A Sierpinski Triangle Data Structure for Efficient Array Value Update and Prefix Sum Calculation

TL;DR

The paper addresses efficient dynamic array operations (updates and prefix sums) with minimal memory, offering a Sierpinski-triangle–based alternative to Fenwick trees. It introduces a triadic data-structure where the encoding is $x_j = n_j + \sum_{k \in F(j)} x_k$, constructed via a sierpinski$(S,E)$ rule and padded to $3^{k}$ nodes before trimming, achieving updates and prefix sums in $O(\log_3 N)$ time while preserving memory usage. The authors prove a tight bound $w_N(j) \leq \lceil \log_3 N \rceil + 1$ for the number of bits touched per operation, establishing the time complexity via induction on triadic subtrees and extending to non-power-of-3 sizes. They also discuss connections to quantum computing through fermion-to-qubit transforms, showing near-optimal Pauli-weight bounds and suggesting refinements by edge deletions, with a companion paper to fully develop the quantum-transform implications.

Abstract

The binary indexed tree, or Fenwick tree, is a data structure that can efficiently update values and calculate prefix sums in an array. It allows both of these operations to be performed in $O(\log_2 N)$ time. Here we present a novel data structure resembling the Sierpinski triangle, which accomplishes these operations with the same memory usage in $O(\log_3 N)$ time instead. We show this order to be optimal by making use of a connection to quantum computing.

Figures (4)

• Figure 1: A data structure based on a seven-node Fenwick tree. Source: havlicek_operator_2017
• Figure 2: The Sierpinski tree for $N = 9$
• Figure 3: The Sierpinski tree for $N = 27$
• Figure 4: Diagram showing the partial sums stored in the Sierpinski tree for $N = 9$.

Theorems & Definitions (4)

• Lemma 3.1
• proof
• Theorem 3.2
• proof