Algorithms and Mathematics

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Questions and Answers

Which of the following best describes the primary goal when designing an algorithm?

  • To minimize the algorithm's memory usage, regardless of its performance.
  • To create a correct algorithm that solves a given problem with effective performance. (correct)
  • To maximize the lines of code to ensure all edge cases are covered.
  • To ensure the algorithm is written in the most modern programming language.

Which criterion is NOT necessarily satisfied by every algorithm?

  • Output: At least one quantity is produced.
  • Complexity: The algorithm must take less than one second to execute. (correct)
  • Finiteness: The algorithm terminates after a finite number of steps.
  • Input: There are zero or more quantities which are externally supplied.

What is the relationship between an algorithm and a program?

  • An algorithm is the expression of a program in a specific programming language.
  • A program is an abstract concept, while an algorithm is a concrete implementation.
  • An algorithm and a program are the same thing.
  • A program is the expression of an algorithm in a programming language. (correct)

What does 'performance analysis' of an algorithm primarily determine?

<p>The amount of computing time and storage space an algorithm requires. (C)</p>
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In the context of algorithm analysis, what does 'correctness' refer to?

<p>The algorithm computes the correct solution for all instances of the input. (D)</p>
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What are the two primary resources considered when evaluating the 'efficiency' of an algorithm?

<p>Time and space. (C)</p>
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Which statement best describes the 'complexity' of an algorithm?

<p>The amount of work the algorithm performs to complete its task. (C)</p>
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An algorithm's running time is most affected by which of the following?

<p>The input size and input quality. (D)</p>
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When analyzing algorithms, which case is typically considered the 'standard' for analysis?

<p>Worst case. (B)</p>
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What does 'worst-case complexity' describe?

<p>The maximum running time over all instances of a given size. (A)</p>
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Under what circumstances might a linear search on a fast computer outperform a binary search on a slow computer?

<p>For small values of input array size. (D)</p>
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What is a major limitation of relying solely on experimental studies to determine the running time of an algorithm?

<p>They can only be done on a limited set of inputs and specific environments. (D)</p>
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Which of the following is a benefit of using a high-level description for algorithm analysis instead of relying on specific implementations?

<p>It allows for evaluating efficiency independent of specific hardware or software. (D)</p>
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What is the primary focus of the theoretical study in 'Analysis of Algorithms'?

<p>The computer program performance and resource usage. (B)</p>
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In the context of algorithm analysis, what is a 'computational model' used for?

<p>Performing analysis with respect to a defined set of assumptions. (D)</p>
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What is the key characteristic of a 'Random Access Machine' (RAM) model used in algorithm analysis?

<p>All memory is equally expensive to access. (A)</p>
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Which of the following is a characteristic of 'primitive operations' in the context of algorithm analysis?

<p>They take a constant amount of time in the RAM model. (C)</p>
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What does 'RT' (Running Time) of an algorithm refer to?

<p>The amount of time it takes for the algorithm to finish execution on a particular input size. (B)</p>
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When counting primitive operations, what are we trying to determine?

<p>The maximum number of primitive operations executed as a function of the input size. (B)</p>
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What is the time complexity of the following code snippet?

x ← y+1

<p>O(1) (B)</p>
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What is the time complexity of the following code snippet?

sum ← 0
x ← y+2
for i ← 1 to n do
  sum ← sum+i

<p>O(n) (B)</p>
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What is the time complexity of the following code snippet?

x=y+1
for(i=1;i<=n;i++) {
  for(j=1;j<=n;j++) {
   x=y+z;
  }
}

<p>O(n^2) (D)</p>
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What is the time complexity of the following code snippet?

for(i=1;i<=n;i++) {
  for(j=1;j<=m;j++) {
   x=y+z; 
  }
}

<p>O(n*m) (A)</p>
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What is the time complexity of the following code snippet?

for(i=1;i<=n;i++) {
  for(j=1;j<=n;j++) {
   for(k=1;k<=n;k++) {
    x=y+z; 
   }
  }
}

<p>O(n^3) (C)</p>
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What is the time complexity of the following code snippet?

for(i=0;i<n;i++) {
  for(j=0;j<n;j++) {
   x=y+z
  }
}
For(k=0;k<n;k++){
 x=y+z
}

<p>O(n^2) (C)</p>
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What is the time complexity of the following code snippet?

while(n >= 1){
 n=n/2
}

<p>O(log n) (C)</p>
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What is the time complexity of the following code snippet?

while(n >= 2){
 n=√n
}

<p>O(log log n) (D)</p>
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What is the time complexity of the following code snippet?

i=1
while(i<=n){
 i=i*2
}

<p>O(log n) (A)</p>
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What is the runtime complexity of the following algorithm?

for(i=1; i<=n; i*=c) {
  // some constant time operations
}

<p>O(log n) (B)</p>
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In general, which case of algorithmic complexity is primarily used to measure and compare algorithms?

<p>Worst-case (B)</p>
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An algorithm with a time complexity of O(1) is said to have:

<p>Constant running time (C)</p>
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An algorithm whose runtime grows logarithmically in proportion to the input size n is known as:

<p>A logarithmic algorithm (B)</p>
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Which of the following is generally considered the 'best' in terms of runtime complexity?

<p>O(log n) (D)</p>
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Which algorithm runtime complexity is considered the 'fastest' possible?

<p>O(1) (A)</p>
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What distinguishes an 'exponential algorithm' from a 'polynomial algorithm' in terms of runtime?

<p>An exponential algorithm grows faster than a polynomial algorithm. (D)</p>
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In the context of space complexity, what does 'auxiliary space' refer to?

<p>The extra space needed beyond the input to execute the program. (A)</p>
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What are the two main components that contribute to the space complexity of an algorithm?

<p>Fixed part and variable part. (B)</p>
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Which factor is included in the 'fixed part' of an algorithm's space complexity?

<p>All of the above (D)</p>
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Flashcards

What is an Algorithm?

A step-by-step procedure for solving a problem in a finite time.

Algorithm definition

A finite set of instructions to accomplish a task.

Algorithm Input

Zero or more quantities which are externally supplied to algorithm.

Algorithm Output

At least one quantity is produced by the algorithm execution.

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Algorithm Definite

Each instruction must be clear and unambiguous.

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Algorithm Finiteness

Algorithm terminates after a finite number of steps.

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Algorithm Effectiveness

Every algorithm instruction must be very basic and feasible.

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What is a program?

Expression of an algorithm in a programming language.

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Program components

Algorithm combined with way of storing data produces a program.

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Algorithm Analysis

Determining computing time and storage an algorithm requires.

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Performance measurement

Executing a program and measuring time and space to compute results.

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Algorithm Correctness

Whether the algorithm computes the correct solution for all instances.

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Algorithm Efficiency

The resources needed by the algorithm to run and finish execution.

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Algorithm Complexity

Amount of work the algorithm performs to complete its task.

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Running time depends on...

How quickly an algorithm calculates its output.

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Worst-case complexity

Maximum running time over all instances of a given size.

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Average complexity

Average of the running times across all instances.

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Best-case complexity

Minimum of running times over all instances of a given size.

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Linear Search (fast machine)

High processing power grows linearly with size n.

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Binary Search (fast machine)

Extremely fast, leverages logarithmic complexity.

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Linear Search (slow machine)

Becomes prohibitively slow for large datasets because of limited resources.

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Binary Search (slow machine)

Maintains good performance due to O(logn).

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Experimental Study - running time

Write, run the code, varying input size. Accurate measure is system time.

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Experimental Studies limitations

Must implement in order to time. Limited set of inputs may be indicative.

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Criteria affect running time

Affects running time like debugging, hardware, coding skill and quality input.

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Theoretical analysis

Uses high-level algorithm description instead of implementations.

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Benefits of theoretical analysis

Evaluates efficiency independently from the hardware and software environment.

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Algorithm analysis

Study of computer program performance and usage.

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Measures efficiency

Measures efficiency of an algorithm/program as input size gets large.

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Algorithm time and space

Analyze time required for an algorithm and space for data structure.

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Algorithm analysis performed with...

Performed with respect to a computational model.

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Uniprocessor random-access machine

All memory equally expensive, with no concurrent operations, unit time.

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Control flow

If, while, repeat, for conditional flow control.

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Function calling

Method/Function call var.method (arg [, arg...]).

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Returning from a function

Return value return expression.

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Algorithm Expressions

Assignment,Equality testing, Superscripts and other mathematical formatting.

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Primitive Operations

Basic computations performed by an algorithm easily identified.

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Examples: Primitive Operations

Evaluating an expression, assign a variable, index to the array.

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Algorithm RT (run time)

The amount of time it takes to finish execution with an input size.

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Algorithm Runtime Measurement

Total primitive operations executed by an algorithm.

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Define Step

To be a unit of work that can be executed in constant amount time.

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Study Notes

  • These notes cover the basics of algorithms and mathematics, including problem-solving approaches, algorithm definitions, analysis, complexity, running time, and space complexity.
  • Includes examples and theoretical analysis

Problem Solving Approach

  • Begins with understanding the problem, describing the input-output relationship.
  • Involves designing an algorithm, a computational sequence transforming input to output.
  • Proceeds with analyzing the algorithm via performance measurement.
  • Requires selecting a proper data structure for organized storage and retrieval.
  • Culminates in implementing the algorithm as a program and solving the problem.

Algorithm: Definition & Criteria

  • An algorithm is a finite set of instructions accomplishing a particular task.
  • Input: Zero or more quantities are externally supplied.
  • Output: At least one quantity is produced.
  • Definite: Each instruction is clear and unambiguous.
  • Finiteness: The algorithm terminates after a finite number of steps for all cases.
  • Effectiveness: Every instruction is very basic and feasible.

What is a Program?

  • A program expresses an algorithm in a programming language.
  • It is a set of instructions for a computer to solve a problem.
  • Key relationship: Algorithm + Data Structure = Program, as highlighted by Niklaus Wirth.

How to Analyze an Algorithm

  • Assess how quickly an algorithm calculates its output.
  • Determine how much memory the algorithm consumes.
  • Algorithm analysis or performance analysis: Task of determining how much computing time and storage an algorithm requires.
  • Performance Measurement: Measuring time and space to compute the results of a correct program on data sets.

Key Needs for Algorithms

  • Correctness: Algorithm computes the correct solution for all instances.
  • Efficiency: Algorithm uses resources effectively.
  • Efficiency is measured by :Time (number of steps) and Space (amount of memory).
  • Performance is measured using the measurement "models": Worst case, average case, and best case.

Complexity

  • Algorithm complexity: The amount of work performed to complete its task.

Running Time

  • Running time depends on input size and quality.
  • Kinds of analysis:
    • Worst case(standard)
    • Average case(sometimes)
    • Best case (never)

Complexity Of an Algorithm

  • Worst-case complexity: The maximum running time over all instances of a given size.
  • Average complexity: The average of the running times.
  • Best-case complexity: The minimum of the running times over all instances of a given size.

Problem Example: Searching a Sorted Array

  • Compare searching an element in a sorted array using:
    • Linear Search (fast computer A)
    • Binary Search (slow computer B)
  • Pick constant values to represent machine performance - how long each machine takes to perform a search.
    • Example: A = 0.2, B = 1000, implying A is 5000 times more powerful than B
  • Key observation: For small input array sizes, the fast computer may take less time; after a certain value, the slower machine with binary search becomes more efficient, even if the binary search is being run on a slow machine.

Observations: Fast vs. Slow Machines

  • Fast Machine:
    • Linear Search: Performs significantly faster due to the processing power, runtime grows linear with n.
    • Binary Search: Extremely fast, leverages logarithmic complexity, high processing speed.
  • Slow Machine:
    • Linear Search: Prohibitively slow for large datasets due to bandwidth.
    • Binary Search: Good performance due to O(log n), slightly slower than the fast machine.

Measuring Running Time: Experimental Study

  • Write a program that implements an algorithm.
  • Run program with datasets of varying size.
  • Use a method like System.currentTimeMillis() to measure the actual running time accurately.

Beyond Experimental Studies

  • Experimental studies limitations :
    • Implementation and testing of the algorithm is necessary.
    • Limited input sets may not indicate running time for other inputs.
    • Comparison of algorithms needs an identical environment.
  • Uses a high-level description of the algorithm instead of one of its implementations.
  • Takes into account all possible inputs.
  • Evaluates efficiency independently of the hardware and software environment.

Analysis of Algorithms

  • Theoretical Computer program performance and resource usage.
  • Measures the efficiency of the algorithm as input size increases.
  • Focuses on time and space requirements.
  • Time : How long is it going to take
  • Space : How much storage will it require

Model for Analysis

  • Conducted with respect to a standard computational model.
  • Assumes a generic uniprocessor random-access machine (RAM) .
  • RAM aspects:
    • All memory is equally accessible.
    • No concurrent operations.
    • All reasonable instructions take unit time.

Pseudocode Details

  • Covers different programming principles that algorithms use:
    • Control flow (if, then, else, while, repeat, for)
    • Method/Function call
    • Return value
    • Expressions
    • Indentation to replace braces
    • Method declarations

Primitive Operations

  • The basic computations are performed by an algorithm
  • Should be identified in the pseudo-code
  • independent to the programming language
  • Assumed to take a constant time

Algorithm Analysis

  • Runtime: Amount of time the algorithm takes to finish.
  • More precisely: number of primitive operations or steps executed.
  • Step Definition: A unit of work executed in a constant amount of time.

Space Complexity

  • The amount of memory cells required to complete a program. The auxiliary space is calculated every time the program tries to execute.
  • Objective: When the input size is n, with what is the extra space needed to execute?

Space Complexity Categories

  • Fixed Part: Independent of input characteristics.
  • Variable Part: Depends on the problem instance being solved

Random Access Machine(RAM)

  • CPU connected to memory cells.
  • Memory cells can store numbers, character strings or addresses.
  • Assumes any primitive operation can be performed in constant time

Time Complexity

  • Time is measured in number of operations performed.
  • T(n) is created for every algorithm to measure the time required for it.
  • n is the size of the problem.

Worst Case Analysis: Guarantees

  • Focuses on the WORST CASE that occurs when the running time with always take longer.
  • Upper bound of performance.
  • Many algorithms perform awfully in most conditions.
  • The average case is not as good

Runtimes of Algorithms

  • O(1) - Constant. Best
  • O(log n) - Logarithmic. Good
  • O(n) - Linear. Fair
  • O(n log n) - N log N. Bad
  • O(n!), O(c^n), O(n^c) - Exponential, Factorial, Polynomial - Worst.

Classifying Algorithms by Growth Rate

  • Logarithmic (O(log n)): Runtime grows logarithmically with input size.
  • Linear (O(n)): Runtime grows directly with input size.
  • Superlinear (O(n log n)): Runtime grows in proportion to n.
  • Polynomial (O(nc)): Runtime grows quicker than all the others.
  • Exponential (O(cn)): Runtime grows even faster based on n.
  • Factorial (O(n!)): Runtime grows the fastest and becomes unusable.

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