On the probe complexity of local computation algorithms

In the Local Computation Algorithms (LCA) model, the algorithm is asked to compute a part of the output by reading as little as possible from the input. For example, an LCA for coloring a graph is given a vertex name (as a “query”), and it should output the color assigned to that vertex after inquiring about some part of the graph topology using “probes”; all outputs must be consistent with the same coloring. LCAs are useful when the input is huge, and the output as a whole is not needed simultaneously. Most previous work on LCAs was limited to bounded-degree graphs, which seems inevitable because probes are of the form “what vertex is at the other end of edge i of vertex v?”. In this work we study LCAs for unbounded-degree graphs. In particular, such LCAs are expected to probe the graph a number of times that is significantly smaller than the maximum, average, or even minimum degree. We show that there are problems that have very e cient LCAs on any graph - specifically, we show that there is an LCA for the weak coloring problem (where a coloring is legal if every vertex has a neighbor with a di erent color) that uses log^∗ n + O(1) probes to reply to any query. As another way of dealing with large degrees, we propose a more powerful type of probe which we call a strong probe: given a vertex name, it returns a list of its neighbors. Lower bounds for strong probes are stronger than ones in the edge probe model (which we call weak probes). Our main result in this model is that roughly (n) strong probes are required to compute a maximal matching. Our findings include interesting separations between closely related problems. For weak probes, we show that while weak 3-coloring can be done with probe complexity log^∗ n + O(1), weak 2-coloring has probe complexity (log n/log log n). For strong probes, our negative result for maximal matching is complemented by an LCA for (1 − )-approximate maximum matching on regular graphs that uses O(1) strong probes, for any constant > 0.