D. Cheriton and R. E. Tarjan, Finding minimum spanning trees, SIAM J. Comput. 5 (1976), 724-742. 10  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of S are vertices, and the elements of V (T ) n S are Steiner points of the SMT. Estimates for the largest degrees of MST s and SMT s have consequences for the complexities of algorithms that nd such trees. For example, it is known that an MST on n points can be calculated in polynomial time [2], while calculating the SMT in the euclidean or rectilinear planes is NP hard [7, 8] Upper bounds for the degrees of vertices and Steiner points are used to reduce the search space of known exponential time algorithms. Distance functions other than euclidean or rectilinear are sometimes used. The ....

D. Cheriton and R. E. Tarjan, Finding minimum spanning trees, SIAM J. Comput. 5 (1976), 724-742. 10

....v; w) is depicted by an arrow from u to v. If the arrow surpasses the subdomain which encapsulates u, u is stored as unsolved according to steps 3.1, 3.2. As it may be observed, only in processor P 0 there is an inner arrow from node 3 to node 4 which, conforming to step 3.3, coalesce. r 3 = root[3] = 3, r 4 = root[4] 4 r 34 = 3 and root[4] 3. Because when merging two trees the new edges list results by a simple concatenation of the two lists associated with each of the trees, internal edges (u; v) having both end nodes in the same tree (find(root; u) find(root; v) invalid ....

....because of the irregular distribution of vertices (basins) and markers to processors. However, in the following, a lower bound estimate of the real performance is given. The analysis heavily relies on the complexity of the local MSF operator for a graph with m edges and n vertices as presented in [3]. The formula follows easily by observing that after every stage in the local MSF the number of remaining trees is at most half the number of trees at the input of the stage. Indeed, by merging two trees, one tree emerges in the next stage. If one of the trees has been already enqueued for the ....

D. Cheriton and R.E. Tarjan, "Finding Minimum Spanning Trees," SIAM J. Comp., vol. 5, no. 4, pp. 724--742, December 1976.

....algorithms in terms of simplicity but did not improve on the O(m log n) time bound first established by Boruvka. Here m and n are the number of edges and vertices in the graph. The m log n barrier was broken in the mid 1970s by O(m log log n) time algorithms by [Yao75] and Cheriton and Tarjan [CT76]. The MST problem saw no new developments until the mid 1980s when Fredman and Tarjan [FT87] used Fibonacci heaps (presented in the same paper) to give an algorithm running in O(m fi(m; n) time ; in the worst case this algorithm runs in O(m log n) time . Soon thereafter Gabow et al. ....

D. Cheriton, R. E. Tarjan. Finding minimum spanning trees. In SIAM J. Comput. 5 (1976), pp. 724--742.

.... homework 6 in [13] Gale and Shapley were the rst to investigate the stable marriage problem and gave an O(n 2 ) solution in [8] 1 Algorithms February 13, 2001 Homework 3 1: Minimum Spanning Trees A linear time algorithm for nding a minimum spanning tree for planar graph was rst given in [5]. The O(m log n) time algorithm for nding a minimum spanning tree in a general graph was described in [7] the paper introducing Fibonacci heaps. The current best dertministic minimum spanning tree algorithms use time O(m (m;n) where is an inverse of Ackerman s function [4, 17] A ....

D. Cheriton and R. E. Tarjan. Finding minimum spanning trees. SIAM Journal of Computing, 5:724742, 1976.

....by Gabow et al. 16] is O(m log #(m, n) for a graph with n vertices and m edges. Here #(m, n) is defined to be min i log (i) n # m n , and log (i) x denotes the log function iterated i times. This is extremely close to linear time. For planar graphs, a MST can be found in linear time [8]. Minimum spanning trees have applications in many areas, including network design, VLSI, and geometric optimization. Yet in many cases what is desired is not necessarily the best spanning tree, but rather a good tree with some other properties that may be di#cult to quantify. For instance, ....

....in any of the k best spanning trees for G. # Putting it all together, we have our result: Theorem 1. The k least weight spanning trees of a graph can be found in time bounded by O(m log #(m, n) k 2 ) for a planar graph they can be found in time O(n k 2 ) Proof: By results of [16] and [8], the MST of the graph can be found in the given bounds. We can use lemma 7 to reduce the graph to one in which there are k vertices, and therefore k 1 tree edges. Then we can use lemma 11 to reduce the graph to one in which there are k 1 non tree edges. Therefore the total number of edges ....

D. Cheriton and R.E. Tarjan, Finding Minimum Spanning Trees, SIAM J. Comput. 5, 1976, 310--313.

....a degree constraint d on the vertices of our spanning trees, then a degree constrained minimum spanning tree (d MST) of a weighted graph G is the minimum spanning tree in G in which the degree of no vertex exceeds d. The problem of finding an MST in a weighted graph has received much attention [1, 2, 3]. Again, the problem is of interest in communications networks where one can imagine that laying wires between pairs of exchanges incurs some cost. The problem is then to find a set of connections which minimises the overall cost. All proposed algorithms for finding an MST are greedy (or ....

....from the connected list U and place it in C. 7. Repeat steps 5 and 6 until there are no vertices left in U . Efficient implementation of Prim s algorithm, making use of good priority queues, leads to a running time of O(m log n) where m is the number of edges and n is the number of vertices [3]. Using Prim s algorithm, and then adding further edges in an appropriately weight biased way, facilitates any search process which needs to find good connected graphs. A simple extension to this which we call dPrim s was first described by [4] This differs from Prim s only in step 5, ....

Cheriton, D., Tarjan, R.E.: Finding Minimum Spanning Trees. SIAM Journal of Computing 5(4), (1976) 724--741

....as shown by Waxman and Imase [17] We fully describe also the following three heuristics because they are not very known and the original sources are not easily accessible. 2 TH (the minimum 2 tuple heuristic of Wang [15] It is similar to PH which is Prim based, but 2 TH is Kruskal based (cf. [3]) Step 1: Begin with the collection F of single vertex trees consisting of the p Z vertices. Step 2: If jF j = 1, then the single tree in F is the solution T 2 TH , STOP. Else find two trees T 1 , T 2 2 F with minimal distance d(T 1 ; T 2 ) in G and join them by a shortest path (between T 1 and ....

....successively for every iteration i of ADHF as follows. In the i th iteration of ADHF put c 0 (uw) ff (i) whenever u and w belong to distinct trees of fT (i) 1 ; T (i) r (i) g F (i) Now apply to K(Z) with c 0 the Kruskal minimum spanning tree algorithm (MSTA) see e.g. [3]) Each iteration i of ADHF connecting trees T (i) 1 ; T (i) r (i) can be associated with r (i) Gamma 1 iterations of MSTA as follows. For each j = 1; r (i) choose (arbitrarily) a Z vertex u j in T (i) j . Then add r (i) Gamma 1 edges, one per iteration of ....

Cheriton D. and Tarjan R. E., Finding minimum spanning trees, SIAM J. Computing 5 (1976), 724--742.

.... MST algorithms Algorithm Variety Running Time Auxiliary Storage Kruskal all O(jEj log jV j) O(jEj) Prim binary heap O(jEj log jV j) O(jV j) Prim d heap O (jEj djV j) log d jV j O(jV j) Prim Fibonacci heap relaxed heap O(jEj jV j log jV j) O(jV j) Cheriton and Tarjan [1] O(jEj log log jV j) O(jEj) Fredman and Tarjan [3] O(jEj (jEj; jV j) O(jEj) Gabow et al. O(jEj log (jEj; jV j) O(jEj) and lower order terms; this relative independence from implementation decisions is, of course, what makes asymptotic analysis attractive, but it is also what makes it ....

Cheriton, D., and R.E. Tarjan, Finding minimum spanning trees, SIAM J. Comput. 5 (1976), pp. 724-742.

.... homework 6 in [13] Gale and Shapley were the rst to investigate the stable marriage problem and gave an O(n 2 ) solution in [8] 1 Algorithms January 17, 2000 Homework 3 1: Minimum Spanning Trees A linear time algorithm for nding a minimum spanning tree for planar graph was rst given in [5]. The O(m log n) time algorithm for nding a minimum spanning tree in a general graph was described in [7] the paper introducing Fibonacci heaps. The current best dertministic minimum spanning tree algorithms use time O(m (m;n) where is an inverse of Ackerman s function [4, 17] A ....

D. Cheriton and R. E. Tarjan. Finding minimum spanning trees. SIAM Journal of Computing, 5:724742, 1976.

.... comparison based MST algorithms Algorithm Variety Running Time Auxiliary Storage Kruskal all O(jEj log jV j) O(jEj) Prim binary heap O(jEj log jV j) O(jV j) Prim d heap O( jEj djV j) log d jV j) O(jV j) Prim Fibonacci heap relaxed heap O(jEj jV j log jV j) O(jV j) Cheriton and Tarjan [1] O(jEj log log jV j) O(jEj) Fredman and Tarjan [3] O(jEj Delta fi(jEj; jV j) O(jEj) Gabow et al. O(jEj Delta log fi(jEj; jV j) O(jEj) considering alternative designs for a problem such as the MST, where all algorithms are fast and thus where even modest leading coefficients play a large ....

Cheriton, D., and R.E. Tarjan, "Finding minimum spanning trees," SIAM J. Comput. 5 (1976), pp. 724--742.

....subgraphs of the Delaunay diagram play a crucial role within this approach. The problem of efficiently computing subgraphs of the Delaunay diagram of a set of points is the subject of a rich body of literature in computational geometry. One of the first papers is due to Cheriton and Tarjan [6], who show a O(n) time algorithm to compute an Euclidean minimumspanning tree from a Delaunay diagram with n vertices. The Gabriel graph can be computed from the Delaunay diagram in O(n) time by using the algorithm of Matula and Sokal [19] The first O(n log n) time algorithm to compute a relative ....

D. Cheriton and R. E. Tarjan. Finding minimum spanning trees. SIAM J. Comput., 5:724--742, 1976.

....are Prim s algorithm and Kruskal s algorithm [12, 14] Both have a quadratic time complexity. More ecient algorithms are known. In [15] for example, an alternative method is given which has a lower time bound of O(jEj log log jV j) but which is more intricate. For other improved algorithms see [2, 3, 4, 9, 10, 11]. 3 Properties of optimal spanning trees In this section we give some properties of optimal spanning trees and SFD of the optimal vertex orderings of a graph. 6 q w q v k q v 2 q v 1 q u a) q u q v 1 q v k 1 q v k q w b) Figure 4: ....

D. Cheriton, R. E. Tarjan, Finding minimum spanning-trees, SIAM J. on Comput., 5 (4), 724-742, 1976.

....of the EMST. Thus, since the EMST is the smallest spanning graph, the minimum spanning tree for the Delaunay triangulation has total length at most twice the length of the EMST. We can find the minimum spanning tree for the Delaunay triangulation in linear time since it is a planar 11 graph [CT76]. Thus, after two initial sorts, we can find a spanning tree that is within a factor of two of the EMST in O(n log log n) time. Even Better EMST. Several triangle based Delaunay triangulations can be combined into a single (nonplanar) graph with the property that, for any two vertices A and B, ....

D. Cheriton and R. E. Tarjan, Finding Minimum Spanning Trees, SIAM J. Comput. , 5 (1976), 724--742.

....on the O(m log n) time bound first established by Boruvka. Here m (resp. n) is the number of edges (resp. vertices) in the graph. The m log n barrier was broken by Yao s O(m log log n) time algorithm 1 [Yao75] which was followed quickly by Cheriton and Tarjan s O(m log log d n) time algorithm [CT76], where d = maxf2; m n g. The MST problem saw no new developments until the mid 1980s when Fredman and Tarjan [FT87] used Fibonacci heaps (presented in the same paper) to give an algorithm running in O(m fi(m; n) time 2 . In the worst case, fi(m; n) log n) Before the ink had dried on this ....

D. Cheriton, R. E. Tarjan. Finding minimum spanning trees. In SIAM J. Comput. 5 (1976), pp. 724--742.

....from the connected list U and place it in C. 7 7. Repeat steps 5 and 6 until there are no vertices left in U . Efficient implementation of Prim s algorithm, making good use of priority queues, leads to a running time of O(m log n) where m is the number of edges and n is the number of vertices [14]. There are three methods by which Prim s algorithm can be employed for finding solutions to the d MST. The first method is to make an alteration to Prim s algorithm so that it does not add edges that violate the degree constraint. The second, a dual simplex approach, begins with a superoptimal ....

D. Cherition and R. E. Tarjan, "Finding Minimum Spanning Trees," SIAM Journal of Computing, vol. 5, no. 4, pp. 724--741, December 1976.

....or Yao graph described in Lemma 3 in time O(n log n) Any of the classical graph minimum spanning tree algorithms can then find the minimum spanning tree of these graphs in time O(n log n) or better. The first two graphs are planar, so Boruvka s algorithm or a method of Cheriton and Tarjan [29] can be used to find their minimum spanning trees in linear time. By the lemmas above, the minimum spanning trees of these graphs are exactly the geometric minimum spanning trees of the points. # 2.1.2 Higher dimensional MSTs The methods described above, for constructing graphs containing the ....

D. Cheriton and R. E. Tarjan. Finding minimum spanning trees. SIAM J. Comput.. vol. 5, 1976, pp. 310--313.

....(S) where the weight of an edge joining the two vertices that are dual to the faces VR (S; p) and VR (S; q) resp. of VR (S) is d(p; q) Since the dual graph of VR (S) is a planar graph of size O(n k) and may also be determined from VR (S) in time O(n k) the algorithm of Cheriton Tarjan [4] finds a minimal spanning tree in O(n k) time. This completes the linear time reduction of the minimal spanning tree problem to the Voronoi diagram problem, and gives: Theorem 4 In O( n k) log(n k) log n) time one can determine a minimal spanning tree of S. 2 7 Conclusions For each of the ....

D. Cheriton, R. E. Tarjan, Finding minimum spanning trees, SIAM J. Computing 5 (1976), 724-742.

....on a sequential access operation becomes smaller, the technique may find its uses. Some possibilities for improvement also exist. Through the sequential access connected components algorithm, m n duplicates will emerge in E 00 . These edges may be removed from E 00 as they show up. In [2], interleaved cleanup steps of redundant edges in minimum spanning trees is investigated, and this is reminiscent of duplicate deletion in the connected component problem, though the upper bound will not change. Further, tiled multi way mergesort may be applied to lower the dominating log 2 ....

D. CHERITON AND R. E. TARJAN, Finding minimum spanning trees, SIAM Journal on Computing 5 (1976), 310--313.

....developed starting from the algorithm proposed by Gabow to generate weighted spanning trees in order [5] This algorithm uses a maximum weight spanning tree as a reference tree and exchanges branches to obtain the remaining trees. To obtain a maximum weight tree many algorithms have been reported [6]. The generation of trees in decreasing order of magnitude is based upon the concept of a T exchange [5] Given a spanning tree T of a graph G (the voltage graph in this case) a T exchange is a pair of edges [e,f] such that e T, f T, and T e f is a spanning tree. The weight of exchange [e,f] is ....

D. Cheriton and R.E. Tarjan, "Finding Minimum Spanning Trees", SIAM J. of Computing, Vol. 5, No. 4, pp. 724-742, Dec.1976.

....tree edge e = fv; p(v)g, the cost of critical(v) is greater than or equal to the cost of e. This gives an algorithm for verification of a minimum spanning tree in a planar graph. An alternative method is to run the O(n) time algorithm for computing minimum spanning trees of Cheriton and Tarjan [2]. To analyze the sensitivity of a minimum spanning tree T we determine for each edge e how much its cost can be perturbed before T is no longer minimal. We compute lower and upper bounds [a; b] such that T remains minimal as long as a cost(e) b. If e is an edge in T , then the lower bound is ....

D. Cheriton and R. E. Tarjan. Finding minimum spanning trees. SIAM J. Comput., 5:724--742, 1976.

....a linear time algorithm building a cheap two vertex connected graph, C2VG(V ) for a Euclidean point set V in the plane, V # 3, given DT(V ) The cost of G is c(G) # k c(MST (V ) k = 4. This is the algorithm: 1. Compute MST(V ) from DT(V ) with the algorithm by Cheriton and Tarjan [8]. 2. Build a new graph G # (V ) V, E # ) with edge set E # : v, x) v, w) and (w, x) are edges in MST(V ) 3. Compute the minimum spanning forest MSF(G # ) with the algorithm by Cheriton and Tarjan [8] 4. C2VG(V ) MST(V )# MSF(G # (V ) The next Lemma shows that C2VG(V ) has the ....

....the algorithm: 1. Compute MST(V ) from DT(V ) with the algorithm by Cheriton and Tarjan [8] 2. Build a new graph G # (V ) V, E # ) with edge set E # : v, x) v, w) and (w, x) are edges in MST(V ) 3. Compute the minimum spanning forest MSF(G # ) with the algorithm by Cheriton and Tarjan [8]. 4. C2VG(V ) MST(V )# MSF(G # (V ) The next Lemma shows that C2VG(V ) has the desired properties. Lemma 6.1 1. G # (V ) has O(n) edges. 2. C2VG(V ) is two vertex connected. 3. c(C2VG(V ) # 4 c(MST(V ) 4. Given DT(V ) the algorithm runs in linear time. Proof. Consider the following ....

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Cheriton and Tarjan. Finding minimum spanning trees. SIAM J. Comput., Vol.5, No.4, December 1976, 724--742.

....the textline orientation by analysis of the distribution of the orientations of the edges in the MST(S) Step (1) is done by computing the center of the bounding box of each character and does not concern us here. Computational geometry is used to solve step (2) in two phases. Cheriton and Tarjan [CT76] proposed a simple algorithm for computing the MST of a graph in O(E) time where E is the number of edges in the graph. If we join every point of S to every other we can certainly use the Cheriton Tarjan algorithm on the resulting complete graph. However such a graph has E = n(n 1) 2 edges and ....

....the shape of a set of points. Let S= x 1 , x 2 , x n be a finite set of points in the plane. A proximity graph on a set of points is a graph obtained by connecting two points in the set by an edge if the two points are close, in some sense, to each other. The minimal spanning tree (MST) [CT76], the relative neighborhood graph (RNG) To80a] and the b skeletons [JT92] are three proximity graphs that have been well investigated in this context. The lune of x i and x j , denoted by Lune(x i , x j ) is defined as the intersection of the two discs centered at x i and x j with radius ....

Cheriton, D. and Tarjan, R., "Finding minimum spanning trees," SIAM Journal of Computing, vol. 5, No. 4, December 1976.

....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 ....

....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 ....

D. Cheriton and R.E. Tarjan. Finding minimum spanning trees. SIAM J. Computing, 5, pp.724--742, 1976.

.... and Tarjan s O(m) randomized minimum spanning tree algorithm [KKT94] or Fredman and Willard s deterministic O(m) time minimum spanning tree algorithm, which operates under a less restrictive model of computation [FrWi90] For planar graphs, a minimum spanning tree can be constructed in O(n) time [ChTa76] (see also Section 3) leading to a O(n log n) MRST algorithm. Parametric search has been the subject of a considerable amount of research in recent times because of its numerous applications to optimization and computational geometry [CoMe93, Tol93a, CEGS92, MaSc93] In the context of ....

....the right non parametric algorithm to simulate. In the context of MRST, we need an algorithm that evaluates Z( for any fixed ; i.e. an algorithm for finding minimum spanning trees in planar graphs. As we stated earlier, Cheriton and Tarjan have devised a O(n) time algorithm for this purpose [ChTa76]; unfortunately, it is not clear how to use it directly to devise an efficient MRST procedure. In this section, we give a new linear time (nonparametric) minimum spanning tree algorithm for planar graphs that relies on multilevel divisions and the idea of graph reduction. While our algorithm is ....

D. Cheriton and R.E. Tarjan. Finding minimum spanning trees. SIAM J. Comput., 5:724--742 (1976).

....have slower asymptotic running times, but also tend to be simpler. Kruskal s algorithm runs in O(jEj log jV j) time; Prim s algorithm runs in O(jEj log jV j) time with simple implementations and in O(jEj jV j log jV j) with more sophisticated priority queues; Cheriton and Tarjan s algorithm [1] runs in O(jEj log log jV j) time. Because both Kruskal s and Prim s algorithms rarely attain their worst case bounds and because their data structures can be maintained with very low overhead, one cannot determine the best algorithm solely on the basis of asymptotic analysis and so must resort ....

Cheriton, D., and R.E. Tarjan, "Finding minimum spanning trees," SIAM J. Comput. 5 (1976), pp. 724--742.

....which is a real number. The labels are maintained so that the cost of an edge is the sum of its label, the label of the queue Q(v; b) that contains it, and the label of the queue Q(b) that contains its queue. These labels enable us to quickly update the costs of edges. This idea is borrowed from [3, 2] For example, to implement decreaseCost(b; ffi) we simply decrease by ffi the label of Q(b) The keys of the elements in Q(v; b) are the labels of the edges. Hence the lowest labeled edge in Q(v; b) can be found in O(log n) time. The queues Q(v; b) are themselves assigned keys because these ....

D. Cheriton and R. E. Tarjan, "Finding minimum spanning trees, " SIAM J. Comput. 5 (1976), pp. 724-742.

....joining pairs of points. The (Euclidean) minimum spanning tree problem can be solved to optimality in the plane in time O(n log n) by appealing to the fact that the MST is a subgraph of the (O(n) size) Delaunay diagram; after computation of the Delaunay diagram, results of Cheriton and Tarjan [104] can be applied to find the MST in only O(n) additional time. Proposition 4 An edge in a Euclidean MST is Delaunay. The above proposition remains valid in d , for d 3; however, the result does not lead directly to a subquadratic time algorithm for MST in higher dimensions, since there can ....

D. Cheriton and R. E. Tarjan. Finding minimum spanning trees. SIAM J. Comput., 5:724--742, 1976.

.... Gamma 1) pairs is O(n) The last step is O(1) Therefore, the complexity of the algorithm is O(m) and is optimal. We can reduce the complexity still further if the graph under consideration is planar. Step 1 finding the minimum spanning tree can be done in O(n) time (Cheriton and Tarjan [4]) However, for a planar graph with n vertices, the number of edges is at most 3n Gamma 6 [12, p.70] Thus, the above complexity reduces to O(n) We can prove that all the algorithms presented in this section are correct. The correctness of the algorithms SEQ MVE1 and SEQ MVE3 follow from Lemma ....

....is n= log n. ii) When the graph is planar, we can obtain a cost optimal parallel algorithm as follows. Use the parallel implementation described in section 5. 2 except for Step 1 where we shall instead use a single processor to obtain a minimum spanning tree in O(n) time (Cheriton and Tarjan [4]) Then, it is easy to show that the cost of the algorithm is O(n 2 ) which is optimal. 5.3 Third Parallel Implementation The motivation for this algorithm comes from THEOREM 5. A high level description of our third parallel algorithm which runs on the CRCW PRAM computational model is given ....

D. Cheriton and R.E. Tarjan, Finding the minimum spanning trees, SIAM J. Comput., 5, 724-742 (1986).

....trees. We shall also assume that all edge weights are nonnegative. If not, we can add a positive value to each edge weight to give an equivalent problem with all edge weights nonnegative. We first find a minimum spanning tree of our graph using the fast algorithm of [GGST] for general graphs or [CT] for planar graphs. Then we use Eppstein s technique to reduce the problem to one in which there are O(k) vertices and edges [E] If k m n, this technique identifies and deletes m n k edges that will be in none of the k smallest spanning trees, and if k n, it identifies and contracts n k ....

....[E] T1] 518 GREG N. FREDERICKSON Fig. 9. The first four levels of inclusion exclusion for Fig. 1, with spanning tree costs indicated. T2] BFPRT] and [F1] finding the contracted graph and transforming it into one with maximum degree 3 will take O(m log #(m, n) time and O(m) space. From [CT], E] BW] BFPRT] and [F1] finding the contracted graph of a planar graph and transforming it into one with maximum degree 3 will take O(n) time and space. In addition to setting up R 1 , the algorithm will perform 2(k 1) updates of best swap structures. With regard to the heap, k 1 ....

D. Cheriton and R. E. Tarjan, Finding minimum spanning trees, SIAM J. Comput., 5 (1976), pp. 310--313.

....in O(n) time using one of the algorithms of Hopcroft and Tarjan [16] or Booth and Lueker [4] see Chiba, Nishizeki, Abe, and Ozawa [6] Each connected component gives rise to a planar subdivision. The initial spanning trees can be found in O(n) time with the algorithm of Cheriton and Tarjan [5]. Thus, given O(n) preprocessing time, one can maintain the minimum spanning forest of G in O(log n) amortized time per operation and O(n) space. 3 Edge ordered Trees and a Fully Dynamic Algorithm In this section we present our main result, the fully dynamic algorithm. We first develop the ....

D. Cheriton and R. E. Tarjan. Finding minimum spanning trees. SIAM J. Comput., 5:724--742, 1976.

....of Tokyo Bunkyo ku, Tokyo 113, Japan Finding a spanning tree of minimum weight is one of the best known graph problems. Several efficient algorithms exist for solving this problem [3, 4, 5, 7, 9] This note 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 ....

D. Cheriton and R.E. Tarjan, Finding minimum spanning trees, SIAM J. Computing 5 (1976) 724--742.

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Cheriton, D., and Tarjan, R. E., "Finding Minimum Spanning Trees," SIAM Journal Computing, 5 (1976), 724-742. 161

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David Cheriton and Robert E. Tarjan. Finding minimum spanning trees. SIAM Journal of Computing, 5(4):724--742, December 1976.

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Cheriton, D. and Tarjan, D.E., \Finding minimum spanning trees", SIAM Journal on Computing, 5, 724-742, 1976.

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D. Cheriton and R. E. Tarjan. Finding minimum spanning trees. SIAM J. Comput., pages 724--742, December 1976.

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Cheriton, D., Tarjan, R. E., "Finding Minimum Spanning Trees," SIAM J. Comput., 5 (1976), 724-742.

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D. Cheriton and R. E. Tarjan. Finding minimum spanning trees. SIAM Journal of computing, 5(4):724--741, December 1976.

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