Circulant matrix: Difference between revisions

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{{Infobox Algorithm
|class=[[Search Algorithm]]
|image=
|data=[[Graph (data structure)|Graph]]
|time=<math>O(|V|+|E|)</math>
|space=
|optimal=no
|complete=no
}}
{{Tree search algorithm}}
In [[computer science]] '''depth-limited search''' is an [[algorithm]] to explore the [[vertex (graph theory)|vertices]] of a [[Graph (mathematics)|graph]]. It is  a modification of [[depth-first search]] and is used for example in the [[iterative deepening depth-first search]] algorithm.


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Like the normal depth-first search, depth-limited search is an [[uninformed search]]. It works exactly like depth-first search, but avoids its drawbacks regarding completeness by imposing a maximum limit on the depth of the search. Even if the search could still expand a vertex beyond that depth, it will not do so and thereby it will not follow infinitely deep [[Path (graph theory)|paths]] or get stuck in [[Path (graph theory)|cycles]]. Therefore depth-limited search will find a solution if it is within the depth limit, which guarantees at least completeness on all graphs.
 
== Algorithm (informal) ==
 
# Determine the vertex where the search should start and assign the maximum search depth
# Check if the current vertex is the goal state
#* If not: Do nothing
#* If yes: return
# Check if the current vertex is within the maximum search depth
#* If not: Do nothing
#* If yes:
#*# Expand the vertex and save all of its successors in a [[Stack (data structure)|stack]]
#*# Call DLS recursively for all vertices of the stack and go back to Step 2
 
== Pseudocode ==
 
'''DLS'''(node, goal, depth) {
  if ( depth >= 0 ) {
    if ( node == goal )
      return node
    for each child in expand(node)
      '''DLS'''(child, goal, depth-1)
  }
}
 
== Properties ==
 
=== Space complexity ===
 
Since depth-limited search internally uses [[depth-first search]], the space [[Computational complexity theory|complexity]] is equivalent to that of normal depth-first search.
 
=== Time complexity ===
 
Since depth-limited search internally uses depth-first-search, the time complexity is equivalent to that of normal depth-first search, and is O(<math> \vert V \vert + \vert E \vert </math>) where <math> \vert V \vert </math> stands for the number of vertices and <math> \vert E \vert </math> for the number of edges in the explored [[Graph (mathematics)|graph]]. Note that depth-limited search does not explore the entire graph, but just the part that lies within the specified bound.
 
===Completeness===
Even though depth-limited search cannot follow infinitely long paths, nor can it get stuck in cycles, in general the algorithm is not complete since it does not find any solution that lies beyond the given search depth. But if the maximum search depth is chosen to be greater than the depth of a solution the algorithm becomes complete.
 
===Optimality===
Depth-limited search is not optimal. It still has the problem of depth-first search that it first explores one path to its end, thereby possibly finding a solution that is more expensive than some solution in another path.
 
==Literature==
*{{Russell Norvig 2003}}
 
[[Category:Graph algorithms]]
[[Category:Search algorithms]]

Latest revision as of 04:18, 30 July 2014

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