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

<p>When writing such an expression, we adhere to the following conventions:</p>

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

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

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

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

<p><strong class="example">Example 2:</strong></p>

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

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> x = 100, target = 100000000
<strong>Output:</strong> 3
<strong>Explanation:</strong> 100 * 100 * 100 * 100.
The expression contains 3 operations.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>2 &lt;= x &lt;= 100</code></li>
	<li><code>1 &lt;= target &lt;= 2 * 10<sup>8</sup></code></li>
</ul>

class Solution {
public:
    // map target to op count
    std::unordered_map<int, int> dp;
    int leastOpsExpressTarget(int x, int target) {
        if(dp.contains(target)) return dp[target];

        // either minus (target/target)'s or -target + (target/target)
        if(x > target){
            return min((x-target)*2, (target*2)-1);
        }
        // just add target in previous iteration
        if(x == target){
            return 0;
        }
        // get amount of *'s : e.g 3, 3 * 3, 3 * 3 * 3
        int times = 0;
        long int sum = x;
        while(sum < target){
            times++;
            sum*=x;
        }

        if(sum == target){
            return times;
        }

        // have remainder, you can either
            // a: find operations to achieve remainder, and then subtract it
            // b: or divide by an x and find the remaining sum to add
        int a = INT_MAX, b = INT_MAX;
        int remainder = sum-target;
        if(sum-target < target){ // if larger than target, will cause infinite loop
            a = leastOpsExpressTarget(x, sum-target) + times;
        }
        // subtract one because were cancelling a '*' out by subtracting
        b = leastOpsExpressTarget(x, target-(sum/x)) + times - 1;
        // add one because either way well add a '-' or a '+'
        dp[target] = min(a,b) + 1;
        return min(a,b) + 1;
    }
};
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