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arXiv:2206.02766 · SPIN

Complexity of Eccentricities and All-Pairs Shortest Paths in the Quantum CONGEST Model

Changsheng Wang, Xudong Wu, Penghui Yao

Published Aug 23, 2021• 4 citations
arXivHTMLPDF

Abstract

Computing the distance parameters of a network, including the diameter, radius, eccentricities and the all-pairs shortest paths (APSP) is a central problem in distributed computing. This paper investigates the distance parameters in the quantum CONGEST models and establishes almost linear lower bounds on eccentricities and APSP, which match the classical upper bounds. Our results imply that there is not quantum speedup for these two problems. In contrast with the diameter and radius, exchanging quantum messages is able to save the communication when the networks have low diameters [F. L. Gall and F. Magniez, Sublinear-time quantum computation of the diameter in CONGEST networks, in Proc. 2018 ACM Symp. Principles of Distributed Computing (PODC), (2018), pp. 337–346; F. Magniez and A. Nayak, arXiv:2002.11795 ]. We obtain the lower bounds via a reduction from the two-way quantum communication complexity of the set intersection [A. A. Razborov, Izv. Math. 159 (2003)].

DOI
10.1142/s2010324721400075

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Computing the distance parameters of a network, including the diameter, radius, eccentricities and the all-pairs shortest paths (APSP) is a central problem in distributed computing. This paper investigates the distance parameters in the quantum CONGEST models and establishes almost linear lower bounds on eccentricities and APSP, which match the classical upper bounds. Our results imply that there is not quantum speedup for these two problems. In contrast with the diameter and radius, exchanging quantum messages is able to save the communication when the networks have low diameters [F. L. Gall and F. Magniez, Sublinear-time quantum computation of the diameter in CONGEST networks, in Proc. 2018 ACM Symp. Principles of Distributed Computing (PODC), (2018), pp. 337–346; F. Magniez and A. Nayak, arXiv:2002.11795 ]. We obtain the lower bounds via a reduction from the two-way quantum communication complexity of the set intersection [A. A. Razborov, Izv. Math. 159 (2003)].
The All-Pairs Shortest Path problem (APSP) is one of the most central problems in distributed computation. In the CONGEST-CLIQUE model, in which n nodes communicate with each other over a fully connected network by exchanging messages of O(łog n) bits in synchronous rounds, the best known general algorithm for APSP uses Õ(n1/3) rounds. Breaking this barrier is a fundamental challenge in distributed graph algorithms. In this paper we investigate for the first time quantum distributed algorithms in the CONGEST-CLIQUE model, where nodes can exchange messages of O(log n) quantum bits, and show that this barrier can be broken: we construct a Õ(n1/4)-round quantum distributed algorithm for the APSP over directed graphs with polynomial weights in the CONGEST-CLIQUE model. This speedup in the quantum setting contrasts with the case of the standard CONGEST model, for which Elkin et al. (PODC 2014) showed that quantum communication does not offer significant advantages over classical communication.
The computation of the diameter is one of the most central problems in distributed computation. In the standard CONGEST model, in which two adjacent nodes can exchange O(log n) bits per round (here n denotes the number of nodes of the network), it is known that exact computation of the diameter requires Ω(n) rounds, even in networks with constant diameter. In this paper we investigate quantum distributed algorithms for this problem in the quantum CONGEST model, where two adjacent nodes can exchange O(log n) quantum bits per round. Our main result is a O(√D )-round quantum distributed algorithm for exact diameter computation, where D denotes the diameter. This shows a separation between the computational power of quantum and classical algorithms in the CONGEST model. We also show an unconditional lower bound Ω(√ ) on the round complexity of any quantum algorithm computing the diameter, and furthermore show a tight lower bound Ω(√D ) for any distributed quantum algorithm in which each node can use only poly(log n) quantum bits of memory.
The computation of the diameter is one of the most central problems in distributed computation. In the standard CONGEST model, in which two adjacent nodes can exchange O(log n) bits per round (here n denotes the number of nodes of the network), it is known that exact computation of the diameter requires Ω(n) rounds, even in networks with constant diameter. In this paper we investigate quantum distributed algorithms for this problem in the quantum CONGEST model, where two adjacent nodes can exchange O(log n) quantum bits per round. Our main result is a O(√D )-round quantum distributed algorithm for exact diameter computation, where D denotes the diameter. This shows a separation between the computational power of quantum and classical algorithms in the CONGEST model. We also show an unconditional lower bound Ω(√ ) on the round complexity of any quantum algorithm computing the diameter, and furthermore show a tight lower bound Ω(√D ) for any distributed quantum algorithm in which each node can use only poly(log n) quantum bits of memory.
Given x,y ∈ {0,1}ⁿ, Set Disjointness consists in deciding whether x_i = y_i = 1 for some index i ∈ [n]. We study the problem of computing this function in a distributed computing scenario in which the inputs x and y are given to the processors at the two extremities of a path of length d. Each vertex of the path has a quantum processor that can communicate with each of its neighbours by exchanging O(log n) qubits per round. We are interested in the number of rounds required for computing Set Disjointness with constant probability bounded away from 1/2. We call this problem "Set Disjointness on a Line". Set Disjointness on a Line was introduced by Le Gall and Magniez [Le Gall and Magniez, 2018] for proving lower bounds on the quantum distributed complexity of computing the diameter of an arbitrary network in the CONGEST model. However, they were only able to provide a lower bound when the local memory used by the processors on the intermediate vertices of the path is severely limited. More precisely, their bound applies only when the local memory of each intermediate processor consists of O(log n) qubits. In this work, we prove an unconditional lower bound of Ω̃(∛{n d²} + √n) rounds for Set Disjointness on a Line with d + 1 processors. This is the first non-trivial lower bound when there is no restriction on the memory used by the processors. The result gives us a new lower bound of Ω̃ (∛{nδ²} + √n) on the number of rounds required for computing the diameter δ of any n-node network with quantum messages of size O(log n) in the CONGEST model. We draw a connection between the distributed computing scenario above and a new model of query complexity. In this model, an algorithm computing a bi-variate function f (such as Set Disjointness) has access to the inputs x and y through two separate oracles 𝒪_x and 𝒪_y, respectively. The restriction is that the algorithm is required to alternately make d queries to 𝒪_x and d queries to 𝒪_y, with input-independent computation in between queries. The model reflects a "switching delay" of d queries between a "round" of queries to x and the following "round" of queries to y. The technique we use for deriving the round lower bound for Set Disjointness on a Line also applies to this query model. We provide an algorithm for Set Disjointness in this query model with query complexity that matches the round lower bound stated above, up to a polylogarithmic factor. In this sense, the round lower bound we show for Set Disjointness on a Line is optimal.
The problems of computing eccentricity, radius, and diameter are fundamental to graph theory. These parameters are intrinsically defined based on the distance metric of the graph. In this work, we propose quantum algorithms for the diameter and radius of undirected, weighted graphs in the adjacency list model. The algorithms output diameter and radius with the corresponding paths in O~(nm)\widetilde{O}(n\sqrt{m})O(nm​) time. Additionally, for the diameter, we present a quantum algorithm that approximates the diameter within a 2/32/32/3 ratio in O~(mn3/4)\widetilde{O}(\sqrt{m}n^{3/4})O(m​n3/4) time. We also establish quantum query lower bounds of Ω(nm)Ω(\sqrt{nm})Ω(nm​) for all the aforementioned problems through a reduction from the minima finding problem.
There are two central models considered in (fault-free synchronous) distributed computing: the CONGEST model, in which communication channels have limited bandwidth, and the LOCAL model, in which communication channels have unlimited bandwidth. Very recently, Le Gall and Magniez (PODC 2018) showed the superiority of quantum distributed computing over classical distributed computing in the CONGEST model. In this work we show the superiority of quantum distributed computing in the LOCAL model: we exhibit two computational tasks that can be solved in a constant number of rounds in the quantum setting but require Omega(n) rounds in the classical (randomized) setting, where n denotes the size of the network.