The Pebble GameThe Pebble Game

Geri GrolingerGeri GrolingerYork UniversityYork University

The Pebble GameThe Pebble Game

• Used for studying time-space Used for studying time-space trade-off trade-off

• One player game, played on a DAGOne player game, played on a DAG

Output nodes

Nodes

Input nodes

Formalization:Formalization:• Directed acyclic graphDirected acyclic graph

• Bounded in-degreeBounded in-degree

Three main rules:Three main rules:

2.2. A pebble can be placed on a node A pebble can be placed on a node vv if all predecessors of the node if all predecessors of the node vv are are marked with pebbles marked with pebbles3.3. A pebble can be removed from a A pebble can be removed from a node at any time node at any time

NoteNote: : a pebble removed from the graph can be ‘reused’a pebble removed from the graph can be ‘reused’

1.1. A pebble can be placed A pebble can be placed on any input nodeon any input node

Strategy:Strategy:sequence of legal moves which ends sequence of legal moves which ends in pebbling the distinguished node in pebbling the distinguished node ff

The Goal:The Goal:to place a pebble on some to place a pebble on some previously distinguished node previously distinguished node ff while minimizing the number while minimizing the number of pebbles usedof pebbles used

A move:A move:placing or removing one of the placing or removing one of the pebblespebblesaccording to the three given according to the three given rulesrules

f

30

7060

50

40

20107 moves and 7 pebbles

Example 1

30

7060

50

40

201011 moves and 3 pebbles

Example 2

Interpretation:Interpretation:

1.1. A pebble can be placed A pebble can be placed on any input node ~ LOADon any input node ~ LOAD

2.2. A pebble can be placed on A pebble can be placed on a node a node vv if all predecessors of the node if all predecessors of the node vv are marked with pebbles ~ COMPUTE are marked with pebbles ~ COMPUTE

3.3. A pebble can be removed form a A pebble can be removed form a node at any time ~ DELETE node at any time ~ DELETE• Use as few pebbles as possible ~ # ~ # REGISERSREGISERS• Use as few moves as possible ~ TIME~ TIME

input nodes

nodes

output nodes

In general:In general:

How many pebbles How many pebbles are required to pebble a are required to pebble a

graphgraph with n nodes?with n nodes?

Pyramid graph PPyramid graph Pkk::

Pyramid graph PPyramid graph Pkk::

Fact 1:

Every pebbling strategy for Pk (k > 1)must use AT LEAST k + 1 pebbles.

That is Ω( ) pebbles expressed in number of edges n.

n√

Pyramid graph PPyramid graph Pkk: :

k = 5

We needat least:

k + 1 = 6

Pyramid graph PPyramid graph Pkk: :

Let’s considerhaving

k = 5 pebbles

Arbitrary graph with restricted Arbitrary graph with restricted in-degree (d =2):in-degree (d =2):

Fact 2:

Number of pebbles needed to pebblea graph of in-degree 2 is O(n/log n) (n = # nodes in the graph).

Arbitrary graph with restricted Arbitrary graph with restricted in-degree (d =2):in-degree (d =2):

Proof:

• Recursive pebbling strategy• Cases

• Recursions for each case

• Solutions:P(n) ≤ cn / log n = O(n/log n)

O(n/log n)

References:

1. Gems of theoretical computer science U. Schöning, R. J. Pruim

2. Asymptotically Tight Bounds on Time-Space Trade-offs in a Pebble Game T. Lengauer, R. E. Tarjan

3. Theoretical Models 2002/03 P. van Emde Boas

Thank you for your attention Thank you for your attention

Questions ?Questions ?