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Graph Traversals as Universal Constructions
[article]

2021
*
arXiv
*
pre-print

We exploit a decomposition of

arXiv:2104.14877v1
fatcat:l3tkrmlfuzdvdnmn7pohiwjgqa
*graph**traversals*to give a novel characterization of depth-first and breadth-first*traversals**as**universal**constructions*. ... We show that each functor factors*as*a composition of*universal**constructions*, and that the usual presentation of*traversals**as*linear orders on vertices can be recovered with the addition of an inclusion ...*universal**constructions*. ...##
###
Graph Traversals as Universal Constructions

2021

We exploit a decomposition of

doi:10.4230/lipics.mfcs.2021.17
fatcat:r5o6obydd5cxxll5ytx3kl24de
*graph**traversals*to give a novel characterization of depth-first and breadth-first*traversals*by means of*universal**constructions*. ... We show that each functor factors*as*a composition of*universal**constructions*, and that the usual presentation of*traversals**as*linear orders on vertices can be recovered with the addition of an inclusion ... M F C S 2 0 2 1 17:12*Graph**Traversals**as**Universal**Constructions*▶ Lemma 52. ...##
###
Universal traversal sequences of length nO(log n) for cliques

1988
*
Information Processing Letters
*

Because of the lack of explicit

doi:10.1016/0020-0190(88)90197-4
fatcat:zhe3vlz4evas5cg5rsipr6msre
*constructions*of*universal**traversal*sequences for the class of undirected*graphs*, it is interesting to*construct**universal*sequences for special classes of undirected*graphs*... One method of finding a DsPAcE(log n) algorithm is to*construct**universal**traversal*sequences in logspace for regular undirected*graphs*. ...##
###
Universal traversal sequences for expander graphs

1993
*
Information Processing Letters
*

*Construction*of the

*Universal*Sequence Proof : Let S ⊆ Σ * ,s ∈ Σ * be

*as*in the lemma. ... Theorem 3. 1 1 There is an explicitly

*constructible*

*universal*

*traversal*sequence of length n O((d log d)/c) for all the consistently labeled (d, n) c-expander

*graphs*. 2. α < δ. ...

##
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Constructing a Map of an Anonymous Graph: Applications of Universal Sequences
[chapter]

2010
*
Lecture Notes in Computer Science
*

We also give

doi:10.1007/978-3-642-17653-1_10
fatcat:n5a7x2hq65e5pgsccmsiz24jdi
*universal*algorithms (independent of the size of the*graph*) for map*construction*when only the starting location of the robot is marked. ... Our solutions apply the technique of*universal*exploration sequences to solve the map*construction*problem under various constraints. ... If there are n nodes in the*graph*, then the Map of an agent can contain at most n nodes. The map*construction*process requires O(n 3 d) steps*as*before. ...##
###
Universal traversal sequences with backtracking

2002
*
Journal of computer and system sciences (Print)
*

In this paper we introduce a new notion of

doi:10.1016/s0022-0000(02)00023-5
fatcat:zqixodndd5h5hfpcsnzzvtq3he
*traversal*sequences that we call exploration sequences. Exploration sequences share many properties with the*traversal*sequences defined in ... Explicit*universal**traversal*sequences for 2-regular*graphs*were*constructed*in [BBKLW,B,I,K] ; for cliques in [KPS] , and for expanders in [IIW] . ... If there is a*universal**traversal*sequence for 3-regular undirected*graphs**constructible*in log-space, then the s-t-connectivity problem is in log-space, hence, SL ¼ L [AKLLR] . ...##
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Polynomial universal traversing sequences for cycles are constructible

1988
*
Proceedings of the twentieth annual ACM symposium on Theory of computing - STOC '88
*

We conjecture that

doi:10.1145/62212.62260
dblp:conf/stoc/Istrail88
fatcat:b7nvgssp2nam5dne3autcop4bm
*universal**traversing*sequences for J-regular*graphs*are log space*constructible*. ... We give the first polynomial*construction*in the literature for*universal**traversing*sequences for a-regular*graphs*in lolgn space having size O(n'."). ...##
###
Open Problems In The Universal Graph Theory

2017
*
Zenodo
*

The

doi:10.5281/zenodo.583293
fatcat:4xgpkbf6zfbchak7lny6r65pou
*universal**graph*is a theoretical*construct*capturing the idea that every aspect of reality can be modeled*as*a*graph*composed of vertices and edges and,*as*such, reality is a*graph*. ... This letter presents three open problems in our understanding of the*universal**graph*. ... Harvard*University*Press, 1975. ISBN 0-674-51030-5. ...##
###
Log-Space Constructible Universal Traversal Sequences for Cycles of Length O(n 4.03)
[chapter]

2001
*
Lecture Notes in Computer Science
*

The paper presents a simple

doi:10.1007/3-540-44679-6_2
fatcat:43a64jcst5eqdcybyijd7lksym
*construction*of polynomial length*universal**traversal*sequences for cycles. ... These*universal**traversal*sequences are log-space (even NC 1 )*constructible*and are of length O(n 4:03 ). ... A more direct approach to proving SL = L is to*construct*an explicit log-space*constructible**universal**traversal*sequence for d-regular*graphs*, where d¿3. ...##
###
Log-space constructible universal traversal sequences for cycles of length O(n4.03)

2003
*
Theoretical Computer Science
*

The paper presents a simple

doi:10.1016/s0304-3975(02)00436-x
fatcat:ke2slamjnzh6jggdxyuefan6eu
*construction*of polynomial length*universal**traversal*sequences for cycles. ... These*universal**traversal*sequences are log-space (even NC 1 )*constructible*and are of length O(n 4:03 ). ... A more direct approach to proving SL = L is to*construct*an explicit log-space*constructible**universal**traversal*sequence for d-regular*graphs*, where d¿3. ...##
###
Adjacency Maps and Efficient Graph Algorithms

2022
*
Algorithms
*

*Graph*algorithms that test adjacencies are usually implemented with an adjacency-matrix representation because the adjacency test takes constant time with adjacency matrices, but it takes linear time in ... In this article, we review the adjacency-map representation, which supports adjacency tests in constant expected time, and we show that

*graph*algorithms run faster with adjacency maps than with adjacency ... from ([ 5 ] [Appendix A]), and run the algorithms for

*graph*

*construction*, breadth-first

*traversal*, and

*universal*sink on the 86,856 random directed

*graphs*in the benchmark dataset. ...

##
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Traversability, Reconfiguration, and Reachability in the Gadget Framework
[chapter]

2022
*
Lecture Notes in Computer Science
*

Consider an agent

doi:10.1007/978-3-030-96731-4_5
fatcat:twq6ly7bhrd2jdrh4q7mq242ie
*traversing*a*graph*of "gadgets", each with local state that changes with each*traversal*by the agent. ... We characterize the complexity of*universal**traversal*, where the goal is to*traverse*every gadget at least once, for DAG gadgets, one-state gadgets, and reversible deterministic gadgets. ... If a solution exists in the initial instance, then performing that solution will bring the agent to the start of the verification*traversals*and all of those*traversals*will be possible since the gadgets ...##
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3-connected Planar Graph Isomorphism is in Log-space

2008
*
Foundations of Software Technology and Theoretical Computer Science
*

The algorithm uses the notion of

doi:10.4230/lipics.fsttcs.2008.1749
dblp:conf/fsttcs/DattaLN08
fatcat:kiwbi4xxgjg6jateoetglsmpbi
*universal*exploration sequences from [Kou02] and [Rei05]. To our knowledge, this is a completely new approach to*graph*canonization. ... We consider the isomorphism and canonization problem for 3-connected planar*graphs*. The problem was known to be L -hard and in UL ∩ coUL [TW08]. ... Thus the canonical code for*graph*G can be*constructed*in log-space*as*follows: For each edge (i, j) of colour 2 in σ ′ ,*traverse*along the edges coloured 1 starting from i and find the minimum label ...##
###
Lower bounds on the length of universal traversal sequences

1989
*
Proceedings of the twenty-first annual ACM symposium on Theory of computing - STOC '89
*

*Universal*

*traversal*sequences for d-regular n-vertex

*graphs*require length SZ(d%* + dn' log(n/d)), for 3 d d< n/3 -2. This is nearly tight for d= s(n). ... ., edge-

*universal*

*traversal*sequences, showing how improved lower bounds on these would improve the bounds given above. ...

*UNIVERSAL*

*TRAVERSAL*SEQUENCES

*Universal*

*traversal*sequences were introduced by Cook (see Aleliunas [2] and Aleliunas et al. [3] ), motivated by the complexity of

*graph*

*traversal*. ...

##
###
Lower bounds on the length of universal traversal sequences

1992
*
Journal of computer and system sciences (Print)
*

*Universal*

*traversal*sequences for d-regular n-vertex

*graphs*require length SZ(d%* + dn' log(n/d)), for 3 d d< n/3 -2. This is nearly tight for d= s(n). ... ., edge-

*universal*

*traversal*sequences, showing how improved lower bounds on these would improve the bounds given above. ...

*UNIVERSAL*

*TRAVERSAL*SEQUENCES

*Universal*

*traversal*sequences were introduced by Cook (see Aleliunas [2] and Aleliunas et al. [3] ), motivated by the complexity of

*graph*

*traversal*. ...

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