Algorithm Design and Analysis 4/6/2008
Lecture 32: Reductions
Instructor: Mike Kowalczyk
Today we saw the solutions to some homework problems and introduced the formal notion of a
mapping reduction.
1 Mapping reduction
We say that decision problem Amapping reduces to decision problem Bif there exists an algorithm
which, on any input x(a problem instance of A) outputs y(a problem instance of B), and xis
a “yes” instance of Aif and only if yis a “yes” instance of B. If this is this case then we write
A≤mB.
Reductions are important because they allow us to relate the difficulty of solving different
problems. For example, if A≤mBand we have an algorithm that solves B, then we can use the
reduction to solve A. Conversely, if A≤mBand Ais known to be undecidable, we know that B
must also be undecidable.
Let’s do such a reduction with the halting problem. Let’s try to show Halt ≤mBwhere Bis
the problem of determining if a given line of given program will be executed, given the input to
that program. If we can show this reduction, then since the halting problem is undecidable, we can
conclude that Bis undecidable as well. Here’s the reduction:
reduceHaltToLineReach(Program P, input i) {
Let program P’ =
"1) Run P on input i
2) i = i"
//Return an instance of line reach: "does program P’ on input i reach line 2?"
Return (P’, i, 2)
}
Let’s prove that this is indeed a mapping reduction. Suppose the input (P, i) is a “yes” instance
of the halting problem, i.e. Phalts on input i. Then if one were to run P0on input i, it would
reach line 2 so (P0, i, 2) is a “yes” instance of the line reach problem. On the other hand, if (P , i) is
a “no” instance of the halting problem, i.e. Pdoesn’t halt on input i, then P0on input idoesn’t
reach line 2 so (P0, i, 2) is a “no” instance of the line reach problem. Thus the halting problem
reduces to the line reach problem via the reduction above, and we conclude that the line reach
problem is undecidable.
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