Pebbling Meets Coloring: Reversible Pebble Game On Trees

TL;DR

This paper establishes a polynomial-time method to compute the reversible pebble game number for trees, relates it to edge rank coloring, and improves pebbling step bounds for binary trees, with implications for space-efficient reversible algorithms.

Contribution

It proves that for rooted trees, the reversible pebble game number equals the edge rank coloring number plus one, enabling polynomial-time computation and improved pebbling strategies.

Findings

Reversible pebble game number for trees equals edge rank coloring number plus one.

Complete binary trees can be pebbled in $n^{O( ext{log log } n)}$ steps.

Polynomial-time algorithms for pebbling trees and time-space trade-offs for bounded degree trees.

Abstract

The reversible pebble game is a combinatorial game played on rooted DAGs. This game was introduced by Bennett (1989) motivated by applications in designing space efficient reversible algorithms. Recently, Chan (2013) showed that the reversible pebble game number of any DAG is the same as its Dymond-Tompa pebble number and Raz-Mckenzie pebble number. We show, as our main result, that for any rooted directed tree T, its reversible pebble game number is always just one more than the edge rank coloring number of the underlying undirected tree U of T. It is known that given a DAG G as input, determining its reversible pebble game number is PSPACE-hard. Our result implies that the reversible pebble game number of trees can be computed in polynomial time. We also address the question of finding the number of steps required to optimally pebble various families of trees. It is known that trees can be pebbled in $n^{O(\log(n))}$ steps where $n$ is the number of nodes in the tree. Using the equivalence between reversible pebble game and the Dymond-Tompa pebble game (Chan, 2013), we show that complete binary trees can be pebbled in $n^{O(\log\log(n))}$ steps, a substantial improvement over the naive upper bound of $n^{O(\log(n))}$. It remains open whether complete binary trees can be pebbled in polynomial (in $n$) number of steps. Towards this end, we show that almost optimal (i.e., within a factor of $(1 + \epsilon)$ for any constant $\epsilon > 0$) pebblings of complete binary trees can be done in polynomial number of steps. We also show a time-space trade-off for reversible pebbling for families of bounded degree trees by a divide-and-conquer approach: for any constant $\epsilon > 0$, such families can be pebbled using $O(n^\epsilon)$ pebbles in $O(n)$ steps. This generalizes an analogous result of Kralovic (2001) for chains.