We show that the quantum query complexity of detecting if an $n$ -vertex graph contains a triangle is $O(n^{9/7})$ . This improves the previous best algorithm of Belovs making $O(n^{35/27})$ queries. For the problem of determining if an operation $\circ : S \times S \rightarrow S$ is associative, we give an algorithm making $O(|S|^{10/7})$ queries, the first improvement to the trivial $O(|S|^{3/2})$ application of Grover search. Our algorithms are designed using the learning graph framework of Belovs. We give a family of algorithms for detecting constant-sized subgraphs, which can possibly be directed and colored. These algorithms are designed in a simple high-level language; our main theorem shows how this high-level language can be compiled as a learning graph and gives the resulting complexity. The key idea to our improvements is to allow more freedom in the parameters of the database kept by the algorithm. As in our previous work, the edge slots maintained in the database are specified by a graph whose edges are the union of regular bipartite graphs, the overall structure of which mimics that of the graph of the certificate. By allowing these bipartite graphs to be unbalanced and of variable degree we obtain better algorithms.We show that the quantum query complexity of detecting if an $n$-vertex graph contains a triangle is $O(n^{9/7})$. This improves the previous best algorithm of Belovs making $O(n^{35/27})$ queries. For the problem of determining if an operation $\circ : S \times S \rightarrow S$ is associative, we give an algorithm making $O(|S|^{10/7})$ queries, the first improvement to the trivial $O(|S|^{3/2})$ application of Grover search. Our algorithms are designed using the learning graph framework of Belovs. We give a family of algorithms for detecting constant-sized subgraphs, which can possibly be directed and colored. These algorithms are designed in a simple high-level language; our main theorem shows how this high-level language can be compiled as a learning graph and gives the resulting complexity. The key idea to our improvements is to allow more freedom in the parameters of the database kept by the algorithm. As in our previous work, the edge slots maintained in the database are specified by a graph whose edges are the union of regular bipartite graphs, the overall structure of which mimics that of the graph of the certificate. By allowing these bipartite graphs to be unbalanced and of variable degree we obtain better algorithms.

arXiv:1210.1014Oct 3, 20120 citations

Quantum Algorithm for Triangle Finding in Sparse Graphs

François Le Gall, Shogo Nakajima

Lecture notes in computer scienceJan 1, 20152 citations

Quantum Algorithm for Triangle Finding in Sparse Graphs

Le Francois, Shogo Nakajima

IEICE Technical Report; IEICE Tech. Rep.Sep 25, 20150 citations

Improved Quantum Query Algorithms for Triangle Finding and Associativity Testing

Troy Lee, Frédéric Magniez, Miklós Sántha

We show that the quantum query complexity of detecting if an n-vertex graph contains a triangle is O(n9/7). This improves the previous best algorithm of Belovs [2] making O(n35/27) queries. For the problem of determining if an operation o : S × S → S is associative, we give an algorithm making O(|S|10/7) queries, the first improvement to the trivial O(|S|3/2) application of Grover search. Our algorithms are designed using the learning graph framework of Belovs. We give a family of algorithms for detecting constant-sized subgraphs, which can possibly be directed and colored. These algorithms are designed in a simple high-level language; our main theorem shows how this high-level language can be compiled as a learning graph and gives the resulting complexity. The key idea to our improvements is to allow more freedom in the parameters of the database kept by the algorithm. As in our previous work [9], the edge slots maintained in the database are specified by a graph whose edges are the union of regular bipartite graphs, the overall structure of which mimics that of the graph of the certificate. By allowing these bipartite graphs to be unbalanced and of variable degree we obtain better algorithms.We show that the quantum query complexity of detecting if an n-vertex graph contains a triangle is O(n9/7). This improves the previous best algorithm of Belovs [2] making O(n35/27) queries. For the problem of determining if an operation o : S × S → S is associative, we give an algorithm making O(|S|10/7) queries, the first improvement to the trivial O(|S|3/2) application of Grover search. Our algorithms are designed using the learning graph framework of Belovs. We give a family of algorithms for detecting constant-sized subgraphs, which can possibly be directed and colored. These algorithms are designed in a simple high-level language; our main theorem shows how this high-level language can be compiled as a learning graph and gives the resulting complexity. The key idea to our improvements is to allow more freedom in the parameters of the database kept by the algorithm. As in our previous work [9], the edge slots maintained in the database are specified by a graph whose edges are the union of regular bipartite graphs, the overall structure of which mimics that of the graph of the certificate. By allowing these bipartite graphs to be unbalanced and of variable degree we obtain better algorithms.

ACM-SIAM Symposium on Discrete Algorithms (SODA)Jan 6, 201333 citations

Quantum Algorithm for Triangle Finding in Sparse Graphs

François Le Gall, Shogo Nakajima

arXiv:1507.06878AlgorithmicaJan 5, 20179 citations

We show that the quantum query complexity of detecting if an

n

-vertex graph contains a triangle is

O(n^{9/7})

. This improves the previous best algorithm of Belovs making

O(n^{35/27})

queries. For the problem of determining if an operation

\circ : S \times S \rightarrow S

is associative, we give an algorithm making

O(|S|^{10/7})

queries, the first improvement to the trivial

O(|S|^{3/2})

application of Grover search. Our algorithms are designed using the learning graph framework of Belovs. We give a family of algorithms for detecting constant-sized subgraphs, which can possibly be directed and colored. These algorithms are designed in a simple high-level language; our main theorem shows how this high-level language can be compiled as a learning graph and gives the resulting complexity. The key idea to our improvements is to allow more freedom in the parameters of the database kept by the algorithm. As in our previous work, the edge slots maintained in the database are specified by a graph whose edges are the union of regular bipartite graphs, the overall structure of which mimics that of the graph of the certificate. By allowing these bipartite graphs to be unbalanced and of variable degree we obtain better algorithms.