Heather Booth and Je#ery Westbrook. A linear algorithm for analysis of minimum spanning and shortest-path trees of planar graphs. Algorithmica, 11(4):341--352, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that element is the first or last of the deque, it cannot subsequently be deleted. A related data structure is the priority queue with attrition [Sun89] Queues with heap order (or minqueues) are useful in pagination [DF84, HL87, LH85, McC77] and VLSI river routing [CS84] Booth and Westbrook [BW90] use catenable heap ordered queues in the sensitivity analysis of minimum spanning trees, shortest path trees, and minimum cost network flow on planar graphs. Larmore and Hirschberg [LH85] and Cole and Siegal [CS84] independently show how to implement heap ordered queues in O(1) amortized time ....

H. Booth and J. R. Westbrook. Linear algorithms for analysis of minimum spanning and shortest path trees in planar graphs. Technical Report YALEU/DCS/TR-763, Yale University Dept. of Computer Science, February 1990. To appear in Algorithmica.

....must be the MST of G e. # Lemma 4. 21] Given a MST T of a graph G, the replacements r G (e) for all edges e in T can be computed in time O(m#(m,n) This bound is never worse than the O(m log #(m, n) time required for constructing the MST of G. The following version of lemma 4 is given in [5]. Lemma 5. Given a MST T of a planar graph G, the replacements r G (e) for all edges e in T can be computed in time O(n) Lemma 6. Given a MST T of a graph G, and the values of r G for each tree edge, in linear time we can compute a set S of n k tree edges that must be contained in all of ....

H. Booth and J. Westbrook, Linear Algorithms for Analysis of Minimum Spanning and Shortest Path Trees in Planar Graphs, Tech. Rep. TR-763, Department of Computer Science, Yale University, Feb. 1990.

....efficiently, if the underlying data structure has been designed to expect such changes. Even then real time responsivity can be further improved. Sensitivity: The path only has to be re planned if the changes will result in a new path. Each part of the path can have an associated sensitivity (Booth and Westbrook 1992), which determine how big the change must be before re planning. Relaxed Updating: If the discrepancies introduced by the changes does not lead to unacceptable results, then the workload can be spread out over time, by updating a certain part of the data structure in each time step. Further ....

Booth, Heather and Westbrook, Je#ery, 1992. A Linear Algorithm for Analysis of Minimum Spanning and Shortest Path Trees of Planar Graphs. Technical Report, YALE/DCS/TR-763, Department of Computer Science, Yale University, New Haven, Connecticut.

....that element is the first or last of the deque, it cannot subsequently be deleted. A related data structure is the priority queue with attrition [Sun89] Queues with heap order (or minqueues) are useful in pagination [DF84, HL87, LH85, McC77] and VLSI river routing [CS84] Booth and Westbrook [BW90] use catenable heap ordered queues in the sensitivity analysis of minimum spanning trees, shortest path trees, and minimum cost network flow on planar graphs. Larmore and Hirschberg [LH85] and Cole and Siegal [CS84] independently show how to implement heap ordered queues in O(1) amortized time ....

H. Booth and J. R. Westbrook. Linear algorithms for analysis of minimum spanning and shortest path trees in planar graphs. Technical Report YALEU/DCS/TR-763, Yale University Dept. of Computer Science, February 1990. To appear in Algorithmica.

....for each edge fv; wg of G, by how much c(v; w) can change without affecting the minimality of G. Tarjan [18] has extended his verification algorithm to an algorithm that solves the sensitivity analysis problem in O(mff(m;n) time. For the special case of planar graphs, Booth and Westbrook [2] have given an algorithm running in O(m) time. We shall describe a randomized O(m) time algorithm and a deterministic algorithm that runs in time minimum to within a constant factor, although all that we can say for sure about the running time of the latter algorithm is that it is O(mff(m;n) ....

H. Booth and J. Westbrook, Linear Algorithms for Analysis of Minimum Spanning and Shortest Path Trees in Planar Graphs, Yale University, Department of Computer Science, TR-768, Feb. 1990; also Algorithmica, to appear.

....efficiently, if the underlying data structure has been designed to expect such changes. Even then real time responsivity can be further improved. ffl Sensitivity: The path only has to be re planned if the changes will result in a new path. Each part of the path can have an associated sensitivity (Booth and Westbrook 1992), which determine how big the change must be before re planning. ffl Relaxed Updating: If the discrepancies introduced by the changes does not lead to unacceptable results, then the workload can be spread out over time, by updating a certain part of the data structure in each time step. Further ....

Booth, Heather and Westbrook, Jeffery, 1992. A Linear Algorithm for Analysis of Minimum Spanning and Shortest Path Trees of Planar Graphs. Technical Report, YALE/DCS/TR-763, Department of Computer Science, Yale University, New Haven, Connecticut.

....of operations in order to distinguish catenation as a special operation which complicates the implementation of mindeques. 2. 1 Related Work and Applications Queues with heap order (or minques) are useful in pagination [DF84, HL87, LH85, McC77] and VLSI river routing [CS84] Booth and Westbrook [BW90] use catenable minques in the sensitivity analysis of minimum spanning trees, shortest path trees, and mininum cost network flow on planar graphs. Larmore and Hirschberg [LH85] and Cole and Siegal [CS84] independently showed how to implement minques in O(1) amortized time per operation [Tar85] ....

....order of the path compressions seems much more difficult. 7 Acknowledgements Jeff Westbrook initially brought the problem of catenable mindeques to our attention. Milena Mihail and Peter Winkler provided thoughtful criticism of an earlier draft of Sections 2 and 3. Greg Frederickson pointed us to [BW90] ....

H. Booth and J. Westbrook. Linear algorithms for analysis of minimum spanning and shortest path trees in planar graphs. Technical Report YALEU/DCS/TR-763, Yale University Dept. of Computer Science, February 1990. To appear in Algorithmica.

....trees, and if k n, it identifies and contracts n k edges that will be in all of these trees. Identifying these edges uses an algorithm for the sensitivity analysis of minimum spanning trees, either Tarjan s algorithm [T1] T2] for general graphs or the algorithm of Booth and Westbrook [BW] for planar graphs. Also used is the linear time selection algorithm [BFPRT] We call the resulting graph the contracted graph. Note that the k smallest spanning trees of the contracted graph are in one to one correspondence with the k smallest spanning trees of our original graph. Next, we ....

....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 ....

H. Booth and J. Westbrook, A linear algorithm for analysis of minimum spanning and shortest-path trees of planar graphs, Algorithmica, 11 (1994), pp. 341--352.

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Heather Booth and Je#ery Westbrook. A linear algorithm for analysis of minimum spanning and shortest-path trees of planar graphs. Algorithmica, 11(4):341--352, 1994.

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