## Divide and Conquer, Sorting and Searching, and Randomized Algorithms

Start Date: 07/05/2020

 Course Type: Common Course

Explore 1600+ online courses from top universities. Join Coursera today to learn data science, programming, business strategy, and more. The primary topics in this part of the specialization are: asymptotic ("Big-oh") notation, sorting and searching, divide and conquer (master method, integer and matrix multiplication, closest pair), and randomized algorithms (QuickSort, contraction algorithm for min cuts). #### Course Introduction

Divide and Conquer, Sorting and Searching, and Randomized Algorithms The objective of this course is to provide a high-level overview of the divide-and-conquer (D&C) model and its elements, sorting and searching, and randomized algorithms. We will cover topics such as the inception, design, implementation, evaluation, and safety specifications, and we will also introduce the new generation (R&S) algorithms. The course will also introduce the new generation (R&S) algorithms, and we will also introduce the new generation (R&S) algorithms from the perspective of a programmer. The course will also introduce the Linked-List and Linked-Set algorithms, and we will also introduce the Iterable and DataList algorithms. The course will also introduce the new generation (R&S) algorithms and the Linked-List and Linked-Set algorithms, and we will also introduce the Iterable and DataList algorithms. This course requires no prior knowledge in computer science or mathematics. You will learn about the basics of algorithms and data structures in a simple and effective way. We will introduce the divide-and-conquer (D&C) model, and we will introduce the key concepts, and we will also introduce the new generation (R&S) algorithms. We will also introduce the Linked-List and Linked-Set algorithms, and we will also introduce the Iterable and DataList algorithms. We will also introduce the Linked-List algorithm as a sort of randomized

#### Course Tag

Algorithms Randomized Algorithm Sorting Algorithm Divide And Conquer Algorithms

#### Related Wiki Topic

Article Example
Divide and conquer algorithm The divide-and-conquer paradigm often helps in the discovery of efficient algorithms. It was the key, for example, to Karatsuba's fast multiplication method, the quicksort and mergesort algorithms, the Strassen algorithm for matrix multiplication, and fast Fourier transforms.
Divide and conquer algorithm This divide and conquer technique is the basis of efficient algorithms for all kinds of problems, such as sorting (e.g., quicksort, merge sort), multiplying large numbers (e.g. the Karatsuba algorithm), finding the closest pair of points, syntactic analysis (e.g., top-down parsers), and computing the discrete Fourier transform (FFTs).
Sorting algorithm Sorting algorithms are prevalent in introductory computer science classes, where the abundance of algorithms for the problem provides a gentle introduction to a variety of core algorithm concepts, such as big O notation, divide and conquer algorithms, data structures such as heaps and binary trees, randomized algorithms, best, worst and average case analysis, time-space tradeoffs, and upper and lower bounds.
Divide and conquer algorithm The name "divide and conquer" is sometimes applied also to algorithms that reduce each problem to only one sub-problem, such as the binary search algorithm for finding a record in a sorted list (or its analog in numerical computing, the bisection algorithm for root finding). These algorithms can be implemented more efficiently than general divide-and-conquer algorithms; in particular, if they use tail recursion, they can be converted into simple loops. Under this broad definition, however, every algorithm that uses recursion or loops could be regarded as a "divide and conquer algorithm". Therefore, some authors consider that the name "divide and conquer" should be used only when each problem may generate two or more subproblems. The name decrease and conquer has been proposed instead for the single-subproblem class.
Divide and conquer Divide and conquer (derived from the Latin saying "Divide et impera") may refer to:
Divide-and-conquer eigenvalue algorithm Divide-and-conquer eigenvalue algorithms are a class of eigenvalue algorithms for Hermitian or real symmetric matrices that have recently (circa 1990s) become competitive in terms of stability and efficiency with more traditional algorithms such as the QR algorithm. The basic concept behind these algorithms is the divide-and-conquer approach from computer science. An eigenvalue problem is divided into two problems of roughly half the size, each of these are solved recursively, and the eigenvalues of the original problem are computed from the results of these smaller problems.
Divide and conquer algorithm For some problems, the branched recursion may end up evaluating the same sub-problem many times over. In such cases it may be worth identifying and saving the solutions to these overlapping subproblems, a technique commonly known as memoization. Followed to the limit, it leads to bottom-up divide-and-conquer algorithms such as dynamic programming and chart parsing.
Divide and conquer algorithm Early examples of these algorithms are primarily decrease and conquer – the original problem is successively broken down into "single" subproblems, and indeed can be solved iteratively.
Divide-and-conquer eigenvalue algorithm As is common for divide and conquer algorithms, we will use the Master theorem to analyze the running time. Remember that above we stated we choose formula_16. We can write the recurrence relation:
Divide and conquer algorithm The correctness of a divide and conquer algorithm is usually proved by mathematical induction, and its computational cost is often determined by solving recurrence relations.
Divide and conquer algorithm Alternatively, one can employ large base cases that still use a divide-and-conquer algorithm, but implement the algorithm for predetermined set of fixed sizes where the algorithm can be completely unrolled into code that has no recursion, loops, or conditionals (related to the technique of partial evaluation). For example, this approach is used in some efficient FFT implementations, where the base cases are unrolled implementations of divide-and-conquer FFT algorithms for a set of fixed sizes. Source code generation methods may be used to produce the large number of separate base cases desirable to implement this strategy efficiently.
Divide-and-conquer eigenvalue algorithm The eigenvalues and eigenvectors of formula_5 are simply those of formula_6 and formula_7, and it will almost always be faster to solve these two smaller problems than to solve the original problem all at once. This technique can be used to improve the efficiency of many eigenvalue algorithms, but it has special significance to divide-and-conquer.
Divide and conquer algorithm Divide-and-conquer algorithms are naturally implemented as recursive procedures. In that case, the partial sub-problems leading to the one currently being solved are automatically stored in the procedure call stack. A recursive function is a function that calls itself within its definition.
Divide and conquer algorithm Divide and conquer algorithms are naturally adapted for execution in multi-processor machines, especially shared-memory systems where the communication of data between processors does not need to be planned in advance, because distinct sub-problems can be executed on different processors.
Divide and conquer algorithm As another example of a divide and conquer algorithm that did not originally involve computers, Donald Knuth gives the method a post office typically uses to route mail: letters are sorted into separate bags for different geographical areas, each of these bags is itself sorted into batches for smaller sub-regions, and so on until they are delivered. This is related to a radix sort, described for punch-card sorting machines as early as 1929.
Divide and Conquer (film) Divide and Conquer (1943) is the third film of Frank Capra's "Why We Fight" propaganda film series, dealing with the Nazi conquest of Western Europe in 1940.
Divide and conquer algorithm Divide and conquer is a powerful tool for solving conceptually difficult problems: all it requires is a way of breaking the problem into sub-problems, of solving the trivial cases and of combining sub-problems to the original problem. Similarly, divide and conquer only requires reducing the problem to a single smaller problem, such as the classic Tower of Hanoi puzzle, which reduces moving a tower of height "n" to moving a tower of height "n" − 1.
Divide and conquer algorithm In computer and IT science, divide and conquer (D&C) is an algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
Sorting algorithm Related problems include partial sorting (sorting only the "k" smallest elements of a list, or alternatively computing the "k" smallest elements, but unordered) and selection (computing the "k"th smallest element). These can be solved inefficiently by a total sort, but more efficient algorithms exist, often derived by generalizing a sorting algorithm. The most notable example is quickselect, which is related to quicksort. Conversely, some sorting algorithms can be derived by repeated application of a selection algorithm; quicksort and quickselect can be seen as the same pivoting move, differing only in whether one recurses on both sides (quicksort, divide and conquer) or one side (quickselect, decrease and conquer).
Divide and conquer algorithm Divide-and-conquer algorithms naturally tend to make efficient use of memory caches. The reason is that once a sub-problem is small enough, it and all its sub-problems can, in principle, be solved within the cache, without accessing the slower main memory. An algorithm designed to exploit the cache in this way is called "cache-oblivious", because it does not contain the cache size as an explicit parameter.