The divide-and-swap cube: a new hypercube variant with small network cost

• Published in: The Journal of supercomputing Vol. 75; no. 7; pp. 3621 - 3639
• Main Authors: Kim, Jong-Seok, Kim, Donghyun, Qiu, Ke, Lee, Hyeong-Ok
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2019; Springer Nature B.V
• Subjects: Algorithms; Asymptotic properties; Broadcasting; Compilers; Computer Science; Hypercubes; Interpreters; Processor Architectures; Programming Languages; Routing; Structural hierarchy; Upper bounds; VLSI; Interconnection network; Folded divide-and-swap cube; Hamiltonian cycle; Broadcasting; Divide-and-swap cube; Routing; Bisection width; Network cost; Diameter
• ISSN: 0920-8542; 1573-0484
• DOI: 10.1007/s11227-018-2712-z

The hypercube is one of the most popular interconnection networks. Its network cost is O ( n 2 ) . In this paper, we propose a new hypercube variant, the divide-and-swap cube DSC ( n ) ( n = 2 d , d ≥ 1 ) , which reduces the network cost to O ( n log n ) while maintaining the same number of nodes and the same asymptotic performances for fundamental algorithms such as the broadcasting. The new network has nice hierarchical properties. We first show that the diameter of DSC ( n ) is lower than or equal to 5 n 4 - 1 . However, unlike the hypercube of dimension n whose degree is n , the node degree of the network is log n + 1 , resulting in a network cost of O ( n log n ) . We then examine the one-to-all and all-to-all broadcasting times of DSC ( n ) , based on the single-link-available and multiple-link-available models. We also present an upper bound on the bisection width of the DSC ( n ) and show that DSC ( n ) is Hamiltonian. Finally, we introduce the folded divide-and-swap cube, FDSC ( n ) , a variant of the DSC ( n ) and study its many properties including its hierarchical structure, routing algorithm, broadcasting algorithms, bisection width, and its Hamiltonicity. All the broadcasting algorithms presented in this paper are asymptotically optimal.