rithms, the work of [6] also contains algorithms that do not require the knowledge of the arboricity, but have the same asymptotic running time as the ones that require it. For example, this corresponds to algorithms computing forests-decomposition and $O(a)$-coloring. Nevertheless, all their algorithms still require the inputs of all nodes to contain a common upper bound on $n$.
We present general methods 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, our method applies to all state of the art non-uniform algorithms for MIS and Maximal Matching, as well as to several of the best known results for $(\Delta+1)$-coloring.
Our transformations are obtained using a new type of local algorithms termed pruning algorithms. Informally, the basic property of a pruning algorithm is that it allows one to iteratively apply a sequence of local algorithms (whose output may not form a correct global solution) one after the other, in a way that “always progresses” toward a solution. In a sense, a pruning algorithm is a combination of a gluing mechanism and a local checking algorithm (cf., [16,32]). A local checking algorithm for a problem $\Pi$ runs on graphs with an output value at each node (and possibly an input too), and can locally detect whether the output is “legal” with respect to $\Pi$. That is, if the instance is not legal then at least one node detects this, and raises an alarm. (For example, a local checking algorithm for MIS is trivial: each node in the set $S$, which is suspected to be a MIS, checks that none of its neighbors belongs to $S$, and each node not in $S$ checks that at least one of its neighbors belongs to $S$. If the check fails, then the node raises an alarm.) A pruning algorithm needs to satisfy an additional gluing property not required by local checking algorithms. Specifically, if the instance is not legal, then the pruning algorithm must carefully choose the nodes raising the alarm (and possibly modify their input too), so that a solution for the subgraph induced by those alarming nodes can be well glued to the previous output of the non-alarming nodes, in a way such that the combined output is a solution to the problem for the whole initial graph.
We believe that this new type of algorithms may be of independent interest. Indeed, as we show, pruning algorithms have several types of other applications in the theory of local computation, besides the aforementioned issue of designing uniform algorithms. Specifically, they can be used also to transform a local Monte-Carlo algorithm into a Las Vegas one, as well as to obtain an algorithm that runs in the minimum running time of a given (finite) set of uniform algorithms.
1.2 Previous Work
MIS and coloring: There is a long line of research concerning the two related problems of $(\Delta+1)$-coloring and MIS [3, 10, 17, 18, 23, 24, 28]. A k-coloring of a graph is an assignment of an integer in ${1, \dots, k}$ to each node such that no two adjacent vertices are assigned the same integer. Recently, Barenboim and Elkin [4] and independently Kuhn [22] presented two elegant $(\Delta+1)$-coloring and MIS algorithms running in $O(\Delta + \log^* n)$ time on general graphs. This is the best currently-known bound for these problems on low degree graphs. For graphs with a large maximum degree $\Delta$, the best bound is due to Panconesi and Srinivasan [34], who devised an algorithm running in $2^{O(\sqrt{\log n})}$ time. The aforementioned algorithms are not uniform. Specifically, all three algorithms require that the inputs of all nodes contain a common upper bound on $n$ and the first two also require a common upper bound on $\Delta$.
For bounded-independence graphs, Schneider and Wattenhofer [37] designed uniform deterministic MIS and $(\Delta + 1)$-coloring algorithms running in $O(\log^* n)$ time. Barenboim and Elkin [6] devised a deterministic algorithm for the MIS problem on graphs of bounded arboricity that requires time $O(\log n / \log \log n)$. More specifically, for graphs with arboricity $a = o(\sqrt{\log n})$, they show that a MIS can be computed deterministically in $o(\log n)$ time, and whenever $a = O(\log^{1/2-\delta} n)$ for some constant $\delta \in (0, 1/2)$, the same algorithm runs in time $O(\log n / \log \log n)$. At the cost of increasing the running time by a multiplicative factor of $O(\log^* n)$, the authors show how to modify their algorithm so that the value of $a$ need not be part of the inputs of nodes. Nevertheless, all their algorithms require the inputs of all nodes to contain a common upper bound on the value of $n$. Another MIS algorithm which is efficient for graphs with low arboricity was devised by Barenboim and Elkin [5]; this algorithm runs in time $O(a+a^\epsilon \log n)$ for arbitrary constant $\epsilon > 0$.
Concerning the problem of coloring with more than $\Delta + 1$ colors, Linial [27, 28], and subsequently Szegedy and Vishwanathan [38], described $O(\Delta^2)$-coloring algorithms with running time $\theta(\log^* n)$. Barenboim and Elkin [4] and, independently, Kuhn [22] generalized this by presenting a tradeoff between the running time and the number of colors: they devised a $\lambda(\Delta + 1)$-coloring algorithm with running time $O(\Delta/\lambda + \log^* n)$, for any $\lambda \ge 1$. All these algorithms require the inputs of all nodes to contain common upper bounds on both $n$ and $\Delta$.
Barenboim and Elkin [5] devised a $\Delta^{1+\epsilon(1)}$ coloring algorithm running in time $O(f(\Delta) \log \Delta \log n)$, for an arbitrarily slow-growing function $f = \omega(1)$. They