[Paper Review] Dual Polynomials for Collision and Element Distinctness

Mark Bun, Justin Thaler|arXiv (Cornell University)|Mar 25, 2015

Complexity and Algorithms in Graphs39 references3 citations

TL;DR

This paper constructs explicit dual polynomials to reprove tight approximate degree lower bounds for the Collision and Element Distinctness functions, achieving $ ilde{igOmega}(n^{1/3})$ and $ ilde{igOmega}(n^{2/3})$ respectively. By leveraging Chebyshev polynomials and randomized restrictions, the authors provide a constructive, lossless alternative to the non-constructive symmetrization arguments used in prior proofs.

ABSTRACT

The approximate degree of a Boolean function $f: \{-1, 1\}^n o \{-1, 1\}$ is the minimum degree of a real polynomial that approximates $f$ to within error $1/3$ in the $\ell_\infty$ norm. In an influential result, Aaronson and Shi (J. ACM 2004) proved tight $ ildeΩ(n^{1/3})$ and $ ildeΩ(n^{2/3})$ lower bounds on the approximate degree of the Collision and Element Distinctness functions, respectively. Their proof was non-constructive, using a sophisticated symmetrization argument and tools from approximation theory. More recently, several open problems in the study of approximate degree have been resolved via the construction of dual polynomials. These are explicit dual solutions to an appropriate linear program that captures the approximate degree of any function. We reprove Aaronson and Shi's results by constructing explicit dual polynomials for the Collision and Element Distinctness functions.

Motivation & Objective

• To provide a constructive proof of the approximate degree lower bounds for Collision and Element Distinctness, which were previously established via non-constructive symmetrization arguments.
• To resolve longstanding open problems in approximate degree complexity by introducing an explicit dual polynomial method that avoids the information loss inherent in symmetrization.
• To extend the applicability of dual polynomial techniques to fundamental problems in quantum query complexity and circuit complexity.
• To demonstrate that dual polynomials can yield tight bounds for functions in AC⁰, including Element Distinctness, which remains the best-known approximate degree lower bound for any AC⁰ function.
• To unify and simplify prior results on Collision and Element Distinctness by providing a single, explicit construction that works even in the small-range case.

Proposed method

• Construct a dual polynomial $ p_S(x) = \mathbb{I}_{\operatorname{ED},S}(x) \cdot A_d(\mathsf{cross}_S(x)/r) $, where $ A_d $ is a transformed Chebyshev polynomial with specific extremal properties.
• Use randomized restrictions via subsets $ S $ of size $ r = N^{1/3} $ to reduce the problem to a lower-dimensional approximation task.
• Apply properties of Chebyshev polynomials $ T_d $, particularly their oscillatory behavior and node distribution, to design $ A_d $ with controlled values at key points.
• Average over all $ S $ of size $ r $ to define $ p(x) = \mathbb{E}_{|S|=r}[p_S(x)] $, ensuring the resulting polynomial is of degree $ O(N^{1/3} \log R) $.
• Leverage concentration bounds to show that for inputs with $ k = c i^2 + 1 $ collisions, the polynomial $ p(x) $ achieves high error with high probability over $ S $.
• Use the fact that $ \mathbb{I}_{\operatorname{ED},S}(x) = 1 $ implies $ \mathsf{cross}_S(x) = r $ for 2-to-1 inputs and $ \approx c i^2 r $ for $ k $-to-1 inputs to align error peaks with extremal points of $ A_d $.

Experimental results

Research questions

• RQ1 Can explicit dual polynomials be constructed to prove tight approximate degree lower bounds for Collision and Element Distinctness?
• RQ2 Does the dual polynomial method provide a constructive, lossless alternative to the symmetrization arguments used in Aaronson and Shi’s original proofs?
• RQ3 Can the dual polynomial approach be extended to the small-range case of the Collision problem?
• RQ4 What is the degree of the dual polynomial that witnesses the $ \tilde{\Omega}(n^{1/3}) $ lower bound for Collision?
• RQ5 How do randomized restrictions and Chebyshev-based constructions help in identifying maximum-error points for the averaged polynomial?

Key findings

• The authors construct an explicit dual polynomial $ p $ of degree $ O(N^{1/3} \log R) $ that witnesses the $ \tilde{\Omega}(n^{1/3}) $ approximate degree lower bound for the Collision function.
• For 1-to-1 inputs, the dual polynomial $ p(x) $ evaluates to 1, while for 2-to-1 inputs, it evaluates to at most $ -2/3 $, establishing a constant error gap.
• The construction achieves error $ \varepsilon \leq 1/3 $, confirming that the approximate degree of Collision is $ \tilde{\Omega}(n^{1/3}) $, matching the known upper bound.
• The method generalizes to the small-range case, providing a dual polynomial even when the range $ R $ is sublinear in $ N $, extending prior results.
• For Element Distinctness, the dual polynomial construction yields a $ \tilde{\Omega}(n^{2/3}) $ lower bound, matching the tight upper bound and confirming it as the best-known approximate degree lower bound for any AC⁰ function.
• The analysis shows that inputs with $ k = c i^2 + 1 $ collisions achieve nearly maximal error with high probability over the random choice of $ S $, aligning with the extremal behavior of the Chebyshev polynomial.