CSC 244 Midterm 2 Study Guide
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Questions and Answers

What is the recurrence relation for the number of bit strings of length n that do not have two consecutive 0s?

  • an = 2an-1 + an-2
  • an = an-1 + an-2 (correct)
  • an = an-1 + an-3
  • an = an-2 + 2an-1

What is the initial condition a1 for the number of bit strings of length n that do not have two consecutive 0s?

  • 3
  • 4
  • 1
  • 2 (correct)

How many bit strings of length 5 do not have two consecutive 0s?

  • 10
  • 11
  • 8
  • 13 (correct)

What is the value of a4 in the recurrence relation for the number of bit strings without consecutive 0s?

<p>8 (D)</p> Signup and view all the answers

In performing a binary search, what is done after finding the middle element and determining it is less than the target?

<p>Delete half of the right elements (C)</p> Signup and view all the answers

What step follows the evaluation of the middle element if it matches the search key in a binary search algorithm?

<p>Return the location of the middle element (B)</p> Signup and view all the answers

Which of these is a characteristic of the binary search algorithm compared to linear search?

<p>Works in logarithmic time (A)</p> Signup and view all the answers

What is the main advantage of using a recurrence relation in combinatorics?

<p>It simplifies calculations for large n (A)</p> Signup and view all the answers

What is the initial value of $a_1$ in the recurrence relation for the number of bit strings of length n that do not have two consecutive 0s?

<p>2 (B)</p> Signup and view all the answers

Which of the following strings satisfies the condition of not having two consecutive 0s?

<p>01010 (C)</p> Signup and view all the answers

How many bit strings of length 5 do not contain two consecutive 0s?

<p>13 (A)</p> Signup and view all the answers

If $S_n$ represents the set of strings of length n satisfying the condition of no two consecutive 0s, what can be said about $S_{n-1}$?

<p>Strings from $S_{n-1}$ can be used to generate strings in $S_n$ by appending 1. (C)</p> Signup and view all the answers

Which of the following accurately describes a constructive relationship observed in the recurrence relation?

<p>Starting from strings of $S_1$, you can generate new strings for $S_2$ and beyond. (A)</p> Signup and view all the answers

In the context of recurrence relations, what would be a critical initial condition for calculating $a_n$?

<p>$a_0 = 1$ (B)</p> Signup and view all the answers

Which of the following bit strings does not satisfy the condition of having two consecutive 0s?

<p>10101 (C)</p> Signup and view all the answers

If a string ends with 1, what can you append to it to form a valid string in the sequence?

<p>0 only (D)</p> Signup and view all the answers

What is the recurrence relation for the number of bit strings of length n that do not have two consecutive 0s?

<p>an = an-1 + an-2 (B)</p> Signup and view all the answers

What are the initial conditions needed for this recurrence relation?

<p>a0 = 1, a1 = 1 (A)</p> Signup and view all the answers

If the last bit of a string is 0, what must be true about the previous bit?

<p>The second last bit must be 1. (D)</p> Signup and view all the answers

What does the term 'proof by cases' imply in this context?

<p>It means dividing cases based on the last bit of the string. (D)</p> Signup and view all the answers

How many bit strings of length 5 do not contain two consecutive 0s?

<p>8 (D)</p> Signup and view all the answers

What dictates the validity of a string in this problem?

<p>The absence of consecutive 0s. (A)</p> Signup and view all the answers

When constructing strings of length n that meet the criteria, what can be inferred when a string ends with 1?

<p>It can be constructed from any valid string of length n-1. (D)</p> Signup and view all the answers

What is c equal to when discussing the string ending with a 0?

<p>1 (A)</p> Signup and view all the answers

What can be inferred if a string of length n ends with a 0 in the context of the recurrence relation?

<p>The string can be formed by appending 1 only. (B)</p> Signup and view all the answers

What represents the set of strings of length n that meet the condition of no two consecutive 0s?

<p>Sn (C)</p> Signup and view all the answers

What is the initial condition for a2, the number of valid strings of length 2 that do not have two consecutive 0s?

<p>3 (A)</p> Signup and view all the answers

When constructing a valid string of length n, what must be true if it is formed from a valid string of length n-1?

<p>The last digit must be a 1. (D)</p> Signup and view all the answers

Which of the following strings satisfies the condition of no two consecutive 0s?

<p>10101 (C)</p> Signup and view all the answers

What does a recurrence relation provide in the context of counting bit strings?

<p>A method to construct new strings from existing ones. (B)</p> Signup and view all the answers

Which bit string of length 5 does NOT satisfy the condition of having two consecutive 0s?

<p>10000 (C)</p> Signup and view all the answers

What is the total number of bit strings of length 4 that do not contain two consecutive 0s?

<p>8 (A)</p> Signup and view all the answers

What is the significance of the middle element in the binary search algorithm?

<p>It determines if the search can continue in the left or right half. (C)</p> Signup and view all the answers

In the recurrence relation for counting bit strings without two consecutive 0s, what does the term $a_{n-1}$ represent?

<p>The number of strings ending in 1. (D)</p> Signup and view all the answers

Which of the following is a base case in the recurrence for counting bit strings without two consecutive 0s?

<p>$a_2 = 3$ (B)</p> Signup and view all the answers

If a string of length n ends with 0, which of the following must be true about its previous bit?

<p>It must be 1. (C)</p> Signup and view all the answers

When performing a binary search, what happens if the middle element is greater than the target element?

<p>Only the left half of the list is searched next. (B)</p> Signup and view all the answers

Which mathematical operation pertains to the recurrence relation defined as $a_n = a_{n-1} + a_{n-2}$?

<p>Linear combination. (D)</p> Signup and view all the answers

How does the recurrence relation for the number of bit strings without two consecutive 0s evolve as n increases?

<p>It increases according to the Fibonacci sequence. (C)</p> Signup and view all the answers

In the binary search algorithm, if a middle element matches the search key, what is the next step?

<p>Return the index of the middle element. (D)</p> Signup and view all the answers

What can be inferred about a string of length n if it ends with a 0?

<p>The second last bit must be 1. (A)</p> Signup and view all the answers

Which expression correctly represents the relationship between the number of valid bit strings of length n?

<p>$a_n = a_{n-1} + a_{n-2}$ (A)</p> Signup and view all the answers

What initial condition is necessary for the calculation of the bit strings without consecutive 0s?

<p>The initial condition is $a_1 = 1$. (D)</p> Signup and view all the answers

What does 'proof by cases' indicate in the context of recurrence relations for bit strings?

<p>Different conditions for the last bit are analyzed. (D)</p> Signup and view all the answers

If a string of length n ends in 1, which of the following can be appended to create a valid string?

<p>00 (A), 10 (B), 11 (C), 0 (D)</p> Signup and view all the answers

What can be concluded about the relationship between $a_n$ and previous terms for bit strings without consecutive 0s?

<p>$a_n$ is the sum of the previous two terms. (B)</p> Signup and view all the answers

Why is it not possible for a string to end in two consecutive 0s?

<p>Such strings cannot be counted in the valid set. (C), It defines invalid strings. (D)</p> Signup and view all the answers

Flashcards

Recurrence Relation for Bit Strings (no two 0s)

A formula that defines the number of bit strings of length n that do not contain two consecutive 0s in terms of the number of such strings of shorter lengths.

a1

Number of bit strings of length 1 with no consecutive 0s (initially there are two strings which are 1,0.)

a2

Number of bit strings of length 2 with no consecutive 0s. (there are three strings such as 11, 10 and 01).

Mathematical Induction

A proof technique used to prove statements about natural numbers, by showing a base case and inductive step.

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Recursive Definitions

Definitions of a concept that refer to itself in a smaller cases.

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Structural Induction

Using recursive definitions and proving properties by checking base cases and inductive steps that reflects the structure of a recursive definition.

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Counting

The process of determining the number of possible outcomes in a given set or situation.

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Bit String

A sequence of 0s and 1s.

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Recurrence Relation for bit strings

A relationship between the number of bit strings of length n that do not have two consecutive 0s (an) and the number of bit strings of length n-1 (an-1) and n-2 (an-2).

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Recurrence Relation for bit strings

A formula that defines the nth term of a sequence based on previous terms.

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Bit strings with no consecutive 0s

sequences of 0s and 1s in which no two consecutive bits are 0.

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an = an-1 + an-2

The recurrence relation for the number of bit strings of length n without two consecutive 0s is the sum of the strings of length n-1 and n-2.

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Case 1 (String ending with 1)

If the string ends in 1, we look at the previous bit strings without consecutive 0s of length n-1.

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Binary Search Algorithm

A search algorithm that finds the position of a target value within a sorted array.

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Sorted List

An array of elements arranged in ascending or descending order, making search algorithms efficient.

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Case 2 (String ending with 0)

If a string ends in 0, the second to last bit must be a 1. We look at the bit strings of length n-2 without consecutive 0s.

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Middle Element

The element in the middle of a sorted array when searching.

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Initial conditions (an)

The starting values needed to fully define the recurrence relation.

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Consecutive Zeros

Two or more 0s appearing next to each other in a bit string.

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an = an-1 + an-2

Recurrence relation describing the number of bit strings with no two consecutive 0s.

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Initial Conditions (a1, a2)

Starting values for a recurrence relation(bit strings); for the recurrence relation.

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Bit String

A sequence of 0s and 1s.

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Length 5 Example

Finding the number of bit strings of length 5 with no consecutive 0s. It requires use of the recurrence relation.

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Searching Problem

Finding an element (x) in a sorted list of integers, if it exists.

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Bit strings with no consecutive 0s

Sequences of 0s and 1s where no two consecutive bits are 0s

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Recurrence Relation

A formula that describes a sequence's nth term based on its previous terms.

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Consecutive Zeros

Two or more 0s appearing together in a bit string

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Initial Conditions (a1, a2)

The initial values of a sequence to start calculating further elements

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a1 and a2

The number of bit strings of length 1 & 2 without consecutive 0s, a1 = 2, a2 = 3.

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aₙ = aₙ₋₁ + aₙ₋₂

Recurrence relation for bit strings of length n without consecutive 0s; the nth term is the sum of (n-1)th and (n-2)th term.

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Length 5 Example

Finding the number of 5-bit strings without consecutive 0s.

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Bit String

A sequence of 0s and 1s.

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Bit strings with no consecutive 0s

Sequences of 0s and 1s where no two consecutive bits are 0

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Recurrence Relation (an)

A formula relating the number of bit strings of length 'n' (an) to those of length 'n-1' and 'n-2'.

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an = an-1 + an-2

The count of n-bit strings (no consecutive 0s) is the sum of the previous length n-1 and n-2 counts.

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Initial conditions (a1, a2)

Starting values (counts for strings of length 1 and 2 without consecutive zeros)

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Length 5 Example

Finding the number of 5-bit strings without consecutive zeros using the recurrence relation.

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Consecutive Zeros

Two or more 0s appearing one after another in a bit string.

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Case 1: String ends with 1

If the string ends in 1, the number of such strings is the count for length n-1.

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Case 2: Strings end with 0

If the string ends in 0, the preceding bit must be a 1. This reduces the problem to length n-2 cases.

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Recurrence Relation for Bit Strings (no two 0s)

A formula defining the count of bit strings of length n without consecutive 0s, using counts of shorter strings.

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Binary Search Algorithm

A search method finding a target in a sorted list by repeatedly dividing the search interval in half.

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Sorted List

A list of items arranged in increasing or decreasing order.

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Initial Conditions (a1, a2)

Starting values for a sequence in a recurrence relation (bit strings with no consecutive 0s).

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Bit Strings with no consecutive 0s

Sequences of 0s and 1s where no two 0s are adjacent.

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Consecutive Zeros

Two or more zeros appearing together in a bit string.

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Length 5 Example

Finding the count of bit strings of length 5 that do not have two consecutive 0s.

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an = an-1 + an-2

Recurrence relation for bit strings that doesn't have two consecutive 0s, with a_n being the length of bit strings of size n.

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Study Notes

Midterm 2 Study Guide - CSC 244

  • Date: Wednesday, November 13th
  • Time: 2:00 PM
  • Duration: 50 minutes
  • Format: Closed book, closed note
  • Tools: Calculator allowed

Topics

  • Applications of Recurrence Relations

    • Find recurrence relation and initial conditions for bit strings of length n that do not have two consecutive 0s.
      -Example bit strings: 10101 (satisfies condition), 101001 (does not satisfy condition). -Calculate the number of such bit strings of length 5 (13). -Establish a recurrence relation (aₙ= aₙ₋₁ + aₙ₋₂).

  • Mathematical Induction -Prove that 1 + 2 + ... + n = n(n+1)/2 for positive integer n (basis step and inductive step are included in example). -Find a formula for sum of first n positive odd integers and prove it with induction (sum= n²).

  • Recursive Definitions and Structural Induction -Consider recursively defined sets (e.g., subset S; 3 ∈ S; if x ∈ S and y ∈ S, then x + y ∈ S). -Prove that S is the set of all positive integers that are multiples of 3 (a subset argument).

  • Basic Counting Principles -Product Rule -Procedure can be broken down into tasks, where each task has specific ways to be executed (n₁ & n₂). -Total ways of completing the procedure is n₁ * n₂. -Example use-cases: assigning different offices to employees (12 and 11 = 132), labeling chairs of an auditorium. -Example with bit strings. Calculate the number of bit strings of length seven. -Example with English letters and three digits or license plates (26 * 26 * 26 * 10 * 10 * 10) -Sum Rule -If a task can be completed in n₁ ways or in n₂ ways (distinct from each other), then total ways are computed as n₁ + n₂. -Division Rule -Procedure has n ways, and every way is repeated d times.
    -Then the # of distinct ways to do the task = n/d. -Example with seating four people around a circular table.

  • Searching problem and solving with Binary Search

  • Integer multiplication and recursive approach for efficiency in arithmetic

  • Master Theorem (used to derive run times for recursive algorithms)

  • Recurrence Relations and their solutions (methods for solving recursive relations introduced)

  • Examples of recurrence relations and solutions (Fibonacci sequence, factorial computation).

Additional notes

  • Review homeworks 4-8 and quizzes 4-7 (examples are included in similar format for the exam).
  • The exam questions will have a similar structure to the examples in slides.
  • Use the materials available in D2L and Panopto recordings for further study.

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Description

Prepare for your CSC 244 Midterm 2 exam with this comprehensive study guide. It covers key topics such as recurrence relations, mathematical induction, and recursive definitions essential for solving complex problems. Make sure to review the examples and proofs to excel in your exam.

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