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arXiv:1609.07786

Extended Learning Graphs for Triangle Finding

Titouan Carette, Mathieu Laurière, Frédéric Magniez

Published Sep 25, 2016• 1 citation
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Abstract

We present new quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and spare instances.For dense graphs on $n$ vertices, we get a query complexity of $O(n^{5/4})$ without any of the extra logarithmic factors present in the previous algorithm of Le Gall [FOCS'14]. For sparse graphs with $m\geq n^{5/4}$ edges, we get a query complexity of $O(n^{11/12}m^{1/6}\sqrt{\log n})$, which is better than the one obtained by Le Gall and Nakajima [ISAAC'15] when $m \geq n^{3/2}$. We also obtain an algorithm with query complexity ${O}(n^{5/6}(m\log n)^{1/6}+d_2\sqrt{n})$ where $d_2$ is the variance of the degree distribution. Our algorithms are designed and analyzed in a new model of learning graphs that we call extended learning graphs. In addition, we present a framework in order to easily combine and analyze them. As a consequence we get much simpler algorithms and analyses than previous algorithms of Le Gall {\it et al} based on the MNRS quantum walk framework [SICOMP'11].

DOI
10.48550/arxiv.1609.07786

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We present new quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and spare instances.For dense graphs on nnn vertices, we get a query complexity of O(n5/4)O(n^{5/4})O(n5/4) without any of the extra logarithmic factors present in the previous algorithm of Le Gall [FOCS'14]. For sparse graphs with m≥n5/4m\geq n^{5/4}m≥n5/4 edges, we get a query complexity of O(n11/12m1/6log⁡n)O(n^{11/12}m^{1/6}\sqrt{\log n})O(n11/12m1/6logn​), which is better than the one obtained by Le Gall and Nakajima [ISAAC'15] when m≥n3/2m \geq n^{3/2}m≥n3/2. We also obtain an algorithm with query complexity O(n5/6(mlog⁡n)1/6+d2n){O}(n^{5/6}(m\log n)^{1/6}+d_2\sqrt{n})O(n5/6(mlogn)1/6+d2​n​) where d2d_2d2​ is the variance of the degree distribution. Our algorithms are designed and analyzed in a new model of learning graphs that we call extended learning graphs. In addition, we present a framework in order to easily combine and analyze them. As a consequence we get much simpler algorithms and analyses than previous algorithms of Le Gall {\it et al} based on the MNRS quantum walk framework [SICOMP'11].
We present new quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and sparse instances. For dense graphs on n vertices, we get a query complexity of O(n^(5/4)) without any of the extra logarithmic factors present in the previous algorithm of Le Gall [FOCS'14]. For sparse graphs with m >= n^(5/4) edges, we get a query complexity of O(n^(11/12) m^(1/6) sqrt(log n)), which is better than the one obtained by Le Gall and Nakajima [ISAAC'15] when m >= n^(3/2). We also obtain an algorithm with query complexity O(n^(5/6) (m log n)^(1/6) + d_2 sqrt(n)) where d_2 is the variance of the degree distribution. Our algorithms are designed and analyzed in a new model of learning graphs that we call extended learning graphs. In addition, we present a framework in order to easily combine and analyze them. As a consequence we get much simpler algorithms and analyses than previous algorithms of Le Gall based on the MNRS quantum walk framework [SICOMP'11].
We study the quantum query algorithms for simplex finding, a generalization of triangle finding to hypergraphs. This problem satisfies a rank-reduction property: a quantum query algorithm for finding simplices in rank-rrr hypergraphs can be turned into a faster algorithm for finding simplices in rank-(r−1)(r-1)(r−1) hypergraphs. We then show that every nested Johnson graph quantum walk (with any constant number of nested levels) can be converted into an adaptive learning graph. Then, we introduce the concept of ααα-symmetric learning graphs, which is a useful framework for designing and analyzing complex quantum search algorithms. Inspired by the work of Le Gall, Nishimura, and Tani (2016) on 333-simplex finding, we use our new technique to obtain an algorithm for 444-simplex finding in rank-444 hypergraphs with O(n2.46)O(n^{2.46})O(n2.46) quantum query cost, improving the trivial O(n2.5)O(n^{2.5})O(n2.5) algorithm.
In this paper we present a quantum algorithm solving the triangle finding problem in unweighted graphs with query complexity Õ(n5/4), where n denotes the number of vertices in the graph. This improves the previous upper bound O(n9/7) = O(n1.285) recently obtained by Lee, Magniez and Santha. Our result shows, for the first time, that in the quantum query complexity setting unweighted triangle finding is easier than its edge-weighted version, since for finding an edge-weighted triangle Belovs and Rosmanis proved that any quantum algorithm requires O(n9/7/ √log n) queries. Our result also illustrates some limitations of the non-adaptive learning graph approach used to obtain the previous O(n9/7) upper bound since, even over unweighted graphs, any quantum algorithm for triangle finding obtained using this approach requires v(n9/7/ √log n) queries as well. To bypass the obstacles characterized by these lower bounds, our quantum algorithm uses combinatorial ideas exploiting the graph-theoretic properties of triangle finding, which cannot be used when considering edge-weighted graphs or the non-adaptive learning graph approach.
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 nnn-vertex graph contains a triangle is O(n9/7)O(n^{9/7})O(n9/7). This improves the previous best algorithm of Belovs making O(n35/27)O(n^{35/27})O(n35/27) queries. For the problem of determining if an operation ∘:S×S→S\circ : S \times S \rightarrow S∘:S×S→S is associative, we give an algorithm making O(∣S∣10/7)O(|S|^{10/7})O(∣S∣10/7) queries, the first improvement to the trivial O(∣S∣3/2)O(|S|^{3/2})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.
The edge list model is arguably the simplest input model for graphs, where the graph is specified by a list of its edges. In this model, we study the quantum query complexity of three variants of the triangle finding problem. The first asks whether there exists a triangle containing a target edge and raises general questions about the hiding of a problem's input among irrelevant data. The second asks whether there exists a triangle containing a target vertex and raises general questions about the shuffling of a problem's input. The third asks whether there exists a triangle; this problem bridges the 333-distinctness and 333-sum problems, which have been extensively studied by both cryptographers and complexity theorists. We provide tight or nearly tight results for these problems as well as some first answers to the general questions they raise. Furthermore, given any graph with low maximum degree, such as a typical random sparse graph, we prove that the quantum query complexity of finding a length-kkk cycle in its length-mmm edge list is m3/4−1/(2k+2−4)±o(1)m^{3/4-1/(2^{k+2}-4)\pm o(1)}m3/4−1/(2k+2−4)±o(1), which matches the best-known upper bound for the quantum query complexity of kkk-distinctness on length-mmm inputs up to an mo(1)m^{o(1)}mo(1) factor. We prove the lower bound by developing new techniques within Zhandry's recording query framework [CRYPTO '19] as generalized by Hamoudi and Magniez [ToCT '23]. These techniques extend the framework to treat any non-product distribution that results from conditioning a product distribution on the absence of rare events. We prove the upper bound by adapting Belovs's learning graph algorithm for kkk-distinctness [FOCS '12]. Finally, assuming a plausible conjecture concerning only cycle finding, we show that the lower bound can be lifted to an essentially tight lower bound on the quantum query complexity of kkk-distinctness, which is a long-standing open question.