heuristic is just an opening procedure, we will not get a particularly good value of the objective function and therefore no particularly good travelling route itself. But the solution we are going to find will be a more or less good foundation from which you could further improve the solution by using improving procedures. There are no weighted edges between the nodes shown yet despite them existing already.
+
Here we have a classic (strongly simplified) Traveling Salesman Problem which we want to solve by using the nearest neighbor heuristic. As the nearest neighbor heuristic is just an opening procedure, we will not necessarily get the optimal value of the objective function and therefore not the best travelling route itself. However, the solution we are going to find will be a more or less good foundation from which you could further improve it by using improving procedures. There are no weighted edges between the nodes shown yet, despite their existing.
  
 
{|
 
{|
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==== Iteration steps ====
 
==== Iteration steps ====
  
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Here, we randomly pick on of the nodes (in this case node E) to be our starting node from which we will iterate further. The edges from E to any other node is now presented by the lines and the weights are shown as numbers above them. As you can see, the weights differ from edge to edge and because we want to find a preferably short route through all the edges we choose the edge with the smallest weight from E to another node.
+
Here, we randomly pick one of the nodes (in this case node E) to be our starting node from which we will further iterate. The edges from E to any other node are now presented by the lines and the weights as numbers above them. As you can see, the weights differ from edge to edge and because we want to find a preferably short route through all the edges we choose the edge with the smallest weight from E to another node.
 
We are confronted by a ''special case'' of the nearest neighbor heuristic: ''Two edges share the smallest weight''. As the algorithm is not particularly foreshadowing we can now choose randomly between the two smallest edges.
 
We are confronted by a ''special case'' of the nearest neighbor heuristic: ''Two edges share the smallest weight''. As the algorithm is not particularly foreshadowing we can now choose randomly between the two smallest edges.
  
Zeile 154: Zeile 152:
 
|}
 
|}
  
−
We chose node C and are now confronted by new edges between C and all other nodes we have not been to yet. The shortest route beginning from C would now be to A.
+
We chose node C and are now confronted by new edges between C and all other nodes we have not yet visited. The shortest route beginning from C would now be to A.
  
 
{|
 
{|
Zeile 160: Zeile 158:
 
|}
 
|}
  
−
Iterate the last steps:
+
Repeat the last steps:
  
 
{|
 
{|
Zeile 168: Zeile 166:
 
|}
 
|}
  
−
This goes on until we reach the final node that has not been visited yet, in this case node F. Now, as we have a Travelling Salesman Problem, we just need to close the cycle by connecting the last node with the starting node.
+
This goes on until we reach the final node that has not yet been visited, in this case node F. Now, as we have a Travelling Salesman Problem, we just need to close the cycle by connecting the last node with the starting node.
  
 
{|
 
{|
Zeile 177: Zeile 175:
  
  
−
With this, the algorithm from the nearest neighbor heuristic has ended and we found an opening travelling route from which we can further improve our round trip. The last step would now be the accumulation of the weights on the travelled edges and we got the value of our route.
+
With this, the algorithm from the nearest neighbor heuristic has ended and we found an opening travelling route from which we can further improve our cycle. The last step would be the addition of the weights of the travelled edges so we get the value of our route.
  
 
In this case: 2+2+3+5+1+6 = 19
 
In this case: 2+2+3+5+1+6 = 19
  
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The solution we want to present is: <math>E\rightarrow C \rightarrow A \rightarrow B \rightarrow D \rightarrow F\rightarrow E</math>
+
The solution is presented as: <math>E\rightarrow C \rightarrow A \rightarrow B \rightarrow D \rightarrow F\rightarrow E</math>
  
 
== Sources ==
 
== Sources ==
Zeile 205: Zeile 203:
  
 
:: Brought to you by Anna Pfeiffer, Markus Kaiser and Viktor Barie
 
:: Brought to you by Anna Pfeiffer, Markus Kaiser and Viktor Barie
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Comment by group 2
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 +
In your wiki- entry you described the theoretical backgrounds of "search space" and "neighborhood" very detailed! You pointed out all aspects of this topic pretty well and we couldn´t find any mistakes in your description. We could have done this in a more detailed way, too. In difference to our entry you focused on the nearest neighbor algorithm which is the simplest one of of the search algorithms.
 +
We think if we combine both of our entries (your theoretical part + nearest neighbor algorithm + our VNS )  we will cover the whole topic!

Aktuelle Version vom 8. Juli 2013, 18:20 Uhr

Search Problems in General

The primary difference between an optimization problem and a search problem is, that the search problem concentrates on the variables you have to fill in for the optimal function value, whereas the optimization problem only focuses on the optimal value itself.

Search spaces are defined by the following:

  • Search Space S:
  • Set of valid transition operators:
  • Initial state from set S:
  • Subset of goal states in S:

See below.
Function which describes the valid sub-solutions for the successiv steps.
The starting point from which the problem begins to unfold.
Describes the boundaries of a valid solution.

Search Spaces

There are multiple definitions of the Search Space. You can divide it into infinite and finite search spaces and in solution and problem spaces.

Finite Search Spaces

The finite search space is characterized by its finite amount of nodes and edges. You can use algorithms or heuristics to find the best solution, but if you have a finite but larger search space it is often more efficient to use heuristic due to the shorter runtime.

Infinite Search Spaces

The infinite search space is characterized by its infinite amount of nodes and edges. You can only use heuristics to find the optimal solution. If one would use algorithms in infinite search spaces, it would not terminate, as there are infinite solution possibilites. Heuristics have the advantage of not computing every last possibility, but rather to narrow down to the best solutions. So heuristics should be used for infinite search spaces.

Solution Spaces

In the solution space, the objective function defines an order of preference among all the possible solutions. There are given potential solutions (e.g. valid tours of a TSP) which are represented by the nodes of the graph. The edges between the nodes illustrate the transition from a given solution to a new (neighboring) solution.

As you can see below, the problem space can be portrayed as a undirected graph.

Fehler beim Erstellen des Vorschaubildes: Die Miniaturansicht konnte nicht am vorgesehenen Ort gespeichert werden

Problem Spaces

In the problem space the nodes depict the problems and sub-problems. The root of the tree is the initial problem while the interim nodes are the subproblems. The leaves represent the solutions. The edges are the decomposition of a problem into a simpler sub-problem.

As you can see below, the problem space can be portrayed as a spanning tree.

Fehler beim Erstellen des Vorschaubildes: Die Miniaturansicht konnte nicht am vorgesehenen Ort gespeichert werden

Neighborhoods in General (Mathematically)

In directed Graphs

Let Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): G= \left ( V, R, \alpha, \omega \right ) 

be the directed graph

with the functions Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \alpha: R \rightarrow V 

and Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \omega: R \rightarrow V
which define starting and end node.

Let Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): v \in V 

be a node of , then you could define:
  • Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \delta_{G}^{+}(v) = \left \{ r \in R \mid \alpha(r) = v \right \}
set of outgoing arcs of
  • Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \delta_{G}^{-}(v) = \left \{ r \in R \mid \omega(r) = v \right \}
set of incoming arcs of
  • Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): N_{G}^{+}(v) = \left \{ \omega(r)\mid r \in \delta_{G}^{+}(v) \right \}
set of successors of
  • Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): N_{G}^{-}(v) = \left \{ \alpha(r)\mid r \in \delta_{G}^{-}(v) \right \}
set of predecessors of

Two nodes Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): u,v \in V 

are called adjacent, also known as neighbored, if an arc exists with  and Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \omega(r)=v
or  and Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \omega(r)=u

. Therefore you could declare the Neighbordhood of as Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): N(v) = N_{G}^{+}(v) + N_{G}^{-}(v)


In undirected Graphs

Let be the undirected graph with the function Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \gamma : E \rightarrow V * V 

which defines the two end nodes of the edge.

Two nodes Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): u,v \in V 

are called adjacent, also known as neighbored, if an edge exists with .

Therefore you could declare the Neighbordhood of as Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): N(v) = \{ u \in V \mid \gamma(e) = \{ u, v \}\} .

Conclusion

Two nodes are neighbored if an arc or an edge connects both nodes.

If you want the neighborhood of a set of nodes, you take a look at the neighborhoods of single nodes and sum them up.


Heuristics in reference to Search Space and Neighborhoods

Opening procedures

The opening procedures construct a feasible solution with the help of algorithms.

A few examples for opening procedures are:

  • Nearest Neighbor
  • Double-sided nearest Neighbor
  • Nearest Addition
  • Nearest Insert
  • Cheapest Insert
  • Farthest Insert

Improving procedures

The improving procedures take these feasible solutions and try to further improve them.

A few examples for improving procedures are:

  • 2-Opt
  • 3-Opt
  • R-Opt
  • 2-Nodes-Opt

The Nearest-Neighbor-Algorithm

In General

The algorithm belongs to the category of "greedy-algorithms" and has a runtime of .


Let be the undirected and weighted graph and

  1. Set
  2. Choose the edge with the least weight from Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): v \in V \setminus T
to the node you chose previously Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): u \in T
  1. Add to
  2. Repeat step (2) and (3) until
  3. Take the edge which connects the starting node with the previously chosen node

Example

The Search Problem in reference to the example

  • Search Space S:
  • Set of valid transition operators:
  • Initial state from set S:
  • Subset of goal states in S:

All nodes {A,B,C,D,E,F} - therefore it is a finite Problem Space
Take the nearest neighbor of the current node, which has not been visited yet. Otherwise if all nodes have been visited, go back to the starting node E.
E
You have a cycle {E,C,A,B,D,F,E} through all nodes.

Initial state

Here we have a classic (strongly simplified) Traveling Salesman Problem which we want to solve by using the nearest neighbor heuristic. As the nearest neighbor heuristic is just an opening procedure, we will not necessarily get the optimal value of the objective function and therefore not the best travelling route itself. However, the solution we are going to find will be a more or less good foundation from which you could further improve it by using improving procedures. There are no weighted edges between the nodes shown yet, despite their existing.

initial situation

Iteration steps

Here, we randomly pick one of the nodes (in this case node E) to be our starting node from which we will further iterate. The edges from E to any other node are now presented by the lines and the weights as numbers above them. As you can see, the weights differ from edge to edge and because we want to find a preferably short route through all the edges we choose the edge with the smallest weight from E to another node. We are confronted by a special case of the nearest neighbor heuristic: Two edges share the smallest weight. As the algorithm is not particularly foreshadowing we can now choose randomly between the two smallest edges.

after choosing the first node

We chose node C and are now confronted by new edges between C and all other nodes we have not yet visited. The shortest route beginning from C would now be to A.

iteration 2

Repeat the last steps:

iteration 3
iteration 4
iteration 5
iteration 6

This goes on until we reach the final node that has not yet been visited, in this case node F. Now, as we have a Travelling Salesman Problem, we just need to close the cycle by connecting the last node with the starting node.

final solution

Summary

With this, the algorithm from the nearest neighbor heuristic has ended and we found an opening travelling route from which we can further improve our cycle. The last step would be the addition of the weights of the travelled edges so we get the value of our route.

In this case: 2+2+3+5+1+6 = 19

The solution is presented as: Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): E\rightarrow C \rightarrow A \rightarrow B \rightarrow D \rightarrow F\rightarrow E


Sources

Scripts:

  • Operations Research - Prof. Dr. Oliver Wendt (SS 2013)
  • Introduction to Network Optimization - Dr. Clemens Thielen (SS 2013)

Books:

  • Optimierung, Operations Research, Spieltheorie - Karl Heinz Borgwardt
  • Lineare Optimierung und Netzwerkoptimierung - Horst W. Hamacher und Kathrin Klamroth
  • Einführung in Operations Research - W. Domschke und A. Drexl

Internet:

About:

Brought to you by Anna Pfeiffer, Markus Kaiser and Viktor Barie



Comment by group 2

In your wiki- entry you described the theoretical backgrounds of "search space" and "neighborhood" very detailed! You pointed out all aspects of this topic pretty well and we couldn´t find any mistakes in your description. We could have done this in a more detailed way, too. In difference to our entry you focused on the nearest neighbor algorithm which is the simplest one of of the search algorithms. We think if we combine both of our entries (your theoretical part + nearest neighbor algorithm + our VNS ) we will cover the whole topic!

Von „https://www.wiwi.uni-kl.de/bisor-orwiki/index.php?title=Heuristics:_Representation_of_Search_Space_and_Neighborhoods_1&oldid=6807“