Design Ch.12 Test 2

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What defines a problem as tractable in algorithm complexity?

A polynomial-time algorithm exists to solve it. (correct)

A problem that has no known solution.

A problem that can only be solved heuristically.

A problem that can be solved in exponential time.

Which of the following statements best describes problems classified under P?

They are equivalent to NP problems.

They require superpolynomial time for solutions.

They cannot be solved efficiently.

They can be solved in polynomial time. (correct)

What differentiates NP-Complete problems from NP problems?

NP-Complete problems have polynomial-time solutions.

Every NP problem can be transformed into an NP-Complete problem. (correct)

NP-Complete problems can be solved faster than NP problems.

NP-Complete problems cannot be verified in polynomial time.

In the context of algorithm complexity, what is a characteristic of comparison-based sorting algorithms?

They typically have a time complexity of O(n log n) in the average case. Signup and view all the answers

Which of these statements is correct regarding P vs NP problems?

P = NP suggests all decision problems can be solved efficiently. Signup and view all the answers

What do the internal nodes of a decision tree represent?

Comparisons Signup and view all the answers

Which of the following best describes decision trees?

Model algorithms based on comparisons Signup and view all the answers

In a decision tree, what role do the internal nodes play in the overall structure?

They function as criteria for branching decisions Signup and view all the answers

What is a significant characteristic of decision tree models?

They involve hierarchical comparisons Signup and view all the answers

Which statement is true about the function of comparisons in decision trees?

Comparisons determine the structure of entire algorithms Signup and view all the answers

What does the reduction from A to B represent in algorithm complexity?

B being easier or more solvable than A Signup and view all the answers

What is the significance of element uniqueness in pairs?

It ensures every element can be paired without duplication. Signup and view all the answers

In the context of reductions, what is the outcome of a minimum reducing to a sorting problem?

The sorting algorithm will always find the minimum element. Signup and view all the answers

What does it imply if B reduces to A?

A solved implies B is solved. Signup and view all the answers

What does the notation $Ω(f)$ represent in complexity theory?

It defines a lower bound for the running time. Signup and view all the answers

Which of the following is NOT a method for establishing lower bounds?

Comparison-based sorting Signup and view all the answers

What type of lower bound can be established using decision trees?

Time complexity bounds Signup and view all the answers

Which of the following accurately describes trivial lower bounds?

Bounds established without detail on the algorithm Signup and view all the answers

In what context would adversary arguments be used?

To derive lower bounds on computational problems Signup and view all the answers

What is an example of a method that utilizes problem reduction?

Translating a problem into another solvable problem Signup and view all the answers

What does a tight lower bound in algorithm efficiency indicate?

There exists an algorithm with the same efficiency as the lower bound. Signup and view all the answers

Which of the following is an example of a situation where the tight lower bound is relevant?

Sorting an array of size n Signup and view all the answers

How is the efficiency of an algorithm characterized by its tight lower bound?

It shows the minimum number of operations that can be achieved for specific inputs. Signup and view all the answers

Which of the following scenarios does NOT relate to the concept of a tight lower bound?

Calculating the number of distinct elements in an array. Signup and view all the answers

What key feature is represented by a tight lower bound in algorithmic analysis?

An algorithm exists that can achieve this performance level on specific inputs. Signup and view all the answers

Study Notes

Lower Bounds

An estimate of the minimum amount of work needed to solve a problem.
An exact count or an efficiency class (Ω).
A tight lower bound exists when an algorithm has the same efficiency as the lower bound.

Examples of Lower Bounds

Finding the largest element in a set of n numbers.
Sorting an array of size n.
Searching a sorted array.
Multiplying two n x n matrices.

Methods for Establishing Lower Bounds

Information-theoretic arguments (decision trees).
Adversary arguments.
Problem reduction.

Lower Bounds by Problem Reduction

A problem with a known lower bound Ω(f) can be reduced to a problem with an unknown lower bound.
If problem P is at least as hard as problem Q, a lower bound for Q is also a lower bound for P.
Problem reduction may assume the reduction itself is fast, i.e., O(f).

Example of Problem Reduction

Finding the minimum spanning tree (MST) for n points in the Cartesian plane.
Element uniqueness problems are Ω(n log n)
Result: MST reduces to element uniqueness, and both have a lower bound of Ω(n log n).

Trivial Lower Bounds

Based on counting items to be processed in input and output.
Examples include finding the maximum element and evaluating polynomial expressions.

Decision Trees

Algorithms that use comparisons
Internal nodes represent comparisons.
Leaves represent outcomes.
Any comparison-based sorting algorithm can be represented by a decision tree.
The number of leaves (outcomes) is greater than or equal to n!
The height of a binary tree with n! leaves will be greater than or equal to log₂ n!

Decision Trees and Sorting

Minimum comparisons in the worst case is greater than or equal to log₂ n!.
This lower bound is tight (mergesort).

P, NP, and NP-Complete

P: Problems solvable in polynomial time.
NP: Problems whose solutions can be verified in polynomial time.
NP-complete: The hardest NP problems, where solving one solvable in polynomial time, means all NP problems are also solved by polynomial time.

Classifying Problems

Tractable: Problems solvable in polynomial time.
Non-tractable: Problems not solvable in polynomial time.

Classifying Problem Complexity

Is the problem tractable (solvable in polynomial time)?
Possible answers: Yes (e.g., searching), No (e.g., halting problem), or Unknown.

Problem Types

Optimization: Finding a solution that maximizes or minimizes an objective function.
Decision: Answering a yes/no question.

NP-Complete Problems

A decision problem in NP is NP-complete if:
- It's in NP.
- Every problem in NP is reducible to it.

Proving Other Problems are NP-Complete

Prove by polynomial-time reductions from a known NP-complete problem.
Pay attention to the direction of the reductions.

P = NP?

If P = NP, every problem in NP can be solved in polynomial time.

Summary

P: Class of decision problems solvable in polynomial time.
NP: Class of decision problems where solutions can be verified in polynomial time.
NP-Complete: Problems that are reducible to each other in polynomial time.
Cook's Theorem: First proof establishing the NP-completeness of a problem.

Reduction for Lower Bound

Establish a lower bound by reducing a problem with a known lower bound to the problem in question.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.