| Problem | Parameters | Time | Ref. | This paper (uniform) | Corollary 1 |
|---|---|---|---|---|---|
| Det. MIS and (Δ+1)-coloring | n, Δ | O(Δ + log* n) | [4,22] | min {O(Δ + log* n), 2O(√log n)} | (i) |
| n | 2O(√log n) | [34] | (ii) | ||
| Det. MIS (arboricity a = o(√log n)) | n, a | o(log n) | [6] | o(log n) | (i) |
| Det. MIS (arboricity a = O(log1/2-δ n)) | n, a | O(log n / log log n) | [6] | O(log n / log log n) | (i) |
| Det. λ(Δ + 1)-coloring | n, Δ | O(Δ/λ + log* n) | [4,22] | O(Δ/λ + log* n) | (iii) |
| Det. O(Δ)-edge-coloring | n, Δ | O(Δε + log* n) | [7] | O(Δε + log* n) | (v) |
| Det. O(Δ1+ε)-edge-coloring | n, Δ | O(log Δ + log* n) | [7] | O(log Δ + log* n) | (v) |
| Det. Maximal Matching | n or Δ | O(log4 n) | [19] | O(log4 n) | (vi) |
| Rand. (2, 2(c + 1))-ruling set | n | O(2c log1/c n) | [36] | O(2c log1/c n) | (vii) |
| Rand. MIS | uniform | O(log n) | [1,30] |
Table 1 Comparison of LOCAL algorithms with respect to the use of global parameters. “Det.” stands for deterministic, and “Rand.” for randomized.
also produced an $O(\Delta^{1+\epsilon})$-coloring algorithm running in $O(\log \Delta \log n)$-time, for an arbitrarily small constant $\epsilon > 0$, and an $O(\Delta)$-coloring algorithm running in $O(\Delta^{\epsilon} \log \frac{1}{M})$ time, for an arbitrarily small constant $\epsilon > 0$. All these coloring algorithms require the inputs of all nodes to contain the values of both $\Delta$ and $n$. Other deterministic non-uniform coloring algorithms with number of colors and running time corresponding to the arboricity parameter were given by Barenboim and Elkin [5, 6].
Efficient deterministic algorithms for the edge-coloring problem can be found in several papers [5, 7, 33]. In particular, Panconesi and Rizzi [33] designed a simple deterministic local algorithm that finds a $(2\Delta - 1)$-edge-coloring of a graph in time $O(\Delta + \log^* n)$. Recently, Barenboim and Elkin [7], designed an $O(\Delta)$-edge-coloring algorithm running in time $O(\Delta^{\epsilon}) + \log^* n$, for any $\epsilon > 0$, and an $O(\Delta^{1+\epsilon})$-edge-coloring algorithm running in time $O(\log \Delta) + \log^* n$, for any $\epsilon > 0$. All these algorithms require the inputs of all nodes to contain common upper bounds on both $n$ and $\Delta$.
Randomized algorithms for MIS and $(\Delta+1)$-coloring running in expected time $O(\log n)$ were initially given by Luby [30] and, independently, by Alon et al. [1].
Recently, Schneider and Wattenhofer [36] constructed the best currently-known non-uniform $(\Delta+1)$-coloring algorithm, which runs in time $O(\log \Delta + \sqrt{\log n})$. They also provided random algorithms for coloring using more colors. For every positive integer $c$, a randomized algorithm for $(2, 2(c+1))$-ruling set running in time $O(2^c \log^{1/c} n)$ is also presented. (A set $S$ of nodes in a graph being $(\alpha, \beta)$-ruling if every node not in $S$ is at distance at most $\beta$ of a node in $S$ and no two nodes in $S$ are at distance less than $\alpha$.) All these algorithms of Schneider and Wattenhofer [36] are not uniform and require the inputs of all nodes to contain a common upper bound on $n$.
Maximal Matching: A maximal matching of a graph $G$ is a set $M$ of edges of $G$ such that every edge not in $M$ is incident to an edge in $M$ and no two edges in $M$ are incident. Schneider and Wattenhofer [37] designed a uniform deterministic maximal matching algorithm on bounded-independence graphs running in $O(\log^* n)$ time. For general graphs, however, the state of the art maximal matching algorithm is not uniform: Hanckowiak et al. [19] presented a non-uniform deterministic algorithm for maximal matching running in time $O(\log^4 n)$. This algorithm assumes that the inputs of all nodes contain a common upper bound on $n$ (this assumption can be omitted for some parts of the algorithm under the condition that the inputs of all nodes contain the value of $\Delta$).
1.3 Our Results
The main conceptual contribution of the paper is the introduction of a new type of algorithms called pruning algorithms. Informally, the fundamental property of this type of algorithms is to allow one to iteratively run a sequence of algorithms (whose output may not necessarily be correct everywhere) so that the global output does not deteriorate, and it always progresses toward a solution.
Our main application for pruning algorithm concerns the problem of locally computing a global solution while minimizing the necessary global information contained in the inputs of the nodes. Addressing this, we provide a method for transforming a non-uniform local algorithm into a uniform one without increasing the asymptotic running time of the original algorithm. Our method applies to a wide family of both deterministic and randomized algorithms; in particular, it applies to many of the best known results concerning classical problems such as MIS, Coloring, and Maximal Matching. (See Table 1.2 for a summary of some of the uni-