Quantum Algorithm for Triangle Finding in Sparse Graphs

Abstract

This paper presents a quantum algorithm for triangle finding over sparse graphs that improves over the previous best quantum algorithm for this task by Buhrman et al. (SIAM J Comput 34(6):1324–1330, 2005). Our algorithm is based on the recent ${\tilde{O}}(n^{5/4})$-query algorithm given by Le Gall (Proceedings of the 55th IEEE annual symposium on foundations of computer science, pp 216–225, 2014) for triangle finding over dense graphs (here n denotes the number of vertices in the graph). We show in particular that triangle finding can be solved with $O(n^{5/4-\epsilon })$ queries for some constant $\epsilon >0$ whenever the graph has at most $O(n^{2-c})$ edges for some constant $c>0$.

1 Introduction

Background. Triangle finding asks to decide if a given undirected graph $G=(V,E)$ contains a cycle of length three, i.e., whether there exist three vertices $u_1,u_2,u_3\in V$ such that $\{u_1,u_2\}\in E$, $\{u_1,u_3\}\in E$ and $\{u_2,u_3\}\in E$. This problem has received recently a lot of attention, for the following reasons.

First, several new applications of triangle finding have been discovered recently. In particular, Vassilevska Williams and Williams [17] have shown a surprising reduction from Boolean matrix multiplication to triangle finding, which indicates that efficient algorithms for triangle finding may be used to design efficient algorithms for matrix multiplication, and thus also for a vast class of problems related to matrix multiplication. Relations between variants of the standard triangle finding problem (such as triangle finding over weighted graphs) and well-studied algorithmic problems (such as 3SUM) have also been shown in the past few years (see for instance [16, 18]).

Second, triangle finding is one of the most elementary graph theoretical problems whose complexity is unsettled. In the time complexity setting, the best classical algorithm uses a reduction to matrix multiplication [9] and solves triangle finding in time $O(n^{2.38})$, where n denotes the number of vertices in G. In the time complexity setting again, Grover [8] search immediately gives, when applied to triangle finding as a search over the set of triples of vertices of the graph, a quantum algorithm with time complexity ${\tilde{O}}(n^{3/2})$, which is still the best known upper bound for the quantum time complexity of this problem.¹ In the query complexity setting, where an oracle to the adjacency matrix of the graph is given and only the number of calls to this oracle is counted, a surge of activity has lead to quantum algorithms with better complexity. Magniez et al. [14] first presented a quantum algorithm that solves triangle finding with ${\tilde{O}}(n^{1.3})$ queries. This complexity was later improved to $O(n^{1.296\ldots })$ by Belovs [4], then to $O(n^{1.285\ldots })$ by Lee et al. [13] and Jeffery et al. [11], and further improved recently to ${\tilde{O}}(n^{5/4})$ by Le Gall [12]. The main open problem now is to understand whether this ${\tilde{O}}(n^{5/4})$-query upper bound is tight or not. The best known lower bound on the quantum query complexity of triangle finding is the straightforward $\Omega (n)$ lower bound.

Another reason why triangle finding has received much attention from the quantum computing community is that work on the quantum complexity of triangle finding has been central to the development of algorithmic techniques. Indeed, all the improvement mentioned in the previous paragraph have been obtained by introducing either new quantum techniques or new paradigms for the design of quantum algorithms: applications of quantum walks to graph-theoretic problems [14], introduction of the concept of learning graphs [4] and improvements to this technique [13], introduction of quantum walks with quantum data structures [11], association of combinatorial arguments with quantum walks [12].

Triangle Finding in Sparse Graphs. The problem we will consider in this paper is triangle finding over sparse graphs (the graphs considered are, as usual, undirected and unweighted). If we denote m the number of edge of the graph (i.e., $m=|E|$), the goal is to design algorithms with complexity expressed as a function of m and n. Ideally, we would like to show that if $m=n^{2-c}$ for any constant $c>0$ then triangle finding can be solved significantly faster than in the dense case (i.e., $m\approx n^2$). Besides its theoretical interest, this problem is of practical importance since in many applications the graphs considered are sparse.

Classically, Alon et al. [1] constructed an algorithm exploiting the sparsity of the graph and working in time $O(m^{1.41})$, which gives better complexity than the $O(n^{2.38})$-time complexity mentioned above when $m\le n^{1.68}$. Understanding whether an improvement over the dense case is also possible for larger values m is a longstanding open problem. Note that in the classical setting, the query complexity of triangle finding is $\Theta (n^2)$, independently of the value of m (indeed, if the graph contains only one triangle, any randomized algorithm would need to hit the edges of this triangle, which requires $\Theta (n^2)$ queries).

In the quantum setting, using amplitude amplification, Buhrman et al. [6] showed how to construct a quantum algorithm for triangle finding with time and query complexity $O(n+\sqrt{nm})$. This upper bound is tight when $m\le n$ since the $\Omega (n)$-query lower bound for the quantum query complexity of triangle finding already mentioned also holds when m is a constant. Childs and Kothari [7] more recently developed an algorithm, based on quantum walks, that detects the existence of subgraphs in a given graph. Their algorithm works for any constant-size subgraph. For detecting the existence of a triangle, however, the upper bound they obtain is ${\tilde{O}}(n^{2/3}\sqrt{m})$ queries for $m\ge n$, which is worse that the bound obtained in [6]. Buhrman et al.’s result in particular gives an improvement over the ${\tilde{O}}(n^{5/4})$-query quantum algorithm algorithm whenever $m\le n^{3/2}$. A natural question is whether a similar improvement can be obtained for larger values of m. For instance, can we obtain query complexity ${\tilde{O}}(n^{5/4-\epsilon })$ for some constant $\epsilon >0$ when $m\approx n^{1.99}$? A positive answer would show that even a little amount of sparsity can be exploited in the quantum query setting, which is not known to be true in the classical setting as mentioned in the previous paragraph.

Our Results. In this paper we answer positively to the above question. Our main result is as follows.

Theorem 1

There exists a quantum algorithm that solves, with high probability, the triangle finding problem over graphs of n vertices and m edges with query complexity

$$\begin{aligned} {\left\{ \begin{array}{ll}\begin{array}{ll} O(n+\sqrt{nm}) &{} \quad \text { if }\,\, 0 \le m \le n^{7/6}, \\ {\tilde{O}}(nm^{1/14}) &{}\quad \text { if }\,\, n^{7/6} \le m \le n^{7/5}, \\ {\tilde{O}}(n^{1/6}m^{2/3}) &{}\quad \text { if }\,\,n^{7/5} \le m \le n^{3/2}, \\ {\tilde{O}}(n^{23/30}m^{4/15}) &{}\quad \text { if }\,\,n^{3/2} \le m \le n^{13/8}, \\ {\tilde{O}}(n^{59/60}m^{2/15}) &{}\quad \text { if }\,\, n^{13/8} \le m \le n^2. \end{array} \end{array}\right. } \end{aligned}$$

The complexity bounds of Theorem 1 are depicted in Fig. 1. For the dense case (i.e., $m\approx n^2$) we recover the same complexity ${\tilde{O}}(n^{5/4})$ as in [12]—in this case it turns out that our procedure directly applies the algorithm from [12]. Whenever $m=n^{2-c}$ for some constant $c>0$ (in particular, for $m\approx n^{1.99}$), we indeed obtain query complexity ${\tilde{O}}(n^{5/4-\epsilon })$ for some constant $\epsilon >0$ depending on c. The query complexity of our algorithm is better than the query complexity of Buhrman et al.’s algorithm [6] whenever $m\gtrsim n^{7/6}$. When $m\lesssim n^{7/6}$ we obtain the same complexity $O(n+\sqrt{nm})$ as in [6]—in this case it turns out that our procedure directly applies the algorithm from [6].

Fig. 1

Quantum query complexity of triangle finding on a graph with n vertices and m edges

Overview of Our Techniques.

The main idea is to adapt the ${\tilde{O}}(n^{5/4})$-query quantum algorithm for triangle finding [12] to handle sparse graphs. This algorithm works in two steps: a first step based on Grover search that detects the existence of a triangle in a well-chosen small part of the graph G, and a second step based on recursive quantum walks that detects the existence of a triangle in the remaining (large) part of the graph. A first simple observation is that the first step can be implemented faster in the case of sparse graphs by applying the quantum algorithm by Buhrman et al. [6] on the small part of G instead of using Grover search. This observation alone however does not give any interesting speed-up unless sparsity is exploited in the second step as well. The hard part is actually to adapt the recursive quantum walk approach to the case of sparse graphs, and we outline below our main ideas to achieve this goal.

The main issue is that the implementation of the recursive quantum walks described in [12] is really tailored for their application on dense graphs; when trying to use the same implementation for sparse graphs prohibitive intermediate costs (of order ${\tilde{O}}(n^{5/4})$, which is fine for the dense case, but not for the sparse case) appear. To overcome this difficulty, we need to modify partially the original approach in several ways, such as modifying how the inner quantum walk checks if it has found a solution and adjusting how the sets of marked states of the walk are defined, to fully exploit the sparsity of the graph.

Several more technical issues have also to be dealt with. The complexity of a quantum walk basically depends on the complexity of three operations performed by the walk: the set up cost (creating the data structure corresponding to the initial state of the walk), the update cost (updating the database after updating the current state of the walk) and the checking cost (checking if the current state of the walk is marked or not). The sparsity of the graph G can be immediately exploited to reduce the cost of these three operations if the graph is “perfectly balanced”, i.e., if each vertex of the graph has degree $\Theta (m/n)$. However, while the average degree will indeed be $\Theta (m/n)$, in general the graph can have many vertices with degree exceeding this estimate (in particular this can happen for vertices of the triangle we are looking for). This is a significant complication since to analyze a quantum walk one need an upper bound on the worst case (i.e., for the worse state of the walk) complexity of the three operations. Indeed, there is no general technique to analyze quantum walks when only an upper bound on the average update cost or checking cost is available. To overcome this difficulty, our approach is to partition the vertices of V into two sets: the set of vertices with degree larger than $n^d$ (which we call below high-degree vertices) and the set of vertices with degree smaller than $n^d$ (low-degree vertices), where d is a parameter. Obtaining the classification can be done by combining quantum search and quantum counting, but is costly when d is small, which means that we need to be careful when choosing d. Once this classification has been obtained, we only need to search separately for four types of triangles: triangles with three low-degree vertices, triangles with two low-degree vertices and one high-degree vertex, triangles with one low-degree vertex and two high-degree vertices, and triangles with three high-degree vertices. Since we know that each low-degree vertex has degree at most $n^d$, we can derive a worst-case upper bound for the corresponding update costs and checking costs. For high-degree vertices we do not have any upper bound on their degree, but we know that the number of high-degree vertices is at most $m/n^d$, which will be significantly smaller than n if d is well chosen and lead to some improvement for the corresponding complexity of the walks, since the graph has at most m edges. Combined with the ideas described in the previous paragraph, this strategy enables us to obtain the upper bounds given in Theorem 1.

2 Preliminaries

2.1 Query Complexity for Graph-Theoretic Problems

In this paper we adopt the standard model of quantum query complexity for graph-theoretic problems. The presentation given below will follow the description of this notions given in [12].

For any finite set T and any $r\in \{1,\ldots ,|T|\}$ we denote ${\mathcal {S}}(T,r)$ the set of all subsets of r elements of T. We use the notation ${\mathcal {E}}(T)$ to represent ${\mathcal {S}}(T,2)$, i.e., the set of unordered pairs of elements in T.

Let $G=(V,E)$ be an undirected and unweighted graph, where V represents the set of vertices and $E\subseteq {\mathcal {E}}(V)$ represents the set of edges. We write $n=|V|$. In the query complexity setting, we assume that V is known, and that E can be accessed through a quantum unitary operation ${\mathcal {O}}_G$ defined as follows. For any pair $\{u,v\}\in {\mathcal {E}}(V)$, any bit $b\in \{0,1\}$, and any binary string $z\in \{0,1\}^*$, the operation ${\mathcal {O}}_G$ maps the basis state $\vert \{u,v\} \rangle \vert b \rangle \vert z \rangle$ to the state

$$\begin{aligned} {\mathcal {O}}_G\vert \{u,v\} \rangle \vert b \rangle \vert z \rangle =\left\{ \begin{array}{ll} \vert \{u,v\} \rangle \vert b\oplus 1 \rangle \vert z \rangle &{}\quad \mathrm { if }\; \{u,v\}\in E,\\ \vert \{u,v\} \rangle \vert b \rangle \vert z \rangle &{}\quad \mathrm { if }\; \{u,v\}\notin E,\\ \end{array} \right. \end{aligned}$$

where $\oplus$ denotes the bit parity (i.e., the logical XOR). We say that a quantum algorithm computing some property of G uses k queries if the operation ${\mathcal {O}}_G$, given as an oracle, is called k times by the algorithm. We also assume that we know the number of edges of the input graph (i.e., we know $m=|E|$). All the results in this paper can actually be generalized to the case where m is unknown, in the following sense. When m is unknown we can use quantum counting [5] to estimate at negligible cost, and with very high probability, the value m within a constant multiplicative factor (such an estimation is enough for us). This gives an algorithm with expected query complexity similar to the bounds given in Theorem 1. Note that it is unclear how to obtain a similar result for the worst case query complexity when m is unknown, since in the worst case the value output by the quantum counting procedure may be wrong.

Quantum Enumeration. Let $f_G:\{1,\ldots ,N\}\rightarrow \{0,1\}$ be a Boolean function depending on the input graph G, and let us write $M=f^{-1}(1)$. Assume that for any $x\in \{1,\ldots ,N\}$ the value $f_G(x)$ can be computed using at most t queries to ${\mathcal {O}}_G$. Grover search enables us to find an element x such that $f_G(x)=1$, if such an element exists, using ${\tilde{O}}(\sqrt{N/M}\times t)$ queries to ${\mathcal {O}}_G$. A folklore observation is that we can then repeat this procedure to find all the elements $x\in \{1,\ldots ,N\}$ such that $f_G(x)=1$ with ${\tilde{O}}\left( \left( \sqrt{\frac{N}{M}} +\sqrt{\frac{N}{M-1}}+\cdots +\sqrt{\frac{N}{1}} \right) \times t\right) ={\tilde{O}}\left( \sqrt{N\times M}\times t\right)$ queries. We call this procedure quantum enumeration.

Quantum Walk Over Johnson Graphs. Let T be a finite set and r be a positive integer such that $r\le |T|$. Let $f_G:{\mathcal {S}}(T,r)\rightarrow \{0,1\}$ be a Boolean function depending on the input graph G. We say that a set $A\in {\mathcal {S}}(T,r)$ is marked if $f_G(A)=1$. Let us consider the following problem. The goal is to find a marked set, if such a set exists, or otherwise report that there is no marked set. We are interested in the number of calls to ${\mathcal {O}}_G$ to solve this problem. The quantum walk search approach developed by Ambainis [2] solves this problem using a quantum walk over a Johnson graph.

The Johnson graph J(T, r) is the undirected graph with vertex set ${\mathcal {S}}(T,r)$ where two vertices $R_1,R_2\in {\mathcal {S}}(T,r)$ are connected if and only if $|R_1\cap R_2|=r-1$. In a quantum walk over a Johnson graph J(T, r), the state of the walk corresponds to a node of the Johnson graph (i.e., to an element $A\in {\mathcal {S}}(T,r)$). A data structure D(A), which in general depends on G, is associated to each state A. There are three costs to consider: the set up cost ${\mathsf {S}}$ representing the number of queries to ${\mathcal {O}}_G$ needed to construct the data structure of the initial state of the walk, the update cost ${\mathsf {U}}$ representing the number of queries to ${\mathcal {O}}_G$ needed to update the data structure when one step of the quantum walk is performed (i.e., updating D(A) to $D(A')$ for some $A'\in {\mathcal {S}}(T,r)$ such that $|A\cap A'|=r-1$), and the checking cost ${\mathsf {C}}$ representing the number of queries to ${\mathcal {O}}_G$ needed to check if the current state A is marked (i.e., checking whether $f_G(A)=1$). Let $\varepsilon >0$ be such that, for all input graphs G for which at least one marked set exists, the fraction of marked states is at least $\varepsilon$. Ambainis [2] (see also [15]) has shown that the quantum walk search approach outlined above finds with high probability a marked set if such set exists (or otherwise report that there is no marked set) and has query complexity ${\tilde{O}}\left( {\mathsf {S}}+\frac{1}{\sqrt{\varepsilon }}\left( \sqrt{r}\times {\mathsf {U}}+{\mathsf {C}}\right) \right) .$

2.2 Quantum Algorithm for Dense Triangle Finding

In this subsection we outline the ${\tilde{O}}(n^{5/4})$-query quantum algorithm for triangle finding over a dense graph by Le Gall [12]. We actually present a version of this algorithm that solves the following slightly more general version of triangle finding, since this will be more convenient when describing our algorithms for sparse graphs in the next section: given two (non necessarily disjoint) sets $V_1,V_2\subseteq V$, find a triangle $\{v_1,v_2,v_3\}$ of G such that $v_1\in V_1$ and $v_2,v_3\in V_2$, if such a triangle exists. Note that the original triangle finding problem is the special case $V_1=V_2=V$.

Definitions and Lemmas. Let $V_1$ be any subset of V. For any sets $X\subseteq V_1$ and $Y\subseteq V$, we define the set $\Delta _G(X,Y)\subseteq {\mathcal {E}}(Y)$ as follows:

$$\begin{aligned} \Delta _G(X,Y)={\mathcal {E}}(Y)\setminus \bigcup _{u\in X}{\mathcal {E}}(N_G(u)), \end{aligned}$$

where $N_G(u)$ denotes the set of neighbors of u. This set $\Delta _G(X,Y)$ is the set of all pairs of vertices in Y that are not in the neighborhood of a same vertex from X. For any vertex $w\in V$, we define the set $\Delta _G(X,Y,w)\subseteq \Delta _G(X,Y)$ as follows:

$$\begin{aligned} \Delta _G(X,Y,w)&= \Big \{\{u,v\}\in \Delta _G(X,Y)\,|\, \{u,w\}\in E \text { and } \{v,w\}\in E\Big \}. \end{aligned}$$

An important concept used in [12] is the notion of k-good sets.

Definition 2.1

Let k be any constant such that $0\le k\le 1$, and $V_1$ be any subset of V. A set $X\subseteq V_1$ is k-good for $(G,V_1)$ if the inequality $\sum _{w\in V_1}\left| \Delta _G(X,Y,w)\right| \le |Y|^2 |V_1|^{1-k}$ holds for all $Y\subseteq V$.

Note that [12] considered only Definition 2.1 for the case $V_1=V$. In our paper we will need the slightly generalized version described here. The point is that k-good sets can be constructed very easily.

Lemma 2.1

([12]) Let k be any constant such that $0\le k\le 1$. Suppose that X is a set obtained by taking uniformly at random, with replacement, $\left\lceil 3|V_1|^{k}\log n \right\rceil$ elements from $V_1$. Then X is k-good for $(G,V_1)$ with probability at least $1-1/n$.

Lemma 2.1 was proved in [12] only for the case $V_1=V$, but the generalization is straightforward.

Quantum Algorithm for Dense Triangle Finding. Let a, b and k be three constants such that $0<b< a<1$ and $0<k<1$. The values of these constants will be set later. The quantum algorithm in [12] works as follows.

The algorithm first takes a set $X\subseteq V_1$ obtained by choosing uniformly at random $\left\lceil 3|V_1|^{k}\log n \right\rceil$ elements from $V_1$, and checks if there exists a triangle of G with a vertex in X and two vertices in $V_2$. This can be done using Grover search with

$$\begin{aligned} O\left( \sqrt{|X|\times |{\mathcal {E}}(V_2)|}\right) ={\tilde{O}}\left( |V_1|^{k/2}|V_2|\right) \end{aligned}$$

(1)

queries. If no triangle has been reported, we know that any triangle of G with one vertex in $V_1$ and two vertices in $V_2$ must have an edge in $\Delta _G(X,V_2)$.

Now, in order to find a triangle with an edge in $\Delta _G(X,V_2)$, if such a triangle exists, the idea is to search for a set $A\in {\mathcal {S}}(V_2,\left\lceil |V_2|^a \right\rceil )$ such that $\Delta _G(X,A)$ contains an edge of a triangle. To find such a set A, the algorithm performs a quantum walk over the Johnson graph $J(V_2,\left\lceil |V_2|^a \right\rceil )$. The states of this walk correspond to the elements in ${\mathcal {S}}(V_2,\left\lceil |V_2|^{a} \right\rceil )$. The state corresponding to a set $A\in {\mathcal {S}}(V_2,\left\lceil |V_2|^{a} \right\rceil )$ is marked if $\Delta _G(X,A)$ contains an edge of a triangle of G. In case the set of marked states is not empty, the fraction of marked states is

$$\begin{aligned} \varepsilon =\Omega \left( |V_2|^{2(a-1)}\right) . \end{aligned}$$

The data structure of the walk stores the set $\Delta _G(X,A)$. Concretely, this is done by storing the couple $(v,N_G(v)\cap X)$ for each $v\in A$, since this information is enough to construct $\Delta _G(X,A)$ without using any additional query. The setup cost is ${\mathsf {S}}= |A|\times |X|={\tilde{O}}(|V_2|^{a}|V_1|^k)$ queries. The update cost is ${\mathsf {U}}=2|X|={\tilde{O}}(|V_1|^{k})$ queries. The query complexity of the quantum walk is

$$\begin{aligned} {\tilde{O}}\left( {\mathsf {S}}+\sqrt{1/\varepsilon }\left( |V_2|^{a/2}\times {\mathsf {U}}+{\mathsf {C}}\right) \right) , \end{aligned}$$

(2)

where ${\mathsf {C}}$ is the cost of checking if a state is marked.

The checking procedure is done as follows: check if there exists a vertex $w\in V_1$ such that $\Delta _G(X,A)$ contains a pair $\{v_1,v_2\}$ for which $\{v_1,v_2,w\}$ is a triangle of G. For any $w\in V_1$, let Q(w) denote the query complexity of checking if there exists a pair $\{v_1,v_2\}\in \Delta _G(X,A)$ such that $\{v_1,v_2,w\}$ is a triangle of G. Using Ambainis’ variable cost search [3] this checking procedure can be implemented using

$$\begin{aligned} {\mathsf {C}}=\sqrt{\sum _{w\in V_1} Q(w)^2} \end{aligned}$$

queries. It thus remains to give an upper bound on Q(w). Let us fix $w\in V_1$. First, a tight estimator of the size of $\Delta _G(X,A,w)$ is computed: the algorithm computes an integer $\delta (X,A,w)$ such that $|\delta (X,A,w)-|\Delta _G(X,A,w)||\le \frac{1}{10}\times |\Delta _G(X,A,w)|$, which can be done in ${\tilde{O}}(|V_1|^k)$ queries using (classical) sampling. The algorithm then performs a quantum walk over the Johnson graph $J(A,\left\lceil |V_2|^{b} \right\rceil )$. The states of this walk correspond to the elements in ${\mathcal {S}}(A,\left\lceil |V_2|^{b} \right\rceil )$. We now define the set of marked states of the walk. The state corresponding to a set $B\in {\mathcal {S}}(A,\left\lceil |V_2|^{b} \right\rceil )$ is marked if B satisfies the following two conditions:

• (i) there exists a pair $\{v_1,v_2\}\in \Delta _G(X,B,w)$ such that $\{v_1,v_2\}\in E$ (i.e., such that $\{v_1,v_2,w\}$ is a triangle of G);

• (ii) $|\Delta _G(X,B,w)|\le 10\times |V_2|^{2(b-a)}\times \delta (X,A,w)$.

The fraction of marked states is

$$\begin{aligned} \varepsilon '=\Omega \left( |V_2|^{2(b-a)}\right) , \end{aligned}$$

as shown in Lemma 4.3 of [12]. The data structure of the walk will store $\Delta _G(X,B,w)$. Concretely, this is done by storing the couple $(v,e_v)$ for each $v\in B$, where $e_v=1$ if $\{v,w\}\in E$ and $e_v=0$ if $\{v,w\}\notin E$. The setup cost is ${\mathsf {S}}'=\left\lceil |V_2|^b \right\rceil$ queries since it is sufficient to check if $\{v,w\}$ is an edge for all $v\in B$. The update cost is ${\mathsf {U}}'=2$ queries. The checking cost is

$$\begin{aligned} {\mathsf {C}}'_w= & {} O\left( \sqrt{|\Delta _G(X,B,w)|}\right) = O\left( \frac{|V_2|^b}{|V_2|^a}\sqrt{\delta (X,A,w)}\right) \\= & {} O\left( \frac{|V_2|^b}{|V_2|^a}\sqrt{|\Delta (X,A,w)|}\right) . \end{aligned}$$

We thus obtain the bound

$$\begin{aligned} Q(w)={\tilde{O}}\left( |V_1|^k+{\mathsf {S}}'+\sqrt{1/\varepsilon '}\left( |V_2|^{b/2}\times {\mathsf {U}}'+{\mathsf {C}}'_w\right) \right) , \end{aligned}$$

and conclude that

$$\begin{aligned} {\mathsf {C}}={\tilde{O}}\left( \sqrt{|V_1|}\left( |V_1|^k+{\mathsf {S}}'+\frac{|V_2|^{b/2}\times {\mathsf {U}}'}{\sqrt{\varepsilon '}}\right) +\frac{|V_2|^{b-a}}{\sqrt{\varepsilon '}}\times \sqrt{\sum _{w\in V_1} |\Delta (X,A,w)|}\right) . \end{aligned}$$

The final key observation is that, since the set X is k-good for $(G,V_1)$ with high probability, as guaranteed by Lemma 2.1, the term $\sum _{w\in V} |\Delta (X,A,w)|$ in the above expression can be replaced by $O(|V_2|^{2a}|V_1|^{1-k})$, which enables us to express ${\mathsf {C}}$ as a function of a, b and k, and then the complexity of the second part of the algorithm (Expression (2)) as a function of a, b and k. The complexity of the whole algorithm (the maximum of Expressions (1) and (2)) can thus be written as a function of a, b and k as well.

For the original triangle finding problem (i.e., for the case $V_1=V_2=V$), taking $a=\frac{3}{4}$ and $b=k=\frac{1}{2}$ gives query complexity ${\tilde{O}}(n^{5/4})$.

3 Quantum Algorithm for Sparse Triangle Finding

In this section we describe our quantum algorithm for triangle finding in sparse graphs and prove Theorem 1.

Let d be a real number such that $0\le d\le 1$. The value of this parameter will be set later. Define the following two subsets of V:

$$\begin{aligned} {\mathcal {V}}^{d}_h&= \{v \in V \mid \deg (v) \ge \frac{9}{10}\times n^{d}\}, \\ {\mathcal {V}}^{d}_l&= \{v \in V \mid \deg (v) \le \frac{11}{10}\times n^{d}\}. \end{aligned}$$

A crucial observation is that $|{\mathcal {V}}^d_h|=O(m/{n^d})$, since the graph G has m edges. More precisely, we have $|{\mathcal {V}}^d_h|\le \frac{20}{9}\times \frac{m}{n^d}$.

The following proposition shows how to efficiently obtain all the vertices in ${\mathcal {V}}^{d}_h$.

Proposition 3.1

Let $d\in [0,1]$ be any real number such that $1\le m/n^d\le n$. There exists a quantum algorithm using $Q_1 = \tilde{O}(n^{1-d}\sqrt{m})$ queries that partitions the set V into two sets $V^{d}_h$ and $V^{d}_l$ such that with high probability the following two conditions hold:

• (i) $V^d_l \subseteq {\mathcal {V}}^d_l$, and

• (ii) $\frac{m}{n^d} \le |V^d_h| \le \frac{20}{9}\times \frac{m}{n^d}$.

We will see in the proof that the quantum algorithm of Proposition 3.1 actually identifies (with high probability) all the vertices in ${\mathcal {V}}^{d}_h$. We nevertheless only require that it outputs (with high probability) a superset of ${\mathcal {V}}^{d}_h$ since it enables us to impose the lower bound $|V^d_h|=\Omega (\frac{m}{n^d})$ on the size of $V^d_h$, which will be much more convenient in our analysis.

Proof of Proposition 3.1

Let v be any vertex in V. Using quantum counting [5] we can compute, using ${\tilde{O}}\left( \sqrt{\frac{n}{n^{d}}}\right)$ queries, a value a(v) such that $|a(v)-\deg (v)|\le n^d/100$ with probability at least $1-1/\mathrm {poly}(n)$. We use a(v) to classify v as follows: we decide “ v is in ${\mathcal {V}}^{d}_h$ ” if $a(v)\ge n^d$, and decide “ v is in ${\mathcal {V}}^{d}_l$ ” if $a(v)< n^d$. This decision is correct with probability at least $1-1/\mathrm {poly}(n)$.

We can thus apply quantum enumeration as described in Sect. 2.1 to obtain a set $V^{'} \subseteq V$ of vertices such that, with high probability, all the vertices in $V^{'}$ are in ${\mathcal {V}}^{d}_h$ and all the vertices in $V\setminus V^{'}$ are in ${\mathcal {V}}^{d}_l$. We then put all the vertices from $V{'}$ in $V^d_h$. If $|V^{'}| < \frac{m}{n^d}$, we add arbitrary vertices from $V\setminus {V}^{'}$ to $V^d_h$ so that $|V^d_h|$ satisfies Condition (ii). We then set $V_l^d=V\setminus V_h^d$.

The overall complexity of this algorithm is

$$\begin{aligned} \tilde{O}\left( \sqrt{n\times \frac{m}{n^d}}\times \sqrt{\frac{n}{n^{d}}}\right) =\tilde{O}(n^{1-d}\sqrt{m}) \end{aligned}$$

queries. $\square$

In the remaining of the section we assume that the algorithm of Proposition 3.1 outputs a correct classification (i.e., $V^{d}_h$ and $V^{d}_l$ satisfy Conditions (i) and (ii)), which happens with high probability. We will choose d carefully such that $m/n^d=o(n)$, which implies $|V^d_l| = \Theta (n)$ since $V^{d}_h$ and $V^{d}_l$ are a partition of V. We will say that a vertex $v\in V$ is d-high if $v\in V^{d}_h$, and say it is d-low if $v\in V^{d}_l$. Once the vertices have been classified, checking if G has a triangle can be divided into four subproblems: checking if G has a triangle with three d-low vertices, checking if G has a triangle with two d-low vertices and one d-high vertex, checking if G has a triangle with one d-low-degree vertex and two d-high vertices, and checking if G has a triangle with three high-degree vertices. We now present six procedures (summarized in Table 1) to handle these cases (for some cases we present more than one procedure to allow us to choose which procedure to use according to the value of m).

Table 1 Subproblems solved in Proposition 3.2–3.7

Proposition 3.2

Let $a_1$, $k_1$ and $b_1$ be any constants such that $0< a_1, k_1 < 1$ and $0< b_1 < a_1$. There exists a quantum algorithm that finds a triangle of G consisting of three d-low vertices, if such a triangle exists, with high probability using $Q_2 = \tilde{O}(n + n^{k_1/2}m^{1/2} + n^{a_1 + d/2 + k_1 - 1/2} + n^{1/2 + d/2 + k_1 - a_1/2} + n^{3/2 + k_1/2 - a_1} + n^{1 + b_1 + d/2 - a_1} + n^{3/2 - b_1/2} + n^{3/2 - k_1/2})$ queries.

The proof of Proposition 3.2 will use the following key lemma.

Lemma 3.1

Let k be any constant such that $0< k < 1$. Suppose that X is a set of size $|X| = \left\lceil 3n^{k}\log {n} \right\rceil$ obtained by taking uniformly at random vertices from $V' \subseteq V$. Then, with probability at least

$$\begin{aligned} 1 - \frac{|V^d_l|}{n^2\exp (\frac{231}{|V'|}n^{d + k}\log {n})}, \end{aligned}$$

the inequality

$$\begin{aligned} |N_G(v) \cap X| < \frac{231}{|V'|}n^{d + k}\log {n} + 2\log {n} \end{aligned}$$

holds for all vertices $v \in V^{d}_l$.

Proof

For any $v \in V^d_l$, the quantity $|N_G(v) \cap X| \le \frac{11}{10}n^{d}$ (remember that we assume that $V^{d}_l\subseteq {\mathcal {V}}^{d}_l$) is a random variable distributed according to the hypergeometric distribution. The expected value of $|N_G(v) \cap X|$ is $\mu = |N_G(v) \cap V^{d}_l| \times \frac{|X|}{|V'|} \le \frac{11}{10}n^{d} \times \frac{3n^k\log {n} + 1}{|V'|} < \frac{33}{|V'|}\times n^{d + k}\log {n}$. By Corollary 2.4 and Theorem 2.10 of [10], $\Pr [|N_G(v) \cap X| \ge 7\mu + 2\log {n}] < \exp (-7\mu )\frac{1}{n^{2}}$. Thus the inequality

$$\begin{aligned} |N_G(v) \cap X| < \frac{231}{|V'|}n^{d + k}\log {n} + 2\log {n} \end{aligned}$$

holds for all $v \in V^{d}_l$ with probability at least

$$\begin{aligned} 1 - |V^{d}_l| \times \frac{1}{n^{2}\exp (\frac{231}{|V'|}n^{d+k}\log {n})}, \end{aligned}$$

as claimed. $\square$

Proof of Proposition 3.2

We adapt the algorithm for the dense case presented in Sect. 2.2. We take $V_1 = V_2 = V^{d}_l$, and $X \subseteq V_1$ of size $|X|=\left\lceil 3|V_1|^{k_1}\log n \right\rceil$. We replace the first step of the algorithm, which checks if there exists a triangle of G with a vertex in X and two vertices in $V_2$, by the following procedure based on [6]. We take a random edge $\{u, v\} \in {\mathcal {E}}(V_2)\cap E$ and then try to find a vertex w from X such that $\{u,v,w\}$ is a triangle of G. Note that this can be implemented using two Grover searches in ${\tilde{O}}(\sqrt{|{\mathcal {E}}(V_2)|/|{\mathcal {E}}(V_2)\cap E|}+\sqrt{|X|})$ queries, and that in the worst case (i.e., when there is only one triangle) the success probability of this approach is $\Theta (1/|{\mathcal {E}}(V_2)\cap E|)$. Using amplitude amplification we can then check with high probability the existence of such a triangle with total query complexity

$$\begin{aligned} {\tilde{O}}\left( \sqrt{|{\mathcal {E}}(V_2)\cap E|}\times (\sqrt{|{\mathcal {E}}(V_2)|/|{\mathcal {E}}(V_2)\cap E|}+\sqrt{|X|})\right) = \tilde{O}(n + \sqrt{n^{k_1}m}). \end{aligned}$$

(3)

We now show how to adapt the second step of the algorithm presented in Sect. 2.2 to exploit the sparsity of the graph. First, as observed in [12], the cost of estimating the size of $\Delta _G(X,A,w)$ can be reduced to ${\tilde{O}}(\sqrt{n^{k_1}})$ queries by using quantum counting instead of random sampling (quantum counting was not used in [12] since it did not result in any speed-up for the dense case, but for the sparse case this is necessary). We now describe our main ideas to exploit the sparsity of the graph, and show how to reduce the cost of two quantum walks.

First, we describe how to reduce the setup cost ${\mathsf {S}}$ and the update cost ${\mathsf {U}}$ as follows. By Lemma 3.1 and $|V^d_l| = \Theta (n)$, we know that $|N_G(v) \cap X| < t$ for all $v \in V_2$, where $t=\tilde{O}(n^{d + k_1 - 1})$. Therefore we can use quantum enumeration to find all vertices in $N_G(v) \cap X$ with

$$\begin{aligned} {\tilde{O}}\left( \sqrt{\frac{|X|}{t}} + \cdots + \sqrt{\frac{|X|}{1}}\right) = {\tilde{O}}(\sqrt{|X|t}) \end{aligned}$$

queries. The setup cost ${\mathsf {S}}$ is thus

$$\begin{aligned} {\mathsf {S}} = {\tilde{O}}\left( |A| \times \sqrt{|X|t}\right) = {\tilde{O}}(n^{a_1 + d/2 + k_1 - 1/2}) \end{aligned}$$

queries, and the update cost ${\mathsf {U}} = {\tilde{O}}(\sqrt{|X|t}) = {\tilde{O}}(n^{d/2 + k_1 - 1/2})$ queries.

Next, we describe how to reduce the setup cost ${\mathsf {S}}'$. This set up requires to obtain the couple $(v, e_v)$ for each $v \in B$, where w is a fixed vertex in $V_1$, $e_v=1$ if $\{v,w\}\in E$ and $e_v=0$ if $\{v,w\}\notin E$. Let $\mu _w$ be the average of $|N_G(w) \cap B|$ over all B. A key observation is that $\mu _w = O(n^{d} \times \frac{|B|}{|V|}) = O(n^{b_1 + d - 1})$. We use quantum enumeration to find at most $10 \times \mu _w$ vertices in $N_G(w) \cap B$ from B. Thus the cost of this procedure is

$$\begin{aligned} {\mathsf {S}}' = {\tilde{O}}\left( \sqrt{\frac{|B|}{\mu _w}} + \cdots + \sqrt{\frac{|B|}{1}} \right) = {\tilde{O}}(\sqrt{|B|\times \mu _w}) = {\tilde{O}}(n^{b_1 + d/2 - 1/2}) \end{aligned}$$

queries. Note that this procedure will not correctly prepare the database for all B’s (since $|N_G(w) \cap B|$ may exceed $10 \times \mu _w$ for some B’s); it will prepare correctly the database only for a large fraction of the B’s. This is nevertheless not a problem since the initial state of the quantum walk is a uniform superposition of all the B’s: this procedure will thus prepare a state close enough to the ideal state, which will modify only in a negligible way the final success probability of the whole walk.

We also modify the definition of a marked state for the second walk (we add one condition). Namely, the state corresponding to a set $B \in {\mathcal {S}}(A, \left\lceil |V_2|^{b_1} \right\rceil )$ will be marked if B satisfies the following three conditions:

• (i) there exist two vertices $v_1, v_2 \in B$ such that $\{v_1,v_2\}\in E$ (i.e., such that $\{v_1,v_2,w\}$ is a triangle of G);

• (ii) $|\Delta _G(X,B,w)|\le 10\times |V_2|^{2(b_1-a_1)}\times \delta (X,A,w)$;

• (iii) $|N_G(w) \cap B| \le 10 \times \mu _w$.

It is easy to show that adding the third condition does not change significantly the fraction of marked states:

$$\begin{aligned} \varepsilon '= & {} \Pr [v_1 \in B \ {\mathrm{and}} \ v_2 \in B \ \mathrm{and} \ |N_G(w) \cap B| \le 10\mu _w ] \\= & {} \Pr [v_1 \in B] \Pr [v_2 \in B \mid v_1 \in B] \Pr [|N_G(w) \cap B| \le 10\mu _w \mid v_1 \in B \ \mathrm{and} \ v_2 \in B ] \\= & {} \Pr [v_1 \in B] \Pr [v_2 \in B \mid v_1 \in B] \Pr [|N_G(w) \cap B'| \le 10\mu _w - 2 ] \\\ge & {} \left( 1 - \Pr [|N_G(w) \cap B'| \ge 10\mu _w - 2 ]\right) \times \Pr [v_1 \in B] \Pr [v_2 \in B \mid v_1 \in B] \\\ge & {} \left( 1 - \Pr [|N_G(w) \cap B'| \ge 10\mu ^{'}_w - 2 ]\right) \times \Pr [v_1 \in B] \Pr [v_2 \in B \mid v_1 \in B] \\\ge & {} \left( 1 - \Pr [|N_G(w) \cap B'| \ge 2\mu ^{'}_w ]\right) \times \Pr [v_1 \in B] \Pr [v_2 \in B \mid v_1 \in B] \\\ge & {} \left( 1 - \frac{1}{2}\right) \times \Pr [v_1 \in B] \Pr [v_2 \in B \mid v_1 \in B] = \Omega \left( n^{2(b_1 - a_1)}\right) , \end{aligned}$$

where $B' \in {\mathcal {S}}(A\setminus \{v_1,v_2\}, \left\lceil |V_2|^{b_1} \right\rceil - 2)$ and $\mu ^{'}_w$ be the average of $|N_G(w) \cap B'|$ over all $B'$. The checking procedure of the second walk (and thus its cost ${\mathsf {C}}_w'$) is the same as in the dense case. From Expression (2), the query complexity of the first walk is

$$\begin{aligned} \tilde{O}\left( n^{a_1 + d/2 + k_1 - 1/2} + \sqrt{n^{2(1-a_1)}}\left( \sqrt{n^{a_1}} \times n^{d/2 + k_1 - 1/2} + {\mathsf {C}}\right) \right) . \end{aligned}$$

By replacing the quantities ${\mathsf {S}}'$ and the cost of estimating $|\Delta _G(X,A,w)|$ by the expressions we just derived, we obtain the checking cost of the first walk

$$\begin{aligned} {\mathsf {C}}= & {} \tilde{O}\left( \sqrt{|V_1|}\left( |V_1|^{k_1/2}+{\mathsf {S}}'+\frac{|V_2|^{b_1/2}\times {\mathsf {U}}'}{\sqrt{\varepsilon '}}\right) \right. \\&\left. +\,\frac{|V_2|^{b_1-a_1}}{\sqrt{\varepsilon '}}\times \sqrt{\sum _{w\in V_1} |\Delta (X,A,w)|}\right) \\= & {} \tilde{O}\left( n^{1/2 + k_1/2} + n^{1/2} \times {\mathsf {S}}' + n^{1/2 + a_1 - b_1/2} + \sqrt{\sum _{w\in V^d_l} |\Delta (X,A,w)|} \right) \\= & {} \tilde{O}\left( n^{1/2 + k_1/2} + n^{b_1 + d/2} + n^{1/2 + a_1 - b_1/2} + n^{1/2 + a_1 - k_1/2}\right) \end{aligned}$$

queries. This thus gives query complexity

$$\begin{aligned} \tilde{O}\left( n + n^{k_1/2}m^{1/2} + n^{a_1 + d/2 + k_1 - 1/2} + \sqrt{n^{2(1-a_1)}}\left( \sqrt{n^{a_1}} \times n^{d/2 + k_1 - 1/2} + {\mathsf {C}}\right) \right) \end{aligned}$$

for the whole algorithm, as claimed. $\square$

Proposition 3.3

Let $a_2$, $k_2$ and $b_2$ be any constants such that $0< b_2< a_2 < 1$ and $1< n^{k_2} < |V^{d}_h|$. A triangle of G consisting of two d-low vertices and one d-high vertex can be detected with high probability using $Q_3 = \tilde{O}(n + n^{k_2/2}m^{1/2} + n^{a_2 + d + k_2}m^{-1/2} + n^{1 + d + k_2 - a_2/2}m^{-1/2} + n^{1 + k_2/2 - a_2 - d/2}m^{1/2} + n^{1 + b_2 -a_2 - d/2}m^{1/2} + n^{1 - b_2/2 - d/2}m^{1/2}$ $+ n^{1 - d/2- k_2/2}m^{1/2})$ queries.

Proof

We again adapt the algorithm for the dense case presented in Sect. 2.2. We take $V_1 = V^{d}_h$, $V_2 = V^{d}_l$, $a = a_2$, $b = b_2$ and $X \subseteq V_1$ of size $|X|=\left\lceil 3n^{k_2}\log n \right\rceil$ (i.e., choose k such that $|V_1|^k=n^{k_2}$). The algorithm is exactly the same as the algorithm of Proposition 3.2, except that this time we cannot use the sparsity of the graph (since the vertices in $V_1$ are not d-low anymore) to reduce ${\mathsf {S}}'$. We check if there exists a triangle of G with a vertex in X and two vertices in $V_2$ using $\tilde{O}(n + \sqrt{n^{k_2}m})$ queries with high probability. By Lemma 3.1 and $|V^d_h| = \Theta (m/n^d)$, we know that $|N_G(v) \cap X| = \tilde{O}(n^{d + k_2}/|V^d_h|) = {\tilde{O}}(n^{2d + k_2}m^{-1})$. In the first walk, we use the same set up and update procedures as the algorithm of Proposition 3.2. The setup cost is thus ${\mathsf {S}} = {\tilde{O}}(n^{a_2 + d + k_2}m^{-1/2})$ queries and the update cost is ${\mathsf {U}} = {\tilde{O}}(n^{d + k_2}m^{-1/2})$ queries. From Expression (2), the query complexity of the first walk is thus

$$\begin{aligned} \tilde{O}\left( n^{a_2 + d + k_2}m^{-1/2} + \sqrt{n^{2(1-a_2)}}\left( \sqrt{n^{a_2}} \times n^{d + k_2}m^{-1/2} + {\mathsf {C}}\right) \right) . \end{aligned}$$

The analysis of ${\mathsf {C}}$ is similar to the analysis done for the algorithm of Proposition 3.2, but this time we have to use the same set up procedure as for the dense case in the second walk, i.e., we have ${\mathsf {S}}'=\left\lceil |V_2|^{b_2} \right\rceil$ as in Sect. 2.2. The checking cost of the first walk is thus

$$\begin{aligned} {\mathsf {C}}= & {} \tilde{O}\left( \sqrt{|V_1|}\left( |V_1|^{k/2}+{\mathsf {S}}'+\frac{|V_2|^{b_2/2}\times {\mathsf {U}}'}{\sqrt{\varepsilon '}}\right) +\frac{|V_2|^{b_2-a_2}}{\sqrt{\varepsilon '}}\times \sqrt{\sum _{w\in V_1} |\Delta (X,A,w)|}\right) \\= & {} \tilde{O}\left( n^{k_2/2}|V^d_h|^{1/2} + |V^d_h|^{1/2} \times {\mathsf {S}}' + n^{a_2 - b_2/2}|V^d_h|^{1/2} + \sqrt{\sum _{w\in V^d_h} |\Delta (X,A,w)|} \right) \\= & {} \tilde{O}\left( n^{k_2/2 - d/2}m^{1/2} + n^{b_2 - d/2}m^{1/2} + n^{a_2 - b_2/2 - d/2}m^{1/2} + n^{a_2 - d/2 - k_2/2}m^{1/2}\right) \end{aligned}$$

queries. This gives query complexity

$$\begin{aligned} \tilde{O}\left( n + n^{k_2/2}m^{1/2} + n^{a_2 + d + k_2}m^{-1/2} + \sqrt{n^{2(1-a_2)}}\left( \sqrt{n^{a_2}} \times n^{d + k_2}m^{-1/2} + {\mathsf {C}}\right) \right) \end{aligned}$$

for the whole algorithm, as claimed. $\square$

Proposition 3.4

Let $a_3$, $k_3$ and $b_3$ be constants such that $1< n^{b_3}< n^{a_3} < |V^{d}_h|$ and $0< k_3 < 1$. A triangle of G consisting of two d-high vertices and one d-low vertex can be detected with high probability using $Q_4 = \tilde{O}(n + n^{k_3/2}m^{1/2} + n^{a_3 + k_3} + n^{k_3 - a_3/2 - d}m + n^{1/2 + k_3/2 - a_3 - d}m + n^{b_3 - a_3 - d/2}m + n^{1/2 - b_3/2 - d}m + n^{1/2 - d - k_3/2}m )$ queries.

Proof

We adapt the algorithm for the dense case presented in Sect. 2.2 to the case we are considering. This time we choose $V_1 = V^{d}_l$, $V_2 = V^{d}_h$, $k=k_3$, take a such that $|V_2|^a=n^{a_3}$ and b such that $|V_2|^b=n^{b_3}$. The algorithm is again almost the same as the algorithm of Proposition 3.2, except that this time we cannot use the sparsity of the graph to reduce ${\mathsf {S}}$ and ${\mathsf {U}}$ since the vertices in $V_2$ are not d-low anymore (but we can use the sparsity to reduce ${\mathsf {S}}'$ exactly as in the algorithm of Proposition 3.2). We check if there exists a triangle of G with a vertex in X and two vertices in $V_2$ using $\tilde{O}(n + \sqrt{n^{k_3}m})$ queries with high probability. The setup cost is ${\mathsf {S}} = {\tilde{O}}(n^{a_3 + k_3})$ queries and the update cost is ${\mathsf {U}} = {\tilde{O}}(n^{k_3})$ queries. From Expression (2), the query complexity of the first walk is

$$\begin{aligned}&\tilde{O}\left( n^{a_3 + k_3} + \sqrt{\frac{|V^d_h|^2}{n^{2a_3}}}\left( \sqrt{n^{a_3}} \times n^{k_3} + {\mathsf {C}}\right) \right) \\&\quad = \tilde{O}\left( n^{a_3 + k_3} + \frac{m}{n^{a_3+d}}\left( \sqrt{n^{a_3}} \times n^{k_3} + {\mathsf {C}}\right) \right) . \end{aligned}$$

The analysis of ${\mathsf {C}}$ is identical to the analysis done for the algorithm of Proposition 3.2. In particular we have ${\mathsf {S}}' = {\tilde{O}}(n^{b_3 + d/2 - 1/2})$, and obtain

$$\begin{aligned} {\mathsf {C}}= & {} \tilde{O}\left( \sqrt{|V_1|}\left( |V_1|^{k_3/2}+{\mathsf {S}}'+\frac{|V_2|^{b/2}\times {\mathsf {U}}'}{\sqrt{\varepsilon '}}\right) +\frac{|V_2|^{b-a}}{\sqrt{\varepsilon '}}\times \sqrt{\sum _{w\in V_1} |\Delta (X,A,w)|}\right) \\= & {} \tilde{O}\left( n^{1/2 + k_3/2} + n^{1/2} \times {\mathsf {S}}' + n^{1/2 + a_3 - b_3/2} + \sqrt{\sum _{w\in V^d_l} |\Delta (X,A,w)|} \right) \\= & {} \tilde{O}\left( n^{1/2 + k_3/2} + n^{b_3 + d/2} + n^{1/2 + a_3 - b_3/2} + n^{1/2 + a_3 - k_3/2}\right) . \end{aligned}$$

This gives query complexity

$$\begin{aligned} \tilde{O}\left( n + n^{k_3/2}m^{1/2} + n^{a_3 + k_3} + \frac{m}{n^{a_3 + d}}\left( \sqrt{n^{a_3}} \times n^{k_3} + {\mathsf {C}}\right) \right) \end{aligned}$$

for the whole algorithm, as claimed. $\square$

Proposition 3.5

A triangle of G consisting of three d-high vertices can be detected with high probability using $Q_5 = {\tilde{O}}((m/n^d)^{5/4})$ queries.

Proof

We simply apply the original algorithm by Le Gall [12] for dense triangle finding over the subgraph of G induced by the vertices in $V^{d}_h$, and use the bound $|V_h^d|=O(m/n^d)$. $\square$

Proposition 3.6

Let $b_4$ be any constant such that $0< b_4 < 1$. A triangle of G consisting of three d-low vertices can be detected with high probability using $Q_6 = \tilde{O}(n^{b_4 + d/2} + n^{3/2 - b_4/2} + n^{1/2 + d})$ queries.

Proof

We take $V_1 = V_2 = V^{d}_l$, and adapt the algorithm for the dense case as in Proposition 3.2, but we choose $X=\emptyset$ and $a=1$, i.e., we do not perform the first step of the algorithm and do not perform the first walk in the second step. That is, we only perform the second walk of the algorithm, with parameter $b = b_4$. The sparsity of the graph can again be used to reduce ${\mathsf {S}}'$, exactly as in the algorithm of Proposition 3.2. The main difference is that we now use directly the sparsity of the graph for the checking step ${\mathsf {C}}'$: instead of performing a Grover search over $\Delta _G(X,B,w)$, as in the dense case, we simply do a Grover search over ${\mathcal {E}}(N(w)\cap B)$, at cost

$$\begin{aligned} {\tilde{O}}(\sqrt{|{\mathcal {E}}(N(w)\cap B)|})={\tilde{O}}(10\times \mu _w), \end{aligned}$$

which gives a new upper bound ${\mathsf {C}}'_w$. Replacing in the analysis of Sect. 2.2 the quantities ${\mathsf {S}}'$ and ${\mathsf {C}}'_w$ by these upper bounds, we obtain query complexity

$$\begin{aligned} Q_6= & {} \tilde{O}\left( \sqrt{|V_1|}\left( {\mathsf {S}}' + \sqrt{\frac{1}{\varepsilon '}}\left( |V_2|^{b_4/2} \times {\mathsf {U}}' + {\mathsf {C}}'_w\right) \right) \right) \\= & {} \tilde{O}(n^{1/2} \times {\mathsf {S}}' + n^{3/2 - b_4/2} + n^{3/2 - b_4} \times {\mathsf {C}}'_w ) \\= & {} {\tilde{O}}(n^{b_4 + d/2} + n^{3/2 - b_4/2} + n^{3/2 - b_4} \times \mu _w) = {\tilde{O}}(n^{b_4 + d/2} + n^{3/2 - b_4/2} + n^{1/2 + d}), \end{aligned}$$

as claimed. $\square$

Proposition 3.7

A triangle consisting of at least one d-high vertex can be detected with high probability using $Q_7 = O(n + n^{-d/2}m)$ queries.

Proof

We use an algorithm similar to the procedure described in the first part of the proof of Proposition 3.2, based on [6]. We take a random edge $\{u, v\} \in E$ and then try to find a vertex w from $V_h^d$ such that $\{u,v,w\}$ is a triangle of G. This can be implemented using two Grover searches in ${\tilde{O}}(\sqrt{n^2/m}+\sqrt{|V_h^d|})$ queries, and that in the worst case (i.e., when there is only one triangle) the success probability of this approach is $\Theta (1/m)$. Using amplitude amplification we can then check with high probability the existence of such a triangle with total query complexity

$$\begin{aligned} \tilde{O}\left( \sqrt{m}\times \left( \sqrt{n^2/m}+\sqrt{|V_h^d|}\right) \right) = \tilde{O}\left( n + \frac{m}{\sqrt{n^d}}\right) , \end{aligned}$$

since $|V_h^d|=O(m/n^d)$. $\square$

We are now ready to prove Theorem 1.

Proof of Theorem 1

Propositions 3.1–3.7 can be combined in several ways to design an algorithm detecting if the graph has a triangle. We found that for $n^{13/8}\le m\le n^2$ the best combination is Propositions 3.1–3.5 (giving cost $Q_1+Q_2+Q_3+Q_4+Q_5$), for $n^{3/2}\le m\le n^{13/8}$ the best combination is Propositions 3.1 and 3.3–3.6 (giving cost $Q_1+Q_3+Q_4+Q_5+Q_6$), for $n^{7/6}\le m\le n^{3/2}$ the best combination is Propositions 3.1, 3.6 and 3.7 (giving cost $Q_1+Q_6+Q_7$), while for $n\le m\le n^{7/6}$ the best strategy is to use the original algorithm by Buhrman et al. [6] alone.

Let us give an intuitive explanation of why these choices are optimal. Buhrman et al.’s algorithm works very well for sparse graphs; it is not surprising that this algorithm is the best choice for small values of m, either alone (for $n\le m\le n^{7/6}$) or combined with the algorithms of Propositions 3.1 and 3.6 (for $n^{7/6}\le m\le n^{3/2}$). For larger values of m it is better to use the algorithms of either Proposition 3.6 (for $n^{3/2}\le m\le n^{13/8}$) or Proposition 3.2 (for $n^{13/8}\le m\le n^{2}$). The former algorithm is better than the latter when $m\le n^{13/8}$ since in the latter algorithm obtaining the relevant information about $\Delta _G(X,A,w)$ and implementing the checking step of the quantum walk has reasonable cost only when the graph is very dense. We write $m = n^{\ell }$, for $0 \le \ell \le 2$, and optimize below the parameters.

If $\frac{7}{6} \le \ell \le \frac{7}{5}$, the query complexity is upper bounded by

$$\begin{aligned} Q_1+ Q_6 + Q_7={\tilde{O}}(n^{1 + {\ell /2} -d} + n^{{3/2} - {b_4/2}} + n^{b_4 + d/2}), \end{aligned}$$

which is optimized by taking $b_4 = 1 - \frac{\ell }{7}$ and $d = \frac{3\ell }{7}$, giving the upper bound ${\tilde{O}}(n^{1 + \ell /14})$.

If $\frac{7}{5} \le \ell \le \frac{3}{2}$, the query complexity is upper bounded by

$$\begin{aligned} Q_1+ Q_6 + Q_7={\tilde{O}}(n^{3/2 - b_4/2} + n^{1/2 + d} + n^{\ell - d/2}), \end{aligned}$$

which is optimized by taking $b_4 = \frac{8}{3} - \frac{4\ell }{3}$ and $d = \frac{2\ell }{3} - \frac{1}{3}$, giving the upper bound ${\tilde{O}}(n^{1/6 + 2\ell /3})$.

If $\frac{3}{2} \le \ell \le \frac{13}{8}$, the query complexity is upper bounded by

$$\begin{aligned} Q_1 + Q_3 + Q_4 + Q_5 + Q_6= & {} {\tilde{O}}(n^{3/2 - b_4/2} + n^{1/2 + d} + n^{a_2 + d + k_2 - \ell /2} \\&\quad +\, n^{1 + b_2 + \ell /2 - a_2 - d/2} + n^{1 + \ell /2 - b_2/2 - d/2} \\&\quad +\,n^{1 + \ell /2 - d/2 - k_2/2} + n^{b_3 + \ell - a_3 - d/2}\\&\quad +\, n^{1/2 + \ell - b_3/2 - d} + n^{1/2 +\ell - d - k_3/2}), \end{aligned}$$

which is optimized by taking $a_2 = \frac{3}{10} + \frac{3\ell }{10}$, $a_3 = \frac{23\ell }{15} - \frac{59}{30}$, $b_2 = k_2 = \frac{1}{5} + \frac{\ell }{5}$, $b_3 = k_3 = \frac{14\ell }{15} - \frac{16}{15}$, $b_4 = \frac{22}{15} - \frac{8\ell }{15}$ and $d = \frac{4}{15} + \frac{4\ell }{15}$, giving the upper bound ${\tilde{O}}(n^{23/30 + 4\ell /15})$.

If $\frac{13}{8} \le \ell \le 2$, the query complexity is upper bounded by

$$\begin{aligned} Q_1 + Q_2 + Q_3 + Q_4 + Q_5= & {} {\tilde{O}}(n^{a_1 + {d/2} + k_1 - {1/2}} + n^{1 + b_1 + {d/2} - a_1} \\&\quad +\, n^{{3/2} - {b_1/2}} + n^{{3/2} - {k_1/2}} \\&\quad +\, n^{a_2 + d + k_2 - {\ell /2}} + n^{1 + b_2 + {\ell /2} - a_2 - {d/2}} \\&\quad +\, n^{1 + {\ell /2} - {b_2/2} - {d/2}} + n^{1 + {\ell /2} - {d/2} - {k_2/2}} \\&\quad +\, n^{b_3 + \ell - a_3 - {d/2}} + n^{{1/2} + \ell - {b_3/2} - d} + n^{{1/2} + \ell - d - {k_3/2}} ), \end{aligned}$$

which is optimized by taking $a_1 = \frac{3}{4}$, $a_2 = \frac{19}{20} - \frac{\ell }{10}$, $a_3 = \frac{3\ell }{5} - \frac{9}{20}$, $b_1 = k_1 = \frac{31}{30} - \frac{4\ell }{15}$, $b_2 = k_2 = \frac{19}{30} - \frac{\ell }{15}$, $b_3 = k_3 = \frac{7}{30} + \frac{2\ell }{15}$ and $d = \frac{4\ell }{5} - \frac{3}{5}$, giving the upper bound ${\tilde{O}}(n^{59/60 + 2\ell /15})$.

For the case $\ell \le \frac{7}{6}$ we can obtain a better upper bound by using the $O(n+\sqrt{nm})$-query algorithm by Buhrman et al. [6]. This upper bound actually corresponds to a degenerate case appearing in our approach: the case $d=0$. Indeed, observe that without loss of generality we can assume that $\deg (v)\ge 1$ for all vertices $v\in V$ (for instance by adding dummy vertices to the graph). In this case we have ${\mathcal {V}}^{d}_h=V$ for $d=0$, which means that we do not need to apply the algorithm of Proposition 3.1 in order to obtain a classification: we simply output $V^{0}_h=V$ and $V^0_l=\emptyset$ (i.e., all the vertices of the graph are 0-high). The only type of triangles we need to consider is triangles with three 0-high vertices, which can be found with complexity $O(n+\sqrt{nm})$ by Proposition 3.7 (in this case the algorithm of Proposition 3.7 is exactly the same as the algorithm in [6]). $\square$