Robustness of the rotor-router mechanism E. Bampas 1 , 2 , L. G - PowerPoint PPT Presentation

Robustness of the rotor-router mechanism E. Bampas 1 , 2 , L. G ˛ asieniec 3 , R. Klasing 1 , A. Kosowski 1 , 4 , T. Radzik 5 1 LaBRI, CNRS / INRIA / Univ. of Bordeaux, France 2 School of Elec. & Comp. Eng., National Technical Univ. of Athens, Greece 3 Dept of Computer Science, Univ. of Liverpool, UK 4 Dept of Algorithms and System Modeling, Gdańsk Univ. of Technology, Poland 5 Dept of Computer Science, King’s College London, UK Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 1/17

Introduction Anonymous graphs (no node labels, but local port numbering) Exploration Agent with no operational memory Fault tolerance Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 2/17

The rotor-router mechanism Each node is equipped with a pointer 3 2 1 d−1 d Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 3/17

The rotor-router mechanism Each node is equipped with a pointer Incoming robot first increments the pointer, 3 2 1 d−1 d Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 3/17

The rotor-router mechanism Each node is equipped with a pointer Incoming robot first increments the pointer, then follows the pointer 3 2 1 d−1 d Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 3/17

Example a b d e c f Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 4/17

Example a b d e c f Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 4/17

Example a b d e c f Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 4/17

Example a b d e c f Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 4/17

Example a b d e c f Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 4/17

Known facts The agent visits each node infinitely many times It takes at most 2 mD steps before the agent stabilizes into an Euler tour of the corresponding symmetric-directed graph After the stabilization period, the agent keeps repeating the same Euler tour No nested repetitions of arc traversals Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 5/17

Known facts The agent visits each node infinitely many times It takes at most 2 mD steps before the agent stabilizes into an Euler tour of the corresponding symmetric-directed graph After the stabilization period, the agent keeps repeating the same Euler tour No nested repetitions of arc traversals ( u, v ) ( u, v ) Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 5/17

Known facts The agent visits each node infinitely many times It takes at most 2 mD steps before the agent stabilizes into an Euler tour of the corresponding symmetric-directed graph After the stabilization period, the agent keeps repeating the same Euler tour No nested repetitions of arc traversals ( u, v ) ( u, v ) possible ( x, y ) ( x, y ) Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 5/17

Known facts The agent visits each node infinitely many times It takes at most 2 mD steps before the agent stabilizes into an Euler tour of the corresponding symmetric-directed graph After the stabilization period, the agent keeps repeating the same Euler tour No nested repetitions of arc traversals ( u, v ) ( u, v ) possible ( x, y ) ( x, y ) Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 5/17

Known facts The agent visits each node infinitely many times It takes at most 2 mD steps before the agent stabilizes into an Euler tour of the corresponding symmetric-directed graph After the stabilization period, the agent keeps repeating the same Euler tour No nested repetitions of arc traversals ( u, v ) ( u, v ) impossible ( x, y ) ( x, y ) Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 5/17

Related work Priezzhev, Dhar, Dhar, Krishnamurthy: Eulerian walkers as a model of self-organized criticality. Phys. Rev. Lett. 77(25), 5079-5082 (1996) introduction of the model Wagner, Lindenbaum, Bruckstein: Smell as a computational resource - a lesson we can learn from the ant. ISTCS 1996, 219-230 Wagner, Lindenbaum, Bruckstein: Distributed covering by ant-robots using evaporating traces. IEEE Trans. Robot. Autom. 15, 918-933 (1999) cover all edges in O ( nm ) steps Bhatt, Even, Greenberg, Tayar: Traversing directed Eulerian mazes. J. Graph Algorithms Appl. 6(2), 157-173 (2002) enter an Euler tour in O ( nm ) steps Yanovski, Wagner, Bruckstein: A distributed ant algorithm for efficiently patrolling a network. Algorithmica 37(3), 165-186 (2003) enter an Euler tour in 2 mD steps Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 6/17

Main question Assuming the rotor-router has stabilized, how long does it take to recover from potential pointer faults or dynamic changes in the graph? Already known answer: the agent will enter a new Euler tour in at most 2 mD steps More refined answer, e.g. in terms of number of faults? Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 7/17

Faults in port pointers initially: after fault: 3 3 2 2 1 1 4 4 5 5 pointers may be changed to point to arbitrary edges Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 8/17

Addition of new edges initially: after fault: 3 3 2 2 4 1 1 4 5 5 6 edges may be added in arbitrary positions in the cyclic orders maybe combined with a pointer change Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 9/17

Deletion of an edge initially: after fault: 3 2 2 1 1 4 3 5 4 an edge may be deleted maybe combined with a pointer change Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 10/17

Leading tree r v Current set of pointers: F = { π v : v ∈ V } Focus on the structure of F r = F \ π r ( r : current node) Component of F r containing r is an in-bound tree rooted at r (leading tree) Other components: in-bound trees with root cycles Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 11/17

Properties of the leading tree (I) r v A node never leaves the leading tree A node v joins the leading tree at the moment the agent visits an ancestor of v Thm. All nodes in the neighborhood of the leading tree are visited within the next 2 m steps Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 12/17

Properties of the leading tree (I) r u v A node never leaves the leading tree A node v joins the leading tree at the moment the agent visits an ancestor of v Thm. All nodes in the neighborhood of the leading tree are visited within the next 2 m steps Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 12/17

Properties of the leading tree (II) r Cor. Assuming the leading tree spans all nodes, an Euler tour is traversed in the next 2 m steps The inverse also holds Def. Stabilization time: the first step when the leading tree spans all nodes Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 13/17

Main theorem r v Def. Length of an arc e is 0 iff e ∈ F r , otherwise 1 Thm. If the distance from a node v to the current node r is k , then node v joins the leading tree within 2 km steps Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 14/17

Main theorem r v Def. Length of an arc e is 0 iff e ∈ F r , otherwise 1 Thm. If the distance from a node v to the current node r is k , then node v joins the leading tree within 2 km steps w v r k − 1 Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 14/17

Robustness properties Assuming the rotor-router has stabilized to some Euler tour: Thm. The system recovers from k pointer faults within 2 m min { k, D } steps Thm. The system recovers from k edge additions within 2 m min { 2 k, D } steps Thm. If the deletion of some edge e does not disconnect the graph, then the system recovers from the deletion of e within 2 γm steps ( γ is the length of the smallest cycle containing e ) Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 15/17

Lower bounds Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 16/17

Conclusions and open problems Structural properties of the rotor-router Application to bounding recovery time from faults/dynamic changes Analyze different fault models? (e.g., changes in local port orders) Extend the leading tree concept to the case of multiple agents? Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 17/17

Conclusions and open problems Structural properties of the rotor-router Application to bounding recovery time from faults/dynamic changes Analyze different fault models? (e.g., changes in local port orders) Extend the leading tree concept to the case of multiple agents? Thank you Evangelos Bampas---OPODIS 2009, Nîmes, France, December 15-18, 2009 17/17