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Cartesian Tree
In computer science, a Cartesian tree is a binary tree derived from a sequence of distinct numbers. To construct the Cartesian tree, set its root to be the minimum number in the sequence, and recursively construct its left and right subtrees from the subsequences before and after this number. It is uniquely defined as a min-heap whose Tree traversal, symmetric (in-order) traversal returns the original sequence. Cartesian trees were introduced by in the context of geometric range searching data structures. They have also been used in the definition of the treap and randomized binary search tree data structures for binary search problems, in comparison sort algorithms that perform efficiently on nearly-sorted inputs, and as the basis for pattern matching algorithms. A Cartesian tree for a sequence can be constructed in linear time. Definition Cartesian trees are defined using binary trees, which are a form of rooted tree. To construct the Cartesian tree for a given sequence of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Data Structure
In computer science, a data structure is a data organization and storage format that is usually chosen for Efficiency, efficient Data access, access to data. More precisely, a data structure is a collection of data values, the relationships among them, and the Function (computer programming), functions or Operator (computer programming), operations that can be applied to the data, i.e., it is an algebraic structure about data. Usage Data structures serve as the basis for abstract data types (ADT). The ADT defines the logical form of the data type. The data structure implements the physical form of the data type. Different types of data structures are suited to different kinds of applications, and some are highly specialized to specific tasks. For example, Relational database, relational databases commonly use B-tree indexes for data retrieval, while compiler Implementation, implementations usually use hash tables to look up Identifier (computer languages), identifiers. Data s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Amortized Time
In computer science, amortized analysis is a method for analyzing a given algorithm's complexity, or how much of a resource, especially time or memory, it takes to execute. The motivation for amortized analysis is that looking at the worst-case run time can be too pessimistic. Instead, amortized analysis averages the running times of operations in a sequence over that sequence. As a conclusion: "Amortized analysis is a useful tool that complements other techniques such as worst-case and average-case analysis." For a given operation of an algorithm, certain situations (e.g., input parametrizations or data structure contents) may imply a significant cost in resources, whereas other situations may not be as costly. The amortized analysis considers both the costly and less costly operations together over the whole sequence of operations. This may include accounting for different types of input, length of the input, and other factors that affect its performance. History Amortiz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lazy Deletion
In computer science, lazy deletion refers to a method of deleting elements from a hash table that uses open addressing Open addressing, or closed hashing, is a method of Hash table#Collision resolution, collision resolution in hash tables. With this method a hash collision is resolved by probing, or searching through alternative locations in the array (the ''prob .... In this method, deletions are done by marking an element as deleted, rather than erasing it entirely. Deleted locations are treated as empty when inserting and as occupied during a search. The deleted locations are sometimes referred to as ''tombstones.'' The problem with this scheme is that as the number of delete/insert operations increases, the cost of a successful search increases. To improve this, when an element is searched and found in the table, the element is relocated to the first location marked for deletion that was probed during the search. Instead of finding an element to relocate when the deletion occ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Merge Algorithm
Merge algorithms are a family of algorithms that take multiple sorted lists as input and produce a single list as output, containing all the elements of the inputs lists in sorted order. These algorithms are used as subroutines in various sorting algorithms, most famously merge sort. Application The merge algorithm plays a critical role in the merge sort algorithm, a comparison-based sorting algorithm. Conceptually, the merge sort algorithm consists of two steps: # Recursively divide the list into sublists of (roughly) equal length, until each sublist contains only one element, or in the case of iterative (bottom up) merge sort, consider a list of ''n'' elements as ''n'' sub-lists of size 1. A list containing a single element is, by definition, sorted. # Repeatedly merge sublists to create a new sorted sublist until the single list contains all elements. The single list is the sorted list. The merge algorithm is used repeatedly in the merge sort algorithm. An example merge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Divide-and-conquer Algorithm
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), 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 cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Parallel Algorithm
In computer science, a parallel algorithm, as opposed to a traditional serial algorithm, is an algorithm which can do multiple operations in a given time. It has been a tradition of computer science to describe serial algorithms in abstract machine models, often the one known as random-access machine. Similarly, many computer science researchers have used a so-called parallel random-access machine (PRAM) as a parallel abstract machine (shared-memory). Many parallel algorithms are executed concurrently – though in general concurrent algorithms are a distinct concept – and thus these concepts are often conflated, with which aspect of an algorithm is parallel and which is concurrent not being clearly distinguished. Further, non-parallel, non-concurrent algorithms are often referred to as " sequential algorithms", by contrast with concurrent algorithms. Parallelizability Algorithms vary significantly in how parallelizable they are, ranging from easily parallelizable to compl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Stack (data Structure)
In computer science, a stack is an abstract data type that serves as a collection of elements with two main operations: * Push, which adds an element to the collection, and * Pop, which removes the most recently added element. Additionally, a peek operation can, without modifying the stack, return the value of the last element added. The name ''stack'' is an analogy to a set of physical items stacked one atop another, such as a stack of plates. The order in which an element added to or removed from a stack is described as last in, first out, referred to by the acronym LIFO. As with a stack of physical objects, this structure makes it easy to take an item off the top of the stack, but accessing a datum deeper in the stack may require removing multiple other items first. Considered a sequential collection, a stack has one end which is the only position at which the push and pop operations may occur, the ''top'' of the stack, and is fixed at the other end, the ''bottom''. A s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
All Nearest Smaller Values
In computer science, the all nearest smaller values problem is the following task: for each position in a sequence of numbers, search among the previous positions for the last position that contains a smaller value. This problem can be solved efficiently both by parallel and non-parallel algorithms: , who first identified the procedure as a useful subroutine for other parallel programs, developed efficient algorithms to solve it in the Parallel Random Access Machine model; it may also be solved in linear time on a non-parallel computer using a stack-based algorithm. Later researchers have studied algorithms to solve it in other models of parallel computation. Example Suppose that the input is the binary van der Corput sequence :0, 8, 4, 12, 2, 10, 6, 14, 1, 9, 5, 13, 3, 11, 7, 15. The first element of the sequence (0) has no previous value. The nearest (only) smaller value previous to 8 and to 4 is 0. All three values previous to 12 are smaller, but the nearest one is 4. Continui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Potential Method
In computational complexity theory, the potential method is a method used to analyze the amortized time and space complexity of a data structure, a measure of its performance over sequences of operations that smooths out the cost of infrequent but expensive operations.. Definition of amortized time In the potential method, a function Φ is chosen that maps states of the data structure to non-negative numbers. If ''S'' is a state of the data structure, Φ(''S'') represents work that has been accounted for ("paid for") in the amortized analysis but not yet performed. Thus, Φ(''S'') may be thought of as calculating the amount of potential energy stored in that state. The potential value prior to the operation of initializing a data structure is defined to be zero. Alternatively, Φ(''S'') may be thought of as representing the amount of disorder in state ''S'' or its distance from an ideal state. Let ''o'' be any individual operation within a sequence of operations on some data stru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Path (graph Theory)
In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipath) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction. Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See e.g. , , or . cover more advanced algorithmic topics concerning paths in graphs. Definitions Walk, trail, and path * A walk is a finite or infinite sequence of edges which joins a sequence of vertices. : Let be a graph. A finite walk is a sequence of edges for which there is a sequence of vertices such that ''Φ''(''e''''i'') = for . is the ''vertex sequence'' of the walk. The walk is ''closed'' if ''v''1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
