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964. Least Operators to Express Number

  • Time: $O(\log_x \texttt{target})$
  • Space: $O(\log \texttt{target})$
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class Solution {
 public:
  int leastOpsExpressTarget(int x, int target) {
    return dfs(x, target, {});
  }

 private:
  int dfs(int x, int target, unordered_map<int, int>&& memo) {
    if (memo.count(target))
      return memo[target];
    if (x > target)
      return min(2 * target - 1, 2 * (x - target));
    if (x == target)
      return 0;

    long prod = x;
    int n = 0;
    while (prod < target) {
      prod *= x;
      ++n;
    }
    if (prod == target)
      return memo[target] = n;

    int ans = dfs(x, target - prod / x, move(memo)) + n;
    if (prod < 2 * target)
      ans = min(ans, dfs(x, prod - target, move(memo)) + n + 1);
    return memo[target] = ans;
  }
};
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class Solution {
  public int leastOpsExpressTarget(int x, int target) {
    return dfs(x, target);
  }

  private Map<Integer, Integer> memo = new HashMap<>();

  private int dfs(int x, int target) {
    if (memo.containsKey(target))
      return memo.get(target);
    if (x > target)
      return Math.min(2 * target - 1, 2 * (x - target));
    if (x == target)
      return 0;

    long prod = x;
    int n = 0;
    while (prod < target) {
      prod *= x;
      ++n;
    }
    if (prod == target) {
      memo.put(target, n);
      return memo.get(target);
    }

    int ans = dfs(x, target - (int) (prod / (long) x)) + n;
    if (prod < 2 * target)
      ans = Math.min(ans, dfs(x, (int) (prod - (long) target)) + n + 1);
    memo.put(target, ans);
    return ans;
  }
}
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class Solution:
  def leastOpsExpressTarget(self, x: int, target: int) -> int:
    @lru_cache(None)
    def dfs(target):
      if x > target:
        return min(2 * target - 1, 2 * (x - target))
      if x == target:
        return 0

      prod = x
      n = 0
      while prod < target:
        prod *= x
        n += 1
      if prod == target:
        return n

      ans = dfs(target - prod // x) + n
      if prod < 2 * target:
        ans = min(ans, dfs(prod - target) + n + 1)
      return ans

    return dfs(target)
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