ALEX Classroom Resource

  

Complexity and Tractability

  Classroom Resource Information  

Title:

Complexity and Tractability

URL:

https://csfieldguide.org.nz/en/chapters/complexity-and-tractability/

Content Source:

Other
CS Field Guide
Type: Lesson/Unit Plan

Overview:

Are there problems that are too hard even for computers? It turns out that there are. In the chapter on Artificial Intelligence, we'll see that just having a conversation – chatting – is something computers can't do well, not because they can't speak but rather because they can't understand or think of sensible things to say. However, that’s not the kind of hard problem we’re talking about here – it's not that computers couldn’t have conversations, it's more that we don't know just how we do it ourselves and so we can't tell the computer what to do.

In this chapter, we're going to look at problems where it's easy to tell the computer what to do – by writing a program – but the computer can’t do what we want because it takes far too long: millions of centuries, perhaps. Not much good buying a faster computer either: if it were a hundred times faster it would still take millions of years; even one a million times faster would take hundreds of years. That's what you call a hard problem – one where it takes far longer than the lifetime of the fastest computer imaginable to come up with a solution.

Content Standard(s):
Digital Literacy and Computer Science
DLIT (2018)
Grade: 9-12
3) Differentiate between a generalized expression of an algorithm in pseudocode and its concrete implementation in a programming language.

a. Explain that some algorithms do not lead to exact solutions in a reasonable amount of time and thus approximations are acceptable.

b. Compare and contrast the difference between specific control structures such as sequential statements, conditional, iteration, and explain the benefits and drawbacks of choices made.

Examples: Tradeoffs involving implementation, readability, and program performance.

c. Distinguish when a problem solution requires decisions to be made among alternatives, such as selection constructs, or when a solution needs to be iteratively processed to arrive at a result, such as iterative 'loop' constructs or recursion.

d. Evaluate and select algorithms based on performance, reusability, and ease of implementation.

e. Explain how more than one algorithm may solve the same problem and yet be characterized with different priorities.

Examples: All self-driving cars have a common goal of taking a passenger to a designation but may have different priorities such as safety, speed, or conservation; web search engines have their own algorithms for search with their own priorities.

Unpacked Content
Evidence Of Student Attainment:
Students will:
  • compare and contrast pseudocode and programming language.
  • be given pseudocode and code in a programming language to differentiate between the two processes.
a.
  • explain that some solutions cannot be reached in an acceptable timeframe, and therefore solutions must be approximated.
b.
  • identify sequential statements in code.
  • identify conditional statements in code.
  • identify iterations in code.
  • compare and contrast the difference between these types of control structures: sequential statements, conditional statements, and iteration.
  • identify trade-offs associated with using one control structure over another.
c.
  • identify when an iterative loop is needed in a program.
  • identify when selection constructs are needed in a program.
  • identify when recursion is needed in a program.
  • distinguish when a solution requires decisions to be made among alternatives such as an iterative loop, selection constructs, or recursion.
d.
  • evaluate algorithms based on performance.
  • evaluate algorithms based on reusability.
  • evaluate algorithms based on ease of implementation.
  • select the best algorithm based on desired strength: performance, reusability, or ease of implementation.
  • e.
    • explain that algorithms can be designed to operate for a specific priority.
Teacher Vocabulary:
  • pseudocode
  • programming language
a.
  • approximated
b.
  • iteration
  • conditional statements
  • control structures
c.
  • iterative loop
  • selection constructs
  • recursion
Knowledge:
Students know:
  • that differences exist in pseudocode and a programming language.
  • that programming languages have certain requirements for language and syntax.
a.
  • that some programs cannot return a result in a reasonable time frame, therefore approximations must be allowed in those cases.
b.
  • how to identify sequential statements, conditional statements, and/or iterations in code.
  • the differences between sequential statements, conditional statements, and/or iterations.
  • trade-offs exist with using one control structure over another.
c.
  • some decisions in a program will require the use of iterative loops, selection constructs, or recursion.
d.
  • programs can be written to satisfy a number of needs such as performance, reusability, and ease of implementation.
  • that most times, algorithms will differ based on the need of the program; performance, reusability, or ease of implementation.
e.
  • that programs can be written with specific priorities in mind.
  • that there are multiple correct ways to write a program.
  • that solutions are often chosen to meet the priority need of the program.
Skills:
Students are able to:
  • distinguish between a generalized expression of an algorithm in pseudocode and its concrete implementation in a programming language.
  • point out similarities in vocabulary and syntax between pseudocode and an algorithm.
  • point out differences in vocabulary and syntax between pseudocode and an algorithm.
a.
  • explain that some algorithms do not lead to exact solutions in a reasonable amount of time and thus approximations are acceptable.
b.
  • identify sequential statements, conditional statements, and/or iterations in code.
  • identify tradeoffs associated with using one control structure over another.
c.
  • distinguish when a problem solution requires decisions to be made among alternatives or when a solution needs to be iteratively processed to arrive at a result.
d.
  • evaluate and select algorithms based on performance, reusability, and ease of implementation.
e.
  • explain how more than one algorithm may solve the same problem and yet be characterized with different priorities.
Understanding:
Students understand that:
  • similarities and differences exist in pseudocode and programming code.
  • some programming languages more closely resemble pseudocode than do other programming languages.
a.
  • due to time or financial constraints, some programs may return an approximation of a solution.
b.
  • both benefits and drawbacks exist when selecting one control structure over another in a code.
c.
  • programs can use multiple methods to arrive at a solution.
d.
  • there are times when a program needs to be selected for a specific purpose, such as performance, reusability, and/or ease of implementation.
e.
  • multiple algorithms can solve the same problem.
  • algorithms can operate with a specific priority in mind, such as speed, simplicity, and/or safety.
Digital Literacy and Computer Science
DLIT (2018)
Grade: 9-12
39) Identify a problem that cannot be solved by either humans or machines alone and discuss a solution for it by decomposing the task into sub-problems suited for a human or machine to accomplish.

Examples: Forecasting weather, piloting airplanes.

Unpacked Content
Evidence Of Student Attainment:
Students will:
  • identify a problem that cannot be solved by humans or machines alone.
  • discuss possible solutions using decomposition.
  • identify subproblems for either a human or machine to solve.
Knowledge:
Students know:
  • how to identify a problem.
  • how to decompose a problem.
  • how to identify possible solutions to a problem.
Skills:
Students are able to:
  • identify a problem that cannot be solved by humans or machines alone.
  • discuss possible solutions using decomposition.
  • identify subproblems for either a human or machine to solve.
Understanding:
Students understand that:
  • problems exist that cannot be solved by a human or machine alone.
  • identifying subproblems can make a complex problem easier to solve.
  • humans and machines can work together to solve complex problems.
Tags: algorithm, asymptotic complexity, Big O notation, complexity, intractable, NPcomplete, tractability
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Author: Aimee Bates