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Graph Traversals as Universal Constructions [article]

Siddharth Bhaskar, Robin Kaarsgaard
2021 arXiv   pre-print
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

Siddharth Bhaskar, Robin Kaarsgaard, Filippo Bonchi, Simon J. Puglisi
2021
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

Howard J. Karloff, Ramamohan Paturi, Janos Simon
1988 Information Processing Letters  
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

Shlomo Hoory, Avi Wigderson
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.  ... 
doi:10.1016/0020-0190(93)90199-j fatcat:hzyceyremnfr3lnhsbdrnvfvv4

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

Jérémie Chalopin, Shantanu Das, Adrian Kosowski
2010 Lecture Notes in Computer Science  
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

Michal Koucký
2002 Journal of computer and system sciences (Print)  
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

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

Marko A. Rodriguez
2017 Zenodo  
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]

Michal KouckÝ
2001 Lecture Notes in Computer Science  
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)

Michal Koucký
2003 Theoretical Computer Science  
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

Gabriel Valiente
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.  ... 
doi:10.3390/a15020067 fatcat:4akcvztzgjbxpg5hswvavg5b6e

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

Joshua Ani, Erik D. Demaine, Yevhenii Diomidov, Dylan Hendrickson, Jayson Lynch
2022 Lecture Notes in Computer Science  
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

Samir Datta, Nutan Limaye, Prajakta Nimbhorkar, Marc Herbstritt
2008 Foundations of Software Technology and Theoretical Computer Science  
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

A. Borodin, W. L. Ruzzo, M. Tompa
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.  ... 
doi:10.1145/73007.73061 dblp:conf/stoc/BorodinRT89 fatcat:zge6afm32nco5nntbqs53uyd4u

Lower bounds on the length of universal traversal sequences

Allan Borodin, Walter L. Ruzzo, Martin Tompat
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.  ... 
doi:10.1016/0022-0000(92)90046-l fatcat:iysh3mufbrdgvejivh7dyo5ryy
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