# Proving tree algorithms for succinct data structures

## TL;DR

This paper formalizes two tree-based succinct data structure representations in Coq/SSReflect, enabling verified, space-efficient, and performant algorithms for large data handling.

## Contribution

It provides the first formalization of Level-Order Unary Degree Sequence and dynamic bit sequence trees, enhancing provably-safe implementations of big data structures.

## Key findings

- Formalization of Level-Order Unary Degree Sequence in Coq
- Efficient algorithms for dynamic bit sequences with low overhead
- Foundation for verified big data data structures

## Abstract

Succinct data structures give space-efficient representations of large amounts of data without sacrificing performance. They rely one cleverly designed data representations and algorithms. We present here the formalization in Coq/SSReflect of two different tree-based succinct representations and their accompanying algorithms. One is the Level-Order Unary Degree Sequence, which encodes the structure of a tree in breadth-first order as a sequence of bits, where access operations can be defined in terms of Rank and Select, which work in constant time for static bit sequences. The other represents dynamic bit sequences as binary balanced trees, where Rank and Select present a low logarithmic overhead compared to their static versions, and with efficient insertion and deletion. The two can be stacked to provide a dynamic representation of dictionaries for instance. While both representations are well-known, we believe this to be their first formalization and a needed step towards provably-safe implementations of big data.

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Source: https://tomesphere.com/paper/1904.02809