Greedy Random Walk
Tal Orenshtein,Igor Shinkar +1 more
TL;DR: In this article, the authors studied the greedy random walk on graphs and showed that the expected edge cover time is linear in the number of edges for certain natural families of graphs, such as the complete graph, even degree expanders of logarithmic girth.
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Abstract: We study a discrete time self interacting random process on graphs, which we call Greedy Random Walk. The walker is located initially at some vertex. As time evolves, each vertex maintains the set of adjacent edges touching it that have not been crossed yet by the walker. At each step, the walker being at some vertex, picks an adjacent edge among the edges that have not traversed thus far according to some (deterministic or randomized) rule. If all the adjacent edges have already been traversed, then an adjacent edge is chosen uniformly at random. After picking an edge the walk jumps along it to the neighboring vertex. We show that the expected edge cover time of the greedy random walk is linear in the number of edges for certain natural families of graphs. Examples of such graphs include the complete graph, even degree expanders of logarithmic girth, and the hypercube graph. We also show that GRW is transient in $\Z^d$ for all $d \geq 3$.
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Citations
Graphene Quantum Dots with Improved Fluorescence Activity via Machine Learning: Implications for Fluorescence Monitoring
TL;DR: In this paper , the synthesis process of B,N-GQDs by oxidizing 3-aminophenylboronic acid monohydrate and study their core synthesis process parameters (synthesis temperature, H2O2 additional volume, and synthesis time) and corresponding synergic/antagonistic effects in a multidimensional and wide-ranging region.
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Random walks which prefer unvisited edges. Exploring high girth even degree expanders in linear time
TL;DR: In this article, a modified random walk which uses unvisited edges whenever possible, and makes a simple random walk otherwise, is considered, and the vertex cover time of such a walk is shown to be Theta(n).
Vacant Sets and Vacant Nets: Component Structures Induced by a Random Walk
Colin Cooper,Alan Frieze +1 more
TL;DR: In this article, the authors established the threshold value for a phase transition in the vertices and edges of the vacant set of a Gaussian random walk on a finite graph, and showed that for a random Gaussian walk on Gaussian regular graphs, the largest component of the Gaussian vacant set has a unique giant component of size O(log n).
The Power of Two Choices for Random Walks
TL;DR: In this article, the power-of-two-choice paradigm is applied to random walks on a graph, where instead of moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours.
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Once reinforced random walk on Z × Γ
TL;DR: This article revisited the article of Vervoort (2002) and prouvons that the processus is recurrent on tout graphe of the form Z×Γ, ou Γ is un graphe fini, and obtenons egalement un theoreme de forme for l'ensemble des sites visites.
References
Expander graphs and their applications
S Hoory,Nathan Linial +1 more
TL;DR: Expander graphs were first defined by Bassalygo and Pinsker in the early 1970s, and their existence was proved in the late 1970s as discussed by the authors and early 1980s.
Random walks, universal traversal sequences, and the complexity of maze problems
Romas Aleliunas,Richard M. Karp,Richard J. Lipton,László Lovász,Charles Rackoff +4 more
- 29 Oct 1979
TL;DR: Results are derived suggesting that the undirected reachability problem is structurally different from, and easier than, the directed version of NSPACE(logn), an affirmative answer to a question of S. Cook.
830
Non-backtracking random walks mix faster
TL;DR: In this article, the mixing rate of a non-backtracking random walk on a regular expander has been shown to be up to twice as fast as that of a simple random walk.