Max

1
Maxing vs Adding

• Comput 652 Project
• Presented by Yi Li

2
Outline

• Introduction
• Project setting
• Experiment Result
• Further plan

3
Introduction

• It is a common technology to use pattern
  database.
• Examples showed that using multiple pattern
  databases can further benefit the heuristic
  search.
• Two ways to use multiple pattern databases
• Maxing
• Adding

4
Introduction

• Maxing
• it is admissible if every h is admissible
• Adding
• it is admissible only when the pattern databases
  is disjoint.

5
Introducation

• Disjoint Pattern Database
• For sliding-tile puzzle, partition the tiles
  into disjoint groups. No tile belongs to more
  than one group.
• Key different between maxing and adding.
• Maxing count all moves required to reach the
  pattern goal target, including moves of other
  tiles not in the group.
• Adding only count moves of the tiles of the
  group. It means the move involving blank tile and
  dont care tile will cost 0.

6
Introduction

• maxing gt adding? Or other way around?

7
Project setting

• Problem domain 15 sliding-tile puzzle.
• Partition size
• 8-7-1.
• the size 16!/8! 518,918,400. it is too larger
  to store in memory.
• 5-5-5-1.
• the size 16!/10! 5,765,760.

8
Project setting

• How to partition?
• G1 1,2,3,6,7
• G2 4,5,8,9,12
• G3 10,11,13,14,15
• General rule, group together tiles that are near
  each other because they are most likely to
  interact with one another.

0 1 2 3
4 5 6 7
8 9 10 11
12 13 14 15
9
Project setting

• Memory.
• When comparing maxing and adding, we will need
  to store 6 pattern database. Each one has
  5,765,760 patterns.
• We only need store the h value. For each value, a
  byte is enough because no h value is large than
  256.

10
Project setting

• Mapping
• Sparse table k-dimensional array. Size
• Example a pattern of three tiles(x,y,z) and
  they are located at position(2,1,3). The
  array213.
• Compact mapping size 16!/10!. Mapping to an
  index corresponding to its position in a
  lexicographic ordering of all permutations.
• Example (2,1,3) -gt 3 because it is preceded
  by
• (1,2,3) and (1,3,2).

11
Experiment Result

• Using BFS backward search.
• Stop at certain number of generated nodes.

Nodes generated Adding gt Maxing () Adding lt Maxing ()
10,000 18.96 5.34
50,000 32.96 9.30
100,000 38.93 10.47
12
Experiment Result
Nodes generated Adding gt Maxing () Adding lt Maxing ()
200,000 44.22 11.36
400,000 48.56 12.17
800,000 52.34 12.77
2,000,000 58.77 12.23
6,000,000 63.13 12.22
10,000,000 65.85 11.67
13
Experiment Result

• An example of maxing lt adding

4 2 6 3
8 1 0 7
9 5 10 11
12 13 14 15
14
Experiment Result

• An example of Adding lt Maxing

4 2 6 3
0 5 1 7
8 9 10 11
12 13 14 15
15
Further Plan

• For Adding, we can further dont consider the
  blank position. It can further decrease the size
  of the databases.
• The position of the blank is less important in a
  disjoint pattern database
• Using IDA to test the performance by using
  Maxing and Adding.