We develop a new technique for constructing sparse graphs that allow us to prove nearlinear lower bounds on the round complexity of computing distances in the CONGEST model. Specifically, we show an Ω(n) lower bound for computing the diameter in sparse networks, which was previously known only for dense networks [Frishknecht et al., SODA 2012]. In fact, we can even modify our construction to obtain graphs with constant degree, using a simple but powerful degree-reduction technique which we define.Moreover, our technique allows us to show Ω(n) lower bounds for computing ( 3 2 − ε)approximations of the diameter or the radius, and for computing a ( 5 1 The notations Ω and O hide factors that are polylogarithmic in n.

We introduce a novel lower bound technique for distributed graph algorithms under bandwidth limitations. We define the notion of fooling views and exemplify its strength by proving two new lower bounds for triangle membership in the Congest(B) model:1. Any 1-round algorithm requires B ≥ c∆ log n for a constant c > 0.2. If B = 1, even in constant-degree graphs any algorithm must take Ω(log * n) rounds.The implication of the former is the first proven separation between the Local and the Congest models for deterministic triangle membership. The latter result is the first non-trivial lower bound on the number of rounds required, even for triangle detection, under limited bandwidth. All previous known techniques are provably incapable of giving these bounds. We hope that our approach may pave the way for proving lower bounds for additional problems in various settings of distributed computing for which previous techniques do not suffice.

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This paper proves strong lower bounds for distributed computing in the congest model, by presenting the bit-gadget: a new technique for constructing graphs with small cuts.The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can be obtained when using dense graphs.Second, small cuts are essential for proving super-linear lower bounds, none of which were known prior to this work. In fact, they allow us to show near-quadratic lower bounds for several problems, such as exact minimum vertex cover or maximum independent set, as well as for coloring a graph with its chromatic number. Such strong lower bounds are not limited to NP-hard problems, as given by two simple graph problems in P which are shown to require a quadratic and near-quadratic number of rounds. All of the above are optimal up to logarithmic factors. In addition, in this context, the complexity of the all-pairs-shortestpaths problem is discussed.Finally, it is shown that graph constructions for congest lower bounds translate to lower bounds for the semi-streaming model, despite being very different in its nature.