In other words, $f(c \cdot i) = O(f(i))$ for every constant $c$ and every integer $i$, where the constant hidden in the $O$ notation depends only on $c$. An example of a moderately-slow function is given by the logarithm.
A function $f: \mathbf{R}^+ \rightarrow \mathbf{R}^+$ is moderately-increasing if it is non-decreasing and there exists a positive integer $\alpha$ such that
Note that $f(x) = x^{k_1} \log^{k_2}(x)$ is a moderately-increasing function for every two reals $k_1 \ge 1$ and $k_2 \ge 0$. Moreover, every moderately-increasing function is moderately-slow. On the other hand, some functions (such as the constant functions or the logarithm) are moderately-slow but not moderately-increasing.
A function $f: \mathbf{R}^+ \to \mathbf{R}^+$ is moderately-fast if it is moderately-increasing and there exists a polynomial $P$ such that
A function $f: \mathbf{R}^+ \to \mathbf{R}^+$ tends to infinity if
and $f$ is ascending if it is non-decreasing and it tends to infinity. (Note that in this case $\lim_{x \to \infty} f(x) = \infty$.)
A function $f: (\mathbf{R}^+)^{\ell} \to \mathbf{R}^+$ is additive if there exist $\ell$ ascending functions $f_1, \dots, f_\ell$ such that
Problems and instances: Given a set $V$ of nodes, a vector for $V$ is an assignment x of a bit string x(v) to each $v \in V$, i.e., x is a function x: $V \rightarrow {0, 1}^*$. A problem is defined by a collection of triplets: $\Pi = {(G, \mathbf{x}, \mathbf{y})}$, where $G$ is a (not necessarily connected) graph, and x and y are input and output vectors for $V$, respectively. We consider only problems that are closed under disjoint union, i.e., if $G_1$ and $G_2$ are two vertex disjoint graphs and $(G_1, \mathbf{x}_1, \mathbf{y}_1), (G_2, \mathbf{x}_2, \mathbf{y}_2) \in \Pi$ then $(G, \mathbf{x}, \mathbf{y}) \in \Pi$, where $G = G_1 \cup G_2$, x = $\mathbf{x}_1 \cup \mathbf{x}_2$ and $\mathbf{y} = \mathbf{y}_1 \cup \mathbf{y}_2$.
An instance, with respect to a given problem $\Pi$, is a pair $(G, \mathbf{x})$ for which there exists an output vector $\mathbf{y}$ satisfying $(G, \mathbf{x}, \mathbf{y}) \in \Pi$. In what follows, whenever we consider a collection $\mathcal{F}$ of instances, we always assume that $\mathcal{F}$ is closed under inclusion. That is, if $(G, \mathbf{x}) \in \mathcal{F}$ and $(G', \mathbf{x}') \subseteq (G, \mathbf{x})$ (i.e., $G'$ is a subgraph of $G$ and $\mathbf{x}'$ is the input vector $\mathbf{x}$ restricted to $V(G')$ then $(G', \mathbf{x}') \in \mathcal{F}$. Informally, given a problem $\Pi$ and a collection of instances $\mathcal{F}$, the goal is to design an efficient distributed algorithm that takes an instance $(G, \mathbf{x}) \in \mathcal{F}$ as input,
and produces an output vector $\mathbf{y}$ satisfying $(G, \mathbf{x}, \mathbf{y}) \in \Pi$. The reason to require problems to be closed under disjoint union is that a distributed algorithm operating on an instance $(G, \mathbf{x})$ runs separately and independently on each connected component of $G$. Let $\mathcal{G}$ be a family of graphs closed under inclusion. We define $\mathcal{F}(\mathcal{G})$ to be ${\mathcal{G}} \times {0, 1}^*$.
We assume that each node $v \in V$ is provided with a unique integer referred to as the identity of $v$, and denoted $\mathrm{Id}(v)$; by unique identities, we mean that $\mathrm{Id}(u) \neq \mathrm{Id}(v)$ for every two distinct nodes $u$ and $v$. For ease of exposition, we consider the identity of a node to be part of its input.
We consider classical problems such as coloring, maximal matching (MM), Maximal Independent Set (MIS) and the $(\alpha, \beta)$-ruling set problem. Informally, viewing the output of a node as a color, the requirement of coloring is that the colors of two neighboring nodes must be different. In the $(\alpha, \beta)$-ruling set problem, the output at each node is Boolean, and indicates whether the node belongs to a set $S$ that must form an $(\alpha, \beta)$-ruling set. That is, the set $S$ of selected nodes must satisfy: (1) two nodes that belong to $S$ must be at distance at least $\alpha$ from each other, and (2) if a node does not belong to $S$, then there is a node in the set at distance at most $\beta$ from it. MIS is a special case of the ruling set problem, specifically, MIS is precisely (2, 1)-ruling set. Finally, given a triplet $(G, \mathbf{x}, \mathbf{y})$, two nodes $u$ and $v$ are said to be matched if $(u, v) \in E$, $\mathbf{y}(u) = \mathbf{y}(v)$ and $\mathbf{y}(w) \neq \mathbf{y}(u)$ for every $w \in (N_G(u) \cup N_G(v)) \setminus {\mathbf{u}, \mathbf{v}}$. Thus, the MM problem requires that each node $u$ is either matched to one of its neighbors or that every neighbor $v$ of $u$ is matched to one of $v$'s neighbors.
Parameters: Fix a problem $\Pi$ and let $\mathcal{F}$ be a collection of instances for $\Pi$. A parameter $\mathbf{p}$ is a positive valued function $\mathbf{p}: \mathcal{F} \to \mathbb{N}$. The parameter $\mathbf{p}$ is non-decreasing, if $\mathbf{p}(G', \mathbf{x}') \leq \mathbf{p}(G, \mathbf{x})$ whenever $(G', \mathbf{x}') \in \mathcal{F}$ and $(G', \mathbf{x}') \subseteq (G, \mathbf{x})$.
Let $\mathcal{F}$ be a collection of instances. A parameter $\mathbf{p}$ for $\mathcal{F}$ is a graph-parameter if $\mathbf{p}$ is independent of the input, that is, if $\mathbf{p}(G, \mathbf{x}) = \mathbf{p}(G, \mathbf{x}')$ for every two instances $(G, \mathbf{x}), (G, \mathbf{x}') \in \mathcal{F}$ such that the input assignments $\mathbf{x}$ and $\mathbf{x}'$ preserve the identities, i.e., the inputs $\mathbf{x}(v)$ and $\mathbf{x}'(v)$ contain the same identity $\mathrm{Id}(v)$ for every $v \in V(G)$. In what follows, we will consider only non-decreasing graph-parameters (note, not all graph-parameters are non-decreasing, an example being the diameter of a graph). More precisely, we will primarily focus on the following non-decreasing graph-parameters: the number $n$ of nodes of the graph $G$, i.e., $|V(G)|$, the maximum degree $\Delta = \Delta(G)$ of $G$, i.e., $\max{\deg_G(u) : u \in V(G)}$