Reversible Pebble Game on Trees

Abstract

A surprising equivalence between different forms of pebble games on graphs - Dymond-Tompa pebble game (studied in [4]), Raz-McKenzie pebble game (studied in [10]) and reversible pebbling (studied in [1]) - was established recently by Chan[2]. Motivated by this equivalence, we study the reversible pebble game and establish the following results.

• We give a polynomial time algorithm for computing reversible pebbling number of trees. As our main technical contribution, we show that the reversible pebbling number of any tree is exactly one more than the edge rank colouring of the underlying undirected tree.

• By exploiting the connection with the Dymond-Tompa pebble game, we show that complete binary trees have optimal pebblings that take at most $n^{{O(\log \log (n))}}$ steps. This substantially improves the previous bound of $n^{{O(\log (n))}}$ steps.

• Furthermore, we show that almost optimal (within $(1+\epsilon )$ factor for any constant $\epsilon > 0$) pebblings of complete binary trees can be done in polynomial number of steps.

• We also show a time-space tradeoff for reversible pebbling for families of bounded degree trees: for any constant $\epsilon > 0$, such families can be pebbled using $O(n^{\epsilon })$ pebbles in O(n) steps. This generalizes a result of Královic[7] who showed the same for chains.