[Paper Review] The Cover Time of Deterministic Random Walks

Tobias Friedrich, Thomas Sauerwald|arXiv (Cornell University)|Jun 17, 2010

Optimization and Search Problems43 references4 citations

TL;DR

This paper analyzes the cover time of deterministic random walks—where particles follow fixed rotor sequences instead of random choices—on various graphs. It establishes general upper bounds using local divergence and expansion properties, and provides tight bounds for specific graph classes, showing the deterministic walk can be asymptotically faster, slower, or comparable to the classic random walk depending on topology.

ABSTRACT

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 fast 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.

Motivation & Objective

• To understand the cover time of deterministic random walks, where each vertex routes particles in a fixed cyclic order.
• To compare the efficiency of deterministic walks with classic random walks in covering all vertices or edges of a graph.
• To derive general upper bounds for vertex and edge cover time using graph properties like local divergence and expansion.
• To establish tight lower and upper bounds for key graph families such as trees, cycles, expanders, and tori.
• To examine short-term behavior, including the time to visit a fixed small number of vertices or edges.

Proposed method

• Uses rotor router model: each vertex maintains a cyclic sequence of neighbors and routes the particle in turn.
• Applies flow-based and divergence-based techniques to bound the number of times edges or vertices are traversed.
• Employs probabilistic and combinatorial arguments to relate deterministic walk behavior to random walk expectations.
• Derives upper bounds via analysis of local expansion and global graph structure, including spectral and geometric properties.
• Uses contradiction arguments and edge-visit counting to prove that no edge is traversed too frequently.
• Compares deterministic walk performance to known random walk cover time bounds using known results from the literature.

Experimental results

Research questions

• RQ1 How does the vertex cover time of the deterministic random walk compare to that of the classic random walk across different graph classes?
• RQ2 What general upper bounds can be derived for the vertex and edge cover time of deterministic walks using local graph properties?
• RQ3 For which graph families is the deterministic walk asymptotically faster, slower, or equally efficient compared to the random walk?
• RQ4 What is the short-term behavior of the deterministic walk—how quickly does it visit a small, fixed number of vertices or edges?
• RQ5 Can the deterministic walk achieve near-optimal cover times without marking visited nodes or using additional state?

Key findings

• On $k$-ary trees ($k = O(1)$), the deterministic walk has vertex cover time $\Theta(n\log n)$, while the random walk takes $\Theta(n\log^2 n)$.
• On stars, the deterministic walk covers all vertices in $\Theta(n)$ steps, significantly faster than the $\Theta(n\log n)$ of the random walk.
• On two-dimensional tori, the deterministic walk achieves $\Theta(n^{1.5})$ vertex cover time, outperforming the $\Theta(n\log^2 n)$ of the random walk.
• On $d$-dimensional tori ($d \geq 3$), the deterministic walk has vertex cover time $\mathcal{O}(n^{1+1/d})$, which is asymptotically faster than the $\Theta(n\log n)$ of the random walk.
• On hypercubes, the deterministic walk has $\Theta(n\log^2 n)$ vertex cover time, matching the random walk’s asymptotic behavior.
• On complete graphs, the deterministic walk has $\Theta(n^2)$ vertex cover time, while the random walk takes $\Theta(n\log n)$, making the deterministic walk slower in this case.