V. King. A simpler minimum spanning tree verification algorithm. Algorithmica, 18:263--270, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... and a survey of these algorithms is provided in [141] We have chosen to use algorithms based on the minimal cost spanning tree (MCST) because of the natural relationship between MCST and nearest neighbours [9, 140] and because there are a number of good algorithms for constructing MCST s [11, 84, 93]. The cluster information can be represented as a binary tree which can be computed from the MCST of the original codewords [62] Before we discuss the binary tree representation of the cluster information, we need to discuss how the MCST can be used to define the boundaries of the clusters. 5.3 ....

....always taking next the vertex closest to the vertices already taken. In other words, we find the edge of smallest weight among those edges that connect vertices already on the tree, then add to the tree that edge and the vertex it leads to. More recent examples of MCST algorithms can be found in [11, 84, 93]. The MCST representation can be stored in O(n) space. The single link clustering algorithm proposed by Gower and Ross [62, 141] consists of an increasing series of pre defined distance thresholds (ffi 1 ; ffi 2 ; The clusters at level ffi l are constructed by grouping together the ....

V. King. A simpler minimum spanning tree verification algorithm. Algorithmica, 18:263--270, 1997.

....running time. 3.3. Filtering edges via the Filter forest. We will maintain, concurrent with the operation of Boruvka A, a structure called the Filter forest. This collec tion of rooted trees records which vertices merged together and the edge weights involved. This structure appeared first in [K97]. If v is a vertex of the original graph or a new vertex resulting from contracting a set of edges, there is a corresponding vertex #(v) in the Filter forest. During a Boruvka step, if a vertex v becomes dead, a new vertex x is added to the Filter forest, as well as a directed edge (#(v) x) ....

....weights of edges (v 1 ,p(v 1 ) v 2 ,p(v 2 ) v j ,p(v j ) respectively. We make the simple observation that the edge weights on the path from #(u) to root(#(u) are exactly the edge weights of the edges chosen by u (or its representative) in previous Boruvka steps. It is shown in [K97] that the heaviest weight in the path from u to v in the MSF is the same as the heaviest weight in the path from #(u) to #(v) in the Filter forest (if there is such a path) We extend this scheme to handle k Min lightness. Let P f (y, z) be the path from y to z in the Filter forest. If #(u) and ....

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V. King, A simpler minimum spanning tree verification algorithm, Algorithmica, 18 (1997), pp. 263--270.

....that our lower bound is tight to within a factor of 2 in the t parameter. 1 Introduction The theoretically best minimum spanning tree algorithms [23, 10, 33] were made possible by even more fundamental algorithms and data structures, namely Komlos s minimum spanning tree verification algorithm [27, 17, 24, 5] and Chazelle s Soft Heap [11] It has been speculated by some (see, e.g. Chazelle [10, p. 1029] that the key to a faster MST algorithm is some interesting new data structure. In this paper we show that there are no linear solutions to the online minimum spanning tree verification problem, ....

.... the problem of answering interval maximum queries in a 1dimensional array can be done in constant time with linear preprocessing [27] contrast this with the superlinear lower bound in [13] for arbitrary semigroups) Solving MST verification offline on arbitrary trees can be done in linear time [27, 17, 24, 5], and the dual to this problem, MST sensitivity analysis, can be solved in randomized linear time [20, 17] or deterministic O(m log (m; n) time. All these problems have m (m;n) lower bounds when generalized to arbitrary semigroups [13] Given this history it is somewhat startling that the ....

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V. King. A simpler minimum spanning tree verification algorithm. Algorithmica, 18(2):263--270, 1997.

....is a growing literature on correctness checking that aims to rectify this omission. Following early work on program checking and certification [2, 29, 30] several researchers have developed e#cient schemes for checking the results of various data structures [3, 5, 4, 14, 22] graph algorithms [18, 20], and geometric algorithms [11, 23] These schemes are directed mainly at defending the user against an inadvertent error made during implementation. In addition, these previous approaches have primarily assumed that usage is limited to a single user on an individual machine. With the advent of ....

V. King. A simpler minimum spanning tree verification algorithm. In Workshop on Algorithms and Data Structures, pages 440--448, 1995.

....it would seek bounds on an edge s cost or capacity within which that edge remains part of the optimal solution. In the minimum spanning tree problem, deletion of an edge changes the MST by replacing it with another edge, and a fast algorithm is known for computing the sensitivity of all MST edges [8, 12]. On the other hand, deleting a single edge from G can change the shortest path between two nodes by many edges, and in that sense our problem appears to be more difficult. All the known methods for shortest path sensitivity analysis seem to require Omega Gamma m) work per shortest path edge ....

V. King. A simpler minimum spanning tree verification algorithm. Algorithmica, 18(2):263--270, 1997. 10

....second term in the stated running time. 2 3.3 The Filtering Step The Filter Forest Concurrent with each modified Boruvka step, we will maintain a Filter forest, a structure that records which vertices merged together at what time, and the edge weights involved. This structure appeared first in [King97]) If v is a vertex of the original graph, or a new vertex resulting from contracting a set of edges, there is a corresponding vertex OE(v) in the Filter forest. During a Boruvka step, if a vertex v becomes dead, a new vertex w is added to the Filter forest, as well as a directed edge (OE(v) w) ....

....: v j are contracted into a live vertex v, a vertex OE(v) is added to the Filter forest in addition to directed edges (OE(v 1 ) OE(v) OE(v 2 ) OE(v) OE(v j ) OE(v) having the weights of edges (v 1 ;p(v 1 ) v 2 ;p(v 2 ) v j ;p(v j ) respectively. It is shown in [King97] that the heaviest weight in the path from u to v in the MSF is the same as the heaviest weight in the path from OE(u) to OE(v) in the Filter forest (if there is such a path) 7 Hence the measures weight v (w) can be easily computed in the following way. Let P f (x; y) be the path from x to y in ....

V. King. A simpler minimum spanning tree verification algorithm. Algorithmica, vol. 18, 1997, pp. 263-270.

....Aho, Hopcroft, and Ullman [2, pp. 139 141] describe an algorithm for a similar offline priority queue problem, however their problem involves delete minimum operations rather than deletions of particular values. Although the best replacement edge for each non MST edge can be found in linear time [10], the fastest known algorithm for finding the best replacement for each MST edge (without the integer restriction) remains Tarjan s slightly superlinear one [12] LEMMA 3.1. The three problems described above can be solved in linear time. Proof. We consider the minimum spanning tree verification ....

V. King. A simpler minimum spanning tree verification algorithm. Algorithmica, 18(2):263--270, June 1997.

....RAM memory cells can contain (log n) bit integers that may be compared, added, subtracted, multiplied and divided (with rest) Furthermore, the integers can be also used as pointers to other memory cells (indirect addressing) All these operations take constant time. A variant of the RAM model [81, 45, 44, 51] allows also bitwise logical operations on the integers in constant time. In the real RAM model memory cells also can store real numbers. Since a real number can contain an infinite amount of information in its binary expansion, the set of valid operations for real numbers must be carefully ....

V. King. A simpler minimum spanning tree verification algorithm. Algorithmica, 18:263--270, 1997. 96

....slowdown given O(m= log n) processors. 3.3 The Filtering Step The Filter Forest Concurrent with each modified Boruvka step, we will maintain a Filter forest, a structure that records which vertices merged together at what time, and the edge weights involved. This structure appeared first in [King97]) If v is a vertex of the original graph, or a new vertex resulting from contracting a set of edges, there is a corresponding vertex OE(v) in the Filter forest. During a Boruvka step, if a vertex v becomes dead, a new vertex w is added to the Filter forest, as well as a directed edge (OE(v) w) ....

....p(v) If live vertices v 1 ; v j are contracted into a live vertex v, a vertex OE(v) is added to the Filter forest in addition to directed edges (OE(v 1 ) OE(v) OE(v j ) OE(v) having the weights of edges (v 1 ;p(v 1 ) v j ;p(v j ) respectively. It is shown in [King97] that the heaviest weight in the path from u to v in the MSF is the same as the heaviest weight in the path from OE(u) to OE(v) in the Filter forest (if there is such a path) Hence the measures weight v (w) can be easily computed in the following way. Let P f (x; y) be the path from x to y in the ....

V. King. A simpler minimum spanning tree verification algorithm. Algorithmica, vol. 18, 1997, pp. 263-270.

....algorithms are all but linear in m, i.e. linear times a log type factor. b) A randomized Las Vegas algorithm [8] achieves expected time that is truly linear in m, and in fact guarantees linear time performance with all but exponentially small probability. c) Verification of a putative MST [4, 9] can be done in time linear in m. The randomized algorithm above uses this result. d) The MST (more generally, MSF) of a fully dynamic graph can be maintained in time O( p m) per change to the graph [6] While these algorithms are quite different in many respects, it is notable that they ....

....Bor uvka [2] In Bor uvka s algorithm, each vertex of the graph colors its lightest incident edge blue; then the blue edges are contracted simultaneously, and the process is repeated until the graph has shrunk down to a single vertex. The full set of blue edges then constitutes the MST. Both King [9] and Karger et al. 8] make use of this algorithm. The generalized bottom up method x4.1 Strategy. To understand the generalized bottom up strategy for constructing an Allow graph contraction during greedy algorithms MST, consider that the generalized greedy algorithm (x3.14) can be modified ....

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Valerie King. 1993. A simpler minimum spanning tree verification algorithm. Unpublished manuscript, ftp://godot.uvic.ca/pub/Publications/King/Algorithmica-MSTverif.ps. (To appear in Proceedings of the Workshop on Algorithms and Data Structures.)

....showed that a minimum spanning tree can be verified in O(m) binary comparisons of edge weights, but with nonlinear overhead to decide which comparisons to make. Dixon, Rauch and Tarjan [7] combined these algorithms with a table lookup technique to obtain an O(m) time verification algorithm. King [17] recently obtained a simpler O(m) time verification algorithm that combines ideas of Boruvka, Koml os, and Dixon, Rauch, and Tarjan. In this paper we describe a randomized algorithm for finding a minimum spanning tree. It runs in O(m) time with high probability in the restricted random access ....

V. King, "A simpler minimum spanning tree verification algorithm," manuscript, 1993.

....traversal through e and undo the changes we made to # and A. The time for this method is O(log n) per binary search, or O(m log n) total. More precisely it is O(m log d) where d is the depth of T 0 . To improve this to the O(m n log n) bound claimed, we apply a sensitivity analysis algorithm [7, 18, 21]. Such algorithms find for each edge e = u, v)inG T 0 the heaviest edge e # on the corresponding path from u to v in T 0 . We use this information in two ways. First, e is in some MST i# w(e) w(e # ) so we can delete from EG all edges not part of some MST (this entails a modification to our ....

V. King. A simpler minimum spanning tree verification algorithm. 4th Worksh. Algorithms and Data Structures (1995) to appear.

.... the minimum only within a single interval of values of #, we perform this test by computing the values of all intra cluster edges at the parameter value #, and testing whether the given tree is still the minimum spanning tree at that parameter using a minimum spanning tree verification algorithm [8, 25]. # Lemma 10. We can find the first non positive intra cluster swap in a cluster of O(z) vertices in O(z log z) time. Proof: We apply parametric search with the decision oracle described above. For a simulated algorithm, we use sorting, since the sorted order is discontinuous at all swaps. Cole ....

V. King. A simpler minimum spanning tree verification algorithm. In Proc. 4th Worksh. Algorithms and Data Structures, Lecture Notes in Computer Science, vol. 955, Springer-Verlag, 1995, 440--448.

....forest to distinguish them from the vertices of the graphs. 5.1 The merge forest We define a rooted forest M , called the merge forest, that captures the effect of the Boruvka steps performed during a call to FindForest. This structure resembles closely the Boruvka tree defined and used by King [21] for verification of minimum spanning trees. The differences reflect our modification of the Boruvka step. The nodes of M correspond to in edge trees in the graph, and we use OE to denote the mapping from in edge trees to nodes of M . For each in edge tree T arising during a call to FindForest, ....

....if ancv (w) belongs to P (v) then cv(w) the maximum cost on the path from OE(F0(v) to ancv(w) can be determined in constant time by consulting the table for the merge forest and the table for the trunk P (v) 5.3. 5 Determining cv(w) the hard case Use a parallel version of King s algorithm [21] to determine cv(w) for each edge vw in G such that OE(F0(v) and OE(F0(w) occur in the same tree of the merge forest. King s algorithm consists primarily of scanning down the forest, assigning labels to the nodes. The time per node is O(log log n) At each level, the work needs to be rebalanced ....

V. King, " A simpler minimum spanning tree verification algorithm, " Proc. 4th International Workshop on Algorithms and Data Structures, published as Lecture Notes in Computer Science 955, Springer-Verlag, Berlin, 1995, pp. 440448

....log 3 n . The sequential algorithm uses a linear time algorithm due to Dixon, Rauch, and Tarjan for verifying a minimum spanning tree. A parallel version of this algorithm that runs in O(log n) time was discovered by Dixon and Tarjan [7] A simpler such algorithm was recently discovered by King [14, 15]. By using one of these algorithms as a subroutine, we obtain an expected time bound of O(2 log 3 n log n) for the concurrent read concurrent write PRAM. To reduce the depth of recursion from logarithmic to log 3 n, we use the following idea. Whereas each recursive call of the sequential ....

....by a path in F and every edge on that path has weight less than that of e. An F heavy edge is not in the minimum spanning forest (see, e.g. 21] and hence can be discarded. Dixon, Rauch, and Tarjan [6] give a linear time algorithm that can be used to determine the set of F heavy edges. King [14] has recently given a simpler linear time algorithm for this problem. Klein and Tarjan [16] show that the number of edges that are not F heavy is probably not much more than n=p, where p is the sampling probability (in the algorithm above,p = 1=2) It follows that the expected sum of sizes of all ....

V. King, " A simpler minimum spanning tree verification algorithm," manuscript, 1993.

....at a child of v. The K batched queries are performed by sorting them first, so that all queries can be performed by scanning S O(1) times. If K N we process the queries in batches of N at a time. For the minimum spanning tree (MST) verification problem, our technique is based on that of King [14]. We verify that a given tree T is an MST of a graph G by verifying that each edge (u; v) in G has weight at least as large as that of the heaviest edge on the path from u to v in T . First, using O(sort(V ) I Os, we convert T into a balanced tree T 0 of size O(V ) such that the weight of the ....

V. King. A simpler minimum spanning tree verification algorithm, 1994.

....the head H . If x lies in H , then the edge (u; x) is split into two edges (u; r) and (r; x) with the same weight as (u; x) where r is the root of the microtree that contains u. The edges (r; x) can be verified locally in O( n 0 m) p) time using the sequential linear time algorithm given in [Kin95] since every processor has a local copy of H . The edges within the microtrees are verified as follows. First, for every leaf v, we compute the heaviest edge on all paths leading from v to the root r of the microtree that contains v. This is done identical to the macrotree verification ....

Valerie King. A simpler minimum spanning tree verification algorithm. In Workshop on Algorithms and Data Structures, pages 440--448. Springer-Verlag, 1995. LNCS 955.

....MST construction, and computing dominators in a flowgraph for which gaps exist between the best known pointer machine and RAM algorithms, as summarized by Table 1. We present the first linear time pointer machine algorithms for these problems, solving several outstanding open questions [9, 17, 18]. In the MST construction case, the time bound is expected. Additionally, our algorithms are simpler than their RAM counterparts. A pointer machine [28] allows binary comparisons between data, arithmetic operations on data, dereferencing of pointers, and equality tests on pointers. It does not ....

....m is the number of edges arcs, and p is the number of LCA queries. ff(m; n) is the standard functional inverse of the Ackermann function. Problem Previous Pointer Machine Bound Previous RAM Bound Off line LCAs O(pff(p; n) n) 1] O(n p) 16, 25] MST Verification O(mff(m; n) n) 27] O(n m) [9, 18] MST Construction O(mff(m; n) log ff(m; n) n) 7] O(n m) 13, 17] Dominators O(mff(m; n) n) 20] O(n m) 3, 15] level microtrees, and pointer based radix sort to the MST verification (and construction) and dominators problems. 2 Least Common Ancestors Let T = V; E) be a tree with root r, ....

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V. King. A simpler minimum spanning tree verification algorithm. Algorithmica, 18:263--70, 1997.

....1 Introduction The problem of verifying if a given spanning tree in an edge weighted graph is a minimum spanning tree for the graph is an important problem which is closely related to the problem of finding a minimum spanning tree of a graph. Recently Dixon, Rauch and Tarjan ( DRT92] and King ([Kin95]) have developed sequential linear time algorithms for the problem. Dixon and Tarjan have also given an optimal CREW algorithm ( DT94] In this paper, we present a parallel algorithm which runs in optimal time and work bounds on the weaker EREW PRAM model. This resolves an open question posed in ....

....have also given an optimal CREW algorithm ( DT94] In this paper, we present a parallel algorithm which runs in optimal time and work bounds on the weaker EREW PRAM model. This resolves an open question posed in [DT94] The high level structure of the algorithm has been adapted from [DT94] and [Kin95]. We use tree contraction (shunting) KR90, KD88] to convert the given spanning tree to a logarithmic depth Bor ffi uvka tree. As in [DT94] we decompose the given tree into microtrees of size O( p log n) The microtrees are verified in parallel using King s ( Kin95] sequential linear time ....

[Article contains additional citation context not shown here]

V. King. A simpler minimum spanning tree verification algorithm. In Lecture Notes in Compututer Science 955, pages 440--448. Springer-Verlag, Berlin, 1995.

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V. King. A simpler minimum spanning tree verification algorithm. Algorithmica, 18:263--270, 1997.

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V. King. A simpler minimum spanning tree verification algorithm, 1994.

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V. King. A simpler minimum spanning tree verification algorithm. In Workshop on Algorithms and Data Structures, pages 440--448, 1995.

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V. King. A simpler minimum spanning tree verification algorithm. Algorithmica, 18(2):263--270, 1997.

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King, V. A simpler minimum spanning tree verification algorithm. manuscript, 1993.

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V. King, A simpler minimum spanning tree verification algorithm, manuscript 1993.

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