Node and edge averaged complexities of local graph problems

Abstract

Abstract We continue the recently started line of work on the distributed node-averaged complexity of distributed graph algorithms. The node-averaged complexity of a distributed algorithm running on a graph

$$G=(V,E)$$

G = ( V , E )

is the average over the times at which the nodes V of G finish their computation and commit to their outputs. We study the node-averaged complexity for some of the central distributed symmetry breaking problems and provide the following results (among others). As our main result, we show that the randomized node-averaged complexity of computing a maximal independent set (MIS) in n-node graphs of maximum degree

$$\Delta $$

Δ

is at least

$$\Omega \big (\min \big \{\frac{\log \Delta }{\log \log \Delta },\sqrt{\frac{\log n}{\log \log n}}\big \}\big )$$

Ω

(

min

{

log Δ

log log Δ

,

log n

log log n

}

)

. This bound is obtained by a novel adaptation of the well-known lower bound by Kuhn, Moscibroda, and Wattenhofer [JACM’16]. As a side result, we obtain that the worst-case randomized round complexity for computing an MIS in trees is also

$$\Omega \big (\min \big \{\frac{\log \Delta }{\log \log \Delta },\sqrt{\frac{\log n}{\log \log n}}\big \}\big )$$

Ω

(

min

{

log Δ

log log Δ

,

log n

log log n

}

)

—this essentially answers open problem 11.15 in the book by Barenboim and Elkin and resolves the complexity of MIS on trees up to an

$$O(\sqrt{\log \log n})$$

O (

log log n

)

factor. We also show that, perhaps surprisingly, a minimal relaxation of MIS, which is the same as (2, 1)-ruling set, to the (2, 2)-ruling set problem drops the randomized node-averaged complexity to O(1). For maximal matching, we show that while the randomized node-averaged complexity is

$$\Omega \big (\min \big \{\frac{\log \Delta }{\log \log \Delta },\sqrt{\frac{\log n}{\log \log n}}\big \}\big )$$

Ω

(

min

{

log Δ

log log Δ

,

log n

log log n

}

)

, the randomized edge-averaged complexity is O(1). Further, we show that the deterministic edge-averaged complexity of maximal matching is

$$O(\log ^2\Delta + \log ^* n)$$

O (

log 2

Δ +

log ∗

n )

and the deterministic node-averaged complexity of maximal matching is

$$O(\log ^3\Delta + \log ^* n)$$

O (

log 3

Δ +

log ∗

n )

. Finally, we consider the problem of computing a sinkless orientation of a graph. The deterministic worst-case complexity of the problem is known to be

$$\Theta (\log n)$$

Θ ( log n )

, even on bounded-degree graphs. We show that the problem can be solved deterministically with node-averaged complexity

$$O(\log ^* n)$$

O (

log ∗

n )

, while keeping the worst-case complexity in

$$O(\log n)$$

O ( log n )

.