form algorithms we obtain and the corresponding state of the art existing non-uniform algorithms.)
In another application, we show how to transform a Monte-Carlo local algorithm into a Las Vegas one. Fi- nally, given a constant number of uniform algorithms for the same problem whose running times depend on different parameters—which are unknown to nodes— we show a method for constructing a uniform algorithm solving the problem, that on every instance runs asymp- totically as fast as the fastest algorithm among those given algorithms.
Stating our main results requires a number of for- mal definitions, so we defer the precise statements to later parts of the paper. Rather, we provide here some interesting corollaries of our results. References for the corresponding non-uniform algorithms are provided in Table 1.2. (The notion of “moderately-slow function” used in item (iii) below is defined in Section 2.)
Corollary 1
(i) There exists a uniform deterministic algorithm solving MIS on general graphs in time
\begin{equation} \begin{array}{@{}l@{}} \min \{g(n), h(\Delta, n), f(a, n)\}, \\ \quad \text{where } g(n) = 2^{O(\sqrt{\log n})}, h(\Delta, n) = O(\Delta + \log^* n), \\ \quad \text{and } f(a, n) \text{ is bounded as follows. } f(a, n) = o(\log n) \\ \quad \text{for graphs of arboricity } a = o(\sqrt{\log n}), f(a, n) = \\ \quad \text{for } (\log n)/(\log \log n) \text{ for arboricity } a = O(\log^{1/2-\delta} n), \\ \quad \text{for some constant } \delta \in (0, 1/2); \text{ and otherwise: } f(a, n) = \\ \quad \qquad O(a + a^{\epsilon} \log n), \text{ for arbitrary small constant } \epsilon > 0. \end{array} \tag{2} \end{equation}
(ii) There exists a uniform deterministic algorithm solving the $(\Delta + 1)$-coloring problem on general graphs in time $\min{O(\Delta + \log^* n), 2^{O(\sqrt{\log n})}}$.
(iii) There exists a uniform deterministic algorithm solving the $\lambda(\Delta+1)$-coloring problem on general graphs and running in time $O(\Delta/\lambda + \log^* n)$, for any $\lambda \ge 1$, such that $\Delta/\lambda$ is a moderately-slow function. In particular, there exists a uniform deterministic algorithm solving the $O(\Delta^2)$-coloring problem in time $O(\log^* n)$.
(iv) The following uniform deterministic coloring algorithms exist.
A uniform $\Delta^{1+\alpha(1)}$-coloring algorithm running in time $O(f(\Delta) \log \Delta \log n \log \log n)$, for an arbitrarily slow-growing function $f = \omega(1)$.
A uniform $O(\Delta^{1+\epsilon})$-coloring algorithm running in $O(\log \Delta \log n \log \log n)$ time, for any constant $\epsilon > 0$.
A uniform $O(\Delta)$-coloring algorithm running in $O(\Delta^{\epsilon} \log n \log \log n)$ time, for any constant $\epsilon > 0$.
(v) — There exists a uniform deterministic $O(\Delta)$-edge-coloring algorithm for general graphs running in time $O(\Delta^{\epsilon} + \log^* n)$, for any constant $\epsilon > 0$.
- There exists a uniform deterministic $O(\Delta^{1+\epsilon})$-edge-coloring algorithm for general graphs that runs in time $O(\log \Delta + \log^* n)$, for any constant $\epsilon > 0$.
(vi) There exists a uniform deterministic algorithm solving the maximal matching problem in time $O(\log^4 n)$.
(vii) For a constant integer $c \ge 1$, there exists a uniform randomized algorithm solving the $(2, 2(c+1))$-ruling set problem in time $O(2^c \log^{1/c} n)$.
2 Preliminaries
General definitions: For two integers $a$ and $b$, we let $[a, b] = {a, a+1, \dots, b}$. A vector $\underline{x} \in \mathbf{R}^\ell$ is said to dominate a vector $\underline{y} \in \mathbf{R}^\ell$ if $\underline{x}$ is coordinate-wise greater than or equal to $\underline{y}$, that is, $x_k \ge \frac{\underline{y}_k}{c}$ for each $k \in [1, \ell]$.
For a graph $G$, we let $V(G)$ and $E(G)$ be the sets of nodes and edges of $G$, respectively. (Unless mentioned otherwise, we consider only undirected and unweighted graphs.) The degree $\deg_G(v)$ of a node $v \in V(G)$ is the number of neighbors of $v$ in $G$. The maximum degree of $G$ is $\Delta_G = \max{\deg_G(v) : v \in V(G)}$.
Let $u$ and $v$ be two nodes of $G$. The distance $\mathrm{dist}_G(u, v)$ between $u$ and $v$ is the number of edges on a shortest path connecting them. Given an integer $r \ge 0$, the ball of radius $r$ around $u$ is the subgraph $B_G(u, r)$ of $G$ induced by the collection of nodes at distance at most $r$ from $u$. The neighborhood $N_G(u)$ of $u$ is the set of neighbors of $u$, i.e., $N_G(u) = B_G(u, 1) \setminus {u}$. In what follows, we may omit the subscript $G$ from the previous notations when there is no risk of confusion.
Functions: A function $f: \mathbf{R}^{\ell} \rightarrow \mathbf{R}$ is non-decreasing if for every two vectors $\underline{x}$ and $\underline{y}$ such that $\underline{x}$ dominates $\underline{y}$,
$f(\underline{y}) \le f(\underline{x}).$
A function $f: \mathbf{R}^+ \to \mathbf{R}^+$ is moderately-slow if it is non-decreasing and there exists a positive integer $\alpha$ such that
$\forall i \in \mathbb{N} \setminus {1}, \quad \alpha \cdot f(i) \ge f(2i).$