## Partitioning Graphs with Costs and Weights on Vertices: Algorithms and Applications (0)

Venue: | of Lecture Notes in Computer Science |

Citations: | 4 - 0 self |

### BibTeX

@INPROCEEDINGS{Djidjev_partitioninggraphs,

author = {Hristo N. Djidjev},

title = {Partitioning Graphs with Costs and Weights on Vertices: Algorithms and Applications},

booktitle = {of Lecture Notes in Computer Science},

year = {},

pages = {130--143},

publisher = {Springer Verlag}

}

### OpenURL

### Abstract

We prove separator theorems in which the size of the separator is minimized with respect to non-negative vertex costs. We show that for any planar graph G there exists a vertex separator of total vertex cost at most c qP v2V (G) (cost(v)) 2 and that this bound is optimal within a constant factor. Moreover such a separator can be found in linear time. This theorem implies a variety of other separation results discussed in the paper. We describe application of our separator theorems to graph embedding problems, graph pebbling, and multi-- commodity flow problems. 1 Introduction Background. A separator is a small set of vertices or edges whose removal divides a graph into two roughly equal parts. The existence of small separators for some important classes of graphs such as planar graphs can be used in the design of efficient divide-and-conquer algorithms for problems on such graphs. Formally, a separator theorem for a given class of graphs S states that any n-vertex graph from S ca...

### Citations

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(Show Context)
Citation Context ...nto components of no more than ffn vertices, for some constant ff ! 1, by removing a set called an ff separator of no more than f(n) vertices, where f(n) = o(n). A classic result of Lipton and Tarjan =-=[20]-=- states that a separator theorem with f(n) = O( p n) holds for the class of planar graphs and that such a This work is partially supported by National Science Foundation grant CCR--9409191. separator ... |

241 |
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Citation Context ...ree dimensional finite element graphs [23, 28]. Separators with special structure have been studied, including simple cycle separators [22, 9] and separators consisting of edges [5]. Leighton and Rao =-=[17]-=- have studies approximation algorithms for constructing separators of minimum size. Separator theorems have been used for solving large sparse systems of linear equations [19, 12], for developing algo... |

190 |
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Citation Context ...es [5]. Leighton and Rao [17] have studies approximation algorithms for constructing separators of minimum size. Separator theorems have been used for solving large sparse systems of linear equations =-=[19, 12]-=-, for developing algorithms for VLSI layout design [4, 18], for shortest path problems [8], in parallel computing [13], and in computational complexity [21]. Our results. In many applications it is us... |

172 | Improved approximation algorithms for the multicommodity flow problem and local competitive routing in dynamic networks
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Citation Context ...e edges should not exceed the capacity of that edge. Approximation algorithms exist that find whether there exists a feasible flow (flow that satisfies the demands and obeys all capacity constraints) =-=[16]-=-. We consider a related version of the problem, where given the set of demands, one has to construct a network for which a feasible flow exists and whose maximum edge capacity (called a congestion of ... |

167 |
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(Show Context)
Citation Context ... work is partially supported by National Science Foundation grant CCR--9409191. separator can be found in linear time. Lipton and Tarjan also described various applications of their separator theorem =-=[21]-=-. Numerous new results on graph separation and many new application have been found since. Separator theorems exist for the class of graphs of bounded genus [6, 10], for the class of chordal graphs [1... |

165 | Faster shortest-path algorithms for planar graphs
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Citation Context ...d edges, one from v to w of length cost(w) and one from w to v of length cost(v). 2. Choose an arbitrary vertex rt to be the root of T . Run the linear time single source shortest path algorithm from =-=[15]-=- on the resulting planar graph, G 0 , with source any vertex rt of G 0 . 3. Define T to be the resulting shortest path tree of G 0 . Lemma 2.2 Algorithm Steep--tree constructs a steep spanning tree of... |

134 | A framework for solving VLSI graph layout problems
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Citation Context ...gorithms for constructing separators of minimum size. Separator theorems have been used for solving large sparse systems of linear equations [19, 12], for developing algorithms for VLSI layout design =-=[4, 18]-=-, for shortest path problems [8], in parallel computing [13], and in computational complexity [21]. Our results. In many applications it is useful to associate costs with the vertices or the edges of ... |

123 |
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Citation Context ... of minimum size. Separator theorems have been used for solving large sparse systems of linear equations [19, 12], for developing algorithms for VLSI layout design [4, 18], for shortest path problems =-=[8]-=-, in parallel computing [13], and in computational complexity [21]. Our results. In many applications it is useful to associate costs with the vertices or the edges of the graph. For example, the cost... |

94 | A separator theorem for graphs with excluded minor and its applications
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Citation Context ...parator theorems exist for the class of graphs of bounded genus [6, 10], for the class of chordal graphs [11], for the threedimensional graphs [24], for the class of graphs that exclude a given minor =-=[1]-=-, and for some geometrically defined graphs that include many three dimensional finite element graphs [23, 28]. Separators with special structure have been studied, including simple cycle separators [... |

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Citation Context ...he requirements. For example, the demands may correspond to the expected volume of phone calls between corresponding pairs of locations and the goal will be to minimize the maximum bandwidth required =-=[27]-=-. Such a tree T is called a minimum routing tree. If T is fixed (but not its capacities), one can determine the smallest possible capacity load(e) on any edge e of T as follows. Delete e from T and le... |

90 |
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Citation Context ...[11], for the threedimensional graphs [24], for the class of graphs that exclude a given minor [1], and for some geometrically defined graphs that include many three dimensional finite element graphs =-=[23, 28]-=-. Separators with special structure have been studied, including simple cycle separators [22, 9] and separators consisting of edges [5]. Leighton and Rao [17] have studies approximation algorithms for... |

84 |
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Citation Context ...], and for some geometrically defined graphs that include many three dimensional finite element graphs [23, 28]. Separators with special structure have been studied, including simple cycle separators =-=[22, 9]-=- and separators consisting of edges [5]. Leighton and Rao [17] have studies approximation algorithms for constructing separators of minimum size. Separator theorems have been used for solving large sp... |

81 |
A separator theorem for graphs of bounded genus
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(Show Context)
Citation Context ...pplications of their separator theorem [21]. Numerous new results on graph separation and many new application have been found since. Separator theorems exist for the class of graphs of bounded genus =-=[6, 10]-=-, for the class of chordal graphs [11], for the threedimensional graphs [24], for the class of graphs that exclude a given minor [1], and for some geometrically defined graphs that include many three ... |

53 | Planar separators and parallel polygon triangulation
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(Show Context)
Citation Context ... theorems have been used for solving large sparse systems of linear equations [19, 12], for developing algorithms for VLSI layout design [4, 18], for shortest path problems [8], in parallel computing =-=[13]-=-, and in computational complexity [21]. Our results. In many applications it is useful to associate costs with the vertices or the edges of the graph. For example, the costs might correspond to differ... |

43 |
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(Show Context)
Citation Context ...ration and many new application have been found since. Separator theorems exist for the class of graphs of bounded genus [6, 10], for the class of chordal graphs [11], for the threedimensional graphs =-=[24]-=-, for the class of graphs that exclude a given minor [1], and for some geometrically defined graphs that include many three dimensional finite element graphs [23, 28]. Separators with special structur... |

40 |
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- 1982
(Show Context)
Citation Context ...gorithms for constructing separators of minimum size. Separator theorems have been used for solving large sparse systems of linear equations [19, 12], for developing algorithms for VLSI layout design =-=[4, 18]-=-, for shortest path problems [8], in parallel computing [13], and in computational complexity [21]. Our results. In many applications it is useful to associate costs with the vertices or the edges of ... |

35 |
Optimal simulations of tree machines
- Bhatt, Chung, et al.
- 1986
(Show Context)
Citation Context ...embedded into a hypercube, a butterfly graph, a shuffle-exchange graph, de Bruin graph, or a cube-connected-cycles graph with average weighted dilation O(log d) and expansion one. Proof: According to =-=[3, 2, 25]-=-, any binary tree can be embedded into any of the above graphs with average dilation O(1) and expansion one. Similar results hold for embedding a planar graph into a linear, a 2dimensional, or a d-dim... |

35 |
separators: A unified geometric approach to graph partitioning
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- 1992
(Show Context)
Citation Context ...[11], for the threedimensional graphs [24], for the class of graphs that exclude a given minor [1], and for some geometrically defined graphs that include many three dimensional finite element graphs =-=[23, 28]-=-. Separators with special structure have been studied, including simple cycle separators [22, 9] and separators consisting of edges [5]. Leighton and Rao [17] have studies approximation algorithms for... |

34 |
Tarjan, The analysis of a nested dissection algorithm
- Gilbert, E
- 1987
(Show Context)
Citation Context ...es [5]. Leighton and Rao [17] have studies approximation algorithms for constructing separators of minimum size. Separator theorems have been used for solving large sparse systems of linear equations =-=[19, 12]-=-, for developing algorithms for VLSI layout design [4, 18], for shortest path problems [8], in parallel computing [13], and in computational complexity [21]. Our results. In many applications it is us... |

20 |
A.: A separator theorem for chordal graphs
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- 1984
(Show Context)
Citation Context ...1]. Numerous new results on graph separation and many new application have been found since. Separator theorems exist for the class of graphs of bounded genus [6, 10], for the class of chordal graphs =-=[11]-=-, for the threedimensional graphs [24], for the class of graphs that exclude a given minor [1], and for some geometrically defined graphs that include many three dimensional finite element graphs [23,... |

19 |
BOptimal simulations by butterfly networks
- Bhatt, Chung, et al.
- 1986
(Show Context)
Citation Context ...embedded into a hypercube, a butterfly graph, a shuffle-exchange graph, de Bruin graph, or a cube-connected-cycles graph with average weighted dilation O(log d) and expansion one. Proof: According to =-=[3, 2, 25]-=-, any binary tree can be embedded into any of the above graphs with average dilation O(1) and expansion one. Similar results hold for embedding a planar graph into a linear, a 2dimensional, or a d-dim... |

13 | Planar separators and the euclidean norm
- Gazit, Miller
- 1990
(Show Context)
Citation Context ...], and for some geometrically defined graphs that include many three dimensional finite element graphs [23, 28]. Separators with special structure have been studied, including simple cycle separators =-=[22, 9]-=- and separators consisting of edges [5]. Leighton and Rao [17] have studies approximation algorithms for constructing separators of minimum size. Separator theorems have been used for solving large sp... |

11 |
Advances in pebbling
- Pippenger
- 1982
(Show Context)
Citation Context ...e similar to the proof of Theorem 7. 5.2 Pebbling Pebbling a directed graph is a well studied problem in programming with applications to register allocation and to deriving space-time tradeoffs (see =-=[26, 29]-=- for a survey of results). Pebbling is a one-person game that involves placing pebbles on the vertices of a directed acyclic graph G. During any step, one can remove a pebble from a currently pebbled ... |

9 |
A separator theorem
- Djidjev
- 1981
(Show Context)
Citation Context ...pplications of their separator theorem [21]. Numerous new results on graph separation and many new application have been found since. Separator theorems exist for the class of graphs of bounded genus =-=[6, 10]-=-, for the class of chordal graphs [11], for the threedimensional graphs [24], for the class of graphs that exclude a given minor [1], and for some geometrically defined graphs that include many three ... |

9 |
A New Pebble Game that Characterizes Parallel Complexity Classes
- Venkateswaran, Tompa
- 1989
(Show Context)
Citation Context ...e similar to the proof of Theorem 7. 5.2 Pebbling Pebbling a directed graph is a well studied problem in programming with applications to register allocation and to deriving space-time tradeoffs (see =-=[26, 29]-=- for a survey of results). Pebbling is a one-person game that involves placing pebbles on the vertices of a directed acyclic graph G. During any step, one can remove a pebble from a currently pebbled ... |

8 |
Comparing Interconnection Networks
- Monien, Sudborough
- 1994
(Show Context)
Citation Context ...embedded into a hypercube, a butterfly graph, a shuffle-exchange graph, de Bruin graph, or a cube-connected-cycles graph with average weighted dilation O(log d) and expansion one. Proof: According to =-=[3, 2, 25]-=-, any binary tree can be embedded into any of the above graphs with average dilation O(1) and expansion one. Similar results hold for embedding a planar graph into a linear, a 2dimensional, or a d-dim... |

6 | Designing multi-commodity flow trees
- Khuller, Raghavachari, et al.
- 1994
(Show Context)
Citation Context ...r which a feasible flow exists and whose maximum edge capacity (called a congestion of the network) is small. The case where the network is required to be a tree is called the tree congestion problem =-=[14]-=-. Formally, let G = (V; E) be a digraph with non-negative real weights (demands) dem(v; w) on its edges. The goal is to find a spanning tree T of G with edge capacities for which (i) a feasible flow s... |

2 |
Ondrej S'ykora, and Imrich Vr^to. Edge separators of planar and outerplanar graphs with applications
- Diks, Djidjev
- 1993
(Show Context)
Citation Context ...s that include many three dimensional finite element graphs [23, 28]. Separators with special structure have been studied, including simple cycle separators [22, 9] and separators consisting of edges =-=[5]-=-. Leighton and Rao [17] have studies approximation algorithms for constructing separators of minimum size. Separator theorems have been used for solving large sparse systems of linear equations [19, 1... |

1 |
Djidjev and Shankar Venkatesan. Reduced constants for simple cycle graph separation
- Hristo
- 1995
(Show Context)
Citation Context ...ll other steps can be implemented in linear time in a straightforward manner. We note that a more involved proof would yield even a better constant in Theorem 1. For instance, using the techniques of =-=[7]-=- we can replace the constant p 8 by 2 and also guarantee that the separator C is a simple cycle, provided the original graph is triangulated. 3 Tightness of the result In this section we will show tha... |