Abstract
We study the problem of monotonicity testing over the hypercube. As previously observed in several works, a positive answer to a natural question about routing properties of the hypercube network would imply the existence of efficient monotonicity testers. In particular, if any set of source-sink pairs on the directed hypercube (with all sources and all sinks distinct) can be connected with edge-disjoint paths, then monotonicity of functions \(f:{\{0,1\}}^n \rightarrow {\mathcal R}\) can be tested with O(n/ε) queries, for any totally ordered range \({\mathcal R}\). More generally, if at least a μ(n) fraction of the pairs can always be connected with edge-disjoint paths then the query complexity is O(n/(εμ(n)) ).
We construct a family of instances of Ω(2n) pairs in n-dimensional hypercubes such that no more than roughly a \(\frac{1}{\sqrt{n}}\) fraction of the pairs can be simultaneously connected with edge-disjoint paths. This answers an open question of Lehman and Ron [LR01], and suggests that the aforementioned appealing combinatorial approach for deriving query-complexity upper bounds from routing properties cannot yield, by itself, query-complexity bounds better than ≈ n 3/2. Additionally, our construction can also be used to obtain a strong counterexample to Szymanski’s conjecture about routing on the hypercube. In particular, we show that for any δ> 0, the n-dimensional hypercube is not \(n^{\frac 12 -\delta}\)-realizable with shortest paths, while previously it was only known that hypercubes are not 1-realizable with shortest paths.
We also prove a lower bound of Ω(n/ε) queries for one-sided non-adaptive testing of monotonicity over the n-dimensional hypercube, as well as additional bounds for specific classes of functions and testers.
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References
Al-Bashabsheh, A., Yongaçoglu, A.: On the k-pairs problem. CoRR, abs/0805.0050 (2008)
Ailon, N., Chazelle, B.: Information theory in property testing and monotonicity testing in higher dimension. Inf. Comput. 204(11), 1704–1717 (2006)
Adler, M., Harvey, N.J.A., Jain, K., Kleinberg, R.D., Lehman, A.R.: On the capacity of information networks. In: SODA, pp. 241–250 (2006)
Benes, V.E.: Mathematical theory of connecting networks and telephone traffic. Academic Press, New York (1965)
Bhattacharyya, A., Grigorescu, E., Jung, K., Raskhodnikova, S., Woodruff, D.: Transitive-closure spanners. In: SODA 2009: Proceedings of the Nineteenth Annual ACM -SIAM Symposium on Discrete Algorithms, pp. 932–941. Society for Industrial and Applied Mathematics, Philadelphia (2009)
Bhattacharyya, A., Grigorescu, E., Jung, K., Raskhodnikova, S., Woodruff, D.: Transitive-closure spanners of the hypercube and the hypergrid. Electronic Colloquium on Computational Complexity (ECCC) 09(046) (2009)
Bhattacharyya, A.: A note on the distance to monotonicity of boolean functions. Electronic Colloquium on Computational Complexity (ECCC) 15(012) (2008)
Dodis, Y., Goldreich, O., Lehman, E., Raskhodnikova, S., Ron, D., Samorodnitsky, A.: Improved testing algorithms for monotonicity. In: Hochbaum, D.S., Jansen, K., Rolim, J.D.P., Sinclair, A. (eds.) RANDOM 1999 and APPROX 1999. LNCS, vol. 1671, pp. 97–108. Springer, Heidelberg (1999)
Ergün, F., Kannan, S., Kumar, R., Rubinfeld, R., Viswanathan, M.: Spot-checkers. J. Comput. Syst. Sci. 60(3), 717–751 (2000)
Fischer, E.: On the strength of comparisons in property testing. Inf. Comput. 189(1), 107–116 (2004)
Fischer, E., Lehman, E., Newman, I., Raskhodnikova, S., Rubinfeld, R., Samorodnitsky, A.: Monotonicity testing over general poset domains. In: STOC, pp. 474–483 (2002)
Goldreich, O., Goldwasser, S., Lehman, E., Ron, D., Samorodnitsky, A.: Testing monotonicity. Combinatorica 20(3), 301–337 (2000)
Goldreich, O., Goldwasser, S., Lehman, E., Ron, D.: Testing monotonicity. In: FOCS, pp. 426–435 (1998)
Halevy, S., Kushilevitz, E.: Testing monotonicity over graph products. Random Struct. Algorithms 33(1), 44–67 (2008)
Lehman, E., Ron, D.: On disjoint chains of subsets. J. Comb. Theory, Ser. A 94(2), 399–404 (2001)
Lubiw, A.: Counterexample to a conjecture of szymanski on hypercube routing. Inf. Process. Lett. 35(2), 57–61 (1990)
Raskhodnikova, S.: Monotonicity testing. Master’s thesis, Department of Electrical Engineering and Computer Science. MIT, Cambridge, MA (1999)
Rasala-Lehman, A.: Network coding. PhD thesis, Department of Electrical Engineering and Computer Science. MIT, Cambridge, MA (2005)
Szymanski, T.H.: On the permutation capability of a circuit-switched hypercube. In: ICPP (1), pp. 103–110 (1989)
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Briët, J., Chakraborty, S., García-Soriano, D., Matsliah, A. (2010). Monotonicity Testing and Shortest-Path Routing on the Cube. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2010 2010. Lecture Notes in Computer Science, vol 6302. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15369-3_35
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DOI: https://doi.org/10.1007/978-3-642-15369-3_35
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