INFO 150: A Mathematical Foundation for Informatics

David Mix Barrington

Fall, 2017

Lecture #9, 27 September 2017

Brute-Force Search

In Lecture #2 we studied a variety of sequences of numbers, which can be thought of as functions that assign a number f(n) to each natural n. We saw constant, linear, quadratic, exponential, and factorial sequences.

Such sequences are important in the study of computation because they can model the time complexity of an algorithm. If we have a measure of the size of an algorithm's input, we can measure the number of steps that the algorithm takes on the worst input of that size. For example, you will later learn sorting algorithms that can sort a list of n items in about n times log n time. And the truth table method, on a compound proposition with n variables, needs 2ⁿ lines and thus over 2ⁿ steps to determine whether the formula is a tautology.

On the homework you'll take some values of n and some potential time functions, and determine the running time of the algorithms on these input sizes, on the assumption that your computer can perform a billion steps in one second.

We've seen that in some cases we can show a compound proposition to be a tautology without checking all 2ⁿ possible settings. But there is no known way to find a proof of this form for an arbitrary tautology that takes a polynomial number of steps (n^k steps for some constant k). In fact, it is not known whether such a short proof always exists for an arbitrary tautology.

But there are cases where we know algorithms that are a great improvement on brute-force search, and you will see these if you ever take an algorithms course:

• A brute-force approach to sorting a list would check all n! possible orderings to see which was right. But the best standard sorting methods take a number of steps around the logarithm of n!, proportional to n log n.

• If we have a network of nodes and edges, and we want to find a path from one node to another (as a GPS system does where the nodes are street addresses and the edges are streets), brute force would check all possible paths to see whether one succeeds. But you know that a GPS can find not only a path, but a shortest path, in a network with millions of nodes in only seconds. It uses the A^* search algorithm, studied in CS 250.

• If we have a network of pipes carrying stuff, each with a certain capacity, a network flow algorithm can determine whether a given quantity of stuff can be routed from one place to another. Brute force would have to check exponentially many ways to assign an amount of stuff to each pipe, but this problem can also be solved in polynomial time.

• Suppose we have n people, each who want to adopt a dog from a set of n dogs. Each person has a rank order on the dogs, and each dog has a rank order on the people. Our goal is to match each person to a dog, such that there is no person-dog pair where each prefers the other to their given match. There are n! possible matchings, so checking all of them would be impossible even for pretty small values of n. But the Stable Marriage Algorithm uses about n² steps, and is what is actually used to match thousands of medical school graduates with residency programs each year.

However, there are many other searching problems where no method is known that is much better than brute-force search, at least in the worst case. Some of these algorithm have a property called NP-completeness, which means that if they can be solved in polynomial time, essentially all brute force searches can be carried out in polynomial time. It seems unlikely that such a general search algorithm exists, but to prove that it doesn't (assuming that it really doesn't) is currently the greatest unsolved problem in theoretical computer science and one of the greatest in all of mathematics. Here are a few examples of NP-complete problems:

• Determining whether a given compound proposition is a tautology.

• Given a network of nodes and edges, determine the shortest path by which a traveling salesperson can visit all of the nodes and return to their starting point. (Note that we can find the best way to go from any point to any other, but there are n! possible orders in which to visit n points.)

• Given a set of positive integers, and a target number, is there a subset of the integers that add up exactly to the target? This Subset Sum problem is NP-complete in general, but can be solved easily if the numbers are all small.

Last modified 28 September 2017