## I/O-Efficient Algorithms for Shortest Path Related Problems (2002)

Citations: | 13 - 2 self |

### BibTeX

@TECHREPORT{Zeh02i/o-efficientalgorithms,

author = {Norbert Ralf Zeh},

title = {I/O-Efficient Algorithms for Shortest Path Related Problems},

institution = {},

year = {2002}

}

### OpenURL

### Abstract

### Citations

8530 |
Introduction to Algorithms
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(Show Context)
Citation Context .... 5.1 Data Structures We start our review with a discussion of I/O-efficient data structures that can be used to design I/O-efficient graph algorithms. 5.1.1 An I/O-Efficient Queue A queue (e.g., see =-=[50]-=-, Chapter 11) is a simple data structure to be used when an algorithm produces data that it has to process at a later point, and it wants to guarantee that the data is processed in the same order as i... |

1130 |
A Bridging Model for Parallel Computation
- Valiant
- 1990
(Show Context)
Citation Context ...s of parallel computation model message passing systems, where the only means of communication between processors is the exchange of messages between them. Coarse grained models include the BSP model =-=[170]-=-, the BSP \Lambdasmodel [21], and the CGM model [57]. In [56], the EM-BSP, EM-BSP \Lambdas, and EM-CGM models have been proposed as extensions of the BSP, BSP \Lambdas, and CGM models which allow the ... |

992 | Depth first search and linear graph algorithms
- Tarjan
- 1972
(Show Context)
Citation Context ...ting separator algorithms are based on breadthfirst search. Depth-first search has been used to derive linear-time algorithms for a number of fundamental problems such as connectivity, biconnectivity =-=[163]-=-, and triconnectivity [97] of graphs, as well as planar embedding [98]. Unfortunately, the vertex access patterns of these two search strategies are inherently sequential and seem to be inherently ran... |

822 |
A Note on Two Problems in Connection with Graphs
- Dijkstra
- 1959
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Citation Context ...ts only if the graph does not contain a negative cycle which contains a vertex on a path from v to w. This is trivially true if all edges have non-negative weights. In this case, Dijkstra's algorithm =-=[60]-=-, when implemented using Fibonacci heaps [79], solves the single source shortest path problem in O(jV j log 2 jV j + jEj) time. If the graph contains negative edges, the Bellman-Ford algorithm [23, 72... |

765 |
Flows in Networks
- Ford, Fulkerson
- 1962
(Show Context)
Citation Context ...hm [60], when implemented using Fibonacci heaps [79], solves the single source shortest path problem in O(jV j log 2 jV j + jEj) time. If the graph contains negative edges, the Bellman-Ford algorithm =-=[23, 72]-=- solves the single source shortest path problem in O(jV jjEj) time. The all pairs shortest path problem can be solved in O(jV j 3 ) time using the Floyd-Warshall algorithm [71, 173]. For sparse graphs... |

575 |
Fibonacci heaps and their uses in improved network optimization algorithms
- Fredman, Tarjan
- 1987
(Show Context)
Citation Context ...ive cycle which contains a vertex on a path from v to w. This is trivially true if all edges have non-negative weights. In this case, Dijkstra's algorithm [60], when implemented using Fibonacci heaps =-=[79]-=-, solves the single source shortest path problem in O(jV j log 2 jV j + jEj) time. If the graph contains negative edges, the Bellman-Ford algorithm [23, 72] solves the single source shortest path prob... |

537 |
The input/output complexity of sorting and related problems
- Aggarwal, Vitter
- 1988
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Citation Context ...ster of the two algorithms on the given input data. 2.1 The Parallel Disk Model The first widely accepted model for analyzing the I/O-complexity of algorithms was the I/O-model of Aggarwal and Vitter =-=[6]-=-. In this model, a single processor is equipped with a random access (internal) memory capable of holding M data items and a disk (external memory) of unlimited size. The disk is partitioned into bloc... |

531 |
Shortest connection networks and some generalizations
- Prim
- 1957
(Show Context)
Citation Context ...r. The minimum spanning tree in the plane can be computed in O(N log N ) time for instance by computing the Delaunay triangulation [141] and then applying any standard minimum spanning tree algorithm =-=[117, 142]-=- for graphs to the Delaunay triangulation. In general, any sparse graph which is guaranteed to contain the minimum spanning tree as a subgraph can be used instead of the Delaunay triangulation. It has... |

468 |
Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms
- Booth, Lueker
- 1976
(Show Context)
Citation Context ...gorithms culminated in the linear time algorithm of Hopcroft and Tarjan [98]. An earlier algorithm of Lempel, Even, and Cederbaum [122] has later been made to run in linear time using techniques from =-=[30, 70]-=-. In [44], Chiba et al. give the details of using the algorithm of [30] to obtain a planar embedding of the given graph. Mehlhorn and Mutzel [131] provide important implementation details of the embed... |

453 | n introduction to disk drive modeling
- Ruemmler, Wilkes
- 1994
(Show Context)
Citation Context ...n, is often the bottleneck of the algorithm, as the seek time of state-of-the-art hard-drives is about six orders of magnitude larger than the time it takes to access a memory location in main memory =-=[147, 171]-=-. The largest portion of the time it takes to transfer a data item between disk and main memory is spent on positioning the read-write head of the disk. Once the head is positioned, successive data it... |

389 | A Separator Theorem for Planar Graphs
- Lipton, Tarjan
- 1979
(Show Context)
Citation Context ... we assume that every vertex of G has weight one. It is a well-known fact that every tree has a 2 3 -separator of size one. Every outerplanar graph has a 2 3 -separator of size two. Lipton and Tarjan =-=[124]-=- show that every planar graph G = (V; E) has a 2 3 -separator of size O i p N j , where N = jV j. Recursive application of these results implies that every tree or outerplanar graph has an "-separator... |

371 |
Self-adjusting binary search trees
- Sleator, Tarjan
- 1985
(Show Context)
Citation Context ...s and queries of the tree can be performed in O(log N ) time. Frederickson proposed this data structure as an alternative implementation of link-cut trees, which were introduced by Sleator and Tarjan =-=[157, 158]-=-. Callahan et al. [36] argue that the topology tree supports Insert and Delete operations at the same complexity as Link and Cut operations. The topology tree T of a rooted binary tree T is defined as... |

324 |
On a routing problem
- Bellman
- 1958
(Show Context)
Citation Context ...hm [60], when implemented using Fibonacci heaps [79], solves the single source shortest path problem in O(jV j log 2 jV j + jEj) time. If the graph contains negative edges, the Bellman-Ford algorithm =-=[23, 72]-=- solves the single source shortest path problem in O(jV jjEj) time. The all pairs shortest path problem can be solved in O(jV j 3 ) time using the Floyd-Warshall algorithm [71, 173]. For sparse graphs... |

322 | lists: A probabilistic alternative to balanced trees
- Pugh
- 1989
(Show Context)
Citation Context ...e construction of a `-frame due to Yao [176]. The algorithm of [148] takes O \GammasN log d\Gamma 1 N \Deltastime. Arya, Mount, and Smid [19] combine the `-graph of Ruppert and Seidel with skip lists =-=[143]-=- to obtain a t-spanner which has O(N ) edges and spanner diameter O(log N ), with high probability. Arya et al. [18] show how to transform any spanner with bounded out-degree into a spanner of bounded... |

240 | A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields
- CALLAHAN, KOSARAJU
- 1995
(Show Context)
Citation Context ...s, breadth-first search, depth-first search, and single source shortest paths on planar and outerplanar graphs. In the second part of the thesis, we show that the well-separated pair decomposition of =-=[37]-=- can be computed in sorting complexity. We use this decomposition to construct two types of Euclidean spanners of linear size for point sets in d dimensions. The first spanner is derived in a natural ... |

234 | Algorithms for parallel memory I: Two level memories
- Vitter, Shriver
- 1994
(Show Context)
Citation Context ... head is restricted to accessing the data stored on its platter rather than being able to access any data item. Vitter and Shriver propose an extension of the I/O-model, the Parallel Disk Model (PDM) =-=[172]-=-, as a more realistic model to describe existing disk systems which takes the restriction just described into account. This model is now the most widely accepted model for the design and analysis of I... |

226 | Tarjan, Efficient planarity testing
- Hopcroft, E
- 1974
(Show Context)
Citation Context ... graph algorithms. Depthfirst search, for example, is applied in algorithms for solving such fundamental problems as computing the connected, biconnected, and triconnected components of a given graph =-=[98]-=- and deciding whether a given graph is planar [97]. Shortest path problems arise naturally in areas such as robotics, computational graph theory, and computational geometry. Recent applications includ... |

219 | Graph structure in the web
- Broder, Kumar, et al.
- 2000
(Show Context)
Citation Context ...n graph is planar [97]. Shortest path problems arise naturally in areas such as robotics, computational graph theory, and computational geometry. Recent applications include the area of web modelling =-=[32]-=-, where depth-first search, breadth-first search, shortest paths, and connected components are used to explore the structure of the web, and Geographic Information Systems (GIS), where many fundamenta... |

218 |
Storing a sparse table with O(1) worst case access time
- Fredman, Komlós, et al.
- 1984
(Show Context)
Citation Context ...e \Omega (N log N ) lower bound can be beaten. Rabin [144] presents an algorithm that runs in expected linear time when implemented using the perfect hashing scheme of Fredman, Koml'os and Szemer'edi =-=[78]-=-. Another algorithm which takes expected linear time is proposed in [108]. The algorithm is based on sieving out points whose closest neighbors are too far away. Golin et al. [87] propose an algorithm... |

216 |
bounds for algebraic computation trees
- Ben-Or, Lower
- 1983
(Show Context)
Citation Context ... in O(N 2 ) time by examining all possible point pairs and reporting the shortest distance. But this leaves a gap to the \Omega (N log N ) lower bound provable in the algebraic computation tree model =-=[24]-=-. The first optimal algorithms solving this problem in two dimensions are due to Shamos [154] and Shamos and Hoey [155]. In [95], Hinrichs, Nievergelt, and Schorn present a very elegant O(N log N ) ti... |

200 | A theorem on Boolean matrices
- Warshall
- 1962
(Show Context)
Citation Context ...lman-Ford algorithm [23, 72] solves the single source shortest path problem in O(jV jjEj) time. The all pairs shortest path problem can be solved in O(jV j 3 ) time using the Floyd-Warshall algorithm =-=[71, 173]-=-. For sparse graphs, Johnson [103] presents an APSP algorithm which takes O(jEjjV j + jV j 2 log 2 jV j) time. For graphs with non-negative integer weights, Ahuja et al. [7] propose an SSSPalgorithm t... |

187 |
On constructing minimum spanning trees in k-dimensional spaces and related problems
- Yao
- 1982
(Show Context)
Citation Context ...ey [155] that the minimum spanning tree of a point set is a subgraph of the Delaunay triangulation of the point set. In higher dimensions, finding such a sparse graph is harder than in the plane. Yao =-=[176]-=- presents an O(N 2\Gamma "d ) time algorithm for finding a minimum spanning tree in higher dimensions, where " d is a small constant depending on the dimension d. The next improvement has been achieve... |

172 | External-memory graph algorithms
- Chiang, Goodrich, et al.
- 1995
(Show Context)
Citation Context ...d degree O(jV j + jEj) O(sort(jV j + jEj)) [126] Outerplanar graphs Outerplanarity testing, outerplanar embedding O(N ) [135] O(perm(N )) [127] BFS, DFS, SSSP, weighted "-separator O(sort(N ) log N ) =-=[28, 43]-=- O(scan(N )) [127] Planar graphs Planarity testing O(sort(N ) log 2 N ) [112, 43] O(perm(N )) [129] Planar embedding O(sort(N ) log N ) [145, 43] O(perm(N )) [129] BFS, SSSP O(N ) [111] O(perm(N )) [1... |

172 |
Disk Striping
- Salem, Garcia-Molina
- 1986
(Show Context)
Citation Context ... propose a randomized technique to achieve optimality for algorithms designed in the I/O-model when run on a machine with multiple disks. A simple, suboptimal, deterministic technique is diskstriping =-=[149]-=-. Using this technique, the D disks are viewed as one large "virtual disk" of block size DB, where the i-th block of the "virtual disk" contains the i-th block of each of the D disks. 2.2 Relation to ... |

170 |
Dividing a Graph into Triconnected Components
- Hopcroft, Tarjan
- 1973
(Show Context)
Citation Context ... is applied in algorithms for solving such fundamental problems as computing the connected, biconnected, and triconnected components of a given graph [98] and deciding whether a given graph is planar =-=[97]-=-. Shortest path problems arise naturally in areas such as robotics, computational graph theory, and computational geometry. Recent applications include the area of web modelling [32], where depth-firs... |

160 | Faster shortest-path algorithms for planar graphs
- Rauch, Klein, et al.
- 1997
(Show Context)
Citation Context ...t algorithms for these graph classes. This idea is by no means new, as it has for instance been applied to obtain more efficient internal memory algorithms for shortest path problems on planar graphs =-=[73, 74, 111]-=- or linear time algorithms for problems on graphs of bounded treewidth which are NP-hard in general [17, 27]. The first part of this thesis focuses on this idea. We propose I/O-efficient algorithms fo... |

152 | Linear time algorithms for NP-hard problems restricted to partial k-trees - Arnborg, Proskurowski - 1989 |

149 | The bu er tree: A new technique for optimal I/O-algorithms
- Arge
- 1995
(Show Context)
Citation Context ...earch trees to maintain the sweep-line status. In particular, queries on one tree need to be answered immediately to drive the updates of the other tree. This creates problems because the buffer tree =-=[11]-=-, which is the only known search tree which achieves optimal I/Operformance in an offline setting, does not support immediate query responses. We show that the sweep-line status can be maintained in a... |

139 | Spanning trees and spanners
- Eppstein
(Show Context)
Citation Context ...e input. 4.2 Geometric Spanners and Proximity Problems We conclude our survey of previous results with a discussion of results on computing geometric spanners and solving proximity problems. Eppstein =-=[67]-=- and Smid [160, 137] give excellent surveys of results in these areas. We only discuss the mosts4.2 Geometric Spanners and Proximity Problems 42 relevant results here and refer the reader to these pub... |

127 | Faster scaling algorithms for network problems
- Gabow, Tarjan
- 1989
(Show Context)
Citation Context ...-negative integer weights, Ahuja et al. [7] propose an SSSPalgorithm that takes O \GammasjEj + jV j p log 2 W \Deltastime, where W is the maximal edges4.1 Graph Algorithms 35 weight. Gabov and Tarjan =-=[81]-=- present an algorithm that solves the single source shortest path problem in O i p jV jjEj log 2 (jV jW ) j time in the presence of negative edge weights, where W is the absolute value of the edge wei... |

123 |
Multidimensional divide-and-conquer
- Bentley
- 1975
(Show Context)
Citation Context ... O(N log N ) time. In [96], Hinrichs, Nievergelt, and Schorn give an extension of their closest pair algorithm, which solves the all nearest neighbor problem in the plane in O(N log N ) time. Bentley =-=[25]-=- shows how to extend his closest pair algorithm for the d-dimensional case so that it can solve the all nearest neighbor problem in O(N log d\Gamma 1 N ) time. The first O(N log N ) time algorithm to ... |

121 | External-memory computational geometry
- Goodrich, Tsay, et al.
- 1993
(Show Context)
Citation Context ...rest neighbor algorithms discussed in Section 4.2.3. In external memory, the closest pair of a planar point set can be computed in optimal O(sort(N )) I/Os using the all nearest neighbor algorithm of =-=[90]-=-. The optimality of this algorithm is shown in [14]. It can also be computed in the same I/O-bound using the Voronoi diagram construction of [51], deriving the Delaunay triangulation from the Voronoi ... |

121 |
Classes of graphs which approximate the complete euclidean graph
- Keil, Gutwin
- 1992
(Show Context)
Citation Context ...ay triangulation has spanning ratio at least ss=2 in the worst case. Dobkin et al. [66] show that the Euclidean Delaunay triangulation has spanning ratio at most (1+ p 5)ss 2 ss 5:08. Keil and Gutwin =-=[107]-=- prove that the spanning ratio is in fact no more than 2ss 3 cos(ss=6) ss 2:42. Unfortunately, by the result of [42], the Delaunay triangulation cannot be used when a spanning ratio arbitrarily close ... |

120 |
Parallel tree contraction and its applications
- Miller, Reif
- 1985
(Show Context)
Citation Context ...gorithms for this problem. In the PRAM model, Klein and Reif [112] present a planar embedding algorithm which runs in O(log 2 N ) time using O(N ) processors, thereby improving on previous results of =-=[102, 133]-=-. As the algorithm of [30], it uses PQ-trees to maintain the set of valid planar embeddings of the graph, which is reduced step-by-step using careful modifications of the tree as more constraints in t... |

119 |
There are planar graphs almost as good as the complete graph
- Chew
- 1989
(Show Context)
Citation Context ...e. Their algorithm takes O(N log N ) time. Similar results have been achieved in [150, 167].s4.2 Geometric Spanners and Proximity Problems 46 The concept of spanner graphs has been introduced by Chew =-=[42]-=-, who shows that the rectilinear Delaunay triangulation has spanning ratio p 10. He also shows that the Euclidean Delaunay triangulation has spanning ratio at least ss=2 in the worst case. Dobkin et a... |

117 |
Fast Algorithms for Shortest Paths in Planar Graphs, with Applications
- Frederickson
- 1987
(Show Context)
Citation Context ...t algorithms for these graph classes. This idea is by no means new, as it has for instance been applied to obtain more efficient internal memory algorithms for shortest path problems on planar graphs =-=[73, 74, 111]-=- or linear time algorithms for problems on graphs of bounded treewidth which are NP-hard in general [17, 27]. The first part of this thesis focuses on this idea. We propose I/O-efficient algorithms fo... |

112 |
A linear algorithm for embedding planar graphs using PQ-trees
- Chiba, Nishizeki, et al.
- 1985
(Show Context)
Citation Context ...inated in the linear time algorithm of Hopcroft and Tarjan [98]. An earlier algorithm of Lempel, Even, and Cederbaum [122] has later been made to run in linear time using techniques from [30, 70]. In =-=[44]-=-, Chiba et al. give the details of using the algorithm of [30] to obtain a planar embedding of the given graph. Mehlhorn and Mutzel [131] provide important implementation details of the embedding phas... |

109 |
Delaunay graphs are almost as good as complete graphs
- Dobkin, Friedman, et al.
- 1990
(Show Context)
Citation Context ...ho shows that the rectilinear Delaunay triangulation has spanning ratio p 10. He also shows that the Euclidean Delaunay triangulation has spanning ratio at least ss=2 in the worst case. Dobkin et al. =-=[66]-=- show that the Euclidean Delaunay triangulation has spanning ratio at most (1+ p 5)ss 2 ss 5:08. Keil and Gutwin [107] prove that the spanning ratio is in fact no more than 2ss 3 cos(ss=6) ss 2:42. Un... |

104 |
Cederbaum I. An algorithm for planarity testing of graphs
- Lempel, Even
- 1967
(Show Context)
Citation Context ... embedding of the graph. The search for efficient planarity testing algorithms culminated in the linear time algorithm of Hopcroft and Tarjan [98]. An earlier algorithm of Lempel, Even, and Cederbaum =-=[122]-=- has later been made to run in linear time using techniques from [30, 70]. In [44], Chiba et al. give the details of using the algorithm of [30] to obtain a planar embedding of the given graph. Mehlho... |

103 | Tarjan, Faster algorithms for the shortest path problem
- Ahuja, Mehlhorn, et al.
- 1990
(Show Context)
Citation Context ...d-Warshall algorithm [71, 173]. For sparse graphs, Johnson [103] presents an APSP algorithm which takes O(jEjjV j + jV j 2 log 2 jV j) time. For graphs with non-negative integer weights, Ahuja et al. =-=[7]-=- propose an SSSPalgorithm that takes O \GammasjEj + jV j p log 2 W \Deltastime, where W is the maximal edges4.1 Graph Algorithms 35 weight. Gabov and Tarjan [81] present an algorithm that solves the s... |

103 | Euclidean spanners: short, thin, and lanky
- Arya, Das, et al.
- 1995
(Show Context)
Citation Context ...d O(sort(N ) log N ) construction, O(log N ) path reporting [34, 43] O(sort(N )) construction, O(log N ) path reporting [91] Dumbbell spanner in R d O(N log N ) construction, O(log N ) path reporting =-=[18]-=- O(sort(N )) construction, O(log N=(DB)) path reporting [126] Planar Steiner spanner, point sets O(N log N ) construction [16] O(sort(N )) construction [126] Planar Steiner spanner, obstacles O(N log ... |

96 | A new data structure for representing sorted Lists
- Huddleston, Mehlhom
- 1982
(Show Context)
Citation Context ...n-out whose nodes have been augmented with buffers to form batches of update and query operations to be processed w.r.t. the subtrees rooted at these nodes. That is, the buffer tree is an (a; b)-tree =-=[99]-=-, where a = m=4 and b = m, m = M=B. Instead of processing an update or query operation immediately, it is appended to a buffer of size B, which is held in internal memory. When this buffer runs full, ... |

93 |
Geometric spanner networks
- Narasimhan, Smid
- 2007
(Show Context)
Citation Context ...eometric Spanners and Proximity Problems We conclude our survey of previous results with a discussion of results on computing geometric spanners and solving proximity problems. Eppstein [67] and Smid =-=[160, 137]-=- give excellent surveys of results in these areas. We only discuss the mosts4.2 Geometric Spanners and Proximity Problems 42 relevant results here and refer the reader to these publications for a more... |

91 |
Probabilistic algorithms
- Rabin
- 1976
(Show Context)
Citation Context ...s not clear how to avoid the use of indirect addressing in their algorithm. If the use of randomization and non-algebraic operations is allowed, the \Omega (N log N ) lower bound can be beaten. Rabin =-=[144]-=- presents an algorithm that runs in expected linear time when implemented using the perfect hashing scheme of Fredman, Koml'os and Szemer'edi [78]. Another algorithm which takes expected linear time i... |

89 | A functional approach to external graph algorithms
- Abello, Buchsbaum, et al.
- 1998
(Show Context)
Citation Context ...ficiently. Next we discuss the results of each chapter in detail.s1.2 Summary of the Thesis 5 Problem Previous Result Our Result General graphs Maximal matching O i jEj jV j sort(jV j) log 2 jV j M j =-=[1]-=- O(sort(jV j + jEj)) [128] Maximal independent set O(jV j + jEj) O(sort(jV j + jEj)) Coloring graphs of bounded degree O(jV j + jEj) O(sort(jV j + jEj)) [126] Outerplanar graphs Outerplanarity testing... |

89 |
An O(n log n) Algorithm for the All-Nearest-Neighbor Problem
- Vaidya
- 1989
(Show Context)
Citation Context ... be reported in sorted order, an improved O(N log N + K) time algorithm is presented in [59]. Salowe [151] presents another O(N log N + K) time algorithm for this problem, which uses the algorithm of =-=[168]-=- and parametric search. Lenhof and Smid [123] give a much simpler algorithm achieving the same running time; but they use indirect addressing, thereby leaving the algebraic model of computation. The w... |

83 |
Graph Algorithms. Computer Science
- Even
- 1979
(Show Context)
Citation Context ...in the literature. Even though different definitions are used for the same concepts in the literature, they are all similar to the ones presented here. Our definitions are closest to the ones used in =-=[69, 93]-=-. The definition of the triconnected components of a graph is taken from [97]. 3.2.1 Graphs A (multi)graph is an ordered pair G = (V; E) of a set V and a multiset E. Graph G is simple if E is a set, i... |

82 |
Finding Small Simple Cycle Separators for 2-Connected Planar Graphs
- Miller
- 1986
(Show Context)
Citation Context ...ded genus. Other results include results on computing edge separators [61], vertex separators with negative and multiple vertex weights [64], and separators with vertex costs and weights [63]. Miller =-=[134]-=- presents a linear time algorithm to find a simple cycle separator of size O( p f N ) for a biconnected planar graph whose largest face has f edges on its boundary. A number of algorithms for computin... |

81 |
A separator theorem for graphs of bounded genus
- Gilbert, Hutchinson, et al.
- 1984
(Show Context)
Citation Context ...raphs. Lipton and Tarjan [124] were the first to show that every planar graph has a 2 3 -vertex separator of size O i p N j . They also present a linear time algorithm to compute such a separator. In =-=[86]-=-, the result is generalized to embedded graphs of bounded genus. In particular, the paper presents a linear time algorithm to compute a 2 3 -vertex separator of size O \Gamma p gN \Deltas, where g is ... |

79 | New sparseness results on graph spanners - Chandra, Das, et al. - 1995 |