[Paper Review] The Decision Tree Complexity for $k$-SUM is at most Nearly Quadratic

Esther Ezra, Micha Sharir|arXiv (Cornell University)|Jul 14, 2016

Complexity and Algorithms in Graphs14 references3 citations

TL;DR

This paper presents a new algorithm for the $k$-SUM problem using a recursive vertical decomposition approach with $\varepsilon$-cuttings, reducing the decision tree complexity to $O(n^2 \log^2 n)$ linear queries. The method improves upon prior bounds by Cardinal et al. and Meyer auf der Heide, achieving nearly quadratic complexity and extending to the more general $k$-linear degeneracy testing (k-LDT) problem with arbitrary real coefficients.

ABSTRACT

Following a recent improvement of Cardinal et al. on the complexity of a linear decision tree for $k$-SUM, resulting in $O(n^3 \log^3{n})$ linear queries, we present a further improvement to $O(n^2 \log^2{n})$ such queries.

Motivation & Objective

• To close the gap between the best-known upper and lower bounds for the $k$-SUM problem in the linear decision tree model.
• To develop a more efficient point location mechanism in high-dimensional arrangements of hyperplanes, particularly for $k$-SUM and $k$-LDT.
• To improve upon the $O(n^3 \log^3 n)$ bound of Cardinal et al. and the $O(n^4 \log n)$ bound of Meyer auf der Heide for $k$-LDT.
• To analyze the query complexity of $k$-SUM and $k$-LDT under the $s$-linear decision tree model, especially when $s$ is large (e.g., $s = n$).
• To explore the feasibility of achieving near-linear query complexity in the decision tree model, given that the current lower bound is $\Omega(n \log n)$.

Proposed method

• The algorithm uses recursive vertical decomposition of the arrangement of hyperplanes defined by $k$-SUM constraints, maintaining a prism at each recursive level.
• At each recursive step, $\varepsilon$-cuttings are applied to partition the space and reduce the number of active hyperplanes by a constant factor.
• The method relies on projecting the vertical walls of the current prism onto a lower-dimensional space, reducing the problem dimension by one at each step.
• Linear queries are performed at each node of the decision tree to test the sign of a linear function, with the number of queries bounded by $O(n^2 \log n)$ per level.
• The construction ensures that the number of hyperplanes is reduced by a constant factor at each of the $O(\log |H|)$ recursive levels, leading to a total query count of $O(n^2 \log^2 n)$.
• The approach extends to $k$-LDT by ensuring no hyperplane is axis-parallel after coordinate rotation, preserving the general applicability to arbitrary real coefficients.

Experimental results

Research questions

• RQ1 Can the decision tree complexity for $k$-SUM be reduced below the previous $O(n^3 \log^3 n)$ bound?
• RQ2 Is it possible to achieve a nearly quadratic query complexity for $k$-SUM in the linear decision tree model?
• RQ3 How does the recursive vertical decomposition with $\varepsilon$-cuttings compare to Meiser's point location mechanism in terms of query time and space complexity?
• RQ4 Can the algorithm be generalized to $k$-LDT with arbitrary real coefficients while maintaining nearly quadratic query complexity?
• RQ5 What is the tightest possible upper bound on the decision tree complexity of $k$-SUM, given the $\Omega(n \log n)$ lower bound?

Key findings

• The decision tree complexity for $k$-SUM is bounded by $O(n^2 \log^2 n)$ linear queries, representing a significant improvement over the prior $O(n^3 \log^3 n)$ bound.
• The algorithm achieves this bound using recursive vertical decomposition and $\varepsilon$-cuttings, which reduce the number of active hyperplanes by a constant factor at each of $O(\log n)$ levels.
• The method extends to $k$-LDT with arbitrary real coefficients, yielding a query complexity of $O(kn^2 \log^2 n)$, which is nearly quadratic in $n$ for fixed $k$.
• The query time of the resulting data structure is $O(d^4 \log n)$, improving Meiser's $O(d^5 \log n)$ query time by an order of magnitude.
• The space complexity remains $n^{O(d)}$, with a constant in the exponent that is only slightly larger than Meiser's, suggesting potential for further optimization.
• The analysis shows that the constant of proportionality in the space complexity may be improved from double exponential to singly exponential, though this remains an open refinement.