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Graph Traversals as Universal Constructions
[article]
2021
arXiv
pre-print
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We exploit a decomposition of 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. ...
arXiv:2104.14877v1
fatcat:l3tkrmlfuzdvdnmn7pohiwjgqa
Graph Traversals as Universal Constructions
2021
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We exploit a decomposition of 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. ...
doi:10.4230/lipics.mfcs.2021.17
fatcat:r5o6obydd5cxxll5ytx3kl24de
Universal traversal sequences of length nO(log n) for cliques
1988
Information Processing Letters
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Because of the lack of explicit 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. ...
doi:10.1016/0020-0190(88)90197-4
fatcat:zhe3vlz4evas5cg5rsipr6msre
Universal traversal sequences for expander graphs
1993
Information Processing Letters
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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. ...
doi:10.1016/0020-0190(93)90199-j
fatcat:hzyceyremnfr3lnhsbdrnvfvv4
Constructing a Map of an Anonymous Graph: Applications of Universal Sequences
[chapter]
2010
Lecture Notes in Computer Science
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We also give 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. ...
doi:10.1007/978-3-642-17653-1_10
fatcat:n5a7x2hq65e5pgsccmsiz24jdi
Universal traversal sequences with backtracking
2002
Journal of computer and system sciences (Print)
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In this paper we introduce a new notion of 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] . ...
doi:10.1016/s0022-0000(02)00023-5
fatcat:zqixodndd5h5hfpcsnzzvtq3he
Polynomial universal traversing sequences for cycles are constructible
1988
Proceedings of the twentieth annual ACM symposium on Theory of computing - STOC '88
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We conjecture that 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'."). ...
doi:10.1145/62212.62260
dblp:conf/stoc/Istrail88
fatcat:b7nvgssp2nam5dne3autcop4bm
Open Problems In The Universal Graph Theory
2017
Zenodo
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The 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. ...
doi:10.5281/zenodo.583293
fatcat:4xgpkbf6zfbchak7lny6r65pou
Log-Space Constructible Universal Traversal Sequences for Cycles of Length O(n 4.03)
[chapter]
2001
Lecture Notes in Computer Science
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The paper presents a simple 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. ...
doi:10.1007/3-540-44679-6_2
fatcat:43a64jcst5eqdcybyijd7lksym
Log-space constructible universal traversal sequences for cycles of length O(n4.03)
2003
Theoretical Computer Science
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The paper presents a simple 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. ...
doi:10.1016/s0304-3975(02)00436-x
fatcat:ke2slamjnzh6jggdxyuefan6eu
Adjacency Maps and Efficient Graph Algorithms
2022
Algorithms
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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. ...
doi:10.3390/a15020067
fatcat:4akcvztzgjbxpg5hswvavg5b6e
Traversability, Reconfiguration, and Reachability in the Gadget Framework
[chapter]
2022
Lecture Notes in Computer Science
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2022 pre-print arXiv:2204.00600v1 fatcat:daah5s3vyjf6zd3lgtmubpqwp4
Consider an agent 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 ...
doi:10.1007/978-3-030-96731-4_5
fatcat:twq6ly7bhrd2jdrh4q7mq242ie
3-connected Planar Graph Isomorphism is in Log-space
2008
Foundations of Software Technology and Theoretical Computer Science
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The algorithm uses the notion of 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 ...
doi:10.4230/lipics.fsttcs.2008.1749
dblp:conf/fsttcs/DattaLN08
fatcat:kiwbi4xxgjg6jateoetglsmpbi
Lower bounds on the length of universal traversal sequences
1989
Proceedings of the twenty-first annual ACM symposium on Theory of computing - STOC '89
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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. ...
doi:10.1145/73007.73061
dblp:conf/stoc/BorodinRT89
fatcat:zge6afm32nco5nntbqs53uyd4u
Lower bounds on the length of universal traversal sequences
1992
Journal of computer and system sciences (Print)
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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. ...
doi:10.1016/0022-0000(92)90046-l
fatcat:iysh3mufbrdgvejivh7dyo5ryy
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