Expression Add Operators

To solve this coding challenge, the goal is to find all valid combinations of adding the binary operators '+', '-', and '*' to a string of digits

num

such that the resultant expression evaluates to a given

target

value. The primary approach involves a depth-first search (DFS) to explore all possible combinations and backtrack if necessary to find valid expressions. Here's a comprehensive breakdown of our approach:

Explanation

1. Backtracking and Recursive Exploration : We use backtracking to explore every possible combination of operators inserted between the digits. The recursion will help in exploring all potential combinations by:
• Adding the current operator and number to the sequence.
• Keeping track of the cumulative total.
• Adjusting based on the previously added number to account for multiplication (since multiplication has a higher precedence over addition & subtraction).
2. Handling Leading Zeros : A special check is required to ensure that numbers with leading zeros (except for '0' itself) are not considered valid operand segments as per the problem constraints.
3. Intermediate State Variables :
• index
  : This is the current position in the string of digits.
• expression
  : The current formed expression.
• value
  : The current evaluated value of the expression.
• prev
  : The last evaluated number in the expression (useful for correct multiplication handling).
4. Base Case :
• When
  index
  reaches the length of the digit string:
• Check if the current evaluated
  value
  matches the
  target
  .
• If yes, then we append the
  expression
  to the results list.
5. Recursive Case :
• Loop through each possible split of the remaining digits.
• Form new numbers and recursively attempt to add each operator.
• Ensure that partial calculations correctly account for operator precedence, especially the multiplication which slightly alters how previous values are considered.
Here is how this can be represented in pseudocode with relevant comments:

                                            
# Function to generate all valid expressions that evaluate to the target value
function addOperators(num, target):
result = []  # Initialize the result list to store valid expressions

# Helper function to perform the backtrack search
function backtrack(index, expression, value, prev):
# Base Case: If we've reached the end of num
if index == length of num:
# If the current expression equals the target value
if value == target:
result.append(expression)  # Add to the result list
return  # Return to backtrack

# Loop through the string to split into substrings
for i from index to length of num:
# Skip if the number has leading zeros
if i != index and num[index] == '0':
break

# Parse current number segment
current_num = integer value of num substring from index to i+1

# If at index 0, initialize the first number segment without an operator
if index == 0:
backtrack(i + 1, str(current_num), current_num, current_num)
else:
# Try adding current number with addition operator
backtrack(i + 1, expression + '+' + str(current_num), value + current_num, current_num)

# Try adding current number with subtraction operator
backtrack(i + 1, expression + '-' + str(current_num), value - current_num, -current_num)

# Try adding current number with multiplication operator
backtrack(i + 1, expression + '*' + str(current_num), value - prev + prev * current_num, prev * current_num)

# If num isn't empty, initiate the backtracking process
if num:
backtrack(0, '', 0, 0)

# Return all valid expressions
return result

                                        

Detailed Steps in Pseudocode

6. Initialization :
• Define the main function
  addOperators
  that accepts
  num
  and
  target
  .
• Initialize an empty list
  result
  to store the valid expressions.
7. Define backtrack function :
• index
  to track the current digit position in
  num
  .
• expression
  to build the current string expression.
• value
  to retain the evaluated value of the current expression.
• prev
  to hold the last evaluated number (useful for handling multiplication).
8. Handling Base Case :
• If
  index
  is equal to the length of
  num
  :
• Check if
  value
  equals
  target
  . If true, append
  expression
  to
  result
  .
• Return to backtrack.
9. Recursive Exploration :
• Use a loop to explore possible splits.
• Skip invalid segments (leading zeros).
• Parse substring to number
  current_num
  .
• Recursively append combinations (+, -, *) considering appropriate precedence.
10. Return the result list after processing :
• Only invoke backtrack if
  num
  is not empty.
• Return the generated
  result
  list.

By carefully managing recursive states and ensuring correct operator precedence, this approach efficiently explores all potential operator placements to find valid expressions matching the target value.