109 research outputs found
Pebbling in Dense Graphs
A configuration of pebbles on the vertices of a graph is solvable if one can
place a pebble on any given root vertex via a sequence of pebbling steps. The
pebbling number of a graph G is the minimum number pi(G) so that every
configuration of pi(G) pebbles is solvable. A graph is Class 0 if its pebbling
number equals its number of vertices. A function is a pebbling threshold for a
sequence of graphs if a randomly chosen configuration of asymptotically more
pebbles is almost surely solvable, while one of asymptotically fewer pebbles is
almost surely not. Here we prove that graphs on n>=9 vertices having minimum
degree at least floor(n/2) are Class 0, as are bipartite graphs with m>=336
vertices in each part having minimum degree at least floor(m/2)+1. Both bounds
are best possible. In addition, we prove that the pebbling threshold of graphs
with minimum degree d, with sqrt{n} << d, is O(n^{3/2}/d), which is tight when
d is proportional to n.Comment: 10 page
The pebbling threshold of the square of cliques
AbstractGiven an initial configuration of pebbles on a graph, one can move pebbles in pairs along edges, at the cost of one of the pebbles moved, with the objective of reaching a specified target vertex. The pebbling number of a graph is the minimum number of pebbles so that every configuration of that many pebbles can reach any chosen target. The pebbling threshold of a sequence of graphs is roughly the number of pebbles so that almost every (resp. almost no) configuration of asymptotically more (resp. fewer) pebbles can reach any chosen target. In this paper we find the pebbling threshold of the sequence of squares of cliques, improving upon an earlier result of Boyle and verifying an important instance of a probabilistic version of Graham's product conjecture
LIPIcs
We study space complexity and time-space trade-offs with a focus not on peak memory usage but on overall memory consumption throughout the computation. Such a cumulative space measure was introduced for the computational model of parallel black pebbling by [Alwen and Serbinenko ’15] as a tool for obtaining results in cryptography. We consider instead the non- deterministic black-white pebble game and prove optimal cumulative space lower bounds and trade-offs, where in order to minimize pebbling time the space has to remain large during a significant fraction of the pebbling. We also initiate the study of cumulative space in proof complexity, an area where other space complexity measures have been extensively studied during the last 10–15 years. Using and extending the connection between proof complexity and pebble games in [Ben-Sasson and Nordström ’08, ’11] we obtain several strong cumulative space results for (even parallel versions of) the resolution proof system, and outline some possible future directions of study of this, in our opinion, natural and interesting space measure
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