P.M. Spira and A. Pan, "On Finding and Updating Spanning Trees and Shortest Paths," SIAM J. Computing 4 (1975), 375--380.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....it from scratch. 4 The Vertex 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 ....

P. M. Spira and A. Pan. On finding and updating spanning trees and shortest paths. SIAM Journal on Computing, 4(3):375--380, 1975.

.... 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 ....

P.M. Spira and A. Pan, On finding and updating spanning trees and shortest paths, SIAM Journal on Computing , 4(3), 1975, 375--380.

.... surveys motivating the exploration of efficient average case algorithms and their probabilistic analysis see ( 17] Basic good average case graph algorithms were presented in various settings: sequential, parallel, distributed, NP hard and so on (e.g. 2] 5] 18] 19] 20] 21] 23] 26] [29]) We are not aware of any investigation prior to ours concerning fully dynamic graph theoretic problems. 1.3 Average Case Analysis of Dynamic Graph Algorithms The investigation of the average case of fully dynamic graph suggests random graph updates. In this setting we would like to perform any ....

P. Spira and A. Pan, "On finding and updating spanning trees and shortest paths", SIAM J. Comput., 4, pp. 375--380, 1975. R.1 5 APPENDIX 1: The Algorithm (high level description) Input: A stochastic graph process starting from G n;p with p =

....issues in Internet routing have attracted much attention [21, 23] In fact, our interest in dynamic SPT algorithms was partly motivated by the problem of routing instability in the Internet. To our knowledge, the earliest work on dynamic SPT algorithms that appeared in the literature was [17]. However, the algorithm presented is largely inecient and improves the complexity of regular SPT algorithms by only a constant. The rst ecient dynamic SPT algorithm was also discussed in [12] in reference to the Arpanet. This algorithm is basically a generalization of Dijkstra s static SPT ....

P. Spira and A. Pan. \On Finding and Updating Spanning Trees and Shortest Paths," SIAM Journal on Computing, vol. 4, no. 3, Sept. 1975, p.375-380.

....in Internet routing have attracted much attention [15, 19] In fact, our interest in dynamic SPT algorithms was partly motivated by the problem of routing instability in the Internet. To the best of our knowledge, the earliest work on dynamic SPT algorithms that appears in the literature is [26]. The work proves a lower bound complexity for the worst case scenario. The rst ecient dynamic SPT algorithm that we know of is discussed in [18] This algorithm is very similar to the First Incremental Dijkstra algorithm presented in our paper, though the work [18] contains no analytical proofs ....

P. Spira and A. Pan. \On Finding and Updating Spanning Trees and Shortest Paths," SIAM Journal on Computing, vol. 4, no. 3, Sept. 1975, p.375-380. 31

....case and better capturing (via probabilistic methods) its complexity rather than analyzing the worst possible case. Basic good average case graph algorithms were presented in various settings: sequential, parallel, distributed, NP hard and so on (e.g. 2] 5] 17] 18] 19] 21] 22] [28]) For these investigations, various random graph models have been employed. We are not aware of any average case investigation concerning fully dynamic graph theoretic problems which has taken place prior to our effort. The investigation of the average case of fully dynamic graphs suggests ....

P. Spira and A. Pan, "On finding and updating spanning trees and shortest paths", SIAM J. Comput., 4, pp. 375--380, 1975. 18

....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] ....

P.M. Spira and A. Pan. On finding and updating spanning trees and shortest paths. SIAM J. Comput. 4 (1975) 375--380. 11

.... 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 ....

P.M. Spira and A. Pan, On finding and updating spanning trees and shortest paths, SIAM Journal on Computing , 4(3), 1975, 375--380. 16

....incremental algorithms that have been proposed for updating the solution to the (various versions of the) shortest path problem after the deletion of a single edge run in time asymptotically no better, in the worst case, than the time required to perform the computation from scratch. Spira and Pan [26], in fact, show that no incremental algorithm for this problem can do better than the best batch algorithm, under the assumption that the incremental algorithm retains only the shortest paths information. In other words, with the usual way of analyzing incremental algorithms worst case analysis ....

.... (v) d (v) 16] else [17] if v Heap then Remove v from Heap fi [18] fi [19] od [20] else u is underconsistent [21] d (u) 22] for v (Succ (u) u ) do [23] rhs (v) g v (d (x 1 ) d (x k ) 24] if rhs (v) d (v) then [25] AdjustHeap(Heap, v, min(rhs (v) d (v) [26] else [27] if v Heap then Remove v from Heap fi [28] fi [29] od [30] fi [31] od end postconditions Every vertex in V (G) is consistent ################################################################################################ Figure 1. An algorithm for the dynamic SWSF fixed point ....

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Spira, P.M. and Pan, A., "On finding and updating spanning trees and shortest paths," SIAM J. Computing 4(3) pp. 375-380 (September 1975).

.... 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 ....

P.M. Spira and A. Pan, On finding and updating spanning trees and shortest paths, SIAM Journal on Computing , 4(3), 1975, 375--380.

....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] ....

P. M. Spira and A. Pan. On finding and updating spanning trees and shortest paths. SIAM J. Comput., 4:375--380, 1975.

....between two trees is the minimum distance between a node in one tree and a node in the other tree. The following discussion of the heuristics assumes the availability of shortest path information. Such information can either be computed when necessary using a well known 3 graph algorithm [33, 39], or updated dynamically as topology changes occur[19, 39] 1.1 Evaluation Methodology In each of the following chapters describing new algorithms, we use simulation to verify average case behavior. These simulations were performed on a large set of sparse, randomlygenerated network topologies in ....

....a node in one tree and a node in the other tree. The following discussion of the heuristics assumes the availability of shortest path information. Such information can either be computed when necessary using a well known 3 graph algorithm [33, 39] or updated dynamically as topology changes occur[19, 39]. 1.1 Evaluation Methodology In each of the following chapters describing new algorithms, we use simulation to verify average case behavior. These simulations were performed on a large set of sparse, randomlygenerated network topologies in which the weight of a network link is determined by the ....

P. Spira and A. Pan. "On finding and updating spanning trees and shortest paths," SIAM J. Comput., vol. 4, no. 3, pp. 375--380, Sep. 1975.

....rules are defined to apply locally on the nodes of the existing MST, so we get parallel algorithms (using tree contraction) as well as sequential ones (using, for example, depthfirst search) History. The vertex updating problem of a minimum spanning tree was first addressed by Spira and Pan in [1], where an O(n) sequential algorithm was presented. Another solution using depth first search and having the same time complexity was later given by Chin and Houck in [2] while Pawagi and Ramakrishnan [3] gave a parallel solution to the problem. Their algorithm, which runs in O(lg n) time using ....

....as being either included in the new MST, or excluded from it. We will make use of the following two well known facts which we state here without proof: Fact 1 The edge with minimum weight incident to some node will always be included in the MST. 2 Actually, Prim s and other sequential [1] and parallel [10] algorithms are based on this fact. Edge inclusion in our rules makes use of this observation. G c = G=fu; vg w 1 v w 1 w 2 u uv w 2 minfc 1 ; c 2 g c 3 c 1 c 2 c 3 G Figure 3: Contraction of edge fu; vg in G produces G c = G=fu; vg. Fact 2 Whenever some edge is found to ....

P.M. Spira and A. Pan. On finding and updating spanning trees and shortest paths. SIAM Journal on Computing, 4(3):375--380, September 1975.

....O(n 2 ) time for edge insertion, and O(mn n 2 log n) time for edge deletion, where n is the number of vertices and m is the number of edges. Such perfor1 mance bounds, especially for deletion, are quite unsatisfactory and indicate the difficulty of the problem (for lower bound arguments, see [5,24]) The best known semi dynamic data structure supporting insertions in digraphs with unit edge lengths uses O(n 2 ) space and has O(1) query time; the total time to process all edge insertions is O(n 3 log n) which amortizes to O(n log n) time per insertion for dense digraphs [3,20] ....

P.M. Spira and A. Pan, "On Finding and Updating Spanning Trees and Shortest Paths," SIAM J. Computing 4 (1975), 375--380.

....of computing a solution. The newly arrived data must be incorporated in the solution at hand. The final solution is to be returned by a certain deadline. Real time computations form a subclass of a larger class of problems known variably as on line, incremental , dynamic, and updating computations [7, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 30, 31, 32, 36, 38, 40, 41]. What distinguishes a real time problem from problems in the larger class is the presence of deadlines by which the input is to be processed, by which the output is to be produced, and so on. 3 Modern Cryptography The purpose of contemporary cryptography is the protection of digital ....

P.M. Spira and A. Pan, On finding and updating spanning trees and shortest paths, SIAM Journal on Computing , 4(3), 1975, 375--380.

.... problems [11] Moreover, despite intensive research 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 ....

.... 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 ....

P. M. Spira, and A. Pan, "On finding and updating spanning trees and shortest paths", SIAM J. Comput. 4 (1975), 375--380.

....rules are defined to apply locally on the nodes of the existing MST, so we get parallel algorithms (using tree contraction) as well as sequential ones (using, for example, depth first search) History. The vertex updating problem of a minimum spanning tree was first addressed by Spira and Pan in [16], where an O(n) sequential algorithm was presented. Another solution using depth first search and having the same time complexity was later given by Chin and Houck in [2] while Pawagi and Ramakrishnan [14] gave a parallel solution to the problem. Their algorithm, which runs in O(lg n) time 1 ....

....that accompany this paper, a conditionally included edge is marked with an asterisk ( It is useful to observe that the edge with minimum weight incident to some node will always be included into the MST. Actually, many sequential and parallel algorithms are based on this observation (i.e. [15, 16, 3]) Edge inclusion makes use of this observation. Another useful observation is that whenever some edge is found to correspond to the MaxWE of some cycle it can be removed from the tree without affecting the computation of the remaining graph. Kruskal s MST algorithm makes use of this fact. Edge ....

P. Spira and A. Pan. On finding and updating spanning trees and shortest paths. SIAM J. Comp., 4(3):375-- 380, September 1975.

....Award. y Department of Computer Science, University of Victoria, Victoria, BC. Email: val csr.uvic.ca. This research was supported by an NSERC Grant. 1 Throughout the paper the logarithms are base 2. Previous Work. In recent years a lot of work has been done in fully dynamic algorithms (see [1, 3, 4, 6, 7, 8, 10, 11, 12, 15, 16, 18] for connectivity related work in undirected graphs) There is also a large body of work for restricted classes of graphs and for insertions only algorithms. Currently the best time bounds for fully dynamic algorithms in undirected n node graphs are: O( p n) per update for a minimum spanning ....

P. M. Spira and A. Pan, "On Finding and Updating Spanning Trees and Shortest Paths", SIAM J. Comput., 4 (1975), 375--380.

....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 ....

P.M. Spira and A. Pan, "On Finding and Updating Spanning Trees and Shortest Paths," SIAM J. Computing 4 (1975), 375--380.

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P.M. Spira and A. Pan, "On Finding and Updating Spanning Trees and Shortest Paths," SIAM J. Computing 4 (1975), 375--380.

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Spira, P.M. and Pan, A., "On finding and updating spanning trees and shortest paths," SIAM J. Computing 4(3) pp. 375-380 (September 1975).

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Spira, P.M. and Pan, A., "On finding and updating spanning trees and shortest paths," SIAM J. Computing 4(3) pp. 375-380 (September 1975).

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Spira, P.M. and Pan, A., "On finding and updating spanning trees and shortest paths," SIAM J. Computing 4(3) pp. 375-380 (September 1975).

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P.M. Spira and A. Pan, On finding and updating spanning trees and shortest paths, SIAM Journal on Computing , 4(3), 1975, 375--380.

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Spira, P. M. and A. Pan, On Finding and Updating Spanning Trees and Shortest Paths, SIAM J. Comput. 4 (1975) 375-380.

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