Design Ch.12 Test 2
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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?

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

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

<p>P = NP suggests all decision problems can be solved efficiently. (A), All problems in P are also in NP. (B)</p> Signup and view all the answers

What do the internal nodes of a decision tree represent?

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

Which of the following best describes decision trees?

<p>Model algorithms based on comparisons (A)</p> Signup and view all the answers

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

<p>They function as criteria for branching decisions (B)</p> Signup and view all the answers

What is a significant characteristic of decision tree models?

<p>They involve hierarchical comparisons (D)</p> Signup and view all the answers

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

<p>Comparisons determine the structure of entire algorithms (D)</p> Signup and view all the answers

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

<p>B being easier or more solvable than A (C)</p> Signup and view all the answers

What is the significance of element uniqueness in pairs?

<p>It ensures every element can be paired without duplication. (B)</p> Signup and view all the answers

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

<p>The sorting algorithm will always find the minimum element. (B)</p> Signup and view all the answers

What does it imply if B reduces to A?

<p>A solved implies B is solved. (D)</p> Signup and view all the answers

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

<p>It defines a lower bound for the running time. (C)</p> Signup and view all the answers

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

<p>Comparison-based sorting (B)</p> Signup and view all the answers

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

<p>Time complexity bounds (A)</p> Signup and view all the answers

Which of the following accurately describes trivial lower bounds?

<p>Bounds established without detail on the algorithm (A)</p> Signup and view all the answers

In what context would adversary arguments be used?

<p>To derive lower bounds on computational problems (D)</p> Signup and view all the answers

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

<p>Translating a problem into another solvable problem (B)</p> Signup and view all the answers

What does a tight lower bound in algorithm efficiency indicate?

<p>There exists an algorithm with the same efficiency as the lower bound. (B)</p> Signup and view all the answers

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

<p>Sorting an array of size n (B)</p> Signup and view all the answers

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

<p>It shows the minimum number of operations that can be achieved for specific inputs. (D)</p> Signup and view all the answers

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

<p>Calculating the number of distinct elements in an array. (B)</p> Signup and view all the answers

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

<p>An algorithm exists that can achieve this performance level on specific inputs. (D)</p> Signup and view all the answers

Flashcards

Tractable problem

A problem that can be solved efficiently using a polynomial-time algorithm.

Polynomial time algorithm

An algorithm whose running time is proportional to a polynomial function of the input size (n).

O(p(n))

Big O notation denoting polynomial time complexity.

Problem Complexity

How difficult a problem is to solve when the size of the data changes.

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Algorithm

A set of steps or instructions to solve a problem.

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Decision Tree

A model that makes decisions by asking a series of questions (comparisons).

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Internal Node

A point in a decision tree where a comparison is made. It branches out based on the answer to the comparison.

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Comparison

A question asked in a decision tree to help guide towards a decision.

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Leaf Node

The final outcome in a decision tree, determined by choices made along the way.

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How does a Decision Tree Work?

A decision tree works by starting at the root node, asking a comparison question, and following the branch corresponding to the answer. This process continues until a leaf node is reached, which represents the final decision or prediction.

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Ω(f)

A lower bound on the running time of an algorithm, meaning the algorithm takes at least this much time in the worst case. The function f describes how the running time grows with the input size.

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Sort is Ω(n log n)

Sorting algorithms have a lower bound of n log n, meaning no algorithm can sort faster than n log n in the worst case for a list of n elements.

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Minimum is Ω(n)

Finding the minimum value in a list of n elements requires examining at least n elements, resulting in a lower bound of Ω(n).

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Element uniqueness for pairs is Ω(n^2)

Checking if all elements in a list of pairs are unique in the worst case requires comparing each pair to every other pair, resulting in a lower bound of Ω(n^2).

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B Ω(?) reduces to A

If problem B has a lower bound complexity of Ω(f), and A is a subproblem of B, then A must also have a lower bound of at least Ω(f).

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Tight Lower Bound

An algorithm's efficiency is limited by a lower bound. It means no algorithm can be more efficient than this bound in the worst-case scenario.

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Finding the largest element (Lower Bound)

The minimum number of comparisons needed to find the largest element in a set of 'n' numbers is 'n-1'.

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Sorting an array (Lower Bound)

The minimum number of comparisons required to sort an array of size 'n' is Ω(n log n).

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Lower Bound for Subproblems

If a problem 'B' requires Ω(f) complexity to solve, any subproblem 'A' within 'B' must also take at least Ω(f).

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Lower Bound

The minimum amount of time or resources required by any algorithm to solve a problem, regardless of the specific algorithm used.

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Trivial Lower Bound

A basic lower bound that's often obvious based on the problem's requirements. For example, finding the minimum in an array requires at least looking at all elements.

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Information-Theoretic Argument

Proves lower bounds by analyzing the amount of information an algorithm needs to process to solve the problem.

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Adversary Argument

A technique where an imaginary opponent (adversary) forces an algorithm to perform a certain number of operations in the worst-case scenario.

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Problem Reduction

Proves lower bounds by showing that a problem A can be solved using another problem B with a known lower bound. This implies that A must have at least the same lower bound as B.

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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.

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