## A Linear Time Algorithm for the Minimum Spanning Tree Problem on a Planar Graph (1994)

### BibTeX

@MISC{Matsui94alinear,

author = {Tomomi Matsui},

title = {A Linear Time Algorithm for the Minimum Spanning Tree Problem on a Planar Graph},

year = {1994}

}

### OpenURL

### Abstract

: In this paper, we propose a linear time algorithm for finding a minimum spanning tree on a planar graph. Keywords: Combinatorial problems; graphs; spanning trees; planar graphs 1 Introduction Finding a spanning tree of minimum weight is one of the best known graph problems. Several efficient algorithms exist for solving this problem [1, 3, 4, 5, 6, 9, 11, 13]. This paper presents a liner time algorithm for the minimum spanning tree problem on a planar graph. In [1], Cheriton and Tarjan have proposed a linear time algorithm for this problem. The time complexity of our algorithm is the same as that of Cheriton and Tarjan's algorithm. Different from Cheriton and Tarjan's algorithm, our algorithm does not require the clean-up activity. So, the implementation of our algorithm is very easy. Our algorithm maintains a pair of a planar graph and its dual graph and breeds both a minimum spanning tree of original graph and a maximum spanning tree of a dual graph. In each iteration of our algor...

### Citations

1466 | A note on two problems in connexion with graphs, Numerische Mathematik 1
- Dijkstra
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Citation Context ...ms; graphs; spanning trees; planar graphs 1 Introduction Finding a spanning tree of minimum weight is one of the best known graph problems. Several efficient algorithms exist for solving this problem =-=[1, 3, 4, 5, 6, 9, 11, 13]-=-. This paper presents a liner time algorithm for the minimum spanning tree problem on a planar graph. In [1], Cheriton and Tarjan have proposed a linear time algorithm for this problem. The time compl... |

1140 |
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Citation Context ... of v is less than four, we can find a minimum weight edge e in ffi G1 (v) in constant time. If we contract the edge e connecting two vertices v and u; it requires O(minfdG1 (v); d G1 (u)g) time (see =-=[2]-=- for example). Since d G1 (v) ! 4; the edge contraction procedure in Step 4 requires O(1) time. Similarly, we can show that the time complexity of Step 6 is O(1): The above discussion implies that whe... |