BasicCalculator IV

Identifying Problem Isomorphism

“Basic Calculator IV” is unique and significantly more complex compared to “Basic Calculator III”.

“Basic Calculator IV” involves not only the basic arithmetic operations (+,-,*,/) and handling parentheses like “Basic Calculator III” but also introduces variables and the handling of these variables in an expression. In this problem, you have an expression string, and an evaluation map that contains the values of certain variables. Your task is to evaluate the expression and return the answer as a list of tokens in lexicographical order.

A problem that relates to the handling of variables and expressions in “Basic Calculator IV” is “Evaluate Division” (LeetCode #399). In “Evaluate Division”, you are given equations in the format A / B = k, where A and B are variables represented as strings, and k is a real number (floating-point number). Your task is to evaluate queries, which are given in the same format, and return the answers. If the query cannot be evaluated, return -1.0.

In both problems, there is the need to evaluate expressions involving variables. However, “Evaluate Division” only involves division and multiplication while “Basic Calculator IV” involves all four basic arithmetic operations. Moreover, the format of the expressions and queries differ significantly. Therefore, this is an approximate mapping with some aspects of the problems being isomorphic (handling variables and evaluating expressions), while other aspects are not. This example serves to show that “Basic Calculator IV” has elements of both “Basic Calculator III” and “Evaluate Division”, and the strategies used to solve both problems may provide insights for solving “Basic Calculator IV”.

The “Basic Calculator IV” problem is a more complex problem that involves parsing and evaluating expressions, and handling variables. The problem requires a good understanding of string manipulation, data structure usage, and potentially recursion. Here are 10 problems of lesser complexity that could be useful for preparing to solve this problem:

  1. 20. Valid Parentheses: This problem introduces the use of stack for checking matching parentheses, which is an essential concept in expression parsing.

  2. 150. Evaluate Reverse Polish Notation: This problem deals with the evaluation of expressions, which is the main topic in “Basic Calculator IV”.

  3. 224. Basic Calculator: This problem is a simpler version of “Basic Calculator IV” and introduces the concept of evaluating arithmetic expressions with parentheses.

  4. 227. Basic Calculator II: This problem extends the previous one by introducing multiplication and division operations.

  5. 772. Basic Calculator III: This is a more complex version of “Basic Calculator” but simpler than “Basic Calculator IV”, it introduces handling the nested parentheses in the expressions.

  6. 394. Decode String: This problem can help you understand how to handle nested expressions in a string, which can be useful for “Basic Calculator IV”.

  7. 438. Find All Anagrams in a String: This problem provides practice for string manipulation, which is an essential skill in expression parsing problems.

  8. 692. Top K Frequent Words: While not directly related to parsing expressions, this problem provides practice in dealing with frequency counts, which may be useful for handling variables in “Basic Calculator IV”.

  9. 720. Longest Word in Dictionary: This problem helps in practicing dealing with strings and their comparisons.

  10. 739. Daily Temperatures: It’s a good problem for practicing stack operations.

These cover how to parse and evaluate expressions, as well as how to handle variables in expressions, which are all essential skills for solving the “Basic Calculator IV” problem.

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from functools import cmp_to_key

class Expression:
    def __init__(self):
        self.coefficient = 0
        self.variables = []

    def getValue(self):
        if not self.coefficient:
            return ""
        if not self.variables:
            return str(self.coefficient)
        return str(self.coefficient) + "*" + "*".join(self.variables)

def multiplyExpressions(expression1, expression2):
    result = Expression()
    result.coefficient = expression1.coefficient * expression2.coefficient
    if result.coefficient == 0:
        return result
    result.variables = list(sorted(expression1.variables + expression2.variables))
    return result

def integrateExpressionIntoStack(expressionStack, operationSigns, expression):
    if not operationSigns[-1]:
        return
    operation = operationSigns[-1][-1]
    match operation:
        case "+":
            expressionStack[-1].append([expression])
        case "-":
            expression.coefficient = - expression.coefficient
            expressionStack[-1].append([expression])
        case "*":
            previousExpressions = expressionStack[-1][-1]
            temporaryExpressions = [multiplyExpressions(previous, expression) for previous in previousExpressions]
            expressionStack[-1][-1] = temporaryExpressions
    operationSigns[-1].pop()

def integrateExpressionGroupIntoStack(expressionStack, operationSigns, group):
    if not operationSigns[-1]:
        return
    operation = operationSigns[-1][-1]
    match operation:
        case "+":
            expressionStack[-1].append(group)    
        case "-":
            updatedGroup = [updateCoefficientForNegation(expression) for expression in group]
            expressionStack[-1].append(updatedGroup)
        case "*":
            previousExpressions = expressionStack[-1].pop()
            multipliedExpressions = [multiplyExpressions(expression1, expression2) for expression1 in previousExpressions for expression2 in group]
            expressionStack[-1].append(multipliedExpressions)
    operationSigns[-1].pop()

def updateCoefficientForNegation(expression):
    expression.coefficient = -expression.coefficient
    return expression

def compareExpressions(expression1, expression2):
    variables1, variables2 = expression1.split("*"), expression2.split("*")
    if len(variables1) != len(variables2):
        return len(variables2) - len(variables1)
    return 1 if variables1 > variables2 else -1

def sumExpressionsAtCurrentLevel(currentLevel):
    aggregatedExpressions = {"":0}
    for groups in currentLevel:
        for expression in groups:
            if not expression.variables:
                aggregatedExpressions[""] += expression.coefficient
            else:
                key = "*".join(expression.variables)
                if key not in aggregatedExpressions:
                    aggregatedExpressions[key] = expression
                else:
                    aggregatedExpressions[key].coefficient += expression.coefficient
    result = [aggregatedExpressions[key] for key in sorted(aggregatedExpressions.keys(), key=cmp_to_key(compareExpressions)) if key != "" and aggregatedExpressions[key].coefficient]

    if aggregatedExpressions[""] != 0:
        constantExpression = Expression()
        constantExpression.coefficient = aggregatedExpressions[""]
        result.append(constantExpression)
    return result

def calculate(s, a, b):
    expressionStack, operationSigns = [[]], [["+"]]
    currentIndex, stringLength = 0, len(s)
    varToValueMapping = {x:y for x, y in zip(a, b)}
    while currentIndex < stringLength:
        if s[currentIndex] == " ":
            currentIndex += 1
            continue
        if s[currentIndex].isalpha():
            expression = generateVariableExpression(s, currentIndex, varToValueMapping)
            currentIndex += len(expression.variables[0]) - 1 if expression.variables else 0
            integrateExpressionIntoStack(expressionStack, operationSigns, expression)
        elif s[currentIndex].isdigit():
            expression = generateNumericExpression(s, currentIndex)
            currentIndex += len(str(expression.coefficient)) - 1
            integrateExpressionIntoStack(expressionStack, operationSigns, expression)
        elif s[currentIndex] in "+-*":
            operationSigns[-1].append(s[currentIndex])
        elif s[currentIndex] == "(":
            expressionStack.append([])
            operationSigns.append(["+"])
        elif s[currentIndex] == ")":
            currentLevelExpressions = sumExpressionsAtCurrentLevel(expressionStack.pop())
            operationSigns.pop()
            integrateExpressionGroupIntoStack(expressionStack, operationSigns, currentLevelExpressions)
        currentIndex += 1
    resultExpressions = sumExpressionsAtCurrentLevel(expressionStack.pop())
    return [expression.getValue() for expression in resultExpressions]

def generateVariableExpression(s, currentIndex, varToValueMapping):
    expression = Expression()
    variable = s[currentIndex]
    while currentIndex+1 < len(s) and s[currentIndex+1].isalpha():
        variable += s[currentIndex+1]
        currentIndex += 1
    if variable in varToValueMapping:
        expression.coefficient = varToValueMapping[variable]
    else:
        expression.coefficient = 1
        expression.variables = [variable]
    return expression

def generateNumericExpression(s, currentIndex):
    expression = Expression()
    number = int(s[currentIndex])
    while currentIndex+1 < len(s) and s[currentIndex+1].isdigit():
        number = number * 10 + int(s[currentIndex+1])
        currentIndex += 1
    expression.coefficient = number
    return expression

class Solution:
    def basicCalculatorIV(self, expression: str, evalvars: List[str], evalints: List[int]) -> List[str]:
        return calculate(expression, evalvars, evalints)

Problem Classification

This problem falls under the String Manipulation and Mathematical Calculation domain. It involves parsing a mathematical expression given as a string, substituting variables with their corresponding values, performing the operations in the correct order, and then simplifying the result.

Components Identification:

  1. A mathematical expression is given as a string (expression). This expression contains variables, integers, and arithmetic operations, including parentheses.

  2. An evaluation map is provided as two separate lists (evalvars and evalints). Each variable in evalvars corresponds to an integer in evalints.

  3. The task is to evaluate the expression by substitifying variables with their corresponding values and performing the operations. Then, the result is simplified and returned in a specific format.

This problem is a Parsing and Evaluation Problem. The core of this problem lies in parsing a mathematical expression, evaluating it with a given mapping, and then presenting the result in a simplified format.

To solve this problem, skills in string parsing, handling mathematical operations, and managing data structure like a dictionary or a map for variable-to-value mapping are required. There’s also a need for understanding order of operations (BIDMAS/PEDMAS), algebraic simplification, and proper formatting of the result.

The input is a mathematical expression as a string, and the output is a simplified and properly formatted result of the evaluation of the expression, both of which involve significant string manipulation. The required operations include addition, subtraction, and multiplication, hence the mathematical calculation domain. The necessity to evaluate the expression by substituting variables with their respective values and the specific formatting of the result contributes to the problem’s classification as a Parsing and Evaluation Problem.

Clarification Questions

What are the clarification questions we can ask about this problem?

Identifying Problem Isomorphism

Can you help me with finding the isomorphism for this problem?

Which problem does this problem map to the corresponding isomorphic problem on Leetcode ?

Thought Process

From the problem statement, the task is to simplify a given mathematical expression. This involves:

  1. Replacing variables with their corresponding values.
  2. Evaluating the expression in accordance with the order of operations: parentheses first, then multiplication, and finally addition and subtraction.
  3. Converting the result to a specific format.

Given this, we can make some preliminary conclusions about how to approach the problem. Firstly, we would need to parse the string. This could potentially involve implementing a parser for the mathematical expression. However, Python already provides us with a built-in function, eval(), that can take a string and evaluate it as a Python expression. This could save us a lot of trouble, but we need to remember to substitute the variables with their values before using this function.

The formatted output should be a list of strings, where each string represents a term in the result. The terms are sorted first by degree (highest first), and then lexicographically.

Given these cues, here’s a step-by-step plan to solve the problem:

  1. Parse the input expression to identify the variables and the corresponding arithmetic operations.
  2. Create a dictionary mapping from the list of variables and integers given.
  3. Replace each variable in the expression with its corresponding value using the dictionary. At this point, we have an expression with only numbers and arithmetic operators.
  4. Evaluate this numerical expression using the eval() function.
  5. Format the output in the required form.

Language Agnostic Coding Drills

  1. Identify and dissect the distinct coding concepts in the Basic Calculator IV code:

  • Variables and Basic Data Structures: Understand and utilize the fundamental coding constructs such as variables, lists, dictionaries, and classes.

  • Control Flow: Use of conditional (if-else) and looping (while) structures for directing the execution of the program.

  • Function Definition and Invocation: Define functions that encapsulate a specific piece of functionality and then call these functions when needed.

  • String and List Manipulation: Process and manipulate strings and lists to extract or combine information, including splitting and joining strings, appending to lists, and iterating over list items.

  • Exception Handling: Detect and handle exceptions or errors that may occur during the program’s execution, particularly index out of range errors.

  • Object-Oriented Programming: Use classes to encapsulate related data and behaviors. This includes understanding how to define a class, instantiate an object, and use object methods and attributes.

  • Recursion and Stack Operations: Use recursive calls and stack data structure to solve problems that have nested structures. This concept includes understanding the stack’s LIFO (last-in, first-out) behavior and how it can be used to keep track of intermediate results in complex calculations.

  • Use of In-built Libraries and Functions: Use of Python’s in-built functions and libraries like functools.

  • Operator Overloading: Overload operators to enable the use of standard Python operators with user-defined objects.

  • Pattern Matching: Use of Python’s pattern matching capabilities for more expressive and readable code.

  1. Order of increasing difficulty:

  • Variables and Basic Data Structures: The building blocks of any programming language, a very fundamental concept.

  • Control Flow: Directs the program flow, also fundamental but requires understanding of conditional logic.

  • Function Definition and Invocation: Functions are crucial for code organization and reusability, introducing a bit more complexity.

  • String and List Manipulation: Requires understanding of how to manipulate these mutable data types.

  • Exception Handling: Important for robust code, but requires anticipation of potential errors.

  • Object-Oriented Programming: Adds a layer of abstraction, which can make the code cleaner but is a step up in complexity.

  • Use of In-built Libraries and Functions: Requires knowledge of existing libraries, their use cases, and functions.

  • Recursion and Stack Operations: Recursive thinking can be a challenge for beginners, and understanding stack operations requires a good understanding of data structures.

  • Operator Overloading: Requires a solid understanding of classes and objects, as well as familiarity with the concept of magic methods in Python.

  • Pattern Matching: Even though this is a powerful feature, it’s somewhat advanced and requires a good understanding of patterns and how to formulate them.

  1. Problem-solving approach:

  • Start by understanding the problem requirements and constraints. Identify the inputs and the expected outputs. From the problem statement, we need a calculator that can evaluate expressions given certain variable evaluations.

  • Create data structures to hold the essential information. In this case, we’ll need a dictionary for variable evaluation and a class to represent expressions.

  • Identify where control flow logic is necessary to handle different situations in the program. For example, different cases for handling spaces, digits, variables, arithmetic operations, and brackets in the input expression.

  • Define functions for repeated operations, like merging expressions and combining expression groups. This will make the code cleaner and more organized.

  • Implement the main function to parse the input string and perform calculations based on the logic defined in the helper functions. Utilize a stack to handle the nested structure of expressions.

  • Be aware of potential exceptions or errors, such as attempting to access an out-of-range index, and handle them properly.

  • Test the solution with various test cases to ensure it handles all possible scenarios correctly.

Each of these “drills” or steps contributes to the overall solution by breaking down the problem into manageable parts. By understanding and implementing each part independently, you can construct a complex solution without becoming overwhelmed by the whole problem. The key is to identify patterns, define reusable functions, and keep track of the state of your program using appropriate data structures.

Targeted Drills in Python

  1. Python Coding Drills for General Concepts:
  • Variables and Basic Data Structures:
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# Variables
myVar = 5
print(myVar)

# List
myList = [1, 2, 3, 4, 5]
print(myList[2]) # Accessing elements

# Dictionary
myDict = {'apple': 1, 'banana': 2}
print(myDict['apple']) # Accessing elements
  • Control Flow:
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# If-Else
num = 10
if num > 5:
    print("Greater")
else:
    print("Not Greater")

# While Loop
i = 0
while i < 5:
    print(i)
    i += 1
  • Function Definition and Invocation:
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# Function Definition
def greeting(name):
    return f"Hello, {name}!"

# Function Invocation
print(greeting("John"))
  • String and List Manipulation:
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# String Manipulation
strVar = "Hello, World!"
print(strVar.split(',')) # Splitting a string

# List Manipulation
listVar = [1, 2, 3]
listVar.append(4) # Adding to a list
print(listVar)
  • Exception Handling:
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# Exception Handling
try:
    print(undefVar)
except NameError:
    print("Variable is not defined!")
  • Object-Oriented Programming:
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# Class Definition and Object Instantiation
class Animal:
    def __init__(self, name):
        self.name = name

    def speak(self):
        return f"{self.name} says hello!"

# Object Instantiation
dog = Animal("Dog")
print(dog.speak())
  • Recursion and Stack Operations:
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# Recursion
def factorial(n):
    if n == 1:
        return 1
    else:
        return n * factorial(n-1)

print(factorial(5))

# Stack Operations
stack = []
stack.append('a')
stack.append('b')
stack.append('c')
print(stack.pop())  # Returns 'c'
  • Use of In-built Libraries and Functions:
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# In-built Functions
print(len("Hello"))  # Returns 5
print(sum([1, 2, 3, 4, 5]))  # Returns 15

# In-built Libraries
import math
print(math.sqrt(16))  # Returns 4.0
  • Operator Overloading:
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# Operator Overloading
class Point:
    def __init__(self, x=0, y=0):
        self.x = x
        self.y = y

    def __str__(self):
        return "({0},{1})".format(self.x, self.y)
        
    def __add__(self,other):
        x = self.x + other.x
        y = self.y + other.y
        return Point(x,y)

p1 = Point(1,2)
p2 = Point(3,4)
print(p1 + p2)  # Returns (4,6)
  • Pattern Matching (Python 3.10 feature):
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# Pattern Matching
def http_response_status(code):
    match code:
        case 200:
            return "OK"
        case 404:
            return "Not Found"
        case _:
            return "Unknown"

print(http_response_status(200))  # Returns "OK"
  1. Problem-Specific Concept Drills:
  • Parsing and Evaluating Mathematical Expressions: This is necessary as our problem requires us to evaluate a mathematical

expression represented as a string.

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# Parsing and Evaluating Expressions
from math import *
expression = "2*(3+4)"
print(eval(expression))  # Returns 14
  • Using a Stack to Evaluate Expressions: This is crucial for evaluating mathematical expressions that include parentheses.
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# Stack Evaluation of Expressions
def evaluate(expression):
    stack = []
    for char in expression:
        if char == ')':
            inside_parentheses = ''
            while stack[-1] != '(':
                inside_parentheses = stack.pop() + inside_parentheses
            stack.pop()  # remove '('
            stack.append(str(eval(inside_parentheses)))
        else:
            stack.append(char)
    return eval(''.join(stack))

print(evaluate("(2+3)*4"))  # Returns 20
  1. Integrating the Coding Drills:

Once we’ve implemented the individual components, we need to integrate them to form the complete solution. We start by parsing the expression and identifying variables and their values. We then build the necessary classes to hold and manipulate our mathematical expressions. We implement operator overloading to handle addition, subtraction, and multiplication operations directly on our custom objects. We use a stack-based approach to evaluate the expression and ensure correct precedence of operations.

Throughout this process, we’ll use control flow, functions, string and list manipulation, exception handling, recursion, and Python’s in-built libraries and functions - concepts we’ve practiced through the drills. It’s a complex process, but by breaking it down into these components and tackling them one by one, we can build up to the complete solution.