Writing an algorithm to decide whether a target number can be reached with a set of other numbers and specific operators?

Question

I\'m trying to learn more about algorithm design, and I\'ve set myself the challenge of creating a simple game that presents users with an array of numbers, a target number,

Answer 1

1. Enumerate all possible expressions. For numbers 1,2,3,and operator + and - ,you can get:1+2+3,1+2-3,1-2+3,1-2-3.

2. Evaluate all possible expressions.

Answer 2

You are facing a more generalized problem of the Partition Problem, which is NP-Complete.

The Partition Problem is: Given n numbers, split them into two (distinct) groups A and B such that sum(A) = sum(B). Now, it is easy to see that if you have a problem with +,- operators and target number 0 - this is basically the same problem, and there is an immidiate reduction from Partition Problem to your problem.

From this we can conclude your problem is NP-Hard as well, and there is no known polynomial solution for your problem.

Alternatives are:

1. Brute Force (As suggested by @Sayakiss)
2. Depending on the operators - you might be able to use some branch and bound techniques.
3. For Partition Problem there is pseudo-polynomial Dynamic Programming solution, that might be utilized in here as well, at least for some of the cases.

Sorry if it's bad news -but at least you won't be looking for something that (most computer scientists believe) is not there

Answer 3

Here is a Java solution from programcreek.

public static boolean isReachable(ArrayList<Integer> list, int target) {
    if (list == null || list.size() == 0)
        return false;

    int i = 0;
    int j = list.size() - 1;

    ArrayList<Integer> results = getResults(list, i, j, target);

    for (int num : results) {
        if (num == target) {
            return true;
        }
    }

    return false;
}

public static ArrayList<Integer> getResults(ArrayList<Integer> list,
        int left, int right, int target) {
    ArrayList<Integer> result = new ArrayList<Integer>();

    if (left > right) {
        return result;
    } else if (left == right) {
        result.add(list.get(left));
        return result;
    }

    for (int i = left; i < right; i++) {

        ArrayList<Integer> result1 = getResults(list, left, i, target);
        ArrayList<Integer> result2 = getResults(list, i + 1, right, target);

        for (int x : result1) {
            for (int y : result2) {
                result.add(x + y);
                result.add(x - y);
                result.add(x * y);
                if (y != 0)
                    result.add(x / y);
            }
        }
    }

    return result;
}