On Triangle Estimation Using Tripartite Independent Set Queries

Abstract

Estimating the number of triangles in a graph is one of the most fundamental problems in sublinear algorithms. In this work, we provide an algorithm that approximately counts the number of triangles in a graph using only polylogarithmic queries when the number of triangles on any edge in the graph is polylogarithmically bounded. Our query oracle Tripartite Independent Set (TIS) takes three disjoint sets of vertices A, B and C as inputs, and answers whether there exists a triangle having one endpoint in each of these three sets. Our query model generally belongs to the class of group queries (Ron and Tsur ACM Trans. Comput. Theory 8(4), 15, 2016; Dell and Lapinskas 2018) and in particular is inspired by the Bipartite Independent Set (BIS) query oracle of Beame et al. (2018). We extend the algorithmic framework of Beame et al., with TIS replacing BIS, for approximately counting triangles in graphs.

Appendices

Appendix : A: Scenario Where Δ_E is Bounded

In this Section, we discuss some scenarios where the number of triangles sharing an edge is bounded. An obvious example for such graphs are graphs with bounded degree. We explore some other scenarios.

• (i) Consider a graph G(P,E) such that the vertex set P corresponds to a subset of $\mathbb {R}^{2}$ and (u,v) ∈ E if and only if the distance between u and v is exactly 1. The objective is to compute the number of triples of points from P forming an equilateral triangle having side length 1, that is, the number of triangles in G. Observe that there can be at most two triangles sharing an edge in G, that is, Δ_E ≤ 2.

• (ii) Consider a graph G(P,E) such that the vertex set P corresponds to a set of points inside an N × N square in $\mathbb {R}^{2}$ and (u,v) ∈ E if and only if the distance between u and v is at most 1. The objective is to compute the number of triples of points from P forming a triangle having each side length at most 1, that is, the number of triangles in G. For large enough N there can be bounded number of triangles sharing an edge in G with high probability.

• (iii) Consider a graph G(V,E) representing a community sharing information. Each node has some information and two nodes are connected if and only if there exists an edge between the nodes. Nodes increase their information by sharing information among their neighbors in G. Observe that the information of a node is derived by the set of neighbors. So, if two nodes have large number of common neighbors in G, then there is no need of an edge between the two nodes. So, the number of triangles on any edge in the graph is bounded. The objective is to compute the number of triangles in G, that is, the number of triples of nodes in G such that each pair of vertices are connected.

In (i) and (ii), TIS oracle can be implemented very efficiently. We can report a TIS query by just running a standard plane sweep algorithm in Computational Geometry that takes $\mathcal {O}(n \log n)$ running time.

Appendix : B: Some Probability Results

Proposition 23

Let X be a random variable. Then $\mathbb {E}[X] \leq \sqrt {\mathbb {E}[X^{2}]}$.

Lemma 24

([14, Theorem 7.1]). Let f be a function of n random variables X₁,…,X_n such that

• (i) Each X_i takes values from a set A_i,

• (ii) $\mathbb {E}[f]$ is bounded, i.e., $0 \leq \mathbb {E}[f] \leq M$,

• (iii) ${\mathscr{B}}$ be any event satisfying the following for each i ∈ [n].

  $$ \left| \mathbb{E}[f ~|~X_{1},\dots,X_{i-1},X_{i}=a_{i},\mathcal{B}^{c}] - \mathbb{E}[f ~|~X_{1},\dots,X_{i-1},X_{i}=a^{\prime}_{i},\mathcal{B}^{c}] \right|\leq c_{i}.$$

Then for any δ ≥ 0,

$${\mathbb{P}}\left( \left| f - \mathbb{E}[f] \right| > \delta + M{\mathbb{P}}(\mathcal{B}) \right) \leq e^{-{\delta^{2}}/{\sum\limits_{i=1}^{n} {c_{i}^{2}}}} + \mathbb{P}(\mathcal{B}).$$

Lemma 25 (Hoeffding’s inequality 14)

Let X₁,…,X_n be n independent random variables such that X_i ∈ [a_i,b_i]. Then for $X=\sum \limits _{i=1}^{n} X_{i}$, the following is true for any δ > 0.

$${\mathbb{P}} \left( \left| X - \mathbb{E}[X] \right| \geq \delta \right) \leq 2 \cdot e^{-{2\delta^{2}}/{\sum\limits_{i=1}^{n} (b_{i} - a_{i})^{2}}}.$$

Lemma 26 (Chernoff-Hoeffding bound 14)

Let X₁,…,X_n be independent random variables such that X_i ∈ [0, 1]. For $X=\sum \limits _{i=1}^{n} X_{i}$ and $\mu _{l} \leq \mathbb {E}[X] \leq \mu _{h}$, the followings hold for any δ > 0.

• (i) ${\mathbb {P}} \left (X > \mu _{h} + \delta \right ) \leq e^{-2\delta ^{2}/n}$.

• (ii) ${\mathbb {P}} \left (X < \mu _{l} - \delta \right ) \leq e^{-2\delta ^{2} / n}$.

Lemma 27

[14, Theorem 3.2] Let X₁,…,X_n be random variables such that a_i ≤ X_i ≤ b_i and $X=\sum \limits _{i=1}^{n} X_{i}$. Let $\mathcal {D}$ be the dependent graph, where $V(\mathcal {D})=\{X_{1},\ldots ,X_{n}\}$ and $E(\mathcal {D})= \{(X_{i},X_{j}): X_i \text {and} X_j \text {are dependent}\}$. Then for any δ > 0,

$$ {\mathbb{P}}(\left| X-\mathbb{E}[X] \right| \geq \delta) \leq 2e^{-2\delta^{2} / \chi^{*}(\mathcal{D})\sum\limits_{i=1}^{n}(b_{i}-a_{i})^{2}},$$

where $\chi ^{*}(\mathcal {D})$ denotes the fractional chromatic number of $\mathcal {D}$.

The following lemma directly follows from Lemma 27.

Lemma 28

Let X₁,…,X_n be indicator random variables such that there are at most d many X_j’s on which an X_i depends and $X=\sum \limits _{i=1}^{n} X_{i}$. Then for any δ > 0,

$${\mathbb{P}}(\left| X-\mathbb{E}[X] \right| \geq \delta) \leq 2e^{-2\delta^{2} / (d+1)n}.$$

Lemma 29 (Importance sampling 7)

Let (D₁,w₁,e₁),…, (D_r,w_r,e_r) are the given structures and each D_i has an associated weight c(D_i) satisfying

• (i) w_i,e_i ≥ 1,∀i ∈ [r];

• (ii) $\frac {e_{i}}{\rho } \leq c(D_{i}) \leq e_{i} \rho$ for some ρ > 0 and all i ∈ [r]; and

• (iii) $\sum \limits _{i=1}^{r} {w_{i}\cdot c(D_{i})} \leq M$.

Note that the exact values c(D_i)’s are not known to us. Then there exists an algorithm that finds $(D^{\prime }_{1},w^{\prime }_{1},e^{\prime }_{1}),\ldots , (D^{\prime }_{s},w^{\prime }_{s},e^{\prime }_{s})$ such that, with probability at least 1 − δ, all of the above three conditions hold and

$$ \left| \sum\limits_{i=1}^{t} {w^{\prime}_{i}\cdot c(D^{\prime}_{i})} - \sum\limits_{i=1}^{r} {w_{i}\cdot c(D_{i})} \right| \leq \lambda S, $$

where $S=\sum \limits _{i=1}^{r} {w_{i}\cdot c(D_{i})}$ and λ,δ > 0. The time complexity of the algorithm is $\mathcal {O}(r)$ and $s=\mathcal {O}\left (\frac {\rho ^{4} \log M \left (\log \log M + \log \frac {1}{\delta }\right )}{\lambda ^{2}}\right )$.