1.4 expression tree
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The document discusses expression trees and algorithms for evaluating postfix, infix, and constructing binary expression trees. It explains that a binary expression tree represents a mathematical expression where each leaf node contains an operand and each non-leaf node contains an operator. The tree shows the order of operations based on the level of each node. The algorithms for evaluating infix and constructing the tree are similar except that pointers to nodes are pushed onto the stack instead of values.
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arithmetic expressions that contains any of the five arithmetic operators (+, -, *, /, %).
This exercise requires three distinct steps, namely:-
Verify that the infix arithmetic expression (the original expression), that may contain regular
parentheses, is properly formed as far as parentheses are concerned.
If the parenthesized expression is properly formed, convert the expression from an infix
expression to its equivalent postfix expression, called Reverse Polish Notation (RPN) named
after the Polish Mathematician J. Lukasiewics.
Evaluate the postfix expression, and print the result.
Step 1 - Verify that the expression
Given an arithmetic expression, called an infixed expression, to verify that it is properly formed
as far as parentheses are concerned, do the following:
Create an empty stack to hold left parenthesis ONLY.
Scanned the arithmetic expression from left to right, one character at a time.
While there are more characters in the arithmetic expression
{
If the character is a left parenthesis ‘(‘, push it on to the stack. However if the character is a right
parenthesis, ‘)’, visit the stack and pop the top element from off the stack.
}
If the stack contains any element at the end of reading the arithmetic expression, then the
expression was not properly formed.
Step 2 - Convert infixed expression to postfix
Given that an arithmetic expression is properly form with respect to parentheses, do the
following:
Create an empty stack to hold any arithmetic operators and left parenthesis, ONLY.
A string to contain the postfix expression – the output from this conversion.
Scan the arithmetic expression from left to right.
While the are more symbols in the arithmetic expression,
{
After a symbol is scanned, there are four (4) basic rules to observed and apply accordingly:
If the symbol is an operand (a number), write it to the output string.
If the symbol is an operator and if the stack is empty, push the symbol on the stack.
Otherwise, if the symbol is either ‘(‘ or ‘)’, check for the following conditions:
If the symbol is ‘(‘, push on to the stack,
Otherwise
If the symbol is ‘)’
{
Pop everything from the operator stack down to the first ‘(‘. Write each item
popped from the stack to the output string. Do not write the item ‘)’. Discard it.
}
If the symbol scanned is an arithmetic operator, check for the following and apply accordingly:
If the operator on the top of the stack has higher or equal precedence, that operator is popped
from off the stack, and is written to the to the output string. This process is continues until one of
two things happen:
Either the first ‘(‘ is encountered. When this occurs, the ‘(‘ is removed from the stack and is
discarded, and the recently scanned symbol is placed on the stack
OR
The operator on the stack has lower preced.
2. Stack. Write a program that uses the stack class (you can use.pdf
2. Stack. Write a program that uses the stack class (you can use the stack type defined in the
lectures, or you can define your own stack type) to solve a problem. The problem contains two
tasks, the first one is to convert an infix arithmetic expression into a postfix arithmetic
expression; the second task is to evaluate arithmetic expressions written in postfix form.
In infix form, the operator of an arithmetic statement is in-between every pair of operands. For
example:
1.5 + 2.3
2.0 * (3.3 + 4.5)
(2.1 + 3.0) * 4.8
In postfix form, the operands of an arithmetic statement appear followed by the operator. One of
the virtues of postfix form is that expressions can be written without the need for parentheses.
Here are some examples of arithmetic expressions written in postfix form:
1.5 2.3 + // Equivalent to 1.5 + 2.3
2.0 3.3 4.5 + * // Equivalent to 2.0 * (3.3 + 4.5)
2.1 3.3 + 4.8 * // Equivalent to (2.1 + 3.0) * 4.8
Please write a C++ program that uses an operator stack to convert an infix arithmetic expression
that the user enters into a postfix arithmetic expression, and then use a float stack to evaluate
postfix arithmetic.
Your program should support all five of the arithmetic operators, +, - , *, and /.
When users input the infix arithmetic expressions, the operands, operators and parenthesis
symbols are separated by spaces. The character # marks the end of the expression.
The infix arithmetic expressions only contain operands, operators and parenthesis. The operands
are float numbers, and the operators are +, -, * and /. If there are letters in the infix arithmetic
expressions, the “invalid expression” message should be displayed.
Here is an algorithm for evaluating an arithmetic expression written in postfix form with the aid
of a stack of floats.
• Scan the expression from left to right.
• When encounter an operand or a float, push it on the stack.
• When encounter an arithmetic operator, pop the top two numbers off the stack and use them as
operands for the indicated operation. Push the resulting number back on the stack.
• when you reach the end of the expression, the result of the calculation should be the sole item
on the stack.
Please show that that the program is working correctly by converting the following infix
arithmetic expressions and then computing the obtained postfix arithmetic expressions:
6.5 * 6.5 - 4 * 1.5 * 1.44 #
3.1 * (2.5 + 4.7 * 1.6) + 8.6 #
70.0 - (10.5 + 6) / 0.5 + 18.2 #
The following steps will convert an infix expression into a postfix expression:
1. Create an empty string stack called opStack for keeping operators. Create an empty string
array called postfixEx for postfix expression.
2. read the infix expression left to right.
2.1. If the string is an operand, append it to the postfixEx array.
2.2. If the string is a left parenthesis, push it on the opStack.
2.3. If the string is a right parenthesis, pop the opStack, append each operator to the postfixEx
array, until you see the corresponding left parent.
Unit 2 application of stack
The document discusses arithmetic expressions and how to evaluate them using stacks. It defines arithmetic expressions and their components. It then describes the rules for evaluating expressions, including order of operations. Different notation styles for expressions are introduced - infix, prefix and postfix. An algorithm is provided for converting infix expressions to postfix notation using a stack. Finally, the document explains how to evaluate a postfix expression using a stack.
COMP1603 Stacks and RPN 2023 Recording (2).pptx
This document discusses stacks and their applications, including evaluating postfix expressions and converting infix to postfix notation. It provides examples and algorithms for evaluating postfix expressions using a stack, as well as the infix to postfix conversion algorithm. The algorithm pushes operators onto a stack and displays or pops operators based on precedence. Practice problems are included to convert infix expressions to postfix notation.
Unit 3 Stacks and Queues.pptx
1. The document discusses stacks and queues as linear data structures. It describes stack operations like push and pop and provides algorithms for implementing these operations.
2. An example program for implementing a stack using an array is presented, with functions defined for push, pop, and display operations on the stack.
3. Applications of stacks discussed include reversing a list or string by pushing characters onto a stack and popping them off in reverse order, and converting infix expressions to postfix using a stack.
computer notes - Evaluating postfix expressions
The document discusses evaluating postfix expressions using a stack. It provides an algorithm to evaluate a postfix expression by pushing operands onto a stack and popping operands to apply operators. An example is worked through step-by-step. It also discusses converting infix expressions to postfix by pushing operators onto a stack and handling operator precedence.
Stack application
The document discusses different applications of stacks including parsing in compilers, undo functions, back buttons in browsers, and Postscript printing languages. It then explains how arithmetic expressions are evaluated using value and operator stacks by pushing operands and operators onto the stacks and popping them off according to precedence rules. Finally, it provides an algorithm for converting infix notation expressions to postfix notation using an operator stack.
Stacks Implementation and Examples
The document summarizes lecture 2 of the CIS-122 Data Structures course. It covers applications of stacks like arithmetic expressions, recursion, quicksort, and towers of Hanoi. It also discusses stack implementations using arrays and linked lists and provides examples of infix to postfix conversion and evaluating arithmetic expressions using a stack.
Data structure lab manual
The document describes implementing a queue using an array. It provides algorithms for enQueue() and deQueue() operations. EnQueue() inserts elements at the rear by incrementing rear and checking for full. DeQueue() deletes elements from the front by incrementing front and checking for empty. The queue uses front and rear pointers to manage insertion and deletion of elements based on FIFO principle using an underlying fixed-size array.
7 stacksqueues
This document discusses stacks and queues as data structures. It begins by describing stacks as having a last-in, first-out characteristic and common uses like method call order tracking. It then provides examples of implementing a stack using an array and manipulating elements by pushing and popping. The document also covers queues as having a first-in, first-out characteristic like a waiting line. It proposes a Queue interface and discusses challenges of array-based implementations versus using Java's LinkedList class.
Topic 2_revised.pptx
The document discusses various applications of stacks including reversing data, parsing, postponing operations, and backtracking. It provides examples of converting infix expressions to postfix notation using a stack and evaluating postfix expressions. The key stack operations of push, pop, and accessing the stack top are defined. Implementation of a stack using an array is also mentioned.
MO 2020 DS Stacks 3 AB.ppt
The document discusses evaluating arithmetic expressions in infix notation using stacks. It explains that two stacks will be used - an operand stack to store numbers and an operator stack to store operators. It then provides the step-by-step process for evaluating an expression using these stacks by popping operands, popping operators, and pushing results. An algorithm is given that iterates through the expression character by character, pushing operands and operators to the appropriate stacks according to precedence rules.
STACK AND ITS OPERATIONS IN DATA STRUCTURES.pptx
This document discusses stacks as a data structure. It defines a stack as a Last In First Out (LIFO) structure where the last item inserted is the first out. Key stack operations like push, pop, peek and isEmpty are described. Examples of stack applications include checking for balanced brackets, converting infix to postfix notation, implementing an undo/redo feature and recursion. The document also provides pseudocode for converting an infix expression to postfix using a stack and describes how a postfix calculator evaluates expressions.
Stacks,queues,linked-list
A stack is a linear data structure that follows the LIFO (last in, first out) principle. Elements are inserted and removed from the top of the stack. Common stack operations include push to add an element and pop to remove the top element. Stacks can be implemented using arrays or linked lists. Stacks are useful for operations like converting infix expressions to postfix and evaluating postfix expressions using a stack to hold operands. Queues follow the FIFO (first in, first out) principle with elements added to the rear and removed from the front. Common queue operations are enqueue to add and dequeue to remove elements. Queues can also be implemented using arrays or linked lists. Linked lists store elements in nodes with each node
Using RustBecause postfix expressions are simpler and faster to ev.pdf
The document describes transforming infix expressions to postfix expressions by implementing a stack-based algorithm. It defines the necessary data types like operators and tokens. The algorithm scans the infix tokens, pushing operands and left parentheses to the stack and popping operators to output until a matching right parenthesis is encountered. It also handles operator precedence by popping higher precedence operators during each new operator.
Applications of Stack
Polish & Reverse Polish Notations, Conversion from Infix to Prefix & Postfix, Evaluation of Expresions
Chapter 3.3
This document discusses numeric data types and operations in Java. It introduces common numeric data types like byte, short, int, long, float, and double. It explains arithmetic expressions and operators like addition, subtraction, multiplication, division, and remainder. It covers operator precedence, arithmetic expression evaluation order, and using parentheses to alter evaluation order. It also discusses the Math class for common mathematical functions.
Merge Sort implementation in C++ The implementation for Mergesort gi.pdf
Merge Sort implementation in C++ The implementation for Mergesort given in Section 7.4 takes
an array as input and soils that array. At the beginning of Section 7.4 there is a simple
pseudocode implementation for sorting a linked list using Mergesort. Implement both a linked
list-based version of Mergesort and the array-based version of Mergesort. and compare and
analyze their running times.
Solution
/* Merger sort array-based version */
/* C program for Merge Sort */
#include
#include
// Merges two subarrays of arr[].
// First subarray is arr[l..m]
// Second subarray is arr[m+1..r]
void merge(int arr[], int l, int m, int r)
{
int i, j, k;
int n1 = m - l + 1;
int n2 = r - m;
/* create temp arrays */
int L[n1], R[n2];
/* Copy data to temp arrays L[] and R[] */
for (i = 0; i < n1; i++)
L[i] = arr[l + i];
for (j = 0; j < n2; j++)
R[j] = arr[m + 1+ j];
/* Merge the temp arrays back into arr[l..r]*/
i = 0; // Initial index of first subarray
j = 0; // Initial index of second subarray
k = l; // Initial index of merged subarray
while (i < n1 && j < n2)
{
if (L[i] <= R[j])
{
arr[k] = L[i];
i++;
}
else
{
arr[k] = R[j];
j++;
}
k++;
}
/* Copy the remaining elements of L[], if there
are any */
while (i < n1)
{
arr[k] = L[i];
i++;
k++;
}
/* Copy the remaining elements of R[], if there
are any */
while (j < n2)
{
arr[k] = R[j];
j++;
k++;
}
}
/* l is for left index and r is right index of the
sub-array of arr to be sorted */
void mergeSort(int arr[], int l, int r)
{
if (l < r)
{
// Same as (l+r)/2, but avoids overflow for
// large l and h
int m = l+(r-l)/2;
// Sort first and second halves
mergeSort(arr, l, m);
mergeSort(arr, m+1, r);
merge(arr, l, m, r);
}
}
/* UTILITY FUNCTIONS */
/* Function to print an array */
void printArray(int A[], int size)
{
int i;
for (i=0; i < size; i++)
printf(\"%d \", A[i]);
printf(\"\ \");
}
/* Driver program to test above functions */
int main()
{
int arr[] = {12, 11, 13, 5, 6, 7};
int arr_size = sizeof(arr)/sizeof(arr[0]);
printf(\"Given array is \ \");
printArray(arr, arr_size);
mergeSort(arr, 0, arr_size - 1);
printf(\"\ Sorted array is \ \");
printArray(arr, arr_size);
return 0;
}
--------------------------------------------------------------------------------------------------------------------
/* merge sort link-list version
#include
#include
/* Link list node */
struct node
{
int data;
struct node* next;
};
/* function prototypes */
struct node* SortedMerge(struct node* a, struct node* b);
void FrontBackSplit(struct node* source,
struct node** frontRef, struct node** backRef);
/* sorts the linked list by changing next pointers (not data) */
void MergeSort(struct node** headRef)
{
struct node* head = *headRef;
struct node* a;
struct node* b;
/* Base case -- length 0 or 1 */
if ((head == NULL) || (head->next == NULL))
{
return;
}
/* Split head into \'a\' and \'b\' sublists */
FrontBackSplit(head, &a, &b);
/* Recursively sort the sublists */
MergeSort(&a);
MergeSort(&b);
/* answer = merge the two sorted lists together */
*.
Data structure and algorithm.(dsa)
1. Linear data structures like arrays, stacks, queues, and linked lists process data items in a linear fashion.
2. Stacks follow a last-in, first-out principle where the last item inserted is the first removed.
3. Queues follow a first-in, first-out principle where the first item inserted is the first removed.
Similar to 1.4 expression tree (20)
The concept of stack is extremely important in computer science and .pdf
The concept of stack is extremely important in computer science and .pdf
2. Stack. Write a program that uses the stack class (you can use.pdf
2. Stack. Write a program that uses the stack class (you can use.pdf
Using RustBecause postfix expressions are simpler and faster to ev.pdf
Using RustBecause postfix expressions are simpler and faster to ev.pdf
Merge Sort implementation in C++ The implementation for Mergesort gi.pdf
Merge Sort implementation in C++ The implementation for Mergesort gi.pdf
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5.5 back track
The document discusses various backtracking algorithms and problems. It begins with an overview of backtracking as a general algorithm design technique for problems that involve traversing decision trees and exploring partial solutions. It then provides examples of specific problems that can be solved using backtracking, including the N-Queens problem, map coloring problem, and Hamiltonian circuits problem. It also discusses common terminology and concepts in backtracking algorithms like state space trees, pruning nonpromising nodes, and backtracking when partial solutions are determined to not lead to complete solutions.
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5.3 dynamic programming
Dynamic programming is a technique for solving problems with overlapping subproblems and optimal substructure. It works by breaking problems down into smaller subproblems and storing the results in a table to avoid recomputing them. Examples where it can be applied include the knapsack problem, longest common subsequence, and computing Fibonacci numbers efficiently through bottom-up iteration rather than top-down recursion. The technique involves setting up recurrences relating larger instances to smaller ones, solving the smallest instances, and building up the full solution using the stored results.
5.3 dyn algo-i
This document summarizes a talk on dynamic graph algorithms. It begins with an introduction to dynamic graph algorithms, which involve maintaining a graph structure and answering queries efficiently as the graph undergoes a sequence of edge insertions and deletions. It then discusses several examples of fully dynamic algorithms for problems like connectivity, minimum spanning trees, and graph spanners. A key data structure introduced is the Euler tour tree, which represents a dynamic tree as a one-dimensional structure to support efficient updates and queries. The document concludes by outlining a fully dynamic randomized algorithm for maintaining connectivity under edge updates with polylogarithmic update time, using a hierarchical approach with multiple levels of edge partitions and ET trees.
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5.2 divede and conquer 03
This document discusses divide-and-conquer algorithms and their time complexities. It begins with examples of finding the maximum of a set and binary search. It then presents the general steps of a divide-and-conquer algorithm and analyzes time complexity. Several algorithms are discussed including quicksort, merge sort, 2D maxima finding, closest pair problem, convex hull problem, and matrix multiplication. Strategies like divide, conquer, and merge are used to solve problems recursively in fewer comparisons than brute force methods. Many algorithms have a time complexity of O(n log n).
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The document discusses various optimization problems that can be solved using the greedy method. It begins by explaining that the greedy method involves making locally optimal choices at each step that combine to produce a globally optimal solution. Several examples are then provided to illustrate problems that can and cannot be solved with the greedy method. These include shortest path problems, minimum spanning trees, activity-on-edge networks, and Huffman coding. Specific greedy algorithms like Kruskal's algorithm, Prim's algorithm, and Dijkstra's algorithm are also covered. The document concludes by noting that the greedy method can only be applied to solve a small number of optimization problems.
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4.4 hashing02
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- Separate chaining, which handles collisions by storing elements that hash to the same value in a linked list. Find, insert, and delete take average time of O(1).
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1.4 expression tree
- 2. Algorithm for evaluating postfix expressions Start scanning the postfix expressions from left to right If operand is encountered then push it on to the stack If an operator is encountered, pop last two operands from the stack, apply the operator on the operands and then push the resultant value on to the stack Finally the stack must contain the single value which is the result
- 3. Contd… Converting infix to postfix and then evaluating the postfix expression is not useful It could be useful if we evaluate the postfix expression while it is being constructed Using two stacks Operand stack and Operator stack will be useful Infix to postfix- operand stack Postfix evaluation- operator stack
- 4. Algorithm for evaluating the infix expression Start scanning the infix expression from left to right If an operand is encountered, push it on to the operand stack If an operator is encountered, do the following Pop until operator stack top has higher or equal precedence than current operator, before pushing the current operator on to the stack While popping an operator, pop last two operands from the operand stack, apply the operator on the operands and then push the result on to the operand stack Finally the operand stack has the single value which is the result.
- 5. Algorithm for constructing a binary expression tree Start scanning the infix expression from left to right If an operand is encountered, create a binary node and push its pointer onto the operand stack If an operator is encountered, create a binary node and do the following Pop until operator stack top has higher or equal precedence than current operator, before pushing the pointer on to the operand stack While popping an operator pointer, pop last two pointers from the operand stack, and make the pointers left and right child and then operator pointer on to the operand stack Finally top of the operand stack has the root of the expression tree.
- 6. Note: The only differences between last two algorithms are the following. instead of just pushing the operator / operand, we create a binarynode and we push the pointers instead of applying the operator on two operands, we make the left and right child instead of pushing the resultant value into the stack, we push the operator node pointer into the operand stack.
- 7. A Binary Expression Tree is . . . A special kind of binary tree in which: 1. Each leaf node contains a single operand 2. Each nonleaf node contains a single binary operator 3. The left and right subtrees of an operator node represent subexpressions that must be evaluated before applying the operator at the root of the subtree. 7
- 8. A Four-Level Binary Expression 8 ‘*’ ‘-’ ‘8’ ‘5’ ‘/’ ‘+’ ‘4’ ‘3’ ‘2’
- 9. Levels Indicate Precedence The levels of the nodes in the tree indicate their relative precedence of evaluation (we do not need parentheses to indicate precedence). Operations at higher levels of the tree are evaluated later than those below them. The operation at the root is always the last operation performed. 9
- 10. A Binary Expression Tree 10 What value does it have? ( 4 + 2 ) * 3 = 18 ‘*’ ‘+’ ‘4’ ‘3’ ‘2’
- 11. Easy to generate the infix, prefix, postfix expressions (how?) 11 Infix: ( ( 8 - 5 ) * ( ( 4 + 2 ) / 3 ) ) Prefix: * - 8 5 / + 4 2 3 Postfix: 8 5 - 4 2 + 3 / * ‘*’ ‘-’ ‘8’ ‘5’ ‘/’ ‘+’ ‘4’ ‘3’ ‘2’
- 12. Do by yourself Evaluate Following Postfix Expression: (1) 3 * 2 + 4 * (A + B) (2) A * (B + C) * D (3) A * (B+C–D/E)/F (4) A * B + (C – D/E) (5) (a + b – c) * (e / f) – ( g – h/i) (6) 1*1*8-5*(6-4*3-5*(7-(2*5))) (7) (300+23)*(43-21)/(84+7) (8) (4+8)*(6-5)/((3-2)*(2+2)) (9) 6-6/7-4*5/7-1/4*4 (10) 5*4-2+3-7+2/2*1/8/4
- 13. How many squares can you create in this figure by connecting any 4 dots (the corners of a square must lie upon a grid dot? TRIANGLES: How many triangles are located in the image below?
- 14. There are 11 squares total; 5 small, 4 medium, and 2 large. 27 triangles. There are 16 one-cell triangles, 7 four-cell triangles, 3 nine- cell triangles, and 1 sixteen-cell triangle.
- 15. GUIDED READING
- 16. MIND MAP
- 17. ASSESSMENT 1. The postfix expression for the infix expression A + B*(C+D)/F + D*E is A. AB+CD+*F/D+E* B. ABCD+*F/DE*++ C. A*B+CD/F*DE++ D. A+*BCD/F*DE++
- 18. Contd.. 2. The leaf nodes of binary expression tree contains A. both operators and operands B. only operators C. only operands D. neither operators nor operands
- 19. Contd.. 3. The root of the expression tree of (a/b*c)+(5-4+(d*e+f)*g) is A. + B. * C. - D. /
- 20. Contd.. 4. Which one of the following statement is correct. A. ‘(‘ has highest precedence in the expression and lowest precedence when present in the stack B. ‘)‘ has highest precedence in the expression and lowest precedence when present in the stack C. ‘(‘ has lowest precedence in the expression and lowest precedence when present in the stack D. ‘(‘ has highest precedence in the expression and highest precedence when present in the stack
- 21. Contd.. 5. The traversal technique which output the postfix notation of binary expression tree is A. pre-order traversal B. converse post-order traversal C. post order traversal D. converse pre-order traversal