Tree basics
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This document provides an introduction to tree data structures. It defines a tree recursively as a collection of nodes where one node is the root and other nodes form non-empty subtrees connected to the root. Key terminology is introduced, including leaves, internal nodes, siblings, paths between nodes specifying their length, depth and height of nodes, and depth and height of the entire tree. Binary trees and tree traversal methods are also listed as topics to be covered.
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A tree is a hierarchical structure consisting of nodes connected by edges. It contains a finite set of nodes where each node stores data and links to successor nodes. The key components are the node, edge, parent, child, sibling, root, and leaf. The root is the top node, leaves have no children, and edges connect nodes in paths from the root to other nodes. Additional properties like level, degree, depth, and height further define the structure and relationships between nodes in a tree.
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Describe the tree data structure. What is a root node? What is a child node? What is a leaf node? What is a subtree? What is the height of a tree? When is a tree full? When is a tree balanced? What is breadth-first traversal? What are the common methods for traversing a binary tree and processing the value of each node?
Solution
A tree is a special type of a graph where the tree contains the nodes and connected by means edges and doesnot have loops.
root node :the node or vertex at the top of a tree is termed as root node.
child node :the nodes that are below the rootnode or any particular node is termedas the child node.
leaf node :the node which doesnot possess any children node is defined as leaf node
subtree :the part of a tree is called subtree or the descendents of a tree.
height of a tree:is the number of edges from the leaf node to the root node.
full tree :full tree is binary tree all the nodes have two child nodes except the leaf nodes.
tree balanced :an AVL tree is said to balanced tree if the balancing factor ranges from -1 to +1
breadth first traversal :travelling the nodes of a tree in breadth wise manner that is all the nodes at the same height are travelled sequentially and then their childs are travelled.queue is used to travell in breadth first search
common ways of traversing are :1)breadth first search
2)depth first search
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A tree is a special type of a graph where the tree contains the nodes and connected by means edges and doesnot have loops.
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child node :the nodes that are below the rootnode or any particular node is termedas the child node.
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Greedy algorithms
This document discusses greedy algorithms and dynamic programming techniques for solving optimization problems. It covers the activity selection problem, which can be solved greedily by always selecting the shortest remaining activity. It also discusses the knapsack problem and how the fractional version can be solved greedily while the 0-1 version requires dynamic programming due to its optimal substructure but non-greedy nature. Dynamic programming builds up solutions by combining optimal solutions to overlapping subproblems.
Longest common subsequences in Algorithm Analysis
The document discusses dynamic programming and the longest common subsequence (LCS) problem. It provides an example of using dynamic programming to find the LCS of two strings "ABCB" and "BDCAB" in multiple steps. The algorithm builds up the solution by filling a 2D array c[m,n] where m and n are the lengths of the two strings. Each entry c[i,j] represents the length of the LCS of the prefixes of length i and j of the two strings. The running time is O(mn).
Dynamic programming in Algorithm Analysis
The document discusses dynamic programming and amortized analysis. It covers:
1) An example of amortized analysis of dynamic tables, where the worst case cost of an insert is O(n) but the amortized cost is O(1).
2) Dynamic programming can be used when a problem breaks into recurring subproblems. Longest common subsequence is given as an example that is solved using dynamic programming in O(mn) time rather than a brute force O(2^m*n) approach.
3) The dynamic programming algorithm for longest common subsequence works by defining a 2D array c where c[i,j] represents the length of the LCS of the first i elements
Average case Analysis of Quicksort
Quicksort has worst case running time of Θ(n^2) when the partition is always unbalanced, but average case running time of Θ(n log n). The document derives the average case running time rigorously using a recurrence relation approach. It involves assuming the partition splits the array randomly and solving the recurrence using an inductive hypothesis that the running time is O(n log n).
Np completeness
1) NP-Completeness refers to problems that are in NP (can be verified in polynomial time) and are as hard as any problem in NP.
2) The first problem proven to be NP-Complete was the Circuit Satisfiability problem, which asks whether there exists an input assignment that makes a Boolean circuit output 1.
3) To prove a problem P is NP-Complete, it must be shown that P is in NP and that any problem in NP can be reduced to P in polynomial time. This establishes P as at least as hard as any problem in NP.
computer languages
The document discusses the differences between low-level machine code used by CPUs and high-level computer languages used by programmers. It explains that programmers write source code in high-level languages, which are then compiled into machine code through a compiler or interpreted line-by-line using an interpreter. Compiled code tends to run faster while interpreted languages typically have more programmer-friendly features. Modern dynamic languages often use a just-in-time compiler to bridge performance gaps.
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Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
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Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
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