Advanced DP

Objectives

• Combine bitmask and DP
• Review some methods for solving DP with a limited amount of memory
• Reduce state size by omitting recoverable information

DP Enhancements

Bitmasks

• Example problem:
  • given a collection of objects (up to 20)
  • combining different pairs gives different values
  • want minimal value of combining all objects into pairs
• Technique:
  • Use bits 1 to 20 to store which objects are paired
  • e.g., to pair \(i\) with \(j\), do bitmask = bitmask | (1 << i) | (1 << j)
  • Notice that index is counted from the right
  • Use bitmask as the index to your DP array.

Other Tricks

• Negative Index
  • In C++, negative indices are tricky.
  • Solution: use an offset
• Running out of memory?
  • Consider map instead of vector
  • Check your representation

Dropping one parameter

• If your state has \(n\) parameters, sometimes you can recover one parameter from the others.
• E.g., DP over a rectangle; need height, width, area.
  • Drop height or width