Introduction Moody walk analysis R/A Conclusion How to beat a random walk with a clock ? Locating a Target With an Agent Guided by Unreliable Local Advice Nicolas Hanusse 1 David Ilcinkas 1 Adrian Kosowski 2 , 1 , 3 Nicolas Nisse 2 , 4 1 CNRS-LaBRI, Univ. Bordeaux 2 Inria, France 3 Gdańsk University of Technology, Poland 4 Univ. Nice Sophia Antipolis, CNRS, I3S Nice Workshop on random graphs Nice, France, May 15, 2014 Many thanks to Nicolas Hanusse for his slides Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea How to find a cash machine in NY ? Take advantage of your knowledge Map, sense of direction (left/right, north/south...), topology... Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea How to find a cash machine in NY ? Algorithm A (Advice) Keep on following the advice until the target t is found. Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea How to find a cash machine in NY ? Algorithm A (Advice) Keep on following the advice until the target t is found (LOOP). Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea Searching for information in networks with liars source : where to go ? Numerous adversaries ADVICE the network map and IS IT A GOOD ADVICE ? the target location unknown dynamicity = ⇒ local destination information unreliable Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea Searching for information in networks with liars source : where to go ? Numerous adversaries LIAR ADVICE the network map and IS IT A GOOD ADVICE ? the target location shortest path unknown shortest path dynamicity = ⇒ local destination information unreliable Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea Searching for information in networks with liars source : where to go ? Numerous adversaries LIAR ADVICE the network map and IS IT A GOOD ADVICE ? the target location shortest path unknown shortest path dynamicity = ⇒ local destination information unreliable Some models of searching with errors For every target t , each node gives an advice. If this advice is bad, the node is a liar. Searching with uncertainty , SIROCCO’1999, Kranakis-Krizanc Searching with mobile agents in networks with liars , Disc. Applied Maths. 2004, Hanusse-Kranakis-Krizanc Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea A model of searching with errors Lille Caen Local information advice : For each target t , Rennes Paris every node u points to an Nantes incident link e ; If e is on a shortest path Lyon from u to t , u is a Bordeaux truthteller , otherwise a Nice liar. Toulouse Montpellier Marseille Hypothesis Advice and topology are unchanged during the search; worst-case analysis : The adversary knows the algorithm and chooses the worst configuration of advice. Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea A model of searching with errors Lille Caen Local information advice : For each target t , Rennes Paris every node u points to an Nantes incident link e ; If e is on a shortest path Lyon from u to t , u is a Bordeaux truthteller , otherwise a Nice liar. Toulouse Montpellier Marseille Hypothesis Advice and topology are unchanged during the search; worst-case analysis : The adversary knows the algorithm and chooses the worst configuration of advice. Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea Performance Measures Performance Measures n : # nodes, k : # liars, d : distance to the target Time T : # hops to reach the target Memory M : of the mobile agent 1 2 3 4 5 6 7 8 9 10 11 With sense of direction: T ≤ 2 n et M = Θ( 1 ) - whole exploration T = Θ( d ) et M = Θ( log n ) - zigzag T ≤ d + 4 k + 2 et M = Θ( log k ) - using advice, know k Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea Performance Measures Performance Measures n : # nodes, k : # liars, d : distance to the target Time T : # hops to reach the target Memory M : of the mobile agent d 2k+1 With sense of direction: T ≤ 2 n et M = Θ( 1 ) - whole exploration T = Θ( d ) et M = Θ( log n ) - zigzag T ≤ d + 4 k + 2 et M = Θ( log k ) - using advice, know k Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea State of the art 2 types of results specialized deterministic algorithms: ring, hypercube, complete graph, ... universal randomized algorithms with M = 0: random walk (R), biasest random walk (BR) Examples T = Ω( d + 2 k ) for binary trees for any algorithms; T = O ( d + 2 Θ( k ) ) for bounded degree graphs (BR); T = Ω( d + 2 k ) for the path (BR) Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea State of the art 2 types of results specialized deterministic algorithms: ring, hypercube, complete graph, ... universal randomized algorithms with M = 0: random walk (R), biasest random walk (BR) Examples T = Ω( d + 2 k ) for binary trees for any algorithms; T = O ( d + 2 Θ( k ) ) for bounded degree graphs (BR); T = Ω( d + 2 k ) for the path (BR) Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea State of the art Idea of the lower bound in trees T = Ω( d + 2 k ) for binary trees for any algorithms; source, liar liar k liar liar target Everything is symmetric: Ω( 2 k ) leaves look the same Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea Contribution Goal Designing universal algorithms using the least amount of requirements : no hypothesis on ports/nodes labeling ... light mobile agents ( M = O ( log n ) ) no knowledge on k . New results New algorithm: using the moody walk or R/A and M = O ( log k ) E ( T ) = 2 d + O ( k 5 ) for the path E ( T ) = O ( k 3 log 3 n ) for some expanders (random regular) Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea Contribution Goal Designing universal algorithms using the least amount of requirements : no hypothesis on ports/nodes labeling ... light mobile agents ( M = O ( log n ) ) no knowledge on k . New results New algorithm: using the moody walk or R/A and M = O ( log k ) E ( T ) = 2 d + O ( k 5 ) for the path E ( T ) = O ( k 3 log 3 n ) for some expanders (random regular) Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Searching for information in networks with liars A model of sea Biasest random walk VS moody walk Algorithm BR With probability p , follow the advice (A); Otherwise, the agent chooses a neighbor at random (R). BR R A R A R A R R A R A A R A p=1/2 R A R A L R Algorithm R/A [ L R , L A ] Keep on alternating Algorithm R for L R hops and Algorithm A for L A hops. Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
Introduction Moody walk analysis R/A Conclusion Ring or path Expanders From R/A/E with knwoledge to R/A Ring or path Gain: r.v X - distance reduction to t in one iteration. Phase R during L steps � With probability 1 − 2 / k c , | X R | < L log k ; √ With probability 1 − 2 / e , | X R | < L . s t � L log k Algorithm R/A[ L , L ] � Safe area : whp. X A > L − L log k . Dangerous area : connected component of nodes at � distance at most L log k from liars. Hanusse, Ilcinkas, Kosowski, Nisse How to beat a random walk with a clock ?
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