CSCI-2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2017 March 21, - - PowerPoint PPT Presentation

CSCI-2200 FOUNDATIONS OF COMPUTER SCIENCE

Spring 2017

March 21, 2017

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Logistics

• Prof. Stacy Patterson sep@cs.rpi.edu, pattes3@rpi.edu
• Office hours: Monday 2pm – 3:30pm (or by appointment)

in Lally 301

• Homework 5 is due Friday, March 24 at 11:59pm.
• Today:
• Review of induction and strong induction (5.1-5.2)
• Recursive Definitions and Structural Induction (5.3)
• Trees

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Principle of Mathematical Induction

• To prove that P(n) is true for all positive integers ≥ b, we complete

these steps:

• Basis Step: Show that P(b) is true
• Inductive Step: Show that P(k) → P(k + 1)

is true for all positive integers k ≥ b

• P(k) is the inductive hypothesis.

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Prove that a set with n elements has n(n-1)/2 subsets with exactly 2 elements.

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Why Induction Works?

• This is a valid argument.

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P(1) ∀k(P(k) → P(k + 1)) ∴ ∀nP(n)

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Graphs

• A graph G=(V,E) is a set of vertices V and a set of edges E.
• We study “undirected graphs”.
• A path between vertices i and j is a sequence of edges that connect i and j
• A connected graph is a graph in which there is a path between every pair of

vertices.

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Trees

• A tree is connected graph with no cycles.
• Only a single path between any pair of vertices
• A rooted tree is a special type of graph. It contains a distinguished

vertex called the root.

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Recursive Definition of a Rooted Tree

• Recursive definition of a rooted tree:
• Basis step: A single vertex r is a rooted tree.
• Recursive step: Suppose that T1, T2, …, Tm are disjoint rooted trees. Then

the graph formed by starting with a root r, which is not in any of T1, … , Tm, and adding an edge from the root of each Ti to r is a rooted tree.

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Recursive Definition of a Full Binary Tree

• A full binary tree is a rooted tree in which each vertex has either 0 or 2

children.

• Recursive definition:
• Basis step: A single vertex r is a full binary tree
• Recursive step: If T1 and T2 are full disjoint full binary trees, there is a full

binary tree, denoted T1ŸT2, consisting of a root r with edges connecting to the roots of T1 and T2.

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Number of nodes in a Full Binary Tree

• Recursive definition of FBT:
• Basis step: A single vertex r is a full binary tree.
• Recursive step: If T1 and T2 are disjoint full binary trees, there is a

full binary tree, denoted T1ŸT2, consisting of a root r with edges connecting to the roots of T1 and T2.

• Recursive definition of number of vertices in a FBT:

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Height of a Full Binary Tree

• Recursive definition of FBT:
• Basis step: A single vertex r is a full binary tree
• Recursive step: If T1 and T2 are full disjoint full binary trees, there is a full

binary tree, denoted T1ŸT2, consisting of a root r with edges connecting to the roots of T1 and T2.

• Theorem: If T is a full binary tree, then N(T) ≥ 2H(T) + 1

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Structural Induction

• We can use structural induction to prove results about

recursively defined sets and structures.

• A proof by structural induction has two parts:
• Basis Step: Show the result holds for the structure(s)

specified in the basis step of the recursive definition.

• Recursive Step: Show that if the statement is true for each

structure used to construct the new structure, then it is true for the new structure.

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• Theorem: If T is a full binary tree, then N(T) ≥ 2H(T) + 1

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• Theorem: If T is a full binary tree, then N(T) ≥ 2H(T) + 1

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Good Problems to Review

• Section 5.2: 3, 5, 7, 13, 25, 29
• Section 5.3: 23, 25

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