Motwani, Rajeev. Randomized ·algorithms / Rajeev Motwani, Prabhakar Raghavan. p. em. Includes bibliographical references and index. .SBN Full Text: PDF View colleagues of Rajeev Motwani . Rajeev Motwani, Prabhakar Raghavan, Randomized algorithms, Cambridge. Randomized Algorithms. RAJEEV MOTWANI. Department of Computer Science, Stanford University, Stanford, California. PRABHAKAR RAGHAVAN.
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Classifying randomized algorithms by their goals 8 http://www. cs. nbafinals.info, and http:// Rajeev Motwani and Prabhakar Raghavan, Randomized Algorithms. Authors: Rajeev Motwani, Stanford University, California; Prabhakar Raghavan, the basic concepts in the design and analysis of randomized algorithms. Cambridge Core - Optimization, OR and risk - Randomized Algorithms - by Rajeev Motwani. Rajeev Motwani, Stanford University, California, Prabhakar Raghavan, Google, Inc. Publisher: Cambridge . PDF; Export citation. Contents.
Observe that this implies that the primality problem is in Co- RP.
If n is big, there may be no other test that is practical. The probability of error can be reduced to an arbitrary degree by performing enough independent tests. Therefore, in practice, there is no penalty associated with accepting a small probability of error, since with a little care the probability of error can be made astronomically small.
Indeed, even though a deterministic polynomial-time primality test has since been found see AKS primality test , it has not replaced the older probabilistic tests in cryptographic software nor is it expected to do so for the foreseeable future.
Quicksort[ edit ] Quicksort is a familiar, commonly used algorithm in which randomness can be useful. Any deterministic version of this algorithm requires O n2 time to sort n numbers for some well-defined class of degenerate inputs such as an already sorted array , with the specific class of inputs that generate this behavior defined by the protocol for pivot selection.
Randomized incremental constructions in geometry[ edit ] In computational geometry , a standard technique to build a structure like a convex hull or Delaunay triangulation is to randomly permute the input points and then insert them one by one into the existing structure. The randomization ensures that the expected number of changes to the structure caused by an insertion is small, and so the expected running time of the algorithm can be upper bounded.
This technique is known as randomized incremental construction. Recall that the contraction of two nodes, u and v, in a multi- graph yields a new node u ' with edges that are the union of the edges incident on either u or v, except from any edge s connecting u and v. Figure 1 gives an example of contraction of vertex A and B. Alon, Noga; Spencer, Joel H. Skip to content.
Logistics Course description: Friedrich Eisenbrand Assistant: Chidambaram Annamalai Language: English ECTS credits: Important notes: Please try to send the scribe notes within a week from the lecture! Make sure to send the tex files with the pdf.
The deadline for submitting solutions to the fourth problem set is Dec 17 Use the preamble. Books and references Randomized Algorithms.
Problem Set 2. In the second part of the lecture, we learn about probabilistic analysis of algorithms.
There are a number of important problems and algorithms for which worst-case analysis does not provide useful or empirically accurate results. One prominent example is the simplex method for linear programming whose worst-case running time is exponential while in fact it runs in near-linear time on almost all inputs of interest. Another example is the knapsack problem.
While this problem is NP-hard, it is a very easy optimization problem in practice and even very large instances with millions of items can be solved efficiently. The reason for this discrepancy between worst-case analysis and empirical observations is that for many algorithms worst-case instances have an artificial structure and hardly ever occur in practical applications.
In smoothed analysis, one does not study the worst-case behavior of an algorithm but its expected behavior on random or randomly perturbed inputs. We will prove, for example, that there are algorithms for the knapsack problem whose expected running time is polynomial if the profits or weights are slightly perturbed at random.