Review of Algorithms & NP-Completeness: Dynamic Programming, Greedy Algorithms, Slides of Computer Science

Download Review of Algorithms & NP-Completeness: Dynamic Programming, Greedy Algorithms and more Slides Computer Science in PDF only on Docsity!

Algorithms

NP Completeness

Review: Dynamic Programming

● When applicable:

■ Optimal substructure: optimal solution to problem consists of optimal solutions to subproblems ■ Overlapping subproblems: few subproblems in total, many recurring instances of each ■ Basic approach: ○ Build a table of solved subproblems that are used to solve larger ones ○ What is the difference between memoization and dynamic programming? ○ Why might the latter be more efficient?

Review: Activity-Selection Problem

● The activity selection problem : get your

money’s worth out of a carnival

■ Buy a wristband that lets you onto any ride ■ Lots of rides, starting and ending at different times ■ Your goal: ride as many rides as possible

● Naïve first-year CS major strategy:

■ Ride the first ride, when get off, get on the very next ride possible, repeat until carnival ends

● What is the sophisticated third-year strategy?

Review: Activity-Selection

● Formally:

■ Given a set S of n activities ○ s (^) i = start time of activity i fi = finish time of activity i ■ Find max-size subset A of compatible activities ■ Assume activities sorted by finish time

● What is optimal substructure for this problem?

Review: Greedy Choice Property For Activity Selection

● Dynamic programming? Memoize? Yes, but…

● Activity selection problem also exhibits the

greedy choice property:

■ Locally optimal choice ⇒ globally optimal sol’n ■ Them 17.1: if S is an activity selection problem sorted by finish time, then ∃ optimal solution A ⊆ S such that {1} ∈ A ○ Sketch of proof: if ∃ optimal solution B that does not contain {1}, can always replace first activity in B with {1} ( Why? ). Same number of activities, thus optimal.

Review: The Knapsack Problem

● The 0-1 knapsack problem :

■ A thief must choose among n items, where the i th item worth vi dollars and weighs w (^) i pounds ■ Carrying at most W pounds, maximize value

● A variation, the fractional knapsack problem :

■ Thief can take fractions of items ■ Think of items in 0-1 problem as gold ingots, in fractional problem as buckets of gold dust

● What greedy choice algorithm works for the

fractional problem but not the 0-1 problem? Docsity.com

Polynomial-Time Algorithms

● Are some problems solvable in polynomial

time?

■ Of course: every algorithm we’ve studied provides polynomial-time solution to some problem ■ We define P to be the class of problems solvable in polynomial time

● Are all problems solvable in polynomial time?

■ No: Turing’s “Halting Problem” is not solvable by any computer, no matter how much time is given ■ Such problems are clearly intractable, not in P

NP-Complete Problems

● The NP-Complete problems are an interesting

class of problems whose status is unknown

■ No polynomial-time algorithm has been discovered for an NP-Complete problem ■ No suprapolynomial lower bound has been proved for any NP-Complete problem, either

● We call this the P = NP question

■ The biggest open problem in CS

P and NP

● As mentioned, P is set of problems that can be

solved in polynomial time

● NP ( nondeterministic polynomial time ) is the

set of problems that can be solved in

polynomial time by a nondeterministic

computer

■ What the hell is that?

Nondeterminism

● Think of a non-deterministic computer as a

computer that magically “guesses” a solution,

then has to verify that it is correct

■ If a solution exists, computer always guesses it ■ One way to imagine it: a parallel computer that can freely spawn an infinite number of processes ○ Have one processor work on each possible solution ○ All processors attempt to verify that their solution works ○ If a processor finds it has a working solution ■ So: NP = problems verifiable in polynomial time

NP-Complete Problems

● We will see that NP-Complete problems are

the “hardest” problems in NP:

■ If any one NP-Complete problem can be solved in polynomial time… ■ …then every NP-Complete problem can be solved in polynomial time… ■ …and in fact every problem in NP can be solved in polynomial time (which would show P = NP ) ■ Thus: solve hamiltonian-cycle in O( n^100 ) time, you’ve proved that P = NP. Retire rich & famous.

Reduction

● The crux of NP-Completeness is reducibility

■ Informally, a problem P can be reduced to another problem Q if any instance of P can be “easily rephrased” as an instance of Q, the solution to which provides a solution to the instance of P ○ What do you suppose “easily” means? ○ This rephrasing is called transformation ■ Intuitively: If P reduces to Q, P is “no harder to solve” than Q

Using Reductions

● If P is polynomial-time reducible to Q, we

denote this P ≤p Q

● Definition of NP-Complete:

■ If P is NP-Complete, P ∈ NP and all problems R are reducible to P ■ Formally: R ≤p P ∀ R ∈ NP

● If P ≤p Q and P is NP-Complete, Q is also NP-

Complete

■ This is the key idea you should take away today

Coming Up

● Given one NP-Complete problem, we can

prove many interesting problems NP-Complete

■ Graph coloring (= register allocation) ■ Hamiltonian cycle ■ Hamiltonian path ■ Knapsack problem ■ Traveling salesman ■ Job scheduling with penalities ■ Many, many more