Problem Reduction

Problem Reduction

• A transformation and conquer technique (chapter 6)
• An important part of P and NP analysis (chapter 11)

Problem Reduction: Definition

• To solve an instance of problem A:
• Transform the instance of problem A into an instance of problem B
• Solve the instance of problem B
• Transform the solution to problem B into a solution of problem A


• We say that problem A reduces to problem B

Example 1

• Finding the Least Common Multiple (LCM) of 2 integers
• Example: LCM(24, 36) = 72


• Solution technique: reduce LCM to Greatest Common Divisor (GCD):
• Instance of Problem A: LCM(24, 36)
• Instance of Problem B: GCD(24, 36)
• Solution of B: GCD(24, 36) = 12 (by Euclid's algorithm)
• Transform solution of B into solution of A:
• LCM(24, 36) = 24*36 / GCD(24, 36) = 24*36 / 12 = 72

Example 2

• Solve Element Uniqueness problem: given a set of two dimensional points, are all elements unique?


• Solution technique:
• Reduce Element Uniqueness to Closest Pair
• Solve Closest Pair
• Transform solution of Closest Pair to solution of Uniqueness:
• Answer to Element Uniqueness is yes if Closest Pair distance > 0

Problem Reduction and Performance

• If A reduces to B then
• If A = Ω(f(n)) then B = Ω(f(n))
• In other words, B can have a higher lower bound than A, but not a lower one


• For example:
• Element Uniqueness is known to be Ω(n lg n).
• Therefore, Closest Pair is also Ω(n lg n)


• More detail on Closest Pair:
• Closest Pair = Ω(n lg n) is still true if Closest Pair's lowest bound is actually higher (eg Ω(n^2))
• But the lower bound for Closest Pair can not be lower (eg it can't be Ω(n))
• Actually, the lower bound for Closest Pair can be proved using decision trees, to be Ω(n lg n), which is also the upper and tight bound.


• Truth in advertising: Some fine points about the performance of the reduction should be considered

Careful!

• This is NOT CORRECT:
• If A reduces to B and B=Ω(f(n)), then A=Ω(f(n))

Remember

• Remember this: If A reduces to B, then the lower bound for B can't be less than the lower bound for A


• B's lower bound can be higher, but not lower.