F.Y. Chin and D. Houck, Algorithms for updating minimum spanning trees, Journal of Computer and System Sciences , 16, 1978, 333--344.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Update Problem The problem of maintaining the MST of a graph under the insertion of a single or multiple nodes along with all their incident edges have been well studied either in sequential or in parallel computation. In sequential computation, first Spira and Pan [41] and later Chin and Houck [4] presented an O(n) sequential algorithm for updating the MST when a new vertex is inserted into the graph. We point out that, an O(n) time algorithm for single vertex insertion is optimal , because any such algorithm must examine at least all the edges incident into it, that might be Omega Gamma ....

....how to perform efficiently a group of vertex deletions (see Table 2) Let us consider the simpler task of inserting a group of k vertices along with their incident edges. We could attempt this problem on a PRAM, by inserting one vertex at a time as done for the optimal sequential algorithm of [4]. Although this approach is very simple, it turns out to be inefficient in parallel. For example, using any parallel single vertex insertion algorithm iterated for k times, the problem could be solved within O(k log n) time. But, if k = Theta(n) the time bound for this problem is not ....

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F. Chin and D. Houck. Algorithms for updating minimum spanning trees. Journal of Computer and System Science, 16:333--344, 1978.

.... be returned as soon as it is available [32, 40, 56] It is helpful to note here that, when no deadlines are imposed, computations for which inputs arrive while the algorithm is in progress are referred to as on line [27, 33, 35, 36] incremental [21, 22, 49, 59] dynamic [11, 12, 67] and updating [20, 23, 28, 38, 53, 54, 62, 66]. It is also important to note that our definition, while striving to be as general as possible, is particularly suitable for our purposes in this paper. Many other definitions exist; see, for example, the various interpretations of the notion of real time provided in [10, 42, 65] 2.2 Models of ....

F.Y. Chin and D. Houck, Algorithms for updating minimum spanning trees, Journal of Computer and System Sciences , 16, 1978, 333--344.

....1 Introduction For many years, algorithm researchers have studied problems of maintaining information about a dynamically changing graph. A classical problem in this field is maintaining the minimum spanning tree (MST) of a graph in which the weights of individual edges are subject to change [1, 5, 6, 10]. Each such update causes at most one edge to leave the MST and at most one other edge to take its place; one might expect algorithms for computing these changes to be quite e#cient. Indeed, the best known algorithm takes O( # m) time per update, for a graph with n vertices and m edges [6] ....

F. Chin and D. Houck. Algorithms for updating minimum spanning trees. J. Comput. Syst. Sci. 16 (1978) 333--344.

.... be returned as soon as it is available [31, 39, 55] It is helpful to note here that, when no deadlines are imposed, computations for which inputs arrive while the algorithm is in progress are referred to as on line [26, 32, 34, 35] incremental [20, 21, 48, 58] dynamic [10, 11, 66] and updating [19, 22, 27, 37, 52, 53, 61, 65]. It is also important to note that our definition, while striving to be as general as possible, is particularly suitable for our purposes in this paper. Many other definitions exist; see, for example, the various interpretations of the notion of real time provided in [9, 41, 64] 2.2 Models of ....

F.Y. Chin and D. Houck, Algorithms for updating minimum spanning trees, Journal of Computer and System Sciences, 16, 1978, 333--344.

.... returned as soon as it is available [21, 28, 37] It is helpful to note here that, when no deadlines are imposed, computations for which inputs arrive while the algorithm is in progress are referred to as on line [18, 22, 23, 24] incremental [14, 15, 32, 39] dynamic [7, 8, 44] and updating [13, 16, 19, 26, 34, 35, 41, 43]. 2.2 Real time optimization The first example of a computation for which a parallel solution is consistently better than a sequential one was provided by real time optimization. The realtime weighted spanning tree problem is defined as follows: 1. The minimum weight spanning tree (MST) of an ....

F.Y. Chin and D. Houck, Algorithms for updating minimum spanning trees, Journal of Computer and System Sciences , 16, 1978, 333--344.

....that can replace e in level i form with high probability a sparse cut. These edges are moved to level (i 1) and the same procedure is applied recursively on level (i 1) One particular dynamic graph problem that has been thoroughly investigated is the maintenance of a minimum spanning forest [4, 10, 14, 45]. This is an important problem on its own, but it has also impact on other problems as well. Indeed the data structures and techniques developed for dynamic minimum spanning forests have found applications also in other areas, such as dynamic edge and vertex connectivity [10, 15, 19, 26, 41, 42] ....

F. Chin and D. Houk. Algorithms for updating minimum spanning trees. J. Comp. Syst. Sci., 16:333--344, 1978.

.... on dynamic problems on graphs (such as dynamic maintenance of connectivity [7, 8, 10, 11, 14, 20, 22, 29, 30] 2 and 3 connectivity [7, 12, 29, 30] transitive closure [3, 4, 15, 16, 17, 18, 19, 31] planar graphs [6, 7, 19, 25] shortest paths [2, 9, 21, 24, 31] and minimum spanning trees [5, 8, 11, 24]) there are very few graphtheoretic problems for which a fully dynamic non trivial algorithm is known. As mentioned in [30] the fully dynamic maintenance of the connected components of a graph differs substantially from the fully dynamic maintenance of the biconnected components. Indeed in the ....

F. Chin, and D. Houk, "Algorithms for updating minimum spanning trees", J. Comput. System. Sci. 16 (1978), 333--344.

....dynamic, or on line algorithms. The idea is to develop algorithms that can adjust their answers efficiently in response to changes in the input data. Domains for such algorithms have included: ffl graph theoretic algorithms: connectivity [ES81, Har83, Che84] spanning trees [SP73, CH78, FS84, Fre85] spanning forests [Wes89] shortest paths [Rod68, Che76, GSV78, Fuj81, CC82, Gaz83, EG85, AMSN89, AIMSN90, Ita91] biconnected components [Sac86, WT92, BT90] triconnected components [Ita91, BT90] transitive closure [IK83, Ita86, Ita88, LPv88, YS88, Yel91] ....

F. Chin and D. Houck. Algorithms for updating minimum spanning trees. Journal of Computer and System Sciences, 16:333--344, 1978.

....spanning tree and returning the total weight of a minimum spanning tree. The dynamic maintenance of minimum spanning trees has the interesting property that, after an update operation, at most one edge needs to be replaced in the minimum spanning tree. Early data structures for this problem [19,102] have O(n) space, O(n) time for edge insertions, O(n 2 ) time for edge deletions and the changing of edge weights, and O(1) time queries. The best result for general graphs is [48] which presents a fully dynamic data structure with O( p m) time per update and O(m) space. When only ....

F. Chin and D. Houk, "Algorithms for Updating Minimum Spanning Trees," J. Computer Systems Sciences 16 (1978), 333--344.

....the MST sequentially [5, 13] However, neither of these algorithms can be used here as they would require time on the order of (n k) 2 and (2n 2k Gamma 3) log(2n 2k Gamma 3) time units, respectively. Two sequential algorithms are also known that are capable of updating an existing MST [12, 32]. Unfortunately, these algorithms cannot be used either as they run in time on the order of 2n 2k Gamma 3 time units. The only viable approach for a sequential algorithm is to replace up to n ffl of the existing edges with an equal number of new edges of smaller weight, such that the entire ....

F.Y. Chin and D. Houck, Algorithms for updating minimum spanning trees, Journal of Computer and System Sciences , 16, 1978, 333--344.

....algorithms that build planar subdivisions such as Voronoi diagrams. Algorithms have been proposed for maintaining the embedding of a planar graph [29] and for incremental planarity testing [2, 3] The dynamic minimum spanning tree problem has been considered by Spira and Pan [28] Chin and Houck [7], Frederickson [10] and Gabow and Stallmann [11] Frederickson gives an algorithm based on topology trees that runs in O( p m) time per operation on general graphs, and O( log n) 2 ) time on plane graphs. As Frederickson notes, the minimum spanning tree for a general graph being modified ....

F. Chin and D. Houck. Algorithms for updating minimum spanning trees. J. Comput. System Sci., 16:333--344, 1978.

....with reconstructing the new MST from the current one when a vertex, along with its incident edges, is inserted or deleted from the graph. On the other hand, the edge update problem is that of determining the new MST when an edge is added or deleted, or when the cost of an edge has changed ([5], 11] 19] 24] For the single vertex insertion update problem, the sequential algorithms of [5] and [24] are optimal. The main idea behind our first algorithm is to consider all pairs (e; e 0 ) defined earlier, and select the pair which gives the greatest increase on the cost of the ....

....is inserted or deleted from the graph. On the other hand, the edge update problem is that of determining the new MST when an edge is added or deleted, or when the cost of an edge has changed ( 5] 11] 19] 24] For the single vertex insertion update problem, the sequential algorithms of [5] and [24] are optimal. The main idea behind our first algorithm is to consider all pairs (e; e 0 ) defined earlier, and select the pair which gives the greatest increase on the cost of the minimum spanning tree. Algorithm SEQ MVE1 Input: A connected weighted graph G. Output: The most vital ....

F. Chin and D. Houck, Algorithms for updating minimum spanning trees, J. Comput. System Sci., 16, 333-344 (1978).

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F.Y. Chin and D. Houck, Algorithms for updating minimum spanning trees, Journal of Computer and System Sciences , 16, 1978, 333--344.

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F.Y. Chin and D. Houck, Algorithms for updating minimum spanning trees, Journal of Computer and System Sciences , 16, 1978, 333--344.

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