Skip to content Skip to navigation
University of Warwick
  • Study
  • |
  • Research
  • |
  • Business
  • |
  • Alumni
  • |
  • News
  • |
  • About

University of Warwick
Publications service & WRAP

Highlight your research

  • WRAP
    • Home
    • Search WRAP
    • Browse by Warwick Author
    • Browse WRAP by Year
    • Browse WRAP by Subject
    • Browse WRAP by Department
    • Browse WRAP by Funder
    • Browse Theses by Department
  • Publications Service
    • Home
    • Search Publications Service
    • Browse by Warwick Author
    • Browse Publications service by Year
    • Browse Publications service by Subject
    • Browse Publications service by Department
    • Browse Publications service by Funder
  • Help & Advice
University of Warwick

The Library

  • Login
  • Admin

The complexity of flood filling games

Tools
- Tools
+ Tools

Clifford, Raphaël, Jalsenius, Markus, Montanaro, Ashley and Sach, Ben (2012) The complexity of flood filling games. Theory of Computing Systems, Vol.50 (No.1). pp. 72-92. doi:10.1007/s00224-011-9339-2

Research output not available from this repository, contact author.
Official URL: http://dx.doi.org/10.1007/s00224-011-9339-2

Request Changes to record.

Abstract

We study the complexity of the popular one player combinatorial game known as Flood-It. In this game the player is given an n×n board of tiles where each tile is allocated one of c colours. The goal is to make the colours of all tiles equal via the shortest possible sequence of flooding operations. In the standard version, a flooding operation consists of the player choosing a colour k, which then changes the colour of all the tiles in the monochromatic region connected to the top left tile to k. After this operation has been performed, neighbouring regions which are already of the chosen colour k will then also become connected, thereby extending the monochromatic region of the board. We show that finding the minimum number of flooding operations is NP-hard for c≥3 and that this even holds when the player can perform flooding operations from any position on the board. However, we show that this ‘free’ variant is in P for c=2. We also prove that for an unbounded number of colours, Flood-It remains NP-hard for boards of height at least 3, but is in P for boards of height 2. Next we show how a (c−1) approximation and a randomised 2c/3 approximation algorithm can be derived, and that no polynomial time constant factor, independent of c, approximation algorithm exists unless P=NP. We then investigate how many moves are required for the ‘most demanding’ n×n boards (those requiring the most moves) and show that the number grows as fast as (cn) . Finally, we consider boards where the colours of the tiles are chosen at random and show that for c≥2, the number of moves required to flood the whole board is Ω(n) with high probability.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software
Divisions: Faculty of Science > Computer Science
Journal or Publication Title: Theory of Computing Systems
Publisher: Springer New York LLC
ISSN: 1432-4350
Official Date: January 2012
Dates:
DateEvent
January 2012Published
Volume: Vol.50
Number: No.1
Page Range: pp. 72-92
DOI: 10.1007/s00224-011-9339-2
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access

Request changes or add full text files to a record

Repository staff actions (login required)

View Item View Item
twitter

Email us: wrap@warwick.ac.uk
Contact Details
About Us