Deterministic Lower Bounds for $k$-Edge Connectivity in the Distributed Sketching Model

The question whether AGM sketches are optimal for connectivity problems remained an open problem for several years, until the seminal work by Nelson and Yu [NY19] showed that $\Omega\left(\log^{3}n\right)$ bits are indeed required for computing spanning forests in the distributed graph sketching model. The main idea of their lower bound is to use a reduction to a variant of the universal relation problem [KRW95] in 2-party communication complexity, denoted by $\mathsf{UR}^{\subset}$, where Alice starts with a subset $S$ of some universe $U$, and Bob gets a proper subset $T\subset S$. Alice sends a single message to Bob, who must output some element in $S\setminus T$. In [KNP⁺17a], Kapralov, Nelson, Pachoki, Wang, and Woodruf showed that $\Omega\left(\log^{3}n\right)$ bits are required for solving $\mathsf{UR}^{\subset}$ with high probability. To obtain a reduction to $\mathsf{UR}^{\subset}$ in the distributed sketching model, [NY19] define a tripartite graph on vertex sets $V^{l}$, $V^{m}$, and a much smaller set $V^{r}$. Each node $v\in V^{m}$ has a set of unique neighbors in $V^{l}$ (not shared with any other node in $V^{m}$) as well as some neighbors in $V^{r}$. Since every node in $V^{l}$ is connected to at most one node $v\in V^{m}$, any spanning forest must include some edge between $v$ and a node in $V^{r}$. The neighborhood of a randomly chosen $v^{*}\in V^{m}$ is determined by the input of $\mathsf{UR}^{\subset}$ such that Alice’s set $S$ corresponds to all of $v^{*}$’s neighbors, whereby Bob’s set $T$ specifies the subset of neighbors in $V^{l}$. They show that the players can jointly simulate the nodes in $V^{l}$ and $V^{m}$ (but not $V^{r}$!), whereby Bob also simulates the referee. From the resulting spanning forest, Bob can extract an edge between $v^{*}$ and $V^{r}$, which corresponds to an element in $S\setminus T$, and thus solves $\mathsf{UR}^{\subset}$. Notably absent in their simulation are the nodes in $V^{r}$, which neither Alice or Bob can simulate, due to their lack of knowledge of set $S\setminus T$. In fact, trying to faithfully simulate every node executing a distributed sketching algorithm in a 2-party model poses a major technical challenge since every edge is shared between its neighbors. However, [NY19] show that choosing $V^{r}$ to be polynomially smaller than $V^{m}$ suffices to overcome this obstacle: Since each $w\in V^{r}$ will be a neighbor of many $v\in V^{m}$ and $w$ does not know which of them is the important node $v^{*}$ (as all of these neighborhoods are sampled from the same hard input distribution for $\mathsf{UR}^{\subset}$), it follows that the amount of information that the referee can learn about the neighbors of $v^{*}$ from the sketches of the nodes in $V^{r}$ is only (roughly) $\tilde{O}(\frac{|V^{r}|}{|V^{m}|})=o(1)$, and thus negligible. Hence, Alice and Bob can omit these messages in their simulation altogether, while only causing a small increase in the overall error probability, due to Pinsker’s inequality [Pin64]. We remark that the actual lower bound construction of [NY19] uses multiple copies of this basic building block.

More recently, Robinson [Rob23] extended the approach of [NY19] to constructing a $k$-edge connected spanning subgraph ($k$-ECSS), with the main difference being a modified graph construction that enables a reduction to the multi-output variant of universal relation, called $\mathsf{UR}^{\subset}_{k}$, which has a lower bound of $\Omega\left(k\log^{2}n\right)$ bits [KNP⁺17a] and requires Bob to find $k$ elements of $S\setminus T$. The simulation argument in [Rob23] does not directly use Pinsker’s inequality as in [NY19], and instead obtains a slightly smaller, but nevertheless nonzero, probability of error, by devising a mechanism of reconstructing the output when omitting the sketches of the nodes in $V^{r}$. At the risk of stating the obvious, we point out that this result does not have any implications for the problem of deciding whether a given graph is $k$-edge connected, which is the focus of the current paper.

In [Yu21], Yu proved that the decision problem of graph connectivity is subject to the same lower bound on the message length as computing a spanning forest, which fully resolved the complexity of connectivity in the distributed sketching model. For this purpose, [Yu21] introduced and proved the hardness of a new decision version of universal relation, called $\mathsf{UR}^{\subset}_{\text{dec}}$. Just as for the $\mathsf{UR}^{\subset}$ problem, Alice again gets a set $S\subseteq U$. However, Bob not only gets $T\subset S$ but, in addition, he also knows a partition $(P_{1},P_{2})$ of $U\setminus T$. The inputs adhere to the promise that either $S\setminus T\subseteq P_{1}$ or $S\setminus T\subseteq P_{2}$, and, upon receiving Alice’s message, Bob needs to decide which is the case. In order to embed an instance of $\mathsf{UR}^{\subset}_{\text{dec}}$ in a graph, [Yu21] adapted the construction of [NY19] by partitioning the set $V^{r}$ into $V^{r}_{1}$ and $V^{r}_{2}$, and by sampling the neighborhood of each node in $V^{m}$ according to the hard distribution of $\mathsf{UR}^{\subset}_{\text{dec}}$. Following the overall simulation approach of [NY19], Alice’s input of $\mathsf{UR}^{\subset}_{\text{dec}}$ is embedded at some special node $v^{*}\in V^{m}$. Accordingly, the partition-promise of $\mathsf{UR}^{\subset}_{\text{dec}}$ translates to each $v\in V^{m}$ having all its $V^{r}$-neighbors in either $V^{r}_{1}$ or $V^{r}_{2}$.

We point out that there does not appear to be a natural extension of the connectivity lower bound of [Yu21] to $k$-edge connectivity. Even though $\mathsf{UR}^{\subset}_{\text{dec}}$ can be generalized to $k>1$, proving a lower bound that scales linearly with $k$ for this generalization appears to be far from straightforward. Interestingly, the situation is reversed for the standard “search” version of $\mathsf{UR}^{\subset}$, where the proof of the lower bound for $\mathsf{UR}^{\subset}_{k}$ is technically less involved than the corresponding result for $\mathsf{UR}^{\subset}$, see [KNP⁺17a]. A second obstacle is that, in contrast to the problem of finding a $k$-edge connected spanning subgraph [Rob23], designing a suitable lower bound graph for deciding $k$-edge connectivity is itself nontrivial. This is because choosing the neighborhood of every node in $V^{m}$ according to the hard distribution of the generalized variant of $\mathsf{UR}^{\subset}_{\text{dec}}$ and embedding it in the constructions of [NY19, Yu21] may result in a graph that is very likely to be $k$-edge connected, independently of the neighborhood of the special node $v^{*}$.

Interestingly, none of the above mentioned lower bound results provide a straightforward way towards obtaining stronger bounds for deterministic graph sketching algorithms (that never fail): Since any deterministic one-way algorithm for $\mathsf{UR}^{\subset}$, $\mathsf{UR}^{\subset}_{k}$, or $\mathsf{UR}^{\subset}_{\text{dec}}$ needs to send Alice’s entire input to Bob,²²2While there is no published deterministic lower bound for $\mathsf{UR}^{\subset}_{\text{dec}}$, the same communication complexity bound as for the search version $\mathsf{UR}^{\subset}$ follows from standard fooling set and counting arguments for one-way communication; see, e.g., [KN97]. it may appear that, at first glance, we can simply “plug in” these results for deterministic protocols and directly obtain stronger lower bounds in the graph sketching model. This intuition, however, turns out to be misleading, as a major technical challenge is that the simulations used in [NY19, Yu21, Rob23] are themselves randomized and, more importantly, are necessarily “lossy” in the sense that they introduce a nonzero probability of error, due to forgoing the simulation of some nodes—in particular, all nodes in $V^{r}$. This poses a major technical obstacle towards obtaining lower bounds for algorithms that fail with exponentially small probability, including deterministic ones.³³3We discuss prior work on lower bounds for other graph problems in the sketching model in Section 1.4. To the best of our knowledge, achieving a perfect (i.e., $0$-error) simulation of graph sketching algorithms in a two-party communication complexity model remains a challenging open problem.

In this work, we take the first step toward determining lower bounds for deterministic connectivity algorithms in the graph sketching model. In more detail, we focus on the problem of deciding $k$-edge connectivity, and we show that any deterministic sketching algorithm has a worst case sketch length that is almost linear (in $k$):

The proof of Theorem 1 requires us to overcome several technical challenges:

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  To obtain a graph that is hard for deciding $k$-edge connectivity in the distributed sketching model, we need to find a suitable graph construction. We cannot directly extend the constructions used for deciding graph connectivity in [Yu21], as this crucially rests on the assumption that the neighborhood of every node in $V^{m}$ (see Sec. 1.1) must be chosen from the same distribution. In fact, the straightforward generalization of this idea to $k$-edge connectivity would yield a graph that is very likely to be $k$-edge connected, for large values of $k$. In this work, we propose a new construction that does not preserve the uniformity property of [Yu21] regarding the input distribution. Instead, there is a special node $v_{\sigma}$, with a somewhat larger degree than its peers, which we connect in a way such that the $k$-edge connectivity of the graph entirely rests on the choice of neighborhood for this node. Nevertheless, we ensure that its neighbors have no easy way of distinguishing $v_{\sigma}$ from the vast majority of unimportant nodes. This approach introduces additional technical difficulties that we address in our simulation.

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  As elaborated in more detail in Section 1.2, we need to depart from the well-trodden path of using the “partial simulation” argument pioneered in [NY19] (and also used in the work of [Yu21] and [Rob23]), where some nodes are omitted from being simulated, as this would result in a nonzero error algorithm. To avoid this pitfall, we design a completely different simulation that is suitable for deterministic algorithms. We point out that the main technical difficulty of the distributed sketching model, namely the “sharing” of edges between nodes, also surfaces in our setting and prevents us from devising a simulation, where every node $v$’s neighborhood is part of the input of one of the simulating players. Instead, we exploit some structural properties of the algorithm at hand that must hold for any deterministic protocol. In more detail, the main idea behind our approach is that it suffices to compute a sketch for $v$ that looks “compatible” with our (limited) knowledge of $v$’s neighborhood, and we prove that this indeed ensures an error free simulation.

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  Our simulation works in a simultaneous 3-party model of communication complexity, where two players, Alice and Bob, each send a single message to Charlie who computes the output. For this setting, we introduce the $\mathsf{UniqueOverlap}$ problem, which abstracts the core difficulty present in our lower bound graph construction: That is, nodes are unaware which of their incident edges point to the crucial node $v_{\sigma}$, even though $v_{\sigma}$ itself is aware of its special role. We capture this property in the communication complexity setting by equipping Alice and Bob each with an input vector that has a single common coordinate whose XOR is $1$. While Charlie knows the location of this coordinate, he does not know the values of the respective bits of Alice and Bob, which are needed for the correct output. The hardness of $\mathsf{UniqueOverlap}$ does not appear to follow from reductions to standard problems such as set disjointness or variants of indexing (see Sec. 4.1 for a more detailed discussion), due to the correlation between the inputs of the three players and since we consider ternary input vectors, whose entries may be either $0$, $1$, or $\perp$. A further complication is that we need to assume Charlie knows the entire support (i.e., the indices of all non-$\perp$ entries) of Alice’s and Bob’s inputs. Instead, we devise a purely combinatorial argument based on properties of cross-intersecting set families to show the following result:

The distributed graph sketching model was first introduced by Becker, Matamala, Nisse, Rapaport, Suchan, and Todinca in [BMN⁺11], where they proved the hardness of several fundamental graph problems for deterministic sketching algorithms: In particular, they showed that problems including deciding whether the diameter is at most $3$, and whether the graph contains a certain subgraph such as a triangle or a square, require sketches of large (i.e., super-polylogarithmic) size. Their approach works by showing that the existence of an efficient protocol for any of these problems can be leveraged for reconstructing graphs with certain properties using similar-sized sketches. However, as pointed out in [BMN⁺11], their approach does not extend to graph connectivity problems.

The model was further explored by Becker, Montealegre, Rapaport, and Todinca in [BMRT14], where they show separations between the power of deterministic, private randomness and public randomness algorithms. Apart from the aforementioned works of [NY19, Yu21, Rob23] on connectivity-related problems, Assadi, Kol, and Oshman [AKO20] showed a lower bound of $\Omega\left(n^{1/2-\varepsilon}\right)$ bits on the sketch size for computing a maximal independent set. For the distributed sketching model with private randomness, Holm, King, Thorup, Zamir, and Zwick [HKT⁺19] showed that computing a spanning tree is possible with sketches of $O(\sqrt{n}\log n)$ bits.

We point out that the distributed graph sketching model is known to be equivalent to a single-round variant of the broadcast congested clique, as observed in several earlier works [JLN18, AKO20], where each one of $n$ nodes can broadcast a single message per round that becomes visible to all other nodes at the end of the current round. In that context, the worst case sketch size corresponds to the available bandwidth, i.e., size of the broadcast message. Several works have studied tradeoffs between the number of rounds and the message size in the (multi-round) broadcast congested clique: Pai and Pemmaraju [PP20] showed that $\Omega\left(\frac{\log n}{b}\right)$ rounds are required if nodes can broadcast messages of size $b$. The work of Montealegre and Todinca [MT16] revealed that there is a deterministic $r$-round connectivity algorithm that sends messages of size $O(n^{1/r}\log n)$. Subsequently, Jurdzinski and Nowicki [JN17] obtained an improved round complexity of $O(\log n/\log\log n)$ when considering the standard assumption of $O(\log n)$ bits per message.

In the distributed sketching model each node $v$ is equipped with an ID of $O(\log n)$ bits. We sometimes abuse notation and use $v$ to refer to either the node itself or its ID, depending on the context. Thus, when referring to the neighborhood of a node, we also mean the list of IDs associated to the node’s neighbors.

We use $[m]$ to denote the set of integers from $1$ to $m$. For a set $S$ and some positive integer $m$, we borrow the standard notation from extremal combinatorics (e.g., see [Juk01]) and use $\binom{S}{m}$ to denote the set family of all $m$-element subsets of $S$.

We structure the proof of Theorem 1 as follows: In Section 2, we introduce our new graph construction and analyze how its properties relate to $k$-connectivity. Then, we show in Section 3 that, for any given deterministic algorithm that sends short sketches, there exists an instance of this lower bound graph, such that many nodes send the same message for certain pairs of inputs. After introducing and proving the hardness of the $\mathsf{UniqueOverlap}$ problem in Section 4, we show in Section 5 how to solve it by simulating a $k$-edge connectivity algorithm on our lower bound graph. In Section 6, we combine these results to complete the proof of Theorem 1. Finally, we discuss some open problems in Section 7.

In [AAD⁺23], the authors define the Leave-One Index problem (Problem 3 in [AAD⁺23]), where there are three players Alice, Bob, and Charlie. Alice and Bob hold the same string $x\in\{0,1\}^{N}$ and Charlie knows an index $i^{*}\in[N]$. In addition, Alice has a set $S\subseteq[N]$ and Bob has a set $T\subseteq[N]$ with the property that $S\cup T=[N]\setminus\{i^{*}\}$, and the goal is that the players output $x_{i^{*}}$, whereby the order of communication is Alice, Bob, and then Charlie. We point out that Leave-One Index lacks an important requirement of the $\mathsf{UniqueOverlap}$ problem, namely that Charlie also knows the exact support sets (not just their size) of the sets $S$ and $T$ given to Alice and Bob, which is crucial for the correctness of our simulation. Thus, it is unclear whether there exists a reduction between these problems.

The uniqueness of the shared index $\sigma$ may give the impression, at first, that $\mathsf{UniqueOverlap}$ is related to a (promise) variant of computing the set intersection. However, we doubt the existence of a straightforward reduction, as, for the $\mathsf{UniqueOverlap}$ problem, the input vectors of Alice and Bob are ternary, and Charlie knows the (single) intersecting index as well as the support of the two input sets.

We now show that either Alice or Bob needs to send a message of size $\Omega\left(m\right)$ bits to Charlie in the worst case.

The argument developed for our lower bound proof in Section 4.2 inspires the design of a simple deterministic algorithm that allows Alice and Bob to save a single bit by sending messages of length $s-1$ under certain conditions. As this is not needed for proving our main result, we relegate the details to Appendix B.