In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks 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. The divide-and-conquer technique is the basis of efficient algorithms for many 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), SAT solving, and computing the discrete Fourier transform (FFT). Designing efficient divide-and-conquer algorithms can be difficult. As in mathematical induction, it is often necessary to generalize the problem to make it amenable to a recursive solution. 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 == The divide-and-conquer paradigm is often used to find an optimal solution of a problem. Its basic idea is to decompose a given problem into two or more similar, but simpler, subproblems, to solve them in turn, and to compose their solutions to solve the given problem. Problems of sufficient simplicity are solved directly. For example, to sort a given list of n natural numbers, split it into two lists of about n/2 numbers each, sort each of them in turn, and interleave both results appropriately to obtain the sorted version of the given list (see the picture). This approach is known as the merge sort algorithm. The name "divide and conquer" is sometimes applied 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 analogue 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. An important application of divide and conquer is in optimization, where if the search space is reduced ("pruned") by a constant factor at each step, the overall algorithm has the same asymptotic complexity as the pruning step, with the constant depending on the pruning factor (by summing the geometric series); this is known as prune and search. == Early historical examples == 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. Binary search, a decrease-and-conquer algorithm where the subproblems are of roughly half the original size, has a long history. While a clear description of the algorithm on computers appeared in 1946 in an article by John Mauchly, the idea of using a sorted list of items to facilitate searching dates back at least as far as Babylonia in 200 BC. Another ancient decrease-and-conquer algorithm is the Euclidean algorithm to compute the greatest common divisor of two numbers by reducing the numbers to smaller and smaller equivalent subproblems, which dates to several centuries BC. An early example of a divide-and-conquer algorithm with multiple subproblems is Gauss's 1805 description of what is now called the Cooley–Tukey fast Fourier transform (FFT) algorithm, although he did not analyze its operation count quantitatively, and FFTs did not become widespread until they were rediscovered over a century later. An early two-subproblem D&C algorithm that was specifically developed for computers and properly analyzed is the merge sort algorithm, invented by John von Neumann in 1945. Another notable example is the algorithm invented by Anatolii A. Karatsuba in 1960 that could multiply two n-digit numbers in O ( n log 2 3 ) {\displaystyle O(n^{\log _{2}3})} operations (in Big O notation). This algorithm disproved Andrey Kolmogorov's 1956 conjecture that Ω ( n 2 ) {\displaystyle \Omega (n^{2})} operations would be required for that task. 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. == Advantages == === Solving difficult problems === 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, decrease 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 {\displaystyle n} to move a tower of height n − 1 {\displaystyle n-1} . === Algorithm efficiency === 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. In all these examples, the D&C approach led to an improvement in the asymptotic cost of the solution. For example, if (a) the base cases have constant-bounded size, the work of splitting the problem and combining the partial solutions is proportional to the problem's size n {\displaystyle n} , and (b) there is a bounded number p {\displaystyle p} of sub-problems of size ~ n p {\displaystyle {\frac {n}{p}}} at each stage, then the cost of the divide-and-conquer algorithm will be O ( n log p n ) {\displaystyle O(n\log _{p}n)} . For other types of divide-and-conquer approaches, running times can also be generalized. For example, when a) the work of splitting the problem and combining the partial solutions take c n {\displaystyle cn} time, where n {\displaystyle n} is the input size and c {\displaystyle c} is some constant; b) when n < 2 {\displaystyle n<2} , the algorithm takes time upper-bounded by c {\displaystyle c} , and c) there are q {\displaystyle q} subproblems where each subproblem has size ~ n 2 {\displaystyle {\frac {n}{2}}} . Then, the running times are as follows: if the number of subproblems q > 2 {\displaystyle q>2} , then the divide-and-conquer algorithm's running time is bounded by O ( n log 2 q ) {\displaystyle O(n^{\log _{2}q})} . if the number of subproblems is exactly one, then the divide-and-conquer algorithm's running time is bounded by O ( n ) {\displaystyle O(n)} . If, instead, the work of splitting the problem and combining the partial solutions take c n 2 {\displaystyle cn^{2}} time, and there are 2 subproblems where each has size n 2 {\displaystyle {\frac {n}{2}}} , then the running time of the divide-and-conquer algorithm is bounded by O ( n 2 ) {\displaystyle O(n^{2})} . === Parallelism === 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. === Memory access === 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. Moreover, D&C algorithms can be designed for important algorithms (e.g., sorting, FFTs, and matrix multiplication) to be optimal cache-oblivious algorithms–they use the cache in a probably optimal way, in an asymptotic sense, regardless of the cache size. In contrast, the traditional approach to exploiting the cache is blocking, as in loop nest optimization, where the problem is explicitly divided into chunks of the appropriate size—this can also use the cache optimally, but only when the algorithm is tuned for the specific cache sizes of a particular machine. The same advantage exists with regards to other hierarchical storage systems, such as NUMA or virtual memory, as well as for multip
Circular thresholding
Circular thresholding is an algorithm for automatic image threshold selection in image processing. Most threshold selection algorithms assume that the values (e.g. intensities) lie on a linear scale. However, some quantities such as hue and orientation are a circular quantity, and therefore require circular thresholding algorithms. The example shows that the standard linear version of Otsu's method when applied to the hue channel of an image of blood cells fails to correctly segment the large white blood cells (leukocytes). In contrast the white blood cells are correctly segmented by the circular version of Otsu's method. == Methods == There are a relatively small number of circular image threshold selection algorithms. The following examples are all based on Otsu's method for linear histograms: (Tseng, Li and Tung 1995) smooth the circular histogram, and apply Otsu's method. The histogram is cyclically rotated so that the selected threshold is shifted to zero. Otsu's method and histogram rotation are applied iteratively until several heuristics involving class size, threshold location, and class variance are satisfied. (Wu et al. 2006) smooth the circular histogram until it contains only two peaks. The histogram is cyclically rotated so that the midpoint between the peaks is shifted to zero. Otsu's method and histogram rotation are applied iteratively until convergence of the threshold. (Lai and Rosin 2014) applied Otsu's method to the circular histogram. For the two class circular thresholding task they showed that, for a histogram with an even number of bins, the optimal solution for Otsu's criterion of within-class variance is obtained when the histogram is split into two halves. Therefore the optimal solution can be efficiently obtained in linear rather than quadratic time. == References and further reading == D.-C. Tseng, Y.-F. Li, and C.-T. Tung, Circular histogram thresholding for color image segmentation in Proc. Int. Conf. Document Anal. Recognit., 1995, pp. 673–676. J. Wu, P. Zeng, Y. Zhou, and C. Olivier, A novel color image segmentation method and its application to white blood cell image analysis in Proc. Int. Conf. Signal Process., vol. 2. 2006, pp. 16–20. Y.K. Lai, P.L. Rosin, Efficient Circular Thresholding, IEEE Trans. on Image Processing 23(3), 992–1001 (2014). doi:10.1109/TIP.2013.2297014
Pointer algorithm
In computer science, a pointer algorithm (sometimes called a pointer machine, or a reference machine; see the article Pointer machine for a close but non-identical concept) is a type of algorithm that manages a linked data structure. This concept is used as a model for lower-bound proofs and specific restrictions on the linked data structure and on the algorithm's access to the structure vary. This model has been used extensively with problems related to the disjoint-set data structure. Thus, Tarjan and La Poutré used this model to prove lower bounds on the amortized complexity of a disjoint-set data structure (La Poutré also addressed the interval split-find problem). Blum used this model to prove a lower bound on the single operation worst-case time of disjoint set data structure. Blum and Rochow proved a worst-case lower bound for the interval union-find problem. == Example == In Tarjan's lower bound for the disjoint set union problem, the assumptions on the algorithm are: The algorithm maintains a linked structure of nodes. Each element of the problem is associated with a node. Each set is represented by a node. The nodes of each set constitute a distinct connected component in the structure (this property is called separability). The find operation is performed by following links from the element node to the set node. Under these assumptions, the lower bound of Ω ( m α ( m , n ) ) {\displaystyle \Omega (m\alpha (m,n))} on the cost of a sequence of m operations is proven.
Algorithmic Puzzles
Algorithmic Puzzles is a book of puzzles based on computational thinking. It was written by computer scientists Anany and Maria Levitin, and published in 2011 by Oxford University Press. == Topics == The book begins with a "tutorial" introducing classical algorithm design techniques including backtracking, divide-and-conquer algorithms, and dynamic programming, methods for the analysis of algorithms, and their application in example puzzles. The puzzles themselves are grouped into three sets of 50 puzzles, in increasing order of difficulty. A final two chapters provide brief hints and more detailed solutions to the puzzles, with the solutions forming the majority of pages of the book. Some of the puzzles are well known classics, some are variations of known puzzles making them more algorithmic, and some are new. They include: Puzzles involving chessboards, including the eight queens puzzle, knight's tours, and the mutilated chessboard problem Balance puzzles River crossing puzzles The Tower of Hanoi Finding the missing element in a data stream The geometric median problem for Manhattan distance == Audience and reception == The puzzles in the book cover a wide range of difficulty, and in general do not require more than a high school level of mathematical background. William Gasarch notes that grouping the puzzles only by their difficulty and not by their themes is actually an advantage, as it provides readers with fewer clues about their solutions. Reviewer Narayanan Narayanan recommends the book to any puzzle aficionado, or to anyone who wants to develop their powers of algorithmic thinking. Reviewer Martin Griffiths suggests another group of readers, schoolteachers and university instructors in search of examples to illustrate the power of algorithmic thinking. Gasarch recommends the book to any computer scientist, evaluating it as "a delight".
March algorithm
The March algorithm is a widely used algorithm that tests SRAM memory by filling all its entries test patterns. It carries out several passes through an SRAM checking the patterns and writing new patterns. The SRAM read and write operations performed on each pass are called a March element and each element is repeated for each entry. The March algorithm is often used to find functional faults in SRAM during testing such as: Stuck-at Faults (SAFs) Transition Faults (TFs) Address Decoder Faults (AFs) Coupling Faults (CFs), such as Inversion (CFin), Idempotent (CFid), and State (CFst) coupling faults It has been suggested to test SRAM modules using the algorithm before sale using a built-in self-test mechanism. == Notation == Each pass in a test sequence is represented by an "element". An element consists of a vertical arrow to indicate the direction in which the memory is scanned followed by a list of read/write operations to be applied to each memory cell. Multiple elements can be listed, separated by semicolons, to form a "test". For example, { ⇕ ( w 0 ) ; ⇑ ( r 0 , w 1 ) ; ⇓ ( r 1 , w 0 , r 0 ) } {\displaystyle \{\Updownarrow (w0);\Uparrow (r0,w1);\Downarrow (r1,w0,r0)\}} specifies to: Scan in both directions, writing 0. Scan from lowest to highest address, reading 0 and writing 1. Scan from highest to lowest address, reading 1, writing 0 and reading 0. == Variants == Many variants of the March algorithm exist with different sequences of tests. Each variant makes a different tradeoff between what faults it can detect and the complexity of the algorithm. Several variants have been given names:
Trustworthy computing
The term trustworthy computing (TwC) has been applied to computing systems that are inherently secure, available, and reliable. It is particularly associated with the Microsoft initiative of the same name, launched in 2002. == History == Until 1995, there were restrictions on commercial traffic over the Internet. On, May 26, 1995, Bill Gates sent the "Internet Tidal Wave" memorandum to Microsoft executives assigning "...the Internet this highest level of importance..." but Microsoft's Windows 95 was released without a web browser as Microsoft had not yet developed one. The success of the web had caught them by surprise but by mid 1995, they were testing their own web server, and on August 24, 1995, launched a major online service, The Microsoft Network (MSN). The National Research Council recognized that the rise of the Internet simultaneously increased societal reliance on computer systems while increasing the vulnerability of such systems to failure and produced an important report in 1999, "Trust in Cyberspace". This report reviews the cost of un-trustworthy systems and identifies actions required for improvement. == Microsoft and Trustworthy Computing == Bill Gates launched Microsoft's "Trustworthy Computing" initiative with a January 15, 2002 memo, referencing an internal whitepaper by Microsoft CTO and Senior Vice President Craig Mundie. The move was reportedly prompted by the fact that they "...had been under fire from some of its larger customers–government agencies, financial companies and others–about the security problems in Windows, issues that were being brought front and center by a series of self-replicating worms and embarrassing attacks." such as Code Red, Nimda, Klez and Slammer. Four areas were identified as the initiative's key areas: Security, Privacy, Reliability, and Business Integrity, and despite some initial scepticism, at its 10-year anniversary it was generally accepted as having "...made a positive impact on the industry...". The Trustworthy Computing campaign was the main reason why Easter eggs disappeared from Windows, Office and other Microsoft products.
Pointer jumping
Pointer jumping or path doubling is a design technique for parallel algorithms that operate on pointer structures, such as linked lists and directed graphs. Pointer jumping allows an algorithm to follow paths with a time complexity that is logarithmic with respect to the length of the longest path. It does this by "jumping" to the end of the path computed by neighbors. The basic operation of pointer jumping is to replace each neighbor in a pointer structure with its neighbor's neighbor. In each step of the algorithm, this replacement is done for all nodes in the data structure, which can be done independently in parallel. In the next step when a neighbor's neighbor is followed, the neighbor's path already followed in the previous step is added to the node's followed path in a single step. Thus, each step effectively doubles the distance traversed by the explored paths. Pointer jumping is best understood by looking at simple examples such as list ranking and root finding. == List ranking == One of the simpler tasks that can be solved by a pointer jumping algorithm is the list ranking problem. This problem is defined as follows: given a linked list of N nodes, find the distance (measured in the number of nodes) of each node to the end of the list. The distance d(n) is defined as follows, for nodes n that point to their successor by a pointer called next: If n.next is nil, then d(n) = 0. For any other node, d(n) = d(n.next) + 1. This problem can easily be solved in linear time on a sequential machine, but a parallel algorithm can do better: given n processors, the problem can be solved in logarithmic time, O(log N), by the following pointer jumping algorithm: The pointer jumping occurs in the last line of the algorithm, where each node's next pointer is reset to skip the node's direct successor. It is assumed, as in common in the PRAM model of computation, that memory access are performed in lock-step, so that each n.next.next memory fetch is performed before each n.next memory store; otherwise, processors may clobber each other's data, producing inconsistencies. The following diagram follows how the parallel list ranking algorithm uses pointer jumping for a linked list with 11 elements. As the algorithm describes, the first iteration starts initialized with all ranks set to 1 except those with a null pointer for next. The first iteration looks at immediate neighbors. Each subsequent iteration jumps twice as far as the previous. Analyzing the algorithm yields a logarithmic running time. The initialization loop takes constant time, because each of the N processors performs a constant amount of work, all in parallel. The inner loop of the main loop also takes constant time, as does (by assumption) the termination check for the loop, so the running time is determined by how often this inner loop is executed. Since the pointer jumping in each iteration splits the list into two parts, one consisting of the "odd" elements and one of the "even" elements, the length of the list pointed to by each processor's n is halved in each iteration, which can be done at most O(log N) time before each list has a length of at most one. == Root finding == Following a path in a graph is an inherently serial operation, but pointer jumping reduces the total amount of work by following all paths simultaneously and sharing results among dependent operations. Pointer jumping iterates and finds a successor — a vertex closer to the tree root — each time. By following successors computed for other vertices, the traversal down each path can be doubled every iteration, which means that the tree roots can be found in logarithmic time. Pointer doubling operates on an array successor with an entry for every vertex in the graph. Each successor[i] is initialized with the parent index of vertex i if that vertex is not a root or to i itself if that vertex is a root. At each iteration, each successor is updated to its successor's successor. The root is found when the successor's successor points to itself. The following pseudocode demonstrates the algorithm. algorithm Input: An array parent representing a forest of trees. parent[i] is the parent of vertex i or itself for a root Output: An array containing the root ancestor for every vertex for i ← 1 to length(parent) do in parallel successor[i] ← parent[i] while true for i ← 1 to length(successor) do in parallel successor_next[i] ← successor[successor[i]] if successor_next = successor then break for i ← 1 to length(successor) do in parallel successor[i] ← successor_next[i] return successor The following image provides an example of using pointer jumping on a small forest. On each iteration the successor points to the vertex following one more successor. After two iterations, every vertex points to its root node. == History and examples == Although the name pointer jumping would come later, JáJá attributes the first uses of the technique in early parallel graph algorithms and list ranking. The technique has been described with other names such as shortcutting, but by the 1990s textbooks on parallel algorithms consistently used the term pointer jumping. Today, pointer jumping is considered a software design pattern for operating on recursive data types in parallel. As a technique for following linked paths, graph algorithms are a natural fit for pointer jumping. Consequently, several parallel graph algorithms utilizing pointer jumping have been designed. These include algorithms for finding the roots of a forest of rooted trees, connected components, minimum spanning trees, and biconnected components. However, pointer jumping has also shown to be useful in a variety of other problems including computer vision, image compression, and Bayesian inference.