Introduction to Data Structures and Algorithms, Summaries of Mathematical Physics

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© 2022 by Greg Ozbirn, UT-Dallas, for use with Data Structures book by Mark Allen Weiss 1

Chapter 1

Introduction

Fall 2022

Objectives

This chapter reviews some pre-requisite

material we will need.

• (^) It is assumed you have already learned

much of this material in previous courses.

Exponents

Rule

• (^) XAXB^ = XA+B
• (^) XA^ / XB^ = XA-B
• (^) (XA)B =^ XAB

X

N

+ X

N

= 2X

N

• 2 N^ + 2N^ = 2N+

Example

• (^22) * 2^3 = 22+3^ = 2^5
• (^25) / 2^3 = 25 – 3^ = 2^2
• (^) (2^2 )^3 = 2^6

2

2

2

• (^22) + 2^2 = 2^3

Logarithms

• (^) X A

= B if and only if log

X

B = A

3 = 8 so log 2

• (^) log A

B = log

C

B / log

C

A; A,B,C > 0, A ≠ 1

log 2 16 = log 10 16 / log 10

Logarithms

• (^) log X

A

B

= B log

X

A

log 10

4 = 4 log 10

• (^) log 2

X < X for all X > 0

Logarithms

Normally ln is base e, and log is base 10.

• (^) In our textbook, log is always base 2. log 1 = 0 log 2 = 1 log 1024 = 10 log 1048576 = 20 log 1073741824 = 30

• (^) If 0<A<1: N ∑ A i ≤ 1 / (1-A) i=
• (^) As N →∞, the sum approaches 1/(1-A).
• (^) Proof: S = 1 + A + A 2 + A 3 + A 4 + A 5 + … Multiply through by A: AS = A + A 2

• A 3
• A 4
• A 5
• … Subtract AS from S (permitted if convergent): S – AS = 1 So S = 1/(1-A)

Series

Arithmetic series - the difference between

successive terms is constant.

N ∑ i = N(N+1)/2 ≈ N 2 / i=

To find the sum of 2 + 5 + 8 + … + 3k-

• (^) Write as: 3(1+2+3+…+k) – (1+1+1+…+1)

= 3(k(k+1)/2) – k

Proof by Contradiction

• (^) Assume theorem is false, then show how this assumption leads to a conclusion that something which is known to be true is false, hence the original false assumption cannot be true, proving the theorem is true.
• (^) For example, prove that the sum of two even numbers is always even.
• (^) Assume it is false: given even x and y, then x+y is odd.
• (^) If x+y is odd, then x+y = 2c+1.
• (^) But x=2a and y=2b means 2a+2b = 2c+
• (^) So 2(a+b) = 2c + 1, but this says an even number equals an odd number, which is impossible.
• (^) Therefore, the sum of two even numbers is even.

Induction

Steps:

1. Basis step : Prove for the minimal case. 2. Inductive step: a. Inductive hypothesis: Assume the theorem holds for all cases up to some limit k. b. Prove the next case: for example, k+

3. Conclusion : by induction, the theorem holds for all cases, i.e., the minimal case and all its successors.

Prove: for Fibonacci numbers, F i

i , i>= Basis step: F 1

1 , F 2

2 Inductive step: assume true for i=1, 2, …, k Show F k+

k+ F k+

= F

k +

F

k- F k+

k

• (5/3) k- < (3/5)(5/3) k+
• (3/5) 2 (5/3) k+ < (3/5 + (3/5) 2 ) (5/3) k+ < (24/25) (5/3) k+ < (5/3) k+ Conclusion: by induction, it is proved

Recursion

• (^) A function defined in terms of itself is

recursive.

• (^) Many math problems can be expressed this

way.

• (^) For example: xn^ = x * xn-

x! = x * (x-1)!

• (^) We can see that these examples are

naturally recursive.

Recursion

public static int power(int x, int n) { if (n = = 0) return 1; else return x * power(x, n-1); } power(5,3) = 5*power(5,2) = 5 * (5 * power(5,1)) = 5 * (5 * (5 * power(5,0))) = 5 * (5 * (5 * 1)) = 5 * (5 * 5) = 5 * 25 = 125

Recursion

Consider this recursive function: public static void print(int n) { if (n >= 10) print( n / 10); printDigit( n % 10); } print(6371); print(637) print(63) print(6) printDigit(6%10); printDigit(63%10); printDigit(637%10); printDigit(6371%10);