Stack Applications. parsing a postfix expression evaluating postfix expressions. converting infix sums from infix to postfix
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1 Stack Applications 1 Evaluating Postfix Expressions parsing a postfix expression evaluating postfix expressions 2 Converting Infix to Postfix Expressions converting infix sums from infix to postfix 3 Extended Infix to Postfix infix expressions with brackets MCS 360 Lecture 15 Introduction to Data Structures Jan Verschelde, 2 October 2017 Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
2 Stack Applications 1 Evaluating Postfix Expressions parsing a postfix expression evaluating postfix expressions 2 Converting Infix to Postfix Expressions converting infix sums from infix to postfix 3 Extended Infix to Postfix infix expressions with brackets Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
3 postfix expressions A postfix expression is an arithmetic expression where we first write the two operands and then the operator. Some examples with corresponding infix notation: postfix infix 9 8 * 9 * * (4 + 5) * * (3 + 4) * 5 Advantage of postfix notation: no brackets needed. Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
4 reading from strings Including <sstream> we can read from strings: string expression; getline(cin,expression, \n ); istringstream tokens(expression); char next_character; while(tokens >> next_character) cout << next_character; Observe: while(tokens >> next_character) means: as long as extraction of a character succeeds, tokens >> next_character reads next nonblank character, so we naturally skip spaces. Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
5 parsing operators const string operators = "+-*/"; istringstream arguments(expression); bool valid = true; while(arguments >> next_character) if(!isdigit(next_character)) if(operators.find(next_character) == string::npos) cout << next_character << " is invalid operator" << endl; valid = false; break; else cout << "operator : " << next_character << endl; Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
6 getting operands else arguments.putback(next_character); int value; arguments >> value; cout << "operand : " << value << endl; Observe: the putback() method unreads a character, with arguments >> value we read any integer. Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
7 running parsing expression Reading from a string twice: $ /tmp/parsing Give a postfix expression : * -> your expression : " *" -> your expression : "13445+*" -> parsing expression... operand : 13 operand : 44 operand : 5 operator : + operator : * $ Note: represent -5 as Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
8 Stack Applications 1 Evaluating Postfix Expressions parsing a postfix expression evaluating postfix expressions 2 Converting Infix to Postfix Expressions converting infix sums from infix to postfix 3 Extended Infix to Postfix infix expressions with brackets Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
9 Evaluating a Postfix Expression Given a postfix arithmetical expression, compute its value. Using a stack, the algorithm evaluates expression as: for every item in expression do if (item is operand) then push item on the stack else right = pop from stack left = pop from stack push (left item right) on the stack result = pop from stack Our pop is stack<int>.top() followed by stack<int>.pop() Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
10 Tracing the Evaluation $ /tmp/test_eval Give a postfix expression : * -> your expression : " *" pushing 3 pushing 4 pushing 5 evaluating + popping 5 popping 4 pushing 9 evaluating * popping 9 popping 3 pushing 27 the value of " *" : 27 $ Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
11 an Object-Oriented Implementation #include "Postfix_Evaluator.h" #include <iostream> #include <string> using namespace std; int main() string expression; cout << "Give a postfix expression : "; getline(cin,expression, \n ); Postfix_Evaluator p(expression); cout << "the value of \"" << expression << "\" : " << p.value() << endl; Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
12 data members of class #ifndef POSTFIX_EVALUATOR_H #define POSTFIX_EVALUATOR_H #include <stack> #include <string> class Postfix_Evaluator private: std::string expression; static const std::string operators; std::stack<int> operands; We store given expression, the definition of the operators, and the stack of operands. Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
13 functions in class private: bool is_operator(char c); // returns true if c is an operator int eval_operator(char c); // returns the value of one operator public: Postfix_Evaluator(std::string s); // stores the expression int value(); // returns the value of the expression The functions declared under private are auxiliary to the public value method. Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
14 the file postfix_evaluator.cpp const std::string Postfix_Evaluator::operators = "+-*/"; bool Postfix_Evaluator::is_operator(char c) return (operators.find(c)!= std::string::npos); int Postfix_Evaluator::eval_operator(char c) int right = this->operands.top(); this->operands.pop(); int left = this->operands.top(); this->operands.pop(); switch(c) case + : return left + right; case - : return left - right; case * : return left * right; case / : return left / right; Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
15 the value() method int Postfix_Evaluator::value() std::istringstream tokens(this->expression); char c; while(tokens >> c) if(is_operator(c)) int v = eval_operator(c); this->operands.push(v); else tokens.putback(c); int operand; tokens >> operand; this->operands.push(operand); return operands.top(); Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
16 Stack Applications 1 Evaluating Postfix Expressions parsing a postfix expression evaluating postfix expressions 2 Converting Infix to Postfix Expressions converting infix sums from infix to postfix 3 Extended Infix to Postfix infix expressions with brackets Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
17 from infix to postfix Given a sum of integers, operators + and -, convert: $ /tmp/converting Give an infix sum : > your expression : " " postfix notation of " " \ is " " $ Algorithm stores the previous operator. For every item in the infix expression: if item is operand, write to result; if item is operator, write the previous operator and store item as the previous operator. At end, write the previous operator to result. Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
18 writing to string #include <iostream> #include <sstream> #include <string> using namespace std; int main() string expression; cout << "Give an infix sum : "; getline(cin,expression, \n ); istringstream arguments(expression); char next_character; char previous_op = ; ostringstream postfix_expression; Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
19 converting expression while(arguments >> next_character) if(!isdigit(next_character)) // assume + or - if(previous_op!= ) postfix_expression << previous_op << " "; previous_op = next_character; else arguments.putback(next_character); int value; arguments >> value; postfix_expression << value << " "; postfix_expression << previous_op; Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
20 Stack Applications 1 Evaluating Postfix Expressions parsing a postfix expression evaluating postfix expressions 2 Converting Infix to Postfix Expressions converting infix sums from infix to postfix 3 Extended Infix to Postfix infix expressions with brackets Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
21 Converting Infix to Postfix Consider infix expressions with operators +, -, *, and /, without brackets. Comparing "2 + 3/4" with "2/3 + 4", or equivalently "2 3 4 / +" with "2 3 / 4 +" we see that precedence order of operators matters. We maintain a stack of operators. Comparing the current operator with the top of the stack, we pop operators as long as their precedence is higher or equal than the precedence of the current operator. Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
22 running the conversion $ /tmp/test_convert Give an infix expression : 2 + 3/4 pushing + pushing / popping / popping + the postfix notation of "2 + 3/4" : / + $ /tmp/test_convert Give an infix expression : 2/3 + 4 pushing / popping / pushing + popping + the postfix notation of "2/3 + 4" : 2 3 / 4 + Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
23 The Conversion Algorithm Given expression in infix notation. for every item in expression do if (item is operator) then process the operator; else write item to result; pop all operators from stack and write. To process the current operator: if (the stack is empty) then push current operator on the stack; else if (current operator > operator on top) then push current operator on the stack; else do pop operator on top and write; break if (stack is empty); while(current operator operator on top); push current operator on the stack. Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
24 an Object-Oriented Implementation #include "Infix_to_Postfix.h" #include <iostream> #include <string> using namespace std; int main() string expression; cout << "Give an infix expression : "; getline(cin,expression, \n ); Infix_to_Postfix e(expression); cout << "the postfix notation of \"" << expression << "\" : " << e.convert() << endl; Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
25 data members #ifndef INFIX_TO_POSTFIX_H #define INFIX_TO_POSTFIX_H #include <stack> #include <sstream> #include <string> class Infix_to_Postfix private: std::string expression; static const std::string operators; std::stack<char> operator_stack; std::ostringstream postfix_expression; To share the result postfix_expression between various methods, we make it a data attribute. Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
26 methods of the class private: bool is_operator(char c); // returns true if c is an operator int precedence_operator(char c); // returns precedence of operator c void process_operator(char c); // processes one operator public: Infix_to_Postfix(std::string s); // stores the expression std::string convert(); // returns the equivalent postfix expression Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
27 the file infix_to_postfix.cpp const std::string Infix_to_Postfix::operators = "+-*/"; bool Infix_to_Postfix::is_operator(char c) return (operators.find(c)!= std::string::npos); int Infix_to_Postfix::precedence_operator(char c) if(c == + c == - ) return 1; else return 2; Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
28 the convert() method std::string Infix_to_Postfix::convert() std::istringstream tokens(this->expression); char c; while(tokens >> c) if(is_operator(c)) process_operator(c); else tokens.putback(c); int operand; tokens >> operand; this->postfix_expression << operand << " "; while(!this->operator_stack.empty()) char c = this->operator_stack.top(); this->postfix_expression << c << " "; this->operator_stack.pop(); return this->postfix_expression.str(); Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
29 process_operator void Infix_to_Postfix::process_operator(char c) if(this->operator_stack.empty()) this->operator_stack.push(c); else char top_op = this->operator_stack.top(); if(precedence_operator(c) > precedence_operator(top_op)) this->operator_stack.push(c); else do this->postfix_expression << top_op << " "; this->operator_stack.pop(); if(this->operator_stack.empty()) break; top_op = this->operator_stack.top(); while(precedence_operator(c) <= precedence_operator(top_op)); this->operator_stack.push(c); Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
30 Stack Applications 1 Evaluating Postfix Expressions parsing a postfix expression evaluating postfix expressions 2 Converting Infix to Postfix Expressions converting infix sums from infix to postfix 3 Extended Infix to Postfix infix expressions with brackets Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
31 Infix with Brackets to Postfix To convert "3 * ( )" to " *" we view brackets ( and ) as operators. To process a bracket operator: an opening bracket ( is pushed on the stack at a closing bracket ), we pop all operators up to and including the matching opening bracket. Modifications to Infix_to_Postfix: 1 Add ( and ) to static const string operators. 2 Define precedence of the brackets as zero. 3 Extend process_operator to test for ) and (. Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
32 Summary + Exercises Expression manipulations use stack, we covered 5.4. Exercises: 1 Define exception Stack_Empty to be thrown when postfix evaluator does top or pop on empty stack. Modify the class Postfix_Evaluator and test it on invalid postfix expressions. 2 Adjust one of our own stack applications of L-14 with a to_string() method that sends the content of the stack to a string. Use this to_string() to print the evolution of the stack in Postfix_Evaluator. 3 Extend the Infix_to_Postfix methods so they throw exceptions when the stack turns out empty when an invalid infix expression is given. Provide a handler (try - catch block) in the test program. Introduction to Data Structures (MCS 360) Stack Applications L-15 2 October / 32
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