The Journal of Supercomputing

, Volume 74, Issue 10, pp 5539–5557

# A stochastic link-fault-tolerant routing algorithm in folded hypercubes

• Bui Thi Thuan
• Lam Boi Ngoc
• Keiichi Kaneko
Article

## Abstract

A folded hypercube is obtained by adding complementary links to a hypercube. The diameter of the folded hypercube is almost half of that of the hypercube, while its degree is larger than the degree of the hypercube by only one. In this paper, we propose a stochastic link-fault-tolerant routing algorithm in a folded hypercube by introducing a limited global information called routing probabilities. For an n-dimensional folded hypercube, we have proved that the routing probabilities for all distances can be calculated in $$O(n^2\log n)$$ time at each node and the message can be forwarded to its neighbor node at each node in O(n) time. We also conducted a computer experiment, the results of which show that our algorithm achieves a better performance than the best algorithm for a hypercube.

## Keywords

Interconnection network Dependable computing Fault tolerance Routing probability

## List of symbols

n

Dimension of a folded hypercube

$$\hbox {FQ}_n$$

n-Dimensional folded hypercube

$$\varvec{a}$$, $$\varvec{b}$$, ...

Nodes

$$\varvec{a}^\mathrm{c}$$, $$\varvec{a}^{(i)}$$ ($$1\le i\le n$$)

For a node $$\varvec{a}=(a_1,a_2,\ldots ,a_n)$$, $$\varvec{a}^\mathrm{c}=(\bar{a}_1,\bar{a}_2,\ldots ,\bar{a}_n)$$ and $$\varvec{a}^{(i)}=(a_1,a_2,\ldots ,a_{i-1},\bar{a}_i,a_{i+1},\ldots ,a_n)$$ where $$\bar{a}_i=1-a_i$$

$$(\varvec{a},\varvec{b})$$

Link between two adjacent nodes $$\varvec{a}$$ and $$\varvec{b}$$

$$(\varvec{a},\varvec{a}^\mathrm{c})$$

$$H(\varvec{a},\varvec{b})$$

Hamming distance between two nodes $$\varvec{a}$$ and $$\varvec{b}$$

$$d(\varvec{a},\varvec{b})$$

Distance between two nodes $$\varvec{a}$$ and $$\varvec{b}$$

$$N(\varvec{a})$$

Set of neighbor nodes of $$\varvec{a}$$

$$C(\varvec{a})$$

Singleton set of the node $$\varvec{a}^\mathrm{c}$$

$$N_0(\varvec{a},\varvec{b})$$

Subset of $$N(\varvec{a}){\setminus } C(\varvec{a})$$ such that Hamming distance between each node in $$N_0(\varvec{a},\varvec{b})$$ and $$\varvec{b}$$ is less than that between $$\varvec{a}$$ and $$\varvec{b}$$

$$N_1(\varvec{a},\varvec{b})$$

Subset of $$N(\varvec{a}){\setminus } C(\varvec{a})$$ such that Hamming distance between each node in $$N_1(\varvec{a},\varvec{b})$$ and $$\varvec{b}$$ is more than that between $$\varvec{a}$$ and $$\varvec{b}$$

$$Pre(\varvec{a},\varvec{b})$$

Subset of $$N(\varvec{a})$$, the elements of which are on the shortest paths from $$\varvec{a}$$ to $$\varvec{b}$$

$$Spr(\varvec{a},\varvec{b})$$

Subset of $$N(\varvec{a})$$, the elements of which are on the detour paths from $$\varvec{a}$$ to $$\varvec{b}$$

$$\gamma (\varvec{a},\varvec{b})$$

Function that expresses the status of a link $$(\varvec{a},\varvec{b})$$. If the link is faulty, $$\gamma (\varvec{a},\varvec{b})=0$$. Otherwise, $$\gamma (\varvec{a},\varvec{b})=1$$

$$P_h(\varvec{a})$$

Routing probability of a node $$\varvec{a}$$ with respect to the Hamming distance h

$$n\atopwithdelims ()h$$

Number of combinations of h elements from a set of n elements

$$p^\mathrm{c}$$

$$\gamma (\varvec{a},\varvec{a}^\mathrm{c})P_{n-h}(\varvec{a}^\mathrm{c})$$

$$p_1<p_2<\cdots <p_n$$

Obtained by sorting $$\gamma (\varvec{a},\varvec{a}^{(i)}) P_{h-1}(\varvec{a}^{(i)})$$ ($$1\le i\le n$$) in ascending order

$$q_1<q_2<\cdots <q_n$$

Obtained by sorting $$\gamma (\varvec{a},\varvec{a}^{(i)}) P_{h+1}(\varvec{a}^{(i)})$$ ($$1\le i\le n$$) in ascending order

$$\hat{p}_i$$ ($$1\le i\le n$$)

$$\hat{p}_i=\gamma (\varvec{a},\varvec{a}^{(j)})P_{h-1}(\varvec{a}^{(j)})$$ if $$q_i=\gamma (\varvec{a},\varvec{a}^{(j)})P_{h+1}(\varvec{a}^{(j)})$$

$$\hat{q}_i$$ ($$1\le i\le n$$)

$$\hat{q}_i=\gamma (\varvec{a},\varvec{a}^{(j)})P_{h+1}(\varvec{a}^{(j)})$$ if $$p_i=\gamma (\varvec{a},\varvec{a}^{(j)})P_{h-1}(\varvec{a}^{(j)})$$

$$\delta _\mathrm{p}(i)$$ ($$1\le i\le n$$)

$$\delta _\mathrm{p}(i)=0$$, if $$\exists \hat{q}_j$$ ($$i+1\le j\le n$$) such that $$\hat{q}_j>\max \{p^\mathrm{c},p_i\}$$. Otherwise, $$\delta _\mathrm{p}(i)=1$$

$$\delta _\mathrm{q}(i)$$ ($$1\le i\le n$$)

$$\delta _\mathrm{q}(i)=0$$, if $$\exists \hat{p}_j$$ ($$i+1\le j\le n$$) such that $$\hat{p}_j>\max \{p^\mathrm{c},q_i\}$$. Otherwise, $$\delta _\mathrm{q}(i)=1$$

x(i) ($$1\le i\le n$$)

$$x(i)=|\{\hat{q}_j\mid 1\le j\le i-1,\hat{q}_j>p_i\}|$$

y(i) ($$1\le i\le n$$)

$$y(i)=|\{\hat{p}_j\mid 1\le j\le i-1,\hat{p}_j>q_i\}|$$

## Notes

### Acknowledgements

We appreciate Dr. Bipin Indurkhya for proofreading this paper. We also appreciate all of the reviewers for their insightful comments and suggestions, which contributed to improvement of this paper. This study was partly supported by JSPS Grant-in-Aid for Scientific Research (C), Grant No. 17K00093.

## References

1. 1.
Abraham S, Padmanabhan K (1991) The twisted cube topology for multiprocessors: a study in network asymmetry. J Parallel Distrib Comput 13(1):104–110
2. 2.
Akers SB, Krishnamurthy B (1989) A group-theoretic model for symmetric interconnection networks. IEEE Trans Comput 38(4):555–566
3. 3.
Akl SG, Qiu K (1992) Parallel minimum spanning forest algorithms on the star and pancake interconnection networks. In: Proceedings of Joint Conference on Vector and Parallel Processing, pp 565–570Google Scholar
4. 4.
Al-Sadi J, Day K, Ould-Khaoua M (2001) Fault-tolerant routing in hypercubes using probability vectors. Parallel Comput 27(10):1381–1399
5. 5.
Al-Sadi J, Day K, Ould-Khaoua M (2001) Probability-based fault-tolerant routing in hypercubes. Comput J 44(5):368–373
6. 6.
Arabnia HR, Bhandarkar SM (1996) Parallel stereocorrelation on a reconfigurable multi-ring network. J Supercomput 10(3):243–269
7. 7.
Arabnia HR, Oliver MA (1986) Fast operations on raster images with SIMD machine architectures. Comput Graph Forum 5(3):179–188
8. 8.
Arabnia HR, Taha TR (1998) A parallel numerical algorithm on a reconfigurable multi-ring network. Telecommun Syst 10(1):185–202
9. 9.
Bossard A, Kaneko K (2012) The set-to-set disjoint-path problem in perfect hierarchical hypercubes. Comput J 55(6):769–775
10. 10.
Bossard A, Kaneko K, Peng S (2011) A new node-to-set disjoint-path algorithm in perfect hierarchical hypercubes. Comput J 54(8):1372–1381
11. 11.
Chang CP, Sung TY, Hsu LH (2000) Edge congestion and topological properties of crossed cubes. IEEE Trans Parallel Distrib Syst 11(1):64–80.
12. 12.
Chen HC, Kung TL, Hsu LH (2018) An augmented pancyclicity problem of crossed cubes. Comput J 61(1):54–62.
13. 13.
Chiu GM, Chen KS (1997) Use of routing capability for fault-tolerant routing in hypercube multicomputers. IEEE Trans Comput 46(8):953–958
14. 14.
Chiu GM, Wu SP (1996) A fault-tolerant routing strategy in hypercube multicomputers. IEEE Trans Comput 45(2):143–155
15. 15.
Duong DT, Kaneko K (2014) Fault-tolerant routing based on approximate directed routable probabilities for hypercubes. Future Gener Comput Syst 37:88–96.
16. 16.
Efe K (1992) The crossed cube architecture for parallel computing. IEEE Trans Parallel Distrib Syst 3(5):513–524
17. 17.
El-Amawy A, Latifi S (1990) Bridged hypercube networks. J Parallel Distrib Comput 10(1):90–95.
18. 18.
El-Amawy A, Latifi S (1991) Properties and performance of folded hypercubes. IEEE Trans Parallel Distrib Syst 2(1):31–42.
19. 19.
Fang JF (2007) The bipanconnectivity and m-panconnectivity of the folded hypercube. Theor Comput Sci 385(1–3):286–300.
20. 20.
Fu JS (2008) Fault-free cycles in folded hypercubes with more faulty elements. Inf Process Lett 108(5):261–263.
21. 21.
Hsieh SY (2008) A note on cycle embedding in folded hypercubes with faulty elements. Inf Process Lett 108(2):81.
22. 22.
Hsieh SY, Kuo CN (2007) Hamiltonian-connectivity and strongly Hamiltonian-laceability of folded hypercubes. Comput Math Appl 53(7):1040–1044.
23. 23.
Hsieh SY, Kuo CN, Huang HL (2009) 1-Vertex-fault-tolerant cycles embedding on folded hypercubes. Discrete Appl Math 157(14):3094–3098.
24. 24.
Hsieh SY, Tsai CY, Chen CA (2013) Strong diagnosability and conditional diagnosability of multiprocessor systems and folded hypercubes. IEEE Trans Comput 62(7):1472–1477.
25. 25.
Jayashree HV, Thapliyal H, Arabnia HR, Agrawal VK (2016) Ancilla-input and garbage-output optimized design of a reversible quantum integer multiplier. J Supercomput 72(4):1477–1493
26. 26.
Kaneko K, Ito H (2001) Fault-tolerant routing algorithms for hypercube interconnection networks. IEICE Trans Inf Syst E84–D(1):121–128Google Scholar
27. 27.
Kuo CN, Chou HH, Chang NW, Hsieh SY (2013) Fault-tolerant path embedding in folded hypercubes with both node and edge faults. Theor Comput Sci 475:82–91.
28. 28.
Kuo CN, Hsieh SY (2010) Pancyclicity and bipancyclicity of conditional faulty folded hypercubes. Inf Sci 180(15):2904–2914.
29. 29.
Lai CN, Chen GH (2008) w-Rabin numbers and strong w-Rabin numbers of folded hypercubes. Networks 51(3):171–177
30. 30.
Lai CN, Chen GH, Duh DR (2002) Constructing one-to-many disjoint paths in folded hypercubes. IEEE Trans Comput 51(1):33–45.
31. 31.
Liu H (2007) A performance guaranteed new algorithm for fault-tolerant routing in folded cubes. In: Preparata FP, Fang Q (eds) Frontiers in algorithmics. Springer, Berlin, pp 236–243Google Scholar
32. 32.
Malluhi QM, Bayoumi MA (1994) The hierarchical hypercube: a new interconnection topology for massively parallel systems. IEEE Trans Parallel Distrib Syst 5(1):17–30
33. 33.
Ngoc LB, Thuan BT, Hirai Y, Kaneko K (2016) Stochastic link-fault-tolerant routing in hypercubes. J Adv Comput Netw 4(2):100–106Google Scholar
34. 34.
Quadras J, Solomon SS (2015) Embedding of the folded hypercubes into tori. Math Comput Sci 9(2):177–183.
35. 35.
Sabir E, Meng J (2018) Structure fault tolerance of hypercubes and folded hypercubes. Theor Comput Sci 711:44–55.
36. 36.
Seitz CL (1985) The cosmic cube. Commun ACM 28(1):22–33
37. 37.
Shor PW (1997) Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J Comput 26(5):1484–1509
38. 38.
Thapliyal H, Arabnia HR, Srinivas MB (2009) Efficient reversible logic design of BCD subtractors. Springer, Berlin, pp 99–121Google Scholar
39. 39.
Thapliyal H, Jayashree HV, Nagamani AN, Arabnia HR (2013) Progress in reversible processor design: a novel methodology for reversible carry look-ahead adder. Springer, Berlin, pp 73–97Google Scholar
40. 40.
Valafar H, Arabnia HR, Williams G (2004) Distributed global optimization and its development on the multiring network. Neural Parallel Sci Comput 12(4):465–490
41. 41.
Wang D (2001) Embedding hamiltonian cycles into folded hypercubes with faulty links. J Parallel Distrib Comput 61(4):545–564.
42. 42.
Wani MA, Arabnia HR (2003) Parallel edge-region-based segmentation algorithm targeted at reconfigurable multi-ring network. J Supercomput 25(1):43–62
43. 43.
Wu J (1998) Adaptive fault-tolerant routing in cube-based multicomputers using safety vectors. IEEE Trans Parallel Distrib Syst 9(4):322–334Google Scholar
44. 44.
Xu JM, Ma M (2006) Cycles in folded hypercubes. Appl Math Lett 19(2):140–145.
45. 45.
Yang JS, Chan HC, Chang JM (2011) Broadcasting secure messages via optimal independent spanning trees in folded hypercubes. Discrete Appl Math 159(12):1254–1263.
46. 46.
Yang W, Zhao S, Zhang S (2017) Strong menger connectivity with conditional faults of folded hypercubes. Inf Process Lett 125:30–34.
47. 47.
Zhang M, Zhang L, Feng X, Lai HJ (2018) An $${O}(\log _2({N}))$$ algorithm for reliability evaluation of $$h$$-extra edge-connectivity of folded hypercubes. IEEE Trans Reliab 67(1):297–307.
48. 48.
Zhu Q, Liu SY, Xu M (2008) On conditional diagnosability of the folded hypercubes. Inf Sci 178(4):1069–1077.
49. 49.
Zhu Q, Xu JM, Hou X, Xu M (2007) On reliability of the folded hypercubes. Inf Sci 177(8):1782–1788.

## Authors and Affiliations

• Bui Thi Thuan
• 1
• Lam Boi Ngoc
• 1
• Keiichi Kaneko
• 1
1. 1.Tokyo University of Agriculture and TechnologyTokyoJapan