Undirected Graph Exploration with ⊝(log log n) Pebbles

We consider the fundamental problem of exploring an undirected and initially unknown graph by an agent with little memory. The vertices of the graph are unlabeled, and the edges incident to a vertex have locally distinct labels. In this setting, it is known that Θ(log n) bits of memory are necessary and sufficient to explore any graph with at most n vertices. We show that this memory requirement can be decreased significantly by making a part of the memory distributable in the form of pebbles. A pebble is a device that can be dropped to mark a vertex and can be collected when the agent returns to the vertex. We show that for an agent O(log log n) distinguishable pebbles and bits of memory are sufficient to explore any bounded-degree graph with at most n vertices. We match this result with a lower bound exhibiting that for any agent with sub-logarithmic memory, Ω(log log n) distinguishable pebbles are necessary for exploration.

[1]  Omer Reingold,et al.  Undirected connectivity in log-space , 2008, JACM.

[2]  Adrian Kosowski,et al.  Constructing a Map of an Anonymous Graph: Applications of Universal Sequences , 2010, OPODIS.

[3]  Michal Koucký.  Log-Space Constructible Universal Traversal Sequences for Cycles of Length O(n4.03) , 2001, COCOON.

[4]  Michael Jenkin,et al.  Robotic exploration as graph construction , 1991, IEEE Trans. Robotics Autom..

[5]  Richard J. Lipton,et al.  Random walks, universal traversal sequences, and the complexity of maze problems , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[6]  Eric Allender,et al.  On traversal sequences, exploration sequences and completeness of kolmogorov random strings , 2003 .

[7]  Sébastien Tixeuil,et al.  The Reduced Automata Technique for Graph Exploration Space Lower Bounds , 2006, Essays in Memory of Shimon Even.

[8]  Hans-Anton Rollik,et al.  Automaten in planaren Graphen , 1979, Acta Informatica.

[9]  Gregor von Bochmann,et al.  A brief history of web crawlers , 2013, CASCON.

[10]  Susanne Albers,et al.  Exploring unknown environments , 1997, STOC '97.

[11]  Avi Wigderson,et al.  Universal Traversal Sequences for Expander Graphs , 1993, Inf. Process. Lett..

[12]  Pierre Fraigniaud,et al.  Digraphs Exploration with Little Memory , 2004, STACS.

[13]  Luca Trevisan,et al.  Pseudorandom walks on regular digraphs and the RL vs. L problem , 2006, STOC '06.

[14]  Michael A. Bender,et al.  The power of a pebble: exploring and mapping directed graphs , 1998, STOC '98.

[15]  Jérémie Chalopin,et al.  Mapping Simple Polygons: How Robots Benefit from Looking Back , 2011, Algorithmica.

[16]  Anupam N. Shah.  Pebble automata on arrays , 1974, Comput. Graph. Image Process..

[17]  Manuel Blum,et al.  On the power of the compass (or, why mazes are easier to search than graphs) , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[18]  Krzysztof Diks,et al.  Tree exploration with little memory , 2002, SODA.

[19]  Mona Singh,et al.  Piecemeal Graph Exploration by a Mobile Robot , 1999, Inf. Comput..

[20]  Sorin Istrail,et al.  Polynomial universal traversing sequences for cycles are constructible , 1988, STOC '88.

[21]  Frank Hoffmann.  One Pebble Does Not Suffice to Search Plane Labyrinths , 1981, FCT.

[22]  M. Blum,et al.  Automata on a 2-Dimensional Tape , 1967, SWAT.

[23]  Michal Koucký,et al.  Log-space constructible universal traversal sequences for cycles of length O(n4.03) , 2001, Theor. Comput. Sci..

[24]  Michal Koucký,et al.  Universal traversal sequences with backtracking , 2001, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[25]  Jérémie Chalopin,et al.  Mapping Simple Polygons: The Power of Telling Convex from Reflex , 2015, TALG.

[26]  Andrzej Pelc,et al.  Graph exploration by a finite automaton , 2005, Theor. Comput. Sci..

[27]  Andrzej Pelc,et al.  Tree exploration with logarithmic memory , 2007, SODA '07.