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Computing

, Volume 31, Issue 2, pp 115–139 | Cite as

Bandwidth and pebbling

  • A. L. Rosenberg
  • I. H. Sudborough
Article

Abstract

The main results of this paper establish relationships between the bandwidth of a graphG — which is the minimum over all layouts ofG in a line of the maximum distance between images of adjacent vertices ofG — and the ease of playing various pebble games onG. Three pebble games on graphs are considered: the well-known computational pebble game, the “progressive” (i.e., no recomputation allowed) version of the computational pebble game, both of which are played on directed acyclic graphs, and the quite different “breadth-first” pebble game, that is played on undirected graphs. We consider two costs of a play of a pebble game: the minimum number of pebbles needed to play the game on the graphG, and the maximumlifetime of any pebble in the game, i.e., the maximum number of moves that any pebble spends on the graph. The first set of results of the paper prove that the minimum lifetime cost of a play of either of the second two pebble games on a graphG is precisely the bandwidth ofG. The second set of results establish bounds on the pebble demand of all three pebble games in terms of the bandwidth of the graph being pebbled; for instance, the number of pebbles needed to pebble a graphG of bandwidthk is at most min (2k2+k+1, 2k log2|G|); and, in addition, there are bandwidth-k graphs that require 3k−1 pebbles. The third set of results relate the difficulty of deciding the cost of playing a pebble game on a given input graphG to the bandwidth ofG; for instance, the Pebble Demand problem forn-vertex graphs of bandwidthf(n) is in the class NSPACE (f(n) log2n); and the Optimal Lifetime Problem for either of the second two pebble games is NP-complete.

AMS Subject Classifications

05C99 68E10 94C15 

Key words

Bandwidth graph pebbling 

Bandweite und Markenspiele

Zusammenfassung

Die Hauptergebnisse dieser Arbeit ergeben Beziehungen zwischen der „Bandwidth” eines GraphenG — die das Minimum ist, über alle Projektionen vonG auf eine Linie, von dem maximalen Abstand zwischen Bildern benachbarter Knoten vonG — und der Leichtigkeit, verschiedene „Pebble Games” aufG zu spielen. Es werden drei Pebble Games auf Graphen betrachtet: das wohlbekannte „computational” Pebble Game, die „progressive” (d. h. keine Wiederberechnung erlaubt) Version des computational Pebble Game, von denen beide auf directed acyclic Graphen gespielt werden, und das ziemlich verschiedene „breadth-first” Pebble Game, das auf undirected Graphen gespielt wird. Wir betrachten zwei verschiedene Kosten für das Pebble Game: die minimale Anzahl von Pebbles, die man braucht, um das Pebble Game auf einem GraphenG zu spielen, und die maximaleLebensdauer eines Pebble in einem Spiel, d. h. die maximale Anzahl von Zügen während denen ein Pebble auf dem Graphen verweilt. Die erste Gruppe von Hauptergebnissen in dieser Arbeit zeigt, daß die minimalen Lebensdauer-Kosten eines Spielverlaufs in einem der beiden letzten Pebble Games auf einem Graphen genau die Bandwidth vonG ist. Die zweite Gruppe von Ergebnissen stellt obere Schranken auf für die Anzahl von benötigten Pebbles in Abhängigkeit von der Bandwidth des betrachteten Graphen, z. B. um einen GraphenG mit Bandwidthk zu pebblen, braucht man höchstens min (2k2+k+1, 2klog2|G|) Pebbles; ferner gibt es GraphenG von Bandwidthk für die man 3k−1 Pebbles braucht. Die dritte Gruppe von Ergebnissen setzt die Schwierigkeit, die Kosten eines Pebble Game auf einem gegebenen input-GraphenG festzustellen, in Beziehung zur Bandwidth vonG, z.B. das „Pebble Demand Problem” für Graphen mitn vertices von Bandwidthf(n) ist in der Klasse NSPACE (f(n)log2n); und das „Optimal Lifetime Problem” ist für jedes der beiden letzten Pebble Games NP-vollständig.

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Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • A. L. Rosenberg
    • 1
  • I. H. Sudborough
    • 2
  1. 1.Department of Computer ScienceDuke UniversityDurhamU.S.A.
  2. 2.Department of Electrical Engineering and Computer ScienceNorthwestern UniversityEvanstonU.S.A.

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