## Embedding graphs in books: a layout problem with applications to VLSI design (1987)

Venue: | SIAM J. ALGEBRAIC DISCRETE METHODS |

Citations: | 48 - 0 self |

### BibTeX

@ARTICLE{Chung87embeddinggraphs,

author = {Fan R. K. Chung and Frank Thomson Leighton and Arnold L. Rosenberg},

title = {Embedding graphs in books: a layout problem with applications to VLSI design},

journal = {SIAM J. ALGEBRAIC DISCRETE METHODS},

year = {1987},

volume = {8},

number = {1},

pages = {33--58}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study the graph-theoretic problem of embedding a graph in a book with its vertices in a line along the spine of the book and its edges on the pages in such a way that edges residing on the same page do not cross. This problem abstracts layout problems arising in the routing of multilayer printed circuit boards and in the design of fault-tolerant processor arrays. In devising an embedding, one strives to minimize both the number of pages used and the "cutwidth" of the edges on each page. Our main results (1) present optimal embeddings of a variety of families of graphs; (2) exhibit situations where one can achieve small pagenumber only at the expense of large cutwidth; and (3) establish bounds on the minimum pagenumber of a graph based on various structural properties of the graph. Notable in the last category are proofs that (a) every n-vertex d-valent graph can be embedded using O(dn1/2) pages, and (b) for every d>2 and all large n, there are n-vertex d-valent graphs whose pagenumber is at least log n]"

### Citations

134 | A framework for solving vlsi graph layout problems
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- 1982
(Show Context)
Citation Context ...RG yields a nontrivial upper bound on pagenumber but does not provide any nontrivial lower bound. For our purposes, the simplest measure of the ease of bisecting a graph resides in the Bhatt-Leighton =-=[4]-=- notion of bifurcator: The graph G has a p-bifurcator ofsize B (B an integer and p> 1) either if G has fewer than B edges or if G admits a decomposition tree with the following property. The root of t... |

104 | A complexity theory for VLSI
- Thompson
- 1980
(Show Context)
Citation Context ...have size f(n/log n) for all p> 1. Proof. Games [12] has shown that the n-input Benes network can be embedded in a 3-page book. A straightforward application of Thompson’s lower bound proof technique =-=[25]-=- shows that every p-bifurcator ofthe O(n. log n)-vertex 3-page embeddable graph B(n) has size (n). l-] The bound in Proposition 4.6 has recently been strengthened by Galil, Kannan and Szemeredi 11], b... |

48 |
Optimal rearrangeable multi-stage connecting networks
- Benes
- 1964
(Show Context)
Citation Context ...s inthe Fast FourierTransform algorithm, Banyan networkswhose structure approximates that of the Boolean n-cube while retaining bounded vertex-degrees, and the Benes rearrangeable permutation network =-=[2]-=-, which is shown in Fig. 5(a).We concentrate on the Benes network, since it is a supergraph of the others, hence the hardest of the group to embed efficiently. Let n be a power of 2. The n-input Benes... |

15 | On nontrivial separators for k-page graphs, and simulations by nondeterministic one-tape Turing machines - Galil, Kannan, et al. |

15 |
The complexity of the hamiltonian circuit problem for maximal planar graphs
- Wigderson
- 1982
(Show Context)
Citation Context ...eft to the reader. [3 The final corollary of Theorem 4.1 is a direct consequence of Wigderson’s result that the problem of deciding whether or not a maximal planar graph is hamiltonian is NP-complete =-=[28]-=-. COROLLARY 4.4. The problem ofdeciding 2-page embeddability is NP-complete. 4.2. Bisection width. The next structural property we consider measures the ease of recursively cutting a graph into two eq... |

9 | Embedding planar graphs in seven pages
- Heath
- 1984
(Show Context)
Citation Context ...ion. Buss and Shor [6] were the first to demonstrate that planar graphs can be embedded in a bounded number of pages; their elegant layout technique embeds an arbitrary planar graph in 9 pages. Heath =-=[15]-=-, [16] used a quite different technique that improves this bound to 7 pages. Yannakakis [29] has recently settled the issue by proving coincident upper and lower bounds of 4 pages. THEOREM 4.2 [29]. E... |

9 | Single row routing
- Raghavan, Sahni
- 1983
(Show Context)
Citation Context ... routing each of its rows and each of its columns independently. The variant of this scenario that does not allow a net to run from the top of a row around to its bottom nor to change layers en route =-=[20]-=- corresponds directly to our embedding problem applied to small-valence graphs. Fault-tolerantprocessor arrays. The DIOGENES approach to the design of faulttolerant arrays of identical processing elem... |

3 |
The complexity ofcoloring circular arcs and chords
- GAREY, JOHNSON, et al.
- 1980
(Show Context)
Citation Context ...ring for a circle graph (which is the intersectiongraph for chords of a circle). The correspondence between the two problems is best seen from Formulation 3 ofthe book-embedding problem. Garey et al. =-=[13]-=- show that the coloring problem for circle graphs is NP-complete. THEOREM 1.3 [10], [13]. Thefollowingproblem is NP-complete" Given a graph G, an orderingofthe vertices ofG, andan integer k, decide wh... |

3 |
Graphs that are almost binary trees
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- 1982
(Show Context)
Citation Context ...5 ofoptimal in cumulative pagewidth. Proof. Optimality in number ofpages is immediate since X(d) is not outerplanar for d >-3. The (near-) optimality of the claimed cutwidth follows from the proof in =-=[17]-=- that X(d) has no bisector of size less than d, coupled with the demonstration that this implies a similar bound on cutwidth. It remains only to verify that the widths ofthe pages in the prescribed em... |

2 |
A combinatorialproblem in geometry
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- 1935
(Show Context)
Citation Context ...nputs and outputs on the one hand, and increasing (resp., decreasing) subsequences of an integer sequence on the other hand, we note the following variant of a well-known result of Erdos and Szekeres =-=[9]-=-.s40 FAN R. K. CHUNG, F. T. LEIGHTON AND A. L. ROSENBERG LEMMA 2.4 [9]. LetA andB be orderings ofthe integers {1, 2,. , n}. Ifsequences A andBshare no similarly ordered subsequence oflength greater th... |

1 | Complexity resultsforsingle-row routing - ARNOLD - 1982 |

1 |
The book thickness ofa graph
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(Show Context)
Citation Context ...s to layers so that edges/chords on the same layer do not cross. Formulation 3 combines the insights of 10] and [22], and yields a simple characterization of the 1-page embeddable graphs. THEOREM 1.1 =-=[3]-=-. A graph can be embedded in a one-page book if, and only if, it is outerplanar. Proofsketch. A graph G is outerplanar just when its vertices can be placed on a circle so that its edges become noncros... |

1 |
On the area ofbinary tree layouts
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- 1980
(Show Context)
Citation Context ...n random graphs. 3.1. Trees. In 1.2 we presented an embedding of the complete binary tree that turns out to be optimal in both pagenumber (one) and pagewidth (log n). (Optimality ofwidth follows from =-=[5]-=-.) It is not hard to show that all trees enjoy embeddings that are approximately as efficient as those of complete trees. PROPOSITION 3.1. Every n-vertex d-ary tree can be embedded in onepage ofwidth ... |

1 |
On thepagenumberofplanargraphs
- SHOR
- 1984
(Show Context)
Citation Context ...ing uses only a bounded number of pages is due to its reusing pages as it proceeds down the columns of B(n). This strategy of reusing independent pages is a central feature ofefficientembeddings (cf. =-=[6]-=-, 15], [29]). It is somewhat surprising that any graph capable of "computing" all permutations can be realized with any bounded number, let alone 3, of pages. PROPOSITION 3.4 [12]. The Benes network B... |

1 |
DIOGENESwA methodologyfor designing fault-tolerant processor arrays
- CHUNG, LEIGHTON, et al.
- 1983
(Show Context)
Citation Context ...to our embedding problem applied to small-valence graphs. Fault-tolerantprocessor arrays. The DIOGENES approach to the design of faulttolerant arrays of identical processing elements (PEs, for short) =-=[7]-=-, [21] uses "stacks of wires" to configure around faulty PEs. In broad terms, the approach works as follows. The PEs are laid out in a (logical, if not physical) line, with some number of "bundles" of... |

1 |
Embedding a tree on a line
- DOLEV, TRICKEY
- 1982
(Show Context)
Citation Context ...ethod of constructing the tree and from the fact that the tree has at most n- 1 edges. Details are left to the reader. [q Proposition 3.1 seeks to optimizetheworst-case treeembedding. DolevandTrickey =-=[8]-=- present an algorithm for finding a width-optimal one-page embedding for an individual tree. 3.2. Square grids. Square grids are planar and subhamiltonian, hence 2-page embeddable. (We verify this cla... |

1 | seedlings ofresearch, 6th Berkeley Symp - HAMMERSLEY - 1972 |

1 | 1891), Die Theorie der regulaeren - PETERSEN |

1 |
The Diogenes approach to testablefault-tolerant arrays ofprocessors
- ROSENBERG
- 1983
(Show Context)
Citation Context ...r embedding problem applied to small-valence graphs. Fault-tolerantprocessor arrays. The DIOGENES approach to the design of faulttolerant arrays of identical processing elements (PEs, for short) [7], =-=[21]-=- uses "stacks of wires" to configure around faulty PEs. In broad terms, the approach works as follows. The PEs are laid out in a (logical, if not physical) line, with some number of "bundles" of wires... |

1 |
Some theoretical results on the routing ofmultilayer printed-wiring boards
- So
- 1974
(Show Context)
Citation Context ...alent to embedding Gn in a k-page book, with its vertices embedded in the order a,. , Single-row routing. In an attempt to simplify the problem of routing multilayer printed circuit boards (PCBs), So =-=[22]-=- decomposed the problem in the following way. In his variant, one arranges the circuit elements in a regular grid, with wiring channels separating rows and columns of elements. One then decomposes the... |

1 |
Characterizations ofouterplanar graphs
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(Show Context)
Citation Context ...pagenumber of C(n) is at least as big as the minimum number of outerplanar graphs into which C(n) can be decomposed (Theorem 1.1); (b) an N-vertex outerplanar graph can have at mostN"noncircle" edges =-=[23]-=-; (c) C(n) has n2n-=(1/2)N logN edges. The lower bound on cumulative pagewidth follows from the easily derived fact that C(n) has minimum bisection width 2n-1. The upper bound is seen easily by descri... |

1 |
Sorting using networks ofqueues and stacks
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- 1972
(Show Context)
Citation Context ...uires width n/2, but there are width-2 two-page embeddings for this graph. 1.2. The origins of the problem. The problem has several origins. Sorting with parallel stacks. Even and Itai 10] and Tarjan =-=[24]-=- study the problem of how to realize fixed permutations of {1,..., n} with noncommunicating stacks. Initially each number is PUSHed, in the order 1 to n, onto any one ofthe stacks. After all the numbe... |

1 |
On an estimate ofthe chromatic class ofa p-graph
- VIZING
- 1964
(Show Context)
Citation Context ...o many pages. We begin by decomposing G into at most d+1 matching graphs, Go,’", Gd, each having at most n vertices, by means ofan edge-coloring algorithm (this is always possible by Vizing’s Theorem =-=[26]-=-). Now consider all possible permutations of G’s vertices (or, equivalently, all possible layouts of the vertices in the spine of a book). Focus on an arbitrary permutation 7r and on its "behavior" on... |

1 |
A set of topological invariantsfor graphs
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(Show Context)
Citation Context ...every biconnected outerplanar graph has a unique outerplanar embedding.) In the case of C(k, 3), a direct verification is a bit more difficult; but the result follows immediately from Whitney’s proof =-=[27]-=- that every triconnected planar graph has a unique planar embedding. The existence ofthe claimed small-pagewidth layouts can be verifiedby the reader from the illustrative layouts depicted in Fig. 6. ... |

1 |
Fourpages are necessary and sufficientforplanar graphs
- YANNAKAKIS
- 1986
(Show Context)
Citation Context ...nly a bounded number of pages is due to its reusing pages as it proceeds down the columns of B(n). This strategy of reusing independent pages is a central feature ofefficientembeddings (cf. [6], 15], =-=[29]-=-). It is somewhat surprising that any graph capable of "computing" all permutations can be realized with any bounded number, let alone 3, of pages. PROPOSITION 3.4 [12]. The Benes network B(n) admits ... |