Quadtrees are trees implemented to efficiently store data of points on a two-dimensional space. In this tree, each node has maximum four children.We can build a quadtree from a two-dimensional area implementing the following stepsThe current two dimensional space is divided into four boxes.If a box consists of one or more points in it, build a child object, storing in it the two dimensional space of the box.If a box does not contain any points, do not build a child for it.Perform recursion for each of the children.Quadtrees are implemented in image compression, where each node consists of the average ... Read More

A range tree is defined as an ordered tree data structure to hold a list of points. It permits all points within a given range to be efficiently retrieved, and is typically implemented in two or higher dimensions. It is same to a kd-tree except with faster query times of O(logd n + k) but worse storage of O(n logd-1 n), with d indicating the dimension of the space, n indicating the number of points in the tree, and k indicating the number of points retrieved for a given query. Range trees may be differentiated with interval trees: instead of ... Read More

IntroductionA HDS for template parameters or halfedge data structure (abbreviated as HalfedgeDS) is defined as an edge-centered data structure capable of maintaining incidence information of vertices, edges and faces, such as for planar maps, polyhedra, or other orientable, two-dimensional surfaces embedded in random dimension. Each edge is broken into two halfedges with opposite orientations. Each halfedge stores one incident face and one incident vertex. One incident halfedge is stored for each face and each vertex. Reduced variants of the halfedge data structure can eliminate some of this information, such as the halfedge pointers in faces or the storage of faces ... Read More

In case of computational geometry, a planar straight-line graph, in short PSLG, (or straight-line plane graph, or plane straight-line graph) is defined as a term implemented for an embedding of a planar graph in the plane such that its edges are mapped into straight line segments. Statement of Fáry's theorem (1948) is that every planar graph has this kind of embedding.In case of computational geometry, PSLGs have often been termed planar subdivisions, with an assumption or assertion that subdivisions are polygonal.Without vertices of degree 1, a PSLG defines a subdivision of the plane into polygonal regions and vice versa. Absence ... Read More

Multivariate cross-sectional data (i.e. not time-series or repeated measure) are indicated by rectangular data in which each column is a variable (feature), and each row is a case or record.First procedure of representing rectangle data is to map it onto a higher-dimensional point data and use point-based data structure procedures such as the grid file, PR quadtree, point quadtree, and k-d-tree. Procedure mapping of the rectangular data to a four-dimensional point can be performed in number techniques such as x and y coordinates of the opposite corners, or x and y coordinates of one corner and the width and height, ... Read More

Bucketing builds, the hash table as a 2D array instead of a single dimensional array. Every entry in the array is big, sufficient to hold M items (M is not amount of data. Just a constant).ProblemsLots of wasted space are created.If M is exceeded, another strategy will need to be implemented.Not so good performer for memory based implementations but doable if buckets are disk-based.For bucketing it is ok to have λ>1. However, the larger λ is the higher a chance of collision. λ>1 guarantees there will be minimum 1 collision (pigeon hole principle). That will enhance both the run time ... Read More

The problem of finding optimal prefix-free codes for unequal letter costs consists of computing a minimum cost prefix-free code in which the encoding alphabet consists of unequal cost (length) letters, of lengths α and β, where α ≤ β. We restrict ourselves limited to binary trees.The code is represented by a lopsided tree, in the similar way as a Huffman tree represents the solution of the Huffman coding problem. Despite the similarity, the case of unequal letter costs is much difficult than the classical Huffman problem; no polynomial time algorithm is known or available for general letter costs, despite a ... Read More

The diagram of height limited or depth limited Huffman tree is given belowTree depth limitation is a non-trivial issue that most real-world Huffman implementations must deal with.Huffman construction doesn't limit the height or depth. If it would, it is not possible for it to be "optimal". Granted, the largest depth of a Huffman tree is bounded by the Fibonacci series, but that leave sufficient room for larger depth than wanted.What is the reason to limit Huffman tree depth? Fast Huffman decoders implement lookup tables. It's possible to implement multiple table levels to mitigate the memory cost, but a very fast ... Read More

A simple algorithmA collection of n initial Huffman trees is prepared, each of which is a single leaf node. Keep the n trees onto a priority queue organized by weight (frequency).Remove or delete the first two trees (the ones with smallest weight). Combine these two trees to create a new tree whose root is associated with the two trees as children, and whose weight is the sum of the weights of the two children trees.Keep this new tree into the priority queue.Repeat steps 2-3 until and unless all of the partial Huffman trees have been joined into one.It's a greedy ... Read More

Huffman CodeA Huffman code is defined asa particular type of optimal prefix code that is commonly used for lossless data compression.The process of finding or implementing such a code proceeds by means of Huffman coding, an algorithm which was developed by David A. Huffman while he was a Sc.D. student at MIT, and published in the 1952 paper "A Method for the Construction of Minimum-Redundancy Codes".The output from Huffman's algorithm can be displayed as a variable-length code table for encoding a source symbol (such as a character in a file). The algorithm creates this table from the estimated probability or ... Read More