Abstract
We focus on energy complexity, a Boolean function measure related closely to Boolean circuit design. Given a circuit \(\mathcal {C}\) over the standard basis \(\{\vee _2,\wedge _2,\lnot \}\), the energy complexity of \(\mathcal {C}\), denoted by \({{\,\mathrm{EC}\,}}(\mathcal {C})\), is the maximum number of its activated inner gates over all inputs. The energy complexity of a Boolean function f, denoted by \({{\,\mathrm{EC}\,}}(f)\), is the minimum of \({{\,\mathrm{EC}\,}}(\mathcal {C})\) over all circuits \(\mathcal {C}\) computing f. Recently, K. Dinesh et al. (International computing and combinatorics conference, Springer, Berlin, 738–750, 2018) gave \({{\,\mathrm{EC}\,}}(f)\) an upper bound by the decision tree complexity, \({{\,\mathrm{EC}\,}}(f)=O({{\,\mathrm{D}\,}}(f)^3)\). On the input size n, They also showed that \({{\,\mathrm{EC}\,}}(f)\) is at most \(3n-1\). For the lower bound side, they showed that \({{\,\mathrm{EC}\,}}(f)\ge \frac{1}{3}{{\,\mathrm{psens}\,}}(f)\), where \({{\,\mathrm{psens}\,}}(f)\) is called positive sensitivity. A remained open problem is whether the energy complexity of a Boolean function has a polynomial relationship with its decision tree complexity.
Our results for energy complexity can be listed below.
-
For the lower bound, we prove the equation that \({{\,\mathrm{EC}\,}}(f)=\varOmega (\sqrt{{{\,\mathrm{D}\,}}(f)})\), which answers the above open problem.
-
For upper bounds, \({{\,\mathrm{EC}\,}}(f)\le \min \{\frac{1}{2}{{\,\mathrm{D}\,}}(f)^2+O({{\,\mathrm{D}\,}}(f)),n+2{{\,\mathrm{D}\,}}(f)-2\}\) holds.
-
For non-degenerated functions, we also provide another lower bound \({{\,\mathrm{EC}\,}}(f)=\varOmega (\log {n})\) where n is the input size.
-
We also discuss the energy complexity of two specific function classes, \(\mathtt {OR}\) functions and \(\mathtt {ADDRESS}\) functions, which implies the tightness of our two lower bounds respectively. In addition, the former one answers another open question in Dinesh et al. (International computing and combinatorics conference, Springer, Berlin, 738–750, 2018) asking for non-trivial lower bound for energy complexity of \(\mathtt {OR}\) functions.
Similar content being viewed by others
References
Aaronson S, Ben-David S, Kothari R, Rao S, Tal A (2020) Degree vs. approximate degree and quantum implications of huang’s sensitivity theorem
Amano K, Maruoka A (2005) On the complexity of depth-2 circuits with threshold gates. In: international symposium on mathematical foundations of computer science, Springer, Berlin, 107–118
Antoniadis A, Barcelo N, Nugent M, Pruhs K, Scquizzato M (2014) Energy-efficient circuit design. In: Proceedings of the 5th conference on Innovations in theoretical computer science, ACM 303–312
Barcelo N, Nugent M, Pruhs K, Scquizzato M (2015) Almost all functions require exponential energy. In: international symposium on mathematical foundations of computer science, Springer 90–101
Buhrman H, De Wolf R (2002) Complexity measures and decision tree complexity: a survey. Theoret Comput Sci 288(1):21–43
Dinesh K, Otiv S, Sarma J (2020) New bounds for energy complexity of boolean functions. Theoret Comput Sci 845:59–75
Dinesh K, Otiv S, Sarma J (2018) New bounds for energy complexity of boolean functions. In: International computing and combinatorics conference, Springer, Berlin, 738–750
Du DZ, Ko KI (2001) Theory of computational complexity. Wiley, New York
Gao Y, Mao J, Sun X, Zuo S (2013) On the sensitivity complexity of bipartite graph properties. Theoret Comput Sci 468:83–91
Hajnal A, Maass W, Pudlák P, Szegedy M, Turán G (1993) Threshold circuits of bounded depth. J Comput Syst Sci 46(2):129–154
Håstad J, Goldmann M (1991) On the power of small-depth threshold circuits. Comput Complex 1(2):113–129
Hatami P, Kulkarni R, Pankratov D (2010) Variations on the sensitivity conjecture. arXiv preprint arXiv:1011.0354
Huang H (2019) Induced subgraphs of hypercubes and a proof of the sensitivity conjecture
Kahn J, Saks M, Sturtevant D (1984) A topological approach to evasiveness. Combinatorica 4(4):297–306
Karpas I (2016) Lower bounds for sensitivity of graph properties. arXiv preprint arXiv:1609.05320
Kasim-Zade OM (1992) On a measure of active circuits of functional elements. Math Probl Cybernet 4:218–228
Lovasz L, Young N (2002) Lecture notes on evasiveness of graph properties. arXiv preprint arXiv:cs/0205031
Lozhkin S, Shupletsov M (2015) Switching activity of boolean circuits and synthesis of boolean circuits with asymptotically optimal complexity and linear switching activity. Lobachevskii J Math 36(4):450–460
Nikolaevich VM (1961) On the power of networks of functional elements. In: Proceedings of the USSR academy of sciences. Volume 139., Russian Academy of Sciences 320–323
Nisan N, Szegedy M (1994) On the degree of boolean functions as real polynomials. Comput Complex 4(4):301–313
Razborov A, Wigderson A (1993) \(n^{\Omega (\log n)}\) lower bounds on the size of depth-\(3\) threshold cicuits with AND gates at the bottom. Inf Process Lett 45(6):303–307
Rivest RL, Vuillemin J (1976) On recognizing graph properties from adjacency matrices. Theoret Comput ence 3(3):371–384
Sun X (2011) An improved lower bound on the sensitivity complexity of graph properties. Theoret Comput Sci 412(29):3524–3529
Suzuki A, Uchizawa K, Zhou X (2013) Energy and fan-in of logic circuits computing symmetric boolean functions. Theoret Comput Sci 505:74–80
Turán G (1984) The critical complexity of graph properties. Inf Process Lett 18(3):151–153
Uchizawa K, Takimoto E (2008) Exponential lower bounds on the size of constant-depth threshold circuits with small energy complexity. Theoret Comput Sci 407(1–3):474–487
Uchizawa K, Douglas R, Maass W (2006) On the computational power of threshold circuits with sparse activity. Neural Comput 18(12):2994–3008
Uchizawa K, Nishizeki T, Takimoto E (2010) Energy and depth of threshold circuits. Theoret Comput Sci 411(44–46):3938–3946
Uchizawa K, Takimoto E, Nishizeki T (2011) Size-energy tradeoffs for unate circuits computing symmetric boolean functions. Theoret Comput Sci 412(8–10):773–782
Yao ACC (1988) Monotone bipartite graph properties are evasive. SIAM J Comput 17(3):517–520
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported in part by the National Natural Science Foundation of China Grants Nos. 61433014, 61832003, 61761136014, 61872334, 61502449, 61602440, 61801459, the 973 Program of China Grant No. 2016YFB1000201, and K. C. Wong Education Foundation.
Rights and permissions
About this article
Cite this article
Sun, X., Sun, Y., Wu, K. et al. On the relationship between energy complexity and other boolean function measures. J Comb Optim 43, 1470–1492 (2022). https://doi.org/10.1007/s10878-020-00689-8
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-020-00689-8