Algorithms and Programming · ⁨อัลกอริทึมและการเขียนโปรแกรม⁩

Enduring Understanding (AAP-1): To find specific solutions to generalizable problems, programmers represent and organize data in multiple ways.

Learning Objective AAP-1.A: Represent a value with a variable. [Skill 3.A]

• AAP-1.A.1 A variable is an abstraction inside a program that can hold a value. Each variable has associated data storage that represents one value at a time, but that value can be a list or other collection that in turn contains multiple values.
• AAP-1.A.2 Using meaningful variable names helps with the readability of program code and understanding of what values are represented by the variables.
• AAP-1.A.3 Some programming languages provide types to represent data, which are referenced using variables. These types include numbers, Booleans, lists, and strings.
• AAP-1.A.4 Some values are better suited to representation using one type of datum rather than another.

Learning Objective AAP-1.B: Determine the value of a variable as a result of an assignment. [Skill 4.B]

• AAP-1.B.1 The assignment operator allows a program to change the value represented by a variable.

• AAP-1.B.2 The exam reference sheet provides the "$\leftarrow$" operator to use for assignment. For example,

  Text:

  a ← expression

  Block:

  a ← expression

  evaluates expression and then assigns a copy of the result to the variable a.

• AAP-1.B.3 The value stored in a variable will be the most recent value assigned. For example:

  a ← 1 b ← a a ← 2 display(b)

  still displays 1.

Enduring Understanding (AAP-1): To find specific solutions to generalizable problems, programmers represent and organize data in multiple ways.

Learning Objective AAP-1.C: Represent a list or string using a variable. [Skill 3.A]

• AAP-1.C.1 A list is an ordered sequence of elements. For example,

  [value1, value2, value3, ...]

  describes a list where value1 is the first element, value2 is the second element, value3 is the third element, and so on.

• AAP-1.C.2 An element is an individual value in a list that is assigned a unique index.

• AAP-1.C.3 An index is a common method for referencing the elements in a list or string using natural numbers.

• AAP-1.C.4 A string is an ordered sequence of characters.

Learning Objective AAP-1.D: For data abstraction: a. Develop data abstraction using lists to store multiple elements. [Skill 3.B] b. Explain how the use of data abstraction manages complexity in program code. [Skill 3.C]

• AAP-1.D.1 Data abstraction provides a separation between the abstract properties of a data type and the concrete details of its representation.

• AAP-1.D.2 Data abstractions manage complexity in programs by giving a collection of data a name without referencing the specific details of the representation.

• AAP-1.D.3 Data abstractions can be created using lists.

• AAP-1.D.4 Developing a data abstraction to implement in a program can result in a program that is easier to develop and maintain.

• AAP-1.D.5 Data abstractions often contain different types of elements.

• AAP-1.D.6 The use of lists allows multiple related items to be treated as a single value. Lists are referred to by different names, such as array, depending on the programming language.

  • Exclusion statement (EK AAP-1.D.6): The use of linked lists is outside the scope of this course and the AP Exam.

• AAP-1.D.7 The exam reference sheet provides the notation

  [value1, value2, value3, ...]

  to create a list with those values as the first, second, third, and so on items. For example,

  • Text:

    aList ← [value1, value2, value3, ...]

    Block:

    aList ← value1, value2, value3

    creates a new list that contains the values value1, value2, value3, and ... at indices 1, 2, 3, and ... respectively and assigns it to aList.

  • Text:

    aList ← []

    Block:

    aList ← (empty)

    creates a new empty list and assigns it to aList.

  • Text:

    aList ← bList

    Block:

    aList ← bList

    assigns a copy of the list bList to the list aList. For example, if bList contains [20, 40, 60], then aList will also contain [20, 40, 60] after the assignment.

• AAP-1.D.8 The exam reference sheet describes a list structure whose index values are 1 through the number of elements in the list, inclusive. For all list operations, if a list index is less than 1 or greater than the length of the list, an error message is produced and the program will terminate.

Enduring Understanding (AAP-2): The way statements are sequenced and combined in a program determines the computed result. Programs incorporate iteration and selection constructs to represent repetition and make decisions to handle varied input values.

Learning Objective AAP-2.A: Express an algorithm that uses sequencing without using a programming language. [Skill 2.A]

• AAP-2.A.1 An algorithm is a finite set of instructions that accomplish a specific task.
• AAP-2.A.2 Beyond visual and textual programming languages, algorithms can be expressed in a variety of ways, such as natural language, diagrams, and pseudocode.
• AAP-2.A.3 Algorithms executed by programs are implemented using programming languages.
• AAP-2.A.4 Every algorithm can be constructed using combinations of sequencing, selection, and iteration.

Learning Objective AAP-2.B: Represent a step-by-step algorithmic process using sequential code statements. [Skill 2.B]

• AAP-2.B.1 Sequencing is the application of each step of an algorithm in the order in which the code statements are given.
• AAP-2.B.2 A code statement is a part of program code that expresses an action to be carried out.
• AAP-2.B.3 An expression can consist of a value, a variable, an operator, or a procedure call that returns a value.
• AAP-2.B.4 Expressions are evaluated to produce a single value.
• AAP-2.B.5 The evaluation of expressions follows a set order of operations defined by the programming language.
• AAP-2.B.6 Sequential statements execute in the order they appear in the code segment.
• AAP-2.B.7 Clarity and readability are important considerations when expressing an algorithm in a programming language.

Learning Objective AAP-2.C: Evaluate expressions that use arithmetic operators. [Skill 4.B]

• AAP-2.C.1 Arithmetic operators are part of most programming languages and include addition, subtraction, multiplication, division, and modulus operators.

• AAP-2.C.2 The exam reference sheet provides a MOD b, which evaluates to the remainder when a is divided by b. Assume that a is an integer greater than or equal to 0 and b is an integer greater than 0. For example, 17 MOD 5 evaluates to 2.

• AAP-2.C.3 The exam reference sheet provides the arithmetic operators +, -, *, /, and MOD.

  Text and Block:

  • a + b
  • a - b
  • a * b
  • a / b
  • a MOD b

  These are used to perform arithmetic on a and b. For example, 17 / 5 evaluates to 3.4.

• AAP-2.C.4 The order of operations used in mathematics applies when evaluating expressions. The MOD operator has the same precedence as the * and / operators.

Enduring Understanding (AAP-2): The way statements are sequenced and combined in a program determines the computed result. Programs incorporate iteration and selection constructs to represent repetition and make decisions to handle varied input values.

Learning Objective AAP-2.D: Evaluate expressions that manipulate strings. [Skill 4.B]

• AAP-2.D.1 String concatenation joins together two or more strings end-to-end to make a new string.
• AAP-2.D.2 A substring is part of an existing string.

Enduring Understanding (AAP-2): The way statements are sequenced and combined in a program determines the computed result. Programs incorporate iteration and selection constructs to represent repetition and make decisions to handle varied input values.

Learning Objective AAP-2.E: For relationships between two variables, expressions, or values: a. Write expressions using relational operators. [Skill 2.B] b. Evaluate expressions that use relational operators. [Skill 4.B]

• AAP-2.E.1 A Boolean value is either true or false.

• AAP-2.E.2 The exam reference sheet provides the following relational operators: =, ≠, >, <, ≥, and ≤.

  Text and Block:

  • a = b
  • a ≠ b
  • a > b
  • a < b
  • a ≥ b
  • a ≤ b

  These are used to test the relationship between two variables, expressions, or values. A comparison using a relational operator evaluates to a Boolean value. For example, a = b evaluates to true if a and b are equal; otherwise, it evaluates to false.

Learning Objective AAP-2.F: For relationships between Boolean values: a. Write expressions using logical operators. [Skill 2.B] b. Evaluate expressions that use logic operators. [Skill 4.B]

• AAP-2.F.1 The exam reference sheet provides the logical operators NOT, AND, and OR, which evaluate to a Boolean value.

• AAP-2.F.2 The exam reference sheet provides

  Text:

  NOT condition

  Block:

  NOT condition

  which evaluates to true if condition is false; otherwise it evaluates to false.

• AAP-2.F.3 The exam reference sheet provides

  Text:

  condition1 AND condition2

  Block:

  condition1 AND condition2

  which evaluates to true if both condition1 and condition2 are true; otherwise it evaluates to false.

• AAP-2.F.4 The exam reference sheet provides

  Text:

  condition1 OR condition2

  Block:

  condition1 OR condition2

  which evaluates to true if condition1 is true or if condition2 is true or if both condition1 and condition2 are true; otherwise it evaluates to false.

• AAP-2.F.5 The operand for a logical operator is either a Boolean expression or a single Boolean value.

Enduring Understanding (AAP-2): The way statements are sequenced and combined in a program determines the computed result. Programs incorporate iteration and selection constructs to represent repetition and make decisions to handle varied input values.

Learning Objective AAP-2.G: Express an algorithm that uses selection without using a programming language. [Skill 2.A]

• AAP-2.G.1 Selection determines which parts of an algorithm are executed based on a condition being true or false.

Learning Objective AAP-2.H: For selection: a. Write conditional statements. [Skill 2.B] b. Determine the result of conditional statements. [Skill 4.B]

• AAP-2.H.1 Conditional statements, or "if-statements," affect the sequential flow of control by executing different statements based on the value of a Boolean expression.

• AAP-2.H.2 The exam reference sheet provides

  Text:

  IF(condition) { <block of statements> }

  Block:

  IF condition block of statements

  in which the code in block of statements is executed if the Boolean expression condition evaluates to true; no action is taken if condition evaluates to false.

• AAP-2.H.3 The exam reference sheet provides

  Text:

  IF(condition) { <first block of statements> } ELSE { <second block of statements> }

  Block:

  IF condition first block of statements ELSE second block of statements

  in which the code in first block of statements is executed if the Boolean expression condition evaluates to true; otherwise, the code in second block of statements is executed.

Enduring Understanding (AAP-2): The way statements are sequenced and combined in a program determines the computed result. Programs incorporate iteration and selection constructs to represent repetition and make decisions to handle varied input values.

Learning Objective AAP-2.I: For nested selection: a. Write nested conditional statements. [Skill 2.B] b. Determine the result of nested conditional statements. [Skill 4.B]

• AAP-2.I.1 Nested conditional statements consist of conditional statements within conditional statements.

Enduring Understanding (AAP-2): The way statements are sequenced and combined in a program determines the computed result. Programs incorporate iteration and selection constructs to represent repetition and make decisions to handle varied input values.

Learning Objective AAP-2.J: Express an algorithm that uses iteration without using a programming language. [Skill 2.A]

• AAP-2.J.1 Iteration is a repeating portion of an algorithm. Iteration repeats a specified number of times or until a given condition is met.

Learning Objective AAP-2.K: For iteration: a. Write iteration statements. [Skill 2.B] b. Determine the result or side effect of iteration statements. [Skill 4.B]

• AAP-2.K.1 Iteration statements change the sequential flow of control by repeating a set of statements zero or more times, until a stopping condition is met.

• AAP-2.K.2 The exam reference sheet provides

  Text:

  REPEAT n TIMES { <block of statements> }

  Block:

  REPEAT n TIMES block of statements

  in which the block of statements is executed n times.

• AAP-2.K.3 The exam reference sheet provides

  Text:

  REPEAT UNTIL(condition) { <block of statements> }

  Block:

  REPEAT UNTIL condition block of statements

  in which the code in block of statements is repeated until the Boolean expression condition evaluates to true.

• AAP-2.K.4 In REPEAT UNTIL(condition) iteration, an infinite loop occurs when the ending condition will never evaluate to true.

• AAP-2.K.5 In REPEAT UNTIL(condition) iteration, if the conditional evaluates to true initially, the loop body is not executed at all, due to the condition being checked before the loop.

Enduring Understanding (AAP-2): The way statements are sequenced and combined in a program determines the computed result. Programs incorporate iteration and selection constructs to represent repetition and make decisions to handle varied input values.

Learning Objective AAP-2.L: Compare multiple algorithms to determine if they yield the same side effect or result. [Skill 1.D]

• AAP-2.L.1 Algorithms can be written in different ways and still accomplish the same tasks.
• AAP-2.L.2 Algorithms that appear similar can yield different side effects or results.
• AAP-2.L.3 Some conditional statements can be written as equivalent Boolean expressions.
• AAP-2.L.4 Some Boolean expressions can be written as equivalent conditional statements.
• AAP-2.L.5 Different algorithms can be developed or used to solve the same problem.

Learning Objective AAP-2.M: For algorithms: a. Create algorithms. [Skill 2.A] b. Combine and modify existing algorithms. [Skill 2.B]

• AAP-2.M.1 Algorithms can be created from an idea, by combining existing algorithms, or by modifying existing algorithms.
• AAP-2.M.2 Knowledge of existing algorithms can help in constructing new ones. Some existing algorithms include:
  • determining the maximum or minimum value of two or more numbers
  • computing the sum or average of two or more numbers
  • identifying if an integer is or is not evenly divisible by another integer
  • determining a robot's path through a maze
• AAP-2.M.3 Using existing correct algorithms as building blocks for constructing another algorithm has benefits such as reducing development time, reducing testing, and simplifying the identification of errors.

Enduring Understanding (AAP-2): The way statements are sequenced and combined in a program determines the computed result. Programs incorporate iteration and selection constructs to represent repetition and make decisions to handle varied input values.

Learning Objective AAP-2.N: For list operations: a. Write expressions that use list indexing and list procedures. [Skill 2.B] b. Evaluate expressions that use list indexing and list procedures. [Skill 4.B]

• AAP-2.N.1 The exam reference sheet provides basic operations on lists, including:

  • accessing an element by index

    Text:

    aList[i]

    Block:

    aList i

    accesses the element of aList at index i. The first element of aList is at index 1 and is accessed using the notation aList[1].

  • assigning a value of an element of a list to a variable

    Text:

    x ← aList[i]

    Block:

    x ← aList i

    assigns the value of aList[i] to the variable x.

  • assigning a value to an element of a list

    Text:

    aList[i] ← x

    Block:

    aList i ← x

    assigns the value of x to aList[i].

    Text:

    aList[i] ← aList[j]

    Block:

    aList i ← aList j

    assigns the value of aList[j] to aList[i].

  • inserting elements at a given index

    Text:

    INSERT(aList, i, value)

    Block:

    INSERT aList, i, value

    shifts to the right any values in aList at indices greater than or equal to i. The length of the list is increased by 1, and value is placed at index i in aList.

  • adding elements to the end of the list

    Text:

    APPEND(aList, value)

    Block:

    APPEND aList, value

    increases the length of aList by 1, and value is placed at the end of aList.

  • removing elements

    Text:

    REMOVE(aList, i)

    Block:

    REMOVE aList, i

    removes the item at index i in aList and shifts to the left any values at indices greater than i. The length of aList is decreased by 1.

  • determining the length of a list

    Text:

    LENGTH(aList)

    Block:

    LENGTH aList

    evaluates to the number of elements currently in aList.

• AAP-2.N.2 List procedures are implemented in accordance with the syntax rules of the programming language.

Learning Objective AAP-2.O: For algorithms involving elements of a list: a. Write iteration statements to traverse a list. [Skill 2.B] b. Determine the result of an algorithm that includes list traversals. [Skill 4.B]

• AAP-2.O.1 Traversing a list can be a complete traversal, where all elements in the list are accessed, or a partial traversal, where only a portion of elements are accessed.

  • Exclusion statement (EK AAP-2.O.1): Traversing multiple lists at the same time using the same index for both (parallel traversals) is outside the scope of this course and the AP Exam.

• AAP-2.O.2 Iteration statements can be used to traverse a list.

• AAP-2.O.3 The exam reference sheet provides

  Text:

  FOR EACH item IN aList { <block of statements> }

  Block:

  FOR EACH item IN aList block of statements

  The variable item is assigned the value of each element of aList sequentially, in order, from the first element to the last element. The code in block of statements is executed once for each assignment of item.

• AAP-2.O.4 Knowledge of existing algorithms that use iteration can help in constructing new algorithms. Some examples of existing algorithms that are often used with lists include:

  • determining a minimum or maximum value in a list
  • computing a sum or average of a list of numbers

• AAP-2.O.5 Linear search or sequential search algorithms check each element of a list, in order, until the desired value is found or all elements in the list have been checked.

Enduring Understanding (AAP-2): The way statements are sequenced and combined in a program determines the computed result. Programs incorporate iteration and selection constructs to represent repetition and make decisions to handle varied input values.

Learning Objective AAP-2.P: For binary search algorithms: a. Determine the number of iterations required to find a value in a data set. [Skill 1.D] b. Explain the requirements necessary to complete a binary search. [Skill 1.A]

• AAP-2.P.1 The binary search algorithm starts at the middle of a sorted data set of numbers and eliminates half of the data; this process repeats until the desired value is found or all elements have been eliminated.
  • Exclusion statement (EK AAP-2.P.1): Specific implementations of the binary search are outside the scope of the course and the AP Exam.
• AAP-2.P.2 Data must be in sorted order to use the binary search algorithm.
• AAP-2.P.3 Binary search is often more efficient than sequential/linear search when applied to sorted data.

Enduring Understanding (AAP-3): Programmers break down problems into smaller and more manageable pieces. By creating procedures and leveraging parameters, programmers generalize processes that can be reused. Procedures allow programmers to draw upon existing code that has already been tested, allowing them to write programs more quickly and with more confidence.

Learning Objective AAP-3.A: For procedure calls: a. Write statements to call procedures. [Skill 3.B] b. Determine the result or effect of a procedure call. [Skill 4.B]

• AAP-3.A.1 A procedure is a named group of programming instructions that may have parameters and return values.

• AAP-3.A.2 Procedures are referred to by different names, such as method or function, depending on the programming language.

• AAP-3.A.3 Parameters are input variables of a procedure. Arguments specify the values of the parameters when a procedure is called.

• AAP-3.A.4 A procedure call interrupts the sequential execution of statements, causing the program to execute the statements within the procedure before continuing. Once the last statement in the procedure (or a return statement) has executed, flow of control is returned to the point immediately following where the procedure was called.

• AAP-3.A.5 The exam reference sheet provides

  procName(arg1, arg2, ...)

  as a way to call

  Text:

  PROCEDURE procName(parameter1, parameter2, ...) { <block of statements> }

  Block:

  PROCEDURE procName parameter1, parameter2,... block of statements

  which takes zero or more arguments; arg1 is assigned to parameter1, arg2 is assigned to parameter2, and so on.

• AAP-3.A.6 The exam reference sheet provides the procedure

  Text:

  DISPLAY(expression)

  Block:

  DISPLAY expression

  to display the value of expression, followed by a space.

• AAP-3.A.7 The exam reference sheet provides the

  Text:

  RETURN(expression)

  Block:

  RETURN expression

  statement, which is used to return the flow of control to the point where the procedure was called and to return the value of expression.

• AAP-3.A.8 The exam reference sheet provides

  result ← procName(arg1, arg2, ...)

  to assign to result the "value of the procedure" being returned by calling

  Text:

  PROCEDURE procName(parameter1, parameter2, ...) { <block of statements> RETURN(expression) }

  Block:

  PROCEDURE procName parameter1, parameter2,... block of statements RETURN expression

• AAP-3.A.9 The exam reference sheet provides procedure

  Text:

  INPUT()

  Block:

  INPUT

  which accepts a value from the user and returns the input value.

Enduring Understanding (AAP-3): Programmers break down problems into smaller and more manageable pieces. By creating procedures and leveraging parameters, programmers generalize processes that can be reused. Procedures allow programmers to draw upon existing code that has already been tested, allowing them to write programs more quickly and with more confidence.

Learning Objective AAP-3.B: Explain how the use of procedural abstraction manages complexity in a program. [Skill 3.C]

• AAP-3.B.1 One common type of abstraction is procedural abstraction, which provides a name for a process and allows a procedure to be used only knowing what it does, not how it does it.
• AAP-3.B.2 Procedural abstraction allows a solution to a large problem to be based on the solutions of smaller subproblems. This is accomplished by creating procedures to solve each of the subproblems.
• AAP-3.B.3 The subdivision of a computer program into separate subprograms is called modularity.
• AAP-3.B.4 A procedural abstraction may extract shared features to generalize functionality instead of duplicating code. This allows for program code reuse, which helps manage complexity.
• AAP-3.B.5 Using parameters allows procedures to be generalized, enabling the procedures to be reused with a range of input values or arguments.
• AAP-3.B.6 Using procedural abstraction helps improve code readability.
• AAP-3.B.7 Using procedural abstraction in a program allows programmers to change the internals of the procedure (to make it faster, more efficient, use less storage, etc.) without needing to notify users of the change as long as what the procedure does is preserved.

Learning Objective AAP-3.C: Develop procedural abstractions to manage complexity in a program by writing procedures. [Skill 3.B]

• AAP-3.C.1 The exam reference sheet provides

  Text:

  PROCEDURE procName(parameter1, parameter2, ...) { <block of statements> }

  Block:

  PROCEDURE procName parameter1, parameter2,... block of statements

  which is used to define a procedure that takes zero or more arguments. The procedure contains block of statements.

• AAP-3.C.2 The exam reference sheet provides

  Text:

  PROCEDURE procName(parameter1, parameter2, ...) { <block of statements> RETURN(expression) }

  Block:

  PROCEDURE procName parameter1, parameter2,... block of statements RETURN expression

  which is used to define a procedure that takes zero or more arguments. The procedure contains block of statements and returns the value of expression. The RETURN statement may appear at any point inside the procedure and causes an immediate return from the procedure back to the calling statement.

Enduring Understanding (AAP-3): Programmers break down problems into smaller and more manageable pieces. By creating procedures and leveraging parameters, programmers generalize processes that can be reused. Procedures allow programmers to draw upon existing code that has already been tested, allowing them to write programs more quickly and with more confidence.

Learning Objective AAP-3.D: Select appropriate libraries or existing code segments to use in creating new programs. [Skill 2.B]

• AAP-3.D.1 A software library contains procedures that may be used in creating new programs.
• AAP-3.D.2 Existing code segments can come from internal or external sources, such as libraries or previously written code.
• AAP-3.D.3 The use of libraries simplifies the task of creating complex programs.
• AAP-3.D.4 Application program interfaces (APIs) are specifications for how the procedures in a library behave and can be used.
• AAP-3.D.5 Documentation for an API/library is necessary in understanding the behaviors provided by the API/library and how to use them.

Enduring Understanding (AAP-3): Programmers break down problems into smaller and more manageable pieces. By creating procedures and leveraging parameters, programmers generalize processes that can be reused. Procedures allow programmers to draw upon existing code that has already been tested, allowing them to write programs more quickly and with more confidence.

Learning Objective AAP-3.E: For generating random values: a. Write expressions to generate possible values. [Skill 2.B] b. Evaluate expressions to determine the possible results. [Skill 4.B]

• AAP-3.E.1 The exam reference sheet provides

  Text:

  RANDOM(a, b)

  Block:

  RANDOM a, b

  which generates and returns a random integer from a to b, inclusive. Each result is equally likely to occur. For example, RANDOM(1, 3) could return 1, 2, or 3.

• AAP-3.E.2 Using random number generation in a program means each execution may produce a different result.

Enduring Understanding (AAP-3): Programmers break down problems into smaller and more manageable pieces. By creating procedures and leveraging parameters, programmers generalize processes that can be reused. Procedures allow programmers to draw upon existing code that has already been tested, allowing them to write programs more quickly and with more confidence.

Learning Objective AAP-3.F: For simulations: a. Explain how computers can be used to represent real-world phenomena or outcomes. [Skill 1.A] b. Compare simulations with real-world contexts. [Skill 1.D]

• AAP-3.F.1 Simulations are abstractions of more complex objects or phenomena for a specific purpose.
• AAP-3.F.2 A simulation is a representation that uses varying sets of values to reflect the changing state of a phenomenon.
• AAP-3.F.3 Simulations often mimic real-world events with the purpose of drawing inferences, allowing investigation of a phenomenon without the constraints of the real world.
• AAP-3.F.4 The process of developing an abstract simulation involves removing specific details or simplifying functionality.
• AAP-3.F.5 Simulations can contain bias derived from the choices of real-world elements that were included or excluded.
• AAP-3.F.6 Simulations are most useful when real-world events are impractical for experiments (e.g., too big, too small, too fast, too slow, too expensive, or too dangerous).
• AAP-3.F.7 Simulations facilitate the formulation and refinement of hypotheses related to the objects or phenomena under consideration.
• AAP-3.F.8 Random number generators can be used to simulate the variability that exists in the real world.

Enduring Understanding (AAP-4): There exist problems that computers cannot solve, and even when a computer can solve a problem, it may not be able to do so in a reasonable amount of time.

Learning Objective AAP-4.A: For determining the efficiency of an algorithm: a. Explain the difference between algorithms that run in reasonable time and those that do not. [Skill 1.D] b. Identify situations where a heuristic solution may be more appropriate. [Skill 1.D]

• AAP-4.A.1 A problem is a general description of a task that can (or cannot) be solved algorithmically. An instance of a problem also includes specific input. For example, sorting is a problem; sorting the list (2,3,1,7) is an instance of the problem.
• AAP-4.A.2 A decision problem is a problem with a yes/no answer (e.g., is there a path from A to B?). An optimization problem is a problem with the goal of finding the "best" solution among many (e.g., what is the shortest path from A to B?).
• AAP-4.A.3 Efficiency is an estimation of the amount of computational resources used by an algorithm. Efficiency is typically expressed as a function of the size of the input.
  • Exclusion statement (EK AAP-4.A.3): Formal analysis of algorithms (Big-O) and formal reasoning using mathematical formulas are outside the scope of this course and the AP Exam.
• AAP-4.A.4 An algorithm's efficiency is determined through formal or mathematical reasoning.
• AAP-4.A.5 An algorithm's efficiency can be informally measured by determining the number of times a statement or group of statements executes.
• AAP-4.A.6 Different correct algorithms for the same problem can have different efficiencies.
• AAP-4.A.7 Algorithms with a polynomial efficiency or slower (constant, linear, square, cube, etc.) are said to run in a reasonable amount of time. Algorithms with exponential or factorial efficiencies are examples of algorithms that run in an unreasonable amount of time.
• AAP-4.A.8 Some problems cannot be solved in a reasonable amount of time because there is no efficient algorithm for solving them. In these cases, approximate solutions are sought.
• AAP-4.A.9 A heuristic is an approach to a problem that produces a solution that is not guaranteed to be optimal but may be used when techniques that are guaranteed to always find an optimal solution are impractical.
  • Exclusion statement (AAP-4.A.9): Specific heuristic solutions are outside the scope of this course and the AP Exam.

Enduring Understanding (AAP-4): There exist problems that computers cannot solve, and even when a computer can solve a problem, it may not be able to do so in a reasonable amount of time.

Learning Objective AAP-4.B: Explain the existence of undecidable problems in computer science. [Skill 1.A]

• AAP-4.B.1 A decidable problem is a decision problem for which an algorithm can be written to produce a correct output for all inputs (e.g., "Is the number even?").
• AAP-4.B.2 An undecidable problem is one for which no algorithm can be constructed that is always capable of providing a correct yes-or-no answer.
  • Exclusion statement (EK AAP-4.B.2): Determining whether a given problem is undecidable is outside the scope of this course and the AP Exam.
• AAP-4.B.3 An undecidable problem may have some instances that have an algorithmic solution, but there is no algorithmic solution that could solve all instances of the problem.