1 / 8

Dynamic Programming

Dynamic Programming. Dynamic Programming is a general algorithm design technique Invented by American mathematician Richard Bellman in the 1950s to solve optimization problems “Programming” here means “planning” Main idea: solve several smaller (overlapping) subproblems

lalex
Télécharger la présentation

Dynamic Programming

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Dynamic Programming • Dynamic Programming is a general algorithm design technique • Invented by American mathematician Richard Bellman in the 1950s to solve optimization problems • “Programming” here means “planning” • Main idea: • solve several smaller (overlapping) subproblems • record solutions in a table so that each subproblem is only solved once • final state of the table will be (or contain) solution Design and Analysis of Algorithms - Chapter 8

  2. Example: Fibonacci numbers • Recall definition of Fibonacci numbers: • f(0) = 0 • f(1) = 1 • f(n) = f(n-1) + f(n-2) • Computing the nth Fibonacci number recursively (top-down): • f(n) • f(n-1) + f(n-2) • f(n-2) + f(n-3) f(n-3) + f(n-4) • ... Design and Analysis of Algorithms - Chapter 8

  3. Example: Fibonacci numbers • Computing the nth fibonacci number using bottom-up iteration: • f(0) = 0 • f(1) = 1 • f(2) = 0+1 = 1 • f(3) = 1+1 = 2 • f(4) = 1+2 = 3 • f(5) = 2+3 = 5 • f(n-2) = • f(n-1) = • f(n) = f(n-1) + f(n-2) Design and Analysis of Algorithms - Chapter 8

  4. Examples of Dynamic Programming Algorithms • Computing binomial coefficients • Optimal chain matrix multiplication • Constructing an optimal binary search tree • Warshall’s algorithm for transitive closure • Floyd’s algorithms for all-pairs shortest paths • Some instances of difficult discrete optimization problems: • travelling salesman • knapsack Design and Analysis of Algorithms - Chapter 8

  5. 3 3 1 1 4 4 2 2 Warshall’s Algorithm: Transitive Closure • Computes the transitive closure of a relation • (Alternatively: all paths in a directed graph) • Example of transitive closure: 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0 Design and Analysis of Algorithms - Chapter 8

  6. 3 3 3 3 3 1 1 1 1 1 4 2 4 4 2 2 4 4 2 2 Warshall’s Algorithm • Main idea: a path exists between two vertices i, j, iff • there is an edge from i to j; or • there is a path from i to j going through vertex 1; or • there is a path from i to j going through vertex 1 and/or 2; or • there is a path from i to j going through vertex 1, 2, and/or 3; or • ... • there is a path from i to j going through any of the other vertices R0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0 R1 0 0 1 0 1 01 1 0 0 0 0 0 1 0 0 R2 0 0 1 0 1 0 1 1 0 0 0 0 1 1 1 1 R3 0 0 1 0 1 0 1 1 0 0 0 0 1 1 1 1 R4 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 Design and Analysis of Algorithms - Chapter 8

  7. Warshall’s Algorithm • In the kth stage determine if a path exists between two vertices i, j using just vertices among 1,…,k • R(k-1)[i,j] (path using just 1 ,…,k-1) • R(k)[i,j] = or • (R(k-1)[i,k] and R(k-1)[k,j]) (path from i to k • and from k to i • using just 1 ,…,k-1) { k i kth stage j Design and Analysis of Algorithms - Chapter 8

  8. 4 3 1 1 6 1 5 4 2 3 Floyd’s Algorithm: All pairs shortest paths • In a weighted graph, find shortest paths between every pair of vertices • Same idea: construct solution through series of matrices D(0), D(1), … using an initial subset of the vertices as intermediaries. • Example: Design and Analysis of Algorithms - Chapter 8

More Related