Separating decision tree complexity from subcube partition complexity

The subcube partition model of computation is at least as powerful as decision trees but no separation between these models was known. We show that there exists a function whose deterministic subcube partition complexity is asymptotically smaller than its randomized decision tree complexity, resolving an open problem of Friedgut, Kahn, and Wigderson (2002). Our lower bound is based on the information-theoretic techniques first introduced to lower bound the randomized decision tree complexity of the recursive majority function. We also show that the public-coin partition bound, the best known lower bound method for randomized decision tree complexity subsuming other general techniques such as block sensitivity, approximate degree, randomized certificate complexity, and the classical adversary bound, also lower bounds randomized subcube partition complexity. This shows that all these lower bound techniques cannot prove optimal lower bounds for randomized decision tree complexity, which answers an open question of Jain and Klauck (2010) and Jain, Lee, and Vishnoi (2014).

💡 Research Summary

The paper resolves a long‑standing open problem by proving that deterministic subcube partition complexity can be asymptotically smaller than randomized decision‑tree complexity, and that the same separation holds for zero‑error and bounded‑error randomized models. The authors construct a family of Boolean functions based on the 4‑majority (4‑MAJ) gate, defined as the majority of four bits with ties broken by the first bit. By iterating this gate h times they obtain a function f_h : {0,1}^{4^h} → {0,1}. Because a single 4‑MAJ gate can be decided by fixing at most three inputs, the deterministic subcube partition complexity satisfies D_sc(4‑MAJ) ≤ 3, while the deterministic decision‑tree complexity is D(4‑MAJ)=4. Using the well‑known composition property D(f∘g)=D(f)·D(g) and the analogous inequality D_sc(f∘g) ≤ D_sc(f)·D_sc(g), they derive D_sc(f_h) ≤ 3^h and D(f_h)=4^h, establishing a deterministic gap.

The core technical contribution is a lower bound on the randomized query complexity of f_h. The authors adapt the information‑theoretic technique introduced by Jayram, Kumar, and Sivakumar (JKS03) and later simplified by Landau, Nisan, Peres, and Vanniasegaram (LNPV06). They design a “hard distribution” over inputs that respects the recursive structure of f_h, despite the function’s asymmetry (the first variable of each 4‑MAJ block plays a special role). By analyzing the expected reduction in Shannon entropy per query and setting up a recurrence between the complexities of f_h and its sub‑functions, they prove R_0(f_h) ≥ 3·2^{h‑1} and consequently R(f_h)=Ω(3·2^h). This shows that any randomized algorithm (zero‑error or bounded‑error) must make roughly 3·2^h queries, whereas a deterministic subcube partition needs only 3^h.

In addition, the paper connects these findings to the public‑coin partition bound (PPRT), the strongest known generic lower‑bound method for randomized decision‑tree complexity. PPRT subsumes block sensitivity, approximate degree, randomized certificate complexity, and the classical adversary bound. The authors prove that PPRT(f) ≤ R_sc(f) for every Boolean function f. Since they already have functions with R_sc(f_h) ≪ R(f_h), it follows that PPRT(f_h) is also asymptotically smaller than R(f_h), answering an open question of Jain and Klauck (2010) and Jain, Lee, and Vishnoi (2014) in the negative.

Overall, the paper makes three major contributions: (1) it provides the first explicit separation between decision‑tree and subcube‑partition models in deterministic, zero‑error, and bounded‑error settings; (2) it extends information‑theoretic lower‑bound techniques to asymmetric functions, yielding a tight Ω(3·2^h) bound for the constructed family; and (3) it shows that even the most powerful known generic lower‑bound (the public‑coin partition bound) cannot achieve optimal randomized decision‑tree lower bounds because it is bounded by randomized subcube‑partition complexity. These results deepen our understanding of the relationships among various query‑complexity measures and suggest new directions for exploring separations in quantum query models, multiparty communication, and beyond.