A loopless and branchless $O(1)$ algorithm to generate the next Dyck word

TL;DR

This paper presents a highly efficient, loopless, and branchless algorithm that computes the next Dyck word of the same size in constant time using bitwise operations, suitable for 64-bit integers.

Contribution

The authors introduce a novel O(1) algorithm for generating the next Dyck word without loops or branches, improving performance over previous methods.

Findings

Constant-time computation of next Dyck word for 64-bit integers

Loopless and branchless implementation using bitwise operations

Applicable to unsigned integer types up to 64 bits

Abstract

Let integer be any C/C++ unsigned integer type up to 64-bits long. Given a Dyck word the following code returns the next Dyck word of the same size, provided it exists. integer next_dyck_word(integer w) { integer const a = w & -w; integer const b = w + a; integer c = w ^ b; c = (c / a >> 2) + 1; c = ((c * c - 1) & 0xaaaaaaaaaaaaaaaa) | b; return c; }