Least Operators to Express Number

Problem

Given a single positive integer x, we will write an expression of the form x (op1) x (op2) x (op3) x ... where each operator op1, op2, etc. is either addition, subtraction, multiplication, or division (+, -, *, or /). For example, with x = 3, we might write 3 * 3 / 3 + 3 - 3 which is a value of 3.

When writing such an expression, we adhere to the following conventions:

The division operator (/) returns rational numbers.
There are no parentheses placed anywhere.
We use the usual order of operations: multiplication and division happen before addition and subtraction.
It is not allowed to use the unary negation operator (-). For example, "x - x" is a valid expression as it only uses subtraction, but "-x + x" is not because it uses negation.

We would like to write an expression with the least number of operators such that the expression equals the given target. Return the least number of operators used.

Example 1:

Input: x = 3, target = 19
Output: 5
Explanation: 3 * 3 + 3 * 3 + 3 / 3.
The expression contains 5 operations.

Example 2:

Input: x = 5, target = 501
Output: 8
Explanation: 5 * 5 * 5 * 5 - 5 * 5 * 5 + 5 / 5.
The expression contains 8 operations.

Example 3:

Input: x = 100, target = 100000000
Output: 3
Explanation: 100 * 100 * 100 * 100.
The expression contains 3 operations.

Constraints:

2 <= x <= 100
1 <= target <= 2 * 10^8

Solution

class Solution {
    private Map<String, Integer> map = new HashMap<>();
    private int x;

    public int leastOpsExpressTarget(int x, int target) {
        this.x = x;
        if (x == target) {
            return 0;
        }
        return dfs(0, target) - 1;
    }

    // ax^0 + bx^1 + cx^2 +....
    // think it as base x problem
    private int dfs(int ex, long target) {
        if (target == 0) {
            return 0;
        }
        if (ex > 40) {
            return 10000000;
        }
        String state = ex + "," + target;
        if (map.containsKey(state)) {
            return map.get(state);
        }
        int res = Integer.MAX_VALUE;
        int mod = (int) (target % x);
        if (mod == 0) {
            if (ex == 0) {
                // not use
                res = Math.min(res, dfs(ex + 1, target));
            } else {
                // not use
                res = Math.min(res, dfs(ex + 1, target / x));
            }
        } else {
            // division is needed
            if (ex == 0) {
                res = Math.min(res, 2 * mod + dfs(ex + 1, target - mod));
                res = Math.min(res, 2 * (x - mod) + dfs(ex + 1, target - mod + x));
            } else {
                res = Math.min(res, (ex - 1) * mod + dfs(ex + 1, (target - mod) / x));
                res = Math.min(res, (ex - 1) * (x - mod) + dfs(ex + 1, (target - mod + x) / x));
            }
        }
        map.put(state, res);
        return res;
    }
}

Explain:

nope.

Complexity:

Time complexity : O(n).
Space complexity : O(n).