This paper studies the round complexity of computing the weighted diameter and radius of a graph in the quantum CONGEST model. We present a quantum algorithm that $(1+o(1))$-approximates the diameter and radius with round complexity $\widetilde O\left(\min\left\{n^{9/10}D^{3/10},n\right\}\right)$, where $D$ denotes the unweighted diameter. This exhibits the advantages of quantum communication over classical communication since computing a $(3/2-\varepsilon)$-approximation of the diameter and radius in a classical CONGEST network takes $\widetilde\Omega(n)$ rounds, even if $D$ is constant [Abboud, Censor-Hillel, and Khoury, DISC '16]. We also prove a lower bound of $\widetilde\Omega(n^{2/3})$ for $(3/2-\varepsilon)$-approximating the weighted diameter/radius in quantum CONGEST networks, even if $D=\Theta(\log n)$. Thus, in quantum CONGEST networks, computing weighted diameter and weighted radius of graphs with small $D$ is strictly harder than unweighted ones due to Le Gall and Magniez's $\widetilde O\left(\sqrt{nD}\right)$-round algorithm for unweighted diameter/radius [PODC '18].
翻译:本文研究了计算量子 CONEST 模型中图的加权直径和半径的圆形复杂性。 我们展示了一种量子算法, 美元(1+1) 美元接近直径和半径, 与圆形复杂 $Unitilde Oleft( min\\ left\\ n\\\ 9/ 10} D\\\ 3/ 3/ 10}, n\\\ right\ right} 美元代表未加权直径和半径。 这显示了量子通信相对于经典通信的优势, 因为计算CONEST 古典网络中直径和半径的美元( 3/\ \ varepsilon), $( +1++o) 美元( 美元) 接近直径直径和半径, 即使$Dobledou, 直径和K 'nqral'ral'lexxx, 我们证明量子通信的重量直径/ 美元( 美元) 直径直径为CON- 直径, 直径网络, 直径为C- 直径, 和直径直径直径为C- deqal'直径, 直径, lex 直径网络, 直达, 直径为 Nxx。