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know more about their immediate neighborhoods and
less about the rest of the network.

A standard model for capturing the essence of lo-
cality is the LOCAL model (cf., [35]). In this model,
the network is modeled by a graph G, where the nodes
of G represent the processors and the edges represent
the communication links. To perform a task, nodes are
woken up simultaneously, and computation proceeds in
fault-free synchronous rounds during which every node
exchanges messages with its neighbors, and performs
arbitrary computations on its data. Since many tasks
cannot be solved distributively in an anonymous net-
work in a deterministic way, symmetry breaking must
be addressed. Arguably, there are two typical ways to
address this issue: the first one is to use randomized
algorithms, while the second one is to assume that each
node v in the network is initially provided a unique
identity Id(v). A local algorithm operating in such a
setting must return an output at each node such that
the collection of outputs satisfies the required task. For
example, a Maximal Independent Set (MIS) of a graph
G is a set S of nodes of G such that every node not in S
has a neighbor in S and no two nodes of S are adjacent.
In a local algorithm for the MIS problem, the output at
each node v is a bit b(v) indicating whether v belongs
to a selected set S of nodes, and it is required that S
forms a MIS of G. The running time of a local algo-
rithm is the number of rounds needed for the algorithm
to complete its operation at each node, taken in the
worst case scenario. This is typically evaluated with re-
spect to some parameters of the underlying graph. The
common parameters used are the number of nodes n in
the graph and the maximum degree Δ of a node in the
graph.

To ease the computation, it is often assumed that
some kind of knowledge about the global network is
provided to each node a priori. A typical example of
such knowledge is the number of nodes *n* in the net-
work. It turns out that in some cases, this (common)
assumption can give a lot of power to the distributed
algorithm. This was observed by Fraigniaud et al. [16] in
the context of local decision: they introduced the com-
plexity class of decision problems NLD, which contains
all decision problems that can be verified in constant
time with the aid of a certificate. They proved that, al-
though there exist decision problems that do not belong
to NLD, every (computable) decision problem falls in
NLD if it is assumed that each node is given the value
of *n* as an input.

In general, the amount and type of such informa-
tion may have a profound effect on the design of the
distributed algorithm. Obviously, if the whole graph
is contained in the input of each node, then the dis-

tributed algorithm can be reduced to a central one. In
fact, the whole area of computation with advice [9,12–
15,20,21] is dedicated to studying the amount of infor-
mation contained in the inputs of the nodes and its
effect on the performances of the distributed algorithm.
For instance, Fraigniaud et al. [15] showed that if each
node is provided with only a constant number of bits
then one can locally construct a BFS-tree in constant
time, and can locally construct a MST in O(log n) time,
while both tasks require diameter time if no knowledge
is assumed. As another example, Cohen et al. [9] proved
that O(1) bits, judiciously chosen at each node, can al-
low a finite automaton to distributively explore every
graph. As a matter of fact, from a radical point of view,
for many questions (e.g., MIS and Maximal Matching),
additional information may push the question at hand
into absurdity: even a constant number of bits of ad-
ditional information per node is enough to compute a
solution—simply let the additional information encode
the solution!

When dealing with locality issues, it is desired that
the amount of information regarding the whole network
contained in the inputs of the nodes is minimized. A lo-
cal algorithm that assumes that each node is initially
given merely its own identity is often called uniform.
Unfortunately, there are only few local algorithms in
the literature that are uniform (e.g., [11, 26, 29, 30, 37]).
In contrast, most known local algorithms assume that
the inputs of all nodes contain upper bounds on the
values of some global parameters of the network. More-
over, it is often assumed that all inputs contain the
same upper bounds on the global parameters. Further-
more, typically, not only the correct operation of the
algorithm requires that upper bounds be contained in
the inputs of all nodes, but also the running time of the
algorithm is actually a function of the upper bound esti-
mations and not of the actual values of the parameters.
Hence, it is desired that the upper bounds contained
in the inputs are not significantly larger than the real
values of the parameters.

Some attempts to transform a non-uniform local al-
gorithm into a uniform one were made by examining
the details of the algorithm at hand and modifying it
appropriately. For example, Barenboim and Elkin [6]
first gave a non-uniform MIS algorithm for the family of
graphs with arboricity $a = O(\log^{1/2-\delta} n)$, for any con-
stant $\delta \in (0, 1/2)$, running in time $O(\log n / \log \log n)$.
(The arboricity of a graph being the smallest number of
acyclic subgraphs that together contain all the edges of
the graph.) At the cost of increasing the running time
to $O(\frac{\log n}{\log \log n} \log^*\* n)$, the authors show how to modify
their algorithm so that the value of *a* need not be part
of the inputs of nodes. In addition to the MIS algo-