|
|
| Line 1: |
Line 1: |
| [[File:Boruvka's algorithm (Sollin's algorithm) Anim.gif|thumb|right|500px|An animation, describing Boruvka's (Sollin's) algorithm, for finding a minimum spanning tree in a graph - An example on the runtime of the algorithm]]
| | My name is Luca Wicks. I life in Baulmes (Switzerland).<br><br>Review my webpage :: [http://hemorrhoidtreatmentfix.com/thrombosed-hemorrhoid-treatment thrombosed hemorrhoids treatment] |
| {{graph search algorithm}}
| |
| | |
| '''Borůvka's algorithm''' is an [[algorithm]] for finding a [[minimum spanning tree]] in a graph for which all edge weights are distinct.
| |
| | |
| It was first published in 1926 by [[Otakar Borůvka]] as a method of constructing an efficient [[electricity network]] for [[Moravia]].<ref>{{cite journal | last = Borůvka | first = Otakar | authorlink = Otakar Borůvka | year = 1926 | title = O jistém problému minimálním (About a certain minimal problem) | journal = Práce mor. přírodověd. spol. v Brně III | volume = 3 | pages = 37–58 | language = Czech, German summary }}</ref><ref>{{cite journal | last = Borůvka | first = Otakar | authorlink = Otakar Borůvka | year = 1926 | title = Příspěvek k řešení otázky ekonomické stavby elektrovodních sítí (Contribution to the solution of a problem of economical construction of electrical networks) | journal = Elektronický Obzor | volume = 15 | pages = 153–154 | language = Czech }}</ref><ref>{{cite journal
| |
| | last1 = Nešetřil | first1 = Jaroslav | author1-link = Jaroslav Nešetřil
| |
| | last2 = Milková | first2 = Eva
| |
| | last3 = Nešetřilová | first3 = Helena
| |
| | doi = 10.1016/S0012-365X(00)00224-7
| |
| | issue = 1–3
| |
| | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
| |
| | mr = 1825599
| |
| | pages = 3–36
| |
| | title = Otakar Borůvka on minimum spanning tree problem: translation of both the 1926 papers, comments, history
| |
| | volume = 233
| |
| | year = 2001}}</ref>
| |
| The algorithm was rediscovered by [[Gustave Choquet|Choquet]] in 1938;<ref>{{cite journal | last = Choquet | first = Gustave | authorlink = Gustave Choquet | year = 1938 | title = Étude de certains réseaux de routes | journal = Comptes-rendus de l’Académie des Sciences | volume = 206 | pages = 310–313 | language = French }}</ref> again by [[Kazimierz Florek|Florek]], [[Jan Łukasiewicz|Łukasiewicz]], [[Julian Perkal|Perkal]], [[Hugo Steinhaus|Steinhaus]], and [[Stefan Zubrzycki|Zubrzycki]]<ref>{{cite journal | last = Florek | first = Kazimierz | authorlink = Kazimierz Florek | year = 1951 | title = Sur la liaison et la division des points d'un ensemble fini | journal = Colloquium Mathematicum 2 (1951) | pages = 282–285 | language = French }}</ref> in [[Hugo Steinhaus#graphs1951|1951]]; and again by [[Sollin]]
| |
| <ref>{{cite journal | last = Sollin | first = M. | authorlink = M. Sollin | year = 1965 | title = Le tracé de canalisation | journal = Programming, Games, and Transportation Networks | language = French }}</ref> in 1965. Because [[Sollin]] was the only computer scientist in this list living in an English speaking country, this algorithm is frequently called '''Sollin's algorithm''', especially in the [[parallel computing]] literature.
| |
| | |
| The algorithm begins by first examining each vertex and adding the cheapest edge from that vertex to another in the graph, without regard to already added edges, and continues joining these groupings in a like manner until a tree spanning all vertices is completed.
| |
| | |
| == Pseudocode ==
| |
| Designating each vertex or set of connected vertices a "[[connected component (graph theory)|component]]", pseudocode for Borůvka's algorithm is:
| |
| <source lang="text" line start="0" highlight="1,9">
| |
| Input: A connected graph G whose edges have distinct weights
| |
| Initialize a forest T to be a set of one-vertex trees, one for each vertex of the graph.
| |
| While T has more than one component:
| |
| For each component C of T:
| |
| Begin with an empty set of edges S
| |
| For each vertex v in C:
| |
| Find the cheapest edge from v to a vertex outside of C, and add it to S
| |
| Add the cheapest edge in S to T
| |
| Output: T is the minimum spanning tree of G.
| |
| </source>
| |
| As in [[Kruskal's algorithm]], tracking components of T can be done efficiently using a [[disjoint-set data structure]]. In graphs where edges have identical weights, edges with equal weights can be ordered based on the [[lexicographic order]] of their endpoints.
| |
| | |
| == Complexity ==
| |
| | |
| Borůvka's algorithm can be shown to take [[Big O notation|O]](log ''V'') iterations of the outer loop until it terminates, and therefore to run in time [[Big O notation|O]](''E'' log ''V''), where ''E'' is the number of edges, and ''V'' is the number of vertices in ''G''. In [[planar graph]]s, and more generally in families of graphs closed under [[graph minor]] operations, it can be made to run in linear time, by removing all but the cheapest edge between each pair of components after each stage of the algorithm.<ref>{{Cite book|last=Eppstein|first=David|authorlink=David Eppstein|contribution=Spanning trees and spanners|title=Handbook of Computational Geometry|editor1-first=J.-R.|editor1-last=Sack|editor1-link=Jörg-Rüdiger Sack|editor2-first=J.|editor2-last=Urrutia|publisher=Elsevier|year=1999|pages=425–461|postscript=<!--None-->}}; {{Cite journal|last=Mareš|first=Martin|title=Two linear time algorithms for MST on minor closed graph classes|journal=Archivum mathematicum|volume=40|year=2004|issue=3|pages=315–320|url=http://www.emis.de/journals/AM/04-3/am1139.pdf|postscript=<!--None-->}}.</ref>
| |
| | |
| == Example ==
| |
| | |
| {| border=1 cellspacing=2 cellpadding=5 class="wikitable"
| |
| ! | Image
| |
| ! width="100" | components
| |
| ! | Description
| |
| |-
| |
| |[[File:Borůvka Algorithm 1.svg|200px]]
| |
| | {A}<br/>{B}<br/>{C}<br/>{D}<br/>{E}<br/>{F}<br/>{G}
| |
| |This is our original weighted graph. The numbers near the edges indicate their weight. Initially, every vertex by itself is a component (blue circles).
| |
| |-
| |
| |[[File:Borůvka Algorithm 2.svg|200px]]
| |
| | {A,B,D,F}<br/>{C,E,G}
| |
| |In the first iteration of the outer loop, the minimum weight edge out of every component is added. Some edges are selected twice (AD, CE). Two components remain.
| |
| |-
| |
| |[[File:Borůvka Algorithm 3.svg|200px]]
| |
| | {A,B,C,D,E,F,G}
| |
| |In the second and final iteration, the minimum weight edge out of each of the two remaining components is added. These happen to be the same edge. One component remains and we are done. The edge BD is not considered because both endpoints are in the same component.
| |
| |}
| |
| | |
| == Other algorithms ==
| |
| | |
| Other algorithms for this problem include [[Prim's algorithm]] and [[Kruskal's algorithm]]. Fast parallel algorithms can be obtained by combining Prim's algorithm with Borůvka's.<ref>{{cite journal|last1=Bader|first1=David A.|last2=Cong|first2=Guojing|title=Fast shared-memory algorithms for computing the minimum spanning forest of sparse graphs|journal=Journal of Parallel and Distributed Computing|date=1 November 2006|volume=66|issue=11|pages=1366–1378|doi=10.1016/j.jpdc.2006.06.001}}</ref>
| |
| | |
| A faster randomized minimum spanning tree algorithm based in part on Borůvka's algorithm due to Karger, Klein, and Tarjan runs in expected <math>O(E)</math> time.<ref>{{cite journal|last1=Karger|first1=David R.|last2=Klein|first2=Philip N.|last3=Tarjan|first3=Robert E.|title=A randomized linear-time algorithm to find minimum spanning trees|journal=Journal of the ACM|date=1 March 1995|volume=42|issue=2|pages=321–328|doi=10.1145/201019.201022}}</ref> The best known (deterministic) minimum spanning tree algorithm by [[Bernard Chazelle]] is also based in part on Borůvka's and runs in O(''E'' α(''E'',''V'')) time, where α is the inverse of the [[Ackermann function]].<ref>{{Cite journal|last=Chazelle|first=Bernard|title=A minimum spanning tree algorithm with inverse-Ackermann type complexity|journal=J. ACM|volume=47|year=2000|issue=6|pages=1028–1047|url=http://www.cs.princeton.edu/~chazelle/pubs/mst.pdf}}</ref> These randomized and deterministic algorithms combine steps of Borůvka's algorithm, reducing the number of components that remain to be connected, with steps of a different type that reduce the number of edges between pairs of components.
| |
| | |
| ==Notes==
| |
| <references/>
| |
| | |
| {{DEFAULTSORT:Boruvka's Algorithm}}
| |
| [[Category:Graph algorithms]]
| |
| [[Category:Spanning tree]]
| |
My name is Luca Wicks. I life in Baulmes (Switzerland).
Review my webpage :: thrombosed hemorrhoids treatment