Least Operators to Express Number - Solution & Explanation

Q: Is Least Operators to Express Number easy or hard?

Least Operators to Express Number is classified as a Hard problem on LeetCode. The challenge comes from modeling arithmetic expressions using powers of x and minimizing operators through recursive or dynamic programming state transitions.

Q: Least Operators to Express Number Python/Java solution

Python implementations often use recursion with functools.lru_cache for memoization. Java solutions typically use a HashMap for caching or an iterative DP approach based on base-x digits. Both implementations achieve O(log_x(target)) complexity.

Q: How to solve Least Operators to Express Number in O(log n)?

Convert the target into base-x representation or recursively break it using powers of x. At each step, decide whether to build the current power exactly or overshoot using the next power and subtract the difference. Memoization or DP ensures each state is computed once, resulting in O(log_x(target)) complexity.

Q: What is the best approach for Least Operators to Express Number?

The most common optimal approach uses memoized recursion or dynamic programming based on powers of x. The algorithm repeatedly decomposes the target using the nearest power of x and computes the minimum operator cost for the remaining value. Both strategies run in O(log_x(target)) time because the target shrinks by powers of x at each step.

Q: Is Least Operators to Express Number asked at Google/Amazon/Meta?

This problem is tagged as a hard math and dynamic programming problem and is commonly associated with high-level algorithm interviews at companies like Google and Meta. It tests number decomposition, recursion optimization, and DP state design.

Q: What data structure is used in Least Operators to Express Number?

The key structure is a memoization map or DP array storing the minimum operator cost for intermediate targets. The algorithm also relies on mathematical computation of powers of x while decomposing the target value.

Q: What is the time complexity of Least Operators to Express Number?

The optimal solutions run in O(log_x(target)) time and space. Each recursive or DP state corresponds to a power level of x while decomposing the target value, so the number of states grows with the logarithm of the target in base x.

HardMath Dynamic Programming Memoization17 min readAsked at: Snapchat

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Problem Statement

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 * 10⁸

Approach Overview

Problem Overview: You are given a number x and a target value. The task is to build the target using only the number x and operators +, -, *, and /. The goal is to minimize the number of operators used while respecting normal arithmetic precedence.

Approach 1: Recursive Approach with Memoization (Time: O(logx(target)), Space: O(logx(target)))

This method treats the problem as repeatedly decomposing the target using powers of x. For each target value, compute the closest power x^k. You then explore two choices: represent the target as x^k + remainder or overshoot using x^(k+1) and subtract the difference. A recursive function calculates the minimum operator count for each remainder while storing results in a memo table to avoid recomputation. The recursion depth stays proportional to log_x(target), making it efficient for large targets.

Approach 2: Dynamic Programming Using Base-x Representation (Time: O(logx(target)), Space: O(logx(target)))

This approach converts the target into base x. Each digit represents how many times a power of x contributes to the expression. You iterate through digits from least significant to most significant while maintaining two states: the cost of building the value directly and the cost if you carry to the next digit. Each step updates operator counts based on whether it is cheaper to add multiples of x^k or subtract from the next power. The DP transition captures the tradeoff between constructing the exact digit or adjusting via carry operations. This approach is deterministic and avoids recursion.

Approach 3: Greedy Backtracking with Power Selection (Time: O(logx(target)), Space: O(logx(target)))

The greedy idea selects the largest power of x that does not exceed the remaining target. After choosing that power, you recursively compute the cost of expressing the remainder. A second branch considers overshooting with the next power and subtracting the difference. Backtracking compares both paths and picks the minimum operator count. Although it looks greedy, the branching ensures correctness by evaluating both under- and over-approximation cases.

Recommended for interviews: The memoized recursion or the base-x dynamic programming solution is typically expected. They clearly demonstrate understanding of number decomposition and state reuse. The brute reasoning (choosing powers of x) shows intuition, while the optimized solution using memoization or dynamic programming highlights strong algorithmic design. The problem also relies heavily on number manipulation concepts from math.

Approach 1: Recursive Approach with Memoization

This approach involves recursively breaking down the target number into sums and differences of powers of the base number x. We utilize memoization to store the results of subproblems to avoid redundant calculations.

This Python code recursively finds the minimum number of operations to express the target using powers of x. By storing intermediate results in a memoization table, it improves efficiency significantly.

Code

Python

Complexity

Time Complexity: O(log_xtarget) due to recursive calls and memoization.
Space Complexity: O(log_xtarget) for the memoization table.

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Approach 2: Dynamic Programming Approach

This approach uses dynamic programming to build up a solution iteratively. It constructs an array to store the minimal operators required for each subproblem up to the target.

This Java solution utilizes a dynamic programming map to store intermediate results and handle overlapping subproblems efficiently.

Code

Java Python JavaScript

Java

Python

JavaScript

Complexity

Time Complexity: O(log_xtarget) due to memoization of subproblems.
Space Complexity: O(log_xtarget) for the storage of intermediate results.

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Approach 3: Greedy Backtracking Approach

This method uses backtracking with a greedy approach. The key is to choose the nearest power of x to either over or under target and recursively assess the number of operations needed.

This C++ solution utilizes an unordered_map to cache results. Similar to DP, it recursively exploits the nearest multiples to minimize the number of operators using power and modulo calculations.

The function chooses whether to round the quotient up or down and compares both options.
Each decision minimizes the use of operators by utilizing results stored in a cache for previously computed states.

Code

C++Java

C++

Java

Complexity

Time Complexity: O(log_xtarget) - logarithmic exploration based on division.
Space Complexity: O(log_xtarget) for caching states.

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Approach 4: Memoization Search

We define a function dfs(v), which represents the minimum number of operators needed to compose the number v using x. Then the answer is dfs(target).

The execution logic of function dfs(v) is as follows:

If x geq v, then we can obtain v by adding v instances of x / x, with the number of operators being v times 2 - 1; or we can obtain v by subtracting (x - v) instances of x / x from x, with the number of operators being (x - v) times 2. We take the minimum of the two.

Otherwise, we enumerate x^k starting from k=2 to find the first k where x^k geq v:

If x^k - v geq v at this point, then we can only first obtain x^{k-1}, then recursively calculate dfs(v - x^{k-1}). The number of operators in this case is k - 1 + dfs(v - x^{k-1});
If x^k - v < v at this point, then we can obtain v in the above manner, with the number of operators being k - 1 + dfs(v - x^{k-1}); or we can first obtain x^k, then recursively calculate dfs(x^k - v), with the number of operators being k + dfs(x^k - v). We take the minimum of the two.

To avoid redundant calculations, we implement the dfs function using memoization search.

The time complexity is O(log_{x}{target}) and the space complexity is O(log_{x}{target}).

Code

Python Java C++Go TypeScript

Python

Java

C++

TypeScript

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Complexity Comparison

Approach	Complexity
Recursive Approach with Memoization	Time Complexity: O(log_xtarget) due to recursive calls and memoization. Space Complexity: O(log_xtarget) for the memoization table.
Dynamic Programming Approach	Time Complexity: O(log_xtarget) due to memoization of subproblems. Space Complexity: O(log_xtarget) for the storage of intermediate results.
Greedy Backtracking Approach	Time Complexity: O(log_xtarget) - logarithmic exploration based on division. Space Complexity: O(log_xtarget) for caching states.
Memoization Search	—

Detailed Complexity Analysis

Approach	Time	Space	When to Use
Recursive with Memoization	O(log_x(target))	O(log_x(target))	General case; clean recursive formulation with cached subproblems
Dynamic Programming (Base-x)	O(log_x(target))	O(log_x(target))	Best for iterative implementations and language environments where recursion depth is limited
Greedy Backtracking	O(log_x(target))	O(log_x(target))	Good for understanding the power-selection intuition behind the optimal solution

Video Solution

花花酱 LeetCode 964. Least Operators to Express Number - 刷题找工作 EP237 • Hua Hua • 4,253 views views

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Frequently Asked Questions

Is Least Operators to Express Number easy or hard?

Least Operators to Express Number is classified as a Hard problem on LeetCode. The challenge comes from modeling arithmetic expressions using powers of x and minimizing operators through recursive or dynamic programming state transitions.

Least Operators to Express Number Python/Java solution

Python implementations often use recursion with functools.lru_cache for memoization. Java solutions typically use a HashMap for caching or an iterative DP approach based on base-x digits. Both implementations achieve O(log_x(target)) complexity.

How to solve Least Operators to Express Number in O(log n)?

Convert the target into base-x representation or recursively break it using powers of x. At each step, decide whether to build the current power exactly or overshoot using the next power and subtract the difference. Memoization or DP ensures each state is computed once, resulting in O(log_x(target)) complexity.

What is the best approach for Least Operators to Express Number?

The most common optimal approach uses memoized recursion or dynamic programming based on powers of x. The algorithm repeatedly decomposes the target using the nearest power of x and computes the minimum operator cost for the remaining value. Both strategies run in O(log_x(target)) time because the target shrinks by powers of x at each step.

Is Least Operators to Express Number asked at Google/Amazon/Meta?

This problem is tagged as a hard math and dynamic programming problem and is commonly associated with high-level algorithm interviews at companies like Google and Meta. It tests number decomposition, recursion optimization, and DP state design.

What data structure is used in Least Operators to Express Number?

The key structure is a memoization map or DP array storing the minimum operator cost for intermediate targets. The algorithm also relies on mathematical computation of powers of x while decomposing the target value.

What is the time complexity of Least Operators to Express Number?

The optimal solutions run in O(log_x(target)) time and space. Each recursive or DP state corresponds to a power level of x while decomposing the target value, so the number of states grows with the logarithm of the target in base x.

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Problem Info

DifficultyHard

Acceptance48.3%

Approaches4

Reading time17 min

Asked at

Snapchat

Practice this problem

Open in Editor

Problem Statement

Approach Overview

Approach 1: Recursive Approach with Memoization

Code

Complexity

Approach 2: Dynamic Programming Approach

Code

Complexity

Approach 3: Greedy Backtracking Approach

Code

Complexity

Approach 4: Memoization Search

Code

Complexity Comparison

Detailed Complexity Analysis

Video Solution

Frequently Asked Questions

Related Problems

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