This Article 
 Bibliographic References 
 Add to: 
All Nearest Smaller Values on the Hypercube
May 1996 (vol. 7 no. 5)
pp. 456-462

Abstract—Given a sequence of n elements, the All Nearest Smaller Values (ANSV) problem is to find, for each element in the sequence, the nearest element to the left (right) that is smaller, or to report that no such element exists. Time and work optimal algorithms for this problem are known on all the PRAM models [3], [5] but the running time of the best previous hypercube algorithm [6] is optimal only when the number of processors p satisfies 1 ≤pn/((lg3n)(lg lg n)2). In this paper, we prove that any normal hypercube algorithm requires Ω(n) processors to solve the ANSV problem in O(lg n) time, and we present the first normal hypercube ANSV algorithm that is optimal for all values of n and p. We use our ANSV algorithm to give the first O(lg n)-time n-processor normal hypercube algorithms for triangulating a monotone polygon and for constructing a Cartesian tree.

[1] A. Aggarwal, D. Kravets, J.K. Park, and S. Sen, "Parallel Searching in Generalized Monge Arrays with Applications," Proc. Second Ann. ACM Symp. Parallel Algorithms and Architectures, pp. 259-268, 1990.
[2] M.J. Atallah and D.Z. Chen, "Optimal Parallel Hypercube Algorithms for Polygon Problems," Proc. Fifth IEEE Symp. Parallel and Distributed Processing, pp. 208-215, 1993.
[3] O. Berkman, B. Schieber, and U. Vishkin, "Optimal Doubly Logarithmic Parallel Algorithms Based on Finding Nearest Smaller Values," J. Algorithms, vol. 14, no. 3, pp. 344-370, 1993.
[4] A. Borodin and J.E. Hopcroft,"Routing, merging and sorting on parallel models of comparison," J. Computer and System Science, vol. 30, pp. 130-145, 1985.
[5] D.Z. Chen, "Efficient Geometric Algorithms in the EREW-PRAM," Proc. 28th Ann. Allerton Conf. Comm., Control, and Computing, pp. 818-827, 1990.
[6] J.F. JáJá and K.W. Ryu, "Optimal Algorithms on the Pipelined Hypercube and Related Networks," IEEE Trans. Parallel and Distributed Systems, vol. 4, pp. 582-591, 1993.
[7] C.P. Kruskal, "Searching, Merging, and Sorting," IEEE Trans. Computers, vol. 32, no. 10, pp. 942-946, 1983.
[8] F.T. Leighton,Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes.San Mateo, Calif.: Morgan Kaufmann, 1992.
[9] E.W. Mayr and R. Werchner, "Optimal Routing of Parentheses on the Hypercube," Proc. Fourth Ann. ACM Symp. Parallel Algorithms and Architectures, pp. 109-117, 1992.
[10] L.G. Valiant, "Parallelism in Comparison Problems," SIAM J. Computing, vol. 4, pp. 348-355, 1975.
[11] J. Vuillemin, "A Unified Look at Data Structures," Comm. ACM, vol. 23, pp. 229-239, 1980.

Index Terms:
All nearest smaller values, normal hypercube algorithm, monotone polygon triangulation, Cartesian tree.
Dina Kravets, C. Greg Plaxton, "All Nearest Smaller Values on the Hypercube," IEEE Transactions on Parallel and Distributed Systems, vol. 7, no. 5, pp. 456-462, May 1996, doi:10.1109/71.503770
Usage of this product signifies your acceptance of the Terms of Use.