Expression Add Operators

The problem involves generating all possible expressions by inserting ‘+’, ‘-’, or ‘*’ between the digits of a given string so that the expression evaluates to a target value. Here’s a way to approach this problem:

Use Recursion: You can write a recursive function that tries all three operators between the digits, as well as no operator (to handle numbers with more than one digit).
Handle Multi-digit Numbers: Make sure to handle cases where numbers have more than one digit (e.g., “23” in the string “232”).
Evaluate Expressions Carefully: Multiplication will have a higher precedence than addition or subtraction, so you will need to handle that manually in your recursive function.
Avoid Leading Zeros: Make sure not to create numbers with leading zeros in your expressions.
Collect Valid Expressions: Keep track of expressions that evaluate to the target value.

Here’s a solution:

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class Solution:
    def addOperators(self, num: str, target: int) -> List[str]:
        def dfs(index, prev_operand, current_operand, value, expression):
            if index == len(num):
                if value == target:
                    result.append(expression)
                return

            for i in range(index, len(num)):
                # No leading zeros
                if i != index and num[index] == '0':
                    break

                curr_num = int(num[index: i + 1])
                if index == 0:
                    dfs(i + 1, curr_num, 0, curr_num, str(curr_num))
                else:
                    dfs(i + 1, curr_num, 0, value + curr_num, expression + "+" + str(curr_num))
                    dfs(i + 1, -curr_num, 0, value - curr_num, expression + "-" + str(curr_num))
                    dfs(i + 1, prev_operand * curr_num, current_operand + prev_operand * curr_num - prev_operand, value - prev_operand + (prev_operand * curr_num), expression + "*" + str(curr_num))

        result = []
        dfs(0, 0, 0, 0, "")
        return result

Explanation:

The dfs function is a recursive function that constructs expressions. It keeps track of the current index in the string, the previous operand, the current operand, the current value of the expression, and the expression itself.
The prev_operand and current_operand are used to handle the multiplication operator, considering its higher precedence.
The loop in the dfs function tries all possibilities for the next number, including multi-digit numbers.

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from typing import List

class Solution:
    def addOperators(self, num: str, target: int) -> List[str]:
        if not num:
            return []

        result = []

        def backtrack(start, expression, total, last_addition):
            # If the total equals the target and all digits are processed
            if start == len(num) and total == target:
                result.append(expression)
                return

            for i in range(start + 1, len(num) + 1):
                current_num_str = num[start:i]
                current_num = int(current_num_str)

                # Skip if there are leading zeros
                if i == start + 1 or (i > start + 1 and num[start] != "0"):
                    # Recurse with addition
                    backtrack(i, expression + '+' + current_num_str, total + current_num, current_num)

                    # Recurse with subtraction
                    backtrack(i, expression + '-' + current_num_str, total - current_num, -current_num)

                    # Recurse with multiplication, special case due to precedence of multiplication
                    backtrack(i, expression + '*' + current_num_str, total - last_addition + last_addition * current_num, last_addition * current_num)

        # Start the backtracking process
        for i in range(1, len(num) + 1):
            current_num_str = num[:i]
            if i == 1 or (i > 1 and num[0] != "0"):  # Skip leading zeros
                backtrack(i, current_num_str, int(current_num_str), int(current_num_str))

        return result

Identifying Problem Isomorphism

“Expression Add Operators” can be mapped to “Target Sum”.

In “Expression Add Operators”, you’re given a string that contains only digits 0-9 and a target value. The aim is to add binary operators (not unary) +, -, or * between the digits so that the resultant expression equals the target value. The task is to return all possible combinations.

In “Target Sum”, you’re given a list of non-negative integers, a target, and you can use + and - operators in front of each integer. The goal is to find the total number of different expressions you can create that add up to the target.

Both problems involve dealing with numeric strings or arrays, deciding where to place certain operators (+, -, or * in the first case and +, - in the second case) to achieve a desired target. The problems are isomorphic because they both deal with creating mathematical expressions to reach a target value.

“Target Sum” is simpler as it involves only two operators (+ and -) and the task is to count valid expressions rather than list all possible ones. “Expression Add Operators” has a higher level of complexity due to the inclusion of the multiplication operator and the requirement to output all valid expressions.

10 Prerequisite LeetCode Problems

Here are some related problems to prepare for it:

39. Combination Sum: This problem also involves finding all possible combinations that satisfy a certain criterion. This is a simpler problem and can serve as a good introduction to the concept of backtracking.
40. Combination Sum II: This problem is similar to Combination Sum but also requires handling duplicates, which adds another layer of complexity.
78. Subsets: This problem asks for all possible subsets of a set, which can be solved by backtracking.
216. Combination Sum III: In this problem, you need to find all possible combinations of k numbers that add up to a number n. This problem could help you understand how to use backtracking to find all possible combinations that satisfy certain criteria.
79. Word Search: This problem requires finding a path in a grid, which involves a similar form of backtracking.
46. Permutations: This problem requires generating all permutations of a list, which can be done using backtracking.
77. Combinations: This problem involves generating all combinations of k numbers out of 1 … n.
17. Letter Combinations of a Phone Number: This problem requires generating all possible letter combinations that a phone number could represent.
131. Palindrome Partitioning: This problem also involves backtracking. It requires partitioning a string such that every substring of the partition is a palindrome.
22. Generate Parentheses: This problem asks you to generate all combinations of well-formed parentheses, which also involves a form of backtracking.

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class Solution:
    def addOperators(self, num: str, target: int) -> List[str]:
        L = len(num)
        ans = set()

        def backtrack(i, total, last, expr):
            if i == L:
                if total == target:
                    ans.add(expr)
                return

            for j in range(i, L):
                n = int(num[i:j+1])
                if i == 0:
                    backtrack(j+1, n, n, str(n))
                else:
                    backtrack(j+1, total + n, n, expr + '+' + str(n))
                    backtrack(j+1, total - n, -n, expr + '-' + str(n))
                    backtrack(j+1, total - last + last * n, last * n, expr + '*' + str(n))
                if n == 0:
                    break

        backtrack(0, 0, 0, '')
        return list(ans)

Problem Classification

The problem involves inserting addition, subtraction, and multiplication operators into a number to achieve a target value. This can be classified as an Expression Construction Problem.

Language Agnostic Coding Drills

Basic Mathematical Operations: The ability to perform addition, subtraction, and multiplication operations is fundamental.
String to Integer Conversion: Understanding how to convert a string representation of a number into an integer.
Use of Recursive Functions (Backtracking): This is a key part of the solution as we use recursion to generate and explore all possible expressions.
Manipulating Strings: The algorithm constructs strings that represent mathematical expressions, so you need to be comfortable with operations such as string concatenation.
Control Flow:
- If-Else Statements: The algorithm uses conditional logic to decide when to add a constructed expression to the set of answers.
- Loops: A loop is used to iterate over the string that represents the number.
Working with Collections:
- Arrays: The problem takes an array as an input.
- Sets: The problem uses a set to store unique mathematical expressions that equal the target number.

Now, let’s describe the problem-solving approach in a step-by-step manner:

First, understand the problem requirements. We need to add binary operators (not unary) +, -, or * between the digits so they evaluate to the target value.
We can solve this problem using recursion (also known as backtracking). At each step, we have 4 choices: we can place a ‘+’, ‘-’, ‘*’ or no operator.
Start by creating a recursive function that will keep track of the current index in the number string, the total calculated so far, the last number added or subtracted, and the current expression.
The base case for the recursion is when we reach the end of the number string. If the total calculated so far equals the target, we add the current expression to our set of answers.
If we are not at the end, we iterate over the rest of the string. At each step, we take the next digit (or digits), convert it (or them) to a number, and:
- if it’s the first number, we simply add it to the total and call the recursive function with the new total,
- otherwise, we try adding, subtracting, and multiplying it by the last number.
If the number is 0, we stop (to avoid leading zeroes).
Finally, the function should return the list of all expressions that evaluate to the target.

Let’s breakdown the Python code snippet for the problem statement, without writing any code:

Understanding the problem: The problem is about generating possible expressions by inserting addition, subtraction, or multiplication between the digits of a given string number. We aim to find the expressions that evaluate to a certain target value.
Initial setup: Start by creating a set to store the resultant expressions (to avoid any duplicates). The function named ‘backtrack’ is defined to recursively build the possible expressions and check if they match the target.
Recursion and Backtracking: Recursive function (backtrack) is used to generate all possible combinations and explore all possible expressions. For each combination, the function calls itself until all digits are consumed (base case). If a generated expression evaluates to the target, it’s added to the set of solutions.
Backtracking logic:
- Add the number (if we’re at the start of the string).
- Subtract the number.
- Multiply with the last number.
This is done by tracking the current total, the last number added or subtracted, and the current expression.
Dealing with zero: In scenarios where the number ‘0’ appears at the start of a multi-digit number (like ‘05’), it is treated as a single-digit number since no number starts with ‘0’. Hence, if a ‘0’ is encountered, we break the loop to avoid this scenario.
Termination: When all digits have been processed, and the total equals the target, the expression is added to the set of answers.
Returning the result: Finally, the function will return a list of expressions (in string format) from the set that we’ve accumulated.
Final assembly: To solve the problem, the functions and techniques from the drills are combined. The recursive function is called initially with the first index, a total of zero, last number as zero, and an empty string. After finding all valid expressions, the function converts the set into a list and returns it as the final result.

These are the key components of the problem-solving approach based on the provided code. It would be beneficial to practice these drills in order to solidify understanding and enhance coding skills.

Targeted Drills in Python

Basic Mathematical Operations: Practice performing basic arithmetic operations in Python.

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# Addition
print(5 + 3)  # Output: 8

# Subtraction
print(5 - 3)  # Output: 2

# Multiplication
print(5 * 3)  # Output: 15

String to Integer Conversion: Practice converting string to integer in Python.

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num_str = "123"
num_int = int(num_str)
print(num_int)  # Output: 123

Use of Recursive Functions (Backtracking): Practice writing simple recursive functions.

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def countdown(n):
    if n <= 0:
        print("Blastoff!")
    else:
        print(n)
        countdown(n-1)
        
countdown(5)  # Output: 5 4 3 2 1 Blastoff!

Manipulating Strings: Practice string concatenation in Python.

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str1 = "Hello, "
str2 = "world!"
print(str1 + str2)  # Output: Hello, world!

Control Flow:
- If-Else Statements: Practice writing if-else statements in Python.

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num = 7
if num > 10:
    print("Greater than 10")
else:
    print("Less than or equal to 10")  # Output: Less than or equal to 10

- **Loops**:
  Practice writing loops in Python.

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for i in range(5):
    print(i)  # Output: 0 1 2 3 4

Working with Collections:
- Arrays: Practice working with arrays (lists) in Python.

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numbers = [1, 2, 3, 4, 5]
print(numbers[2])  # Output: 3

- **Sets**:
  Practice working with sets in Python.

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unique_numbers = {1, 2, 2, 3, 4, 4, 4, 5, 5}
print(unique_numbers)  # Output: {1, 2, 3, 4, 5}