scholarly journals The power of two choices for random walks

2021 ◽  
pp. 1-28
Author(s):  
Agelos Georgakopoulos ◽  
John Haslegrave ◽  
Thomas Sauerwald ◽  
John Sylvester
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Graph ◽  
Optimal Strategy ◽  
Cover Time ◽  
Bounded Degree ◽  
Graph Classes ◽  
Algorithmic Question ◽  
Cover Times ◽  
Bounded Degree Trees

Abstract We apply the power-of-two-choices paradigm to a random walk on a graph: rather than moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours. We prove that this allows the controller to significantly accelerate the hitting and cover times in several natural graph classes. In particular, we show that the cover time becomes linear in the number n of vertices on discrete tori and bounded degree trees, of order $${\mathcal O}(n\log \log n)$$ on bounded degree expanders, and of order $${\mathcal O}(n{(\log \log n)^2})$$ on the Erdős–Rényi random graph in a certain sparsely connected regime. We also consider the algorithmic question of computing an optimal strategy and prove a dichotomy in efficiency between computing strategies for hitting and cover times.

The ruin problem and cover times of asymmetric random walks and Brownian motions

2000 ◽  
Vol 32 (01) ◽  
pp. 177-192 ◽  
Author(s):  
K. S. Chong ◽  
Richard Cowan ◽  
Lars Holst
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Cover Time ◽  
Brownian Motions ◽  
Brownian Motion With Drift ◽  
And Brownian Motion ◽  
Cover Times ◽  
Ruin Problem

A simple asymmetric random walk on the integers is stopped when its range is of a given length. When and where is it stopped? Analogous questions can be stated for a Brownian motion. Such problems are studied using results for the classical ruin problem, yielding results for the cover time and the range, both for asymmetric random walks and Brownian motion with drift.

scholarly journals The Cover Time of Deterministic Random Walks

10.37236/439 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Tobias Friedrich ◽  
Thomas Sauerwald
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Cover Time ◽  
Short Term ◽  
Edge Cover ◽  
Graph Classes ◽  
Fixed Order ◽  
Deterministic Random Walk

The rotor router model is a popular deterministic analogue of a random walk on a graph. Instead of moving to a random neighbor, the neighbors are served in a fixed order. We examine how quickly this "deterministic random walk" covers all vertices (or all edges). We present general techniques to derive upper bounds for the vertex and edge cover time and derive matching lower bounds for several important graph classes. Depending on the topology, the deterministic random walk can be asymptotically faster, slower or equally fast as the classic random walk. We also examine the short term behavior of deterministic random walks, that is, the time to visit a fixed small number of vertices or edges.

The ruin problem and cover times of asymmetric random walks and Brownian motions

2000 ◽  
Vol 32 (1) ◽  
pp. 177-192 ◽  
Author(s):  
K. S. Chong ◽  
Richard Cowan ◽  
Lars Holst
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Cover Time ◽  
Brownian Motions ◽  
Brownian Motion With Drift ◽  
And Brownian Motion ◽  
Cover Times ◽  
Ruin Problem

A simple asymmetric random walk on the integers is stopped when its range is of a given length. When and where is it stopped? Analogous questions can be stated for a Brownian motion. Such problems are studied using results for the classical ruin problem, yielding results for the cover time and the range, both for asymmetric random walks and Brownian motion with drift.

scholarly journals Cover Times and Generic Chaining

2014 ◽  
Vol 51 (1) ◽  
pp. 247-261
Author(s):  
Joseph Lehec
Keyword(s):  
Random Walk ◽  
Recent Result ◽  
Hitting Time ◽  
Isomorphism Theorem ◽  
Cover Time ◽  
Time Estimates ◽  
Alternative Approach ◽  
Cover Times ◽  
Generic Chaining ◽  
Commute Distance

A recent result of Ding, Lee and Peres (2012) expressed the cover time of the random walk on a graph in terms of generic chaining for the commute distance. Their argument is based on Dynkin's isomorphism theorem. The purpose of this article is to present an alternative approach to this problem, based only on elementary hitting time estimates and chaining arguments.

scholarly journals Cover Times and Generic Chaining

2014 ◽  
Vol 51 (01) ◽  
pp. 247-261
Author(s):  
Joseph Lehec
Keyword(s):  
Random Walk ◽  
Recent Result ◽  
Hitting Time ◽  
Isomorphism Theorem ◽  
Cover Time ◽  
Time Estimates ◽  
Alternative Approach ◽  
Cover Times ◽  
Generic Chaining ◽  
Commute Distance

A recent result of Ding, Lee and Peres (2012) expressed the cover time of the random walk on a graph in terms of generic chaining for the commute distance. Their argument is based on Dynkin's isomorphism theorem. The purpose of this article is to present an alternative approach to this problem, based only on elementary hitting time estimates and chaining arguments.

PERFORMANCE ANALYSIS AND EVALUATION OF RANDOM WALK ALGORITHMS ON WIRELESS NETWORKS

2012 ◽  
Vol 23 (04) ◽  
pp. 779-802 ◽  
Author(s):  
KEQIN LI
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Network Model ◽  
Simple Random Walk ◽  
Cover Time ◽  
Planar Point ◽  
Random Walk Algorithm ◽  
Node Distribution ◽  
Random Node ◽  
Walk Algorithm

We propose a model of dynamically evolving random networks and give an analytical result of the cover time of the simple random walk algorithm on a dynamic random symmetric planar point graph. Our dynamic network model considers random node distribution and random node mobility. We analyze the cover time of the parallel random walk algorithm on a complete network and show by numerical data that k parallel random walks reduce the cover time by almost a factor of k. We present simulation results for four random walk algorithms on random asymmetric planar point graphs. These algorithms include the simple random walk algorithm, the intelligent random walk algorithm, the parallel random walk algorithm, and the parallel intelligent random walk algorithm. Our random network model considers random node distribution and random battery transmission power. Performance measures include normalized cover time, probability distribution of the length of random walks, and load distribution.

scholarly journals Random Perturbation of Sparse Graphs

10.37236/9510 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Max Hahn-Klimroth ◽  
Giulia Maesaka ◽  
Yannick Mogge ◽  
Samuel Mohr ◽  
Olaf Parczyk
Keyword(s):  
Random Graph ◽  
Minimum Degree ◽  
Random Perturbation ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Bounded Degree ◽  
Sparse Graphs ◽  
Fixed Constant ◽  
Bounded Degree Trees ◽  
Random Part

In the model of randomly perturbed graphs we consider the union of a deterministic graph $\mathcal{G}_\alpha$ with minimum degree $\alpha n$ and the binomial random graph $\mathbb{G}(n,p)$. This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac's theorem and the results by Pósa and Korshunov on the threshold in $\mathbb{G}(n,p)$. In this note we extend this result in $\mathcal{G}_\alpha\cup\mathbb{G}(n,p)$ to sparser graphs with $\alpha=o(1)$. More precisely, for any $\varepsilon>0$ and $\alpha \colon \mathbb{N} \mapsto (0,1)$ we show that a.a.s. $\mathcal{G}_\alpha\cup \mathbb{G}(n,\beta /n)$ is Hamiltonian, where $\beta = -(6 + \varepsilon) \log(\alpha)$. If $\alpha>0$ is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if $\alpha=O(1/n)$ the random part $\mathbb{G}(n,p)$ is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into $\mathbb{G}(n,p)$.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

scholarly journals Current Trends in Random Walks on Random Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1148
Author(s):  
Jewgeni H. Dshalalow ◽  
Ryan T. White
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Time ◽  
Real Space ◽  
Historical Background ◽  
Escape Time ◽  
The Real ◽  
Current Trends ◽  
One Step ◽  
Time Position

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

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