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Articles by Arnab Chakraborty
Page 359 of 377
Optimal Lopsided Trees in Data Structure
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 MoreHeight Limited Huffman Trees in Data Structure
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 MoreHuffman Algorithm for t-ary Trees in Data Structure
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 MoreHuffman Codes and Entropy in Data Structure
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 MoreSplay in Virtual Tree in Data Structure
In virtual tree, some edges are treated as solid and some are treated as dashed. Usual splaying is performed only in the solid trees. To splay at a node y in the virtual tree, following method is implemented.The algorithm looks at the tree three times, once in each pass, and changes it. In first pass, by splaying only in the solidtrees, beginning from the node y, the path from y to the root of the overall tree, becomes dashed. This path is createdsolid by splicing. A final splay at node y will now create y the root of the tree. ...
Read MoreSolid Trees in Data Structure
For the given forest, we create some of the given edges “dashed” and the rest of them are kept solid. Each non-leaf node is associated with only one “solid” edge to one of its children. All other children will be connected with the help of a dashed edge.To be more concrete, in any given tree, the right-most link (to its child) should be kept solid, and all other links to its other children are created “dashed”.As a result, the tree will be broken into a collection of solid paths. The roots of solid paths will be joined to some other ...
Read MoreOptimality of Splay Trees in Data Structure
Dynamic optimality conjectureIn addition to the proven performance guarantees for splay trees there is an unproven conjecture with great interest. Dynamic optimality conjecture denotes this conjecture. Let any binary search tree algorithm such as B accesses an element y by traversing the path from the root to y at a cost of d(y)+1, and that between accesses can make any rotations in the tree at a cost of 1 per rotation. Let B(s) be the cost for B to perform the sequence s of accesses. Then the cost for a splay tree to perform the same accesses is O[n+B(s)].There are ...
Read MoreAdaptive Merging and Sorting in Data Structure
ADAPTIVE MERGE SORTAdaptive Merge Sort performs the merging of sorted sub-list merge sort does. However, the size of initial sub-list is depended upon the existence of ordering among the list of elements rather than having sub-list of size 1. For example, consider list in the figure.It consists of 2 sorted sub-lists.sub-list 1 with elements 16, 15, 14, 13.sub-list 2 with elements 9, 10, 11, 12.The sub-list 1 is sorted but in reverse order. Thus, the sub-list 1 is reversed as shown in the figure.Once the sub-lists are found merging process starts. Adaptive merge sort starts merging the sub-lists. Adaptive merge ...
Read MoreSkip Lists in Data Structure
In a skip list, one can finger search for element a from a node containing the element b by simply continuing the search from this point a.Note that if a < b, then search proceeds at backward direction, and if a > b, then search proceeds at forward direction.The backwards case is symmetric to normal search in a skip list, but the forward case is actually more complicated.Normally, search in a skip list is expected to be fast because the sentinel at the start of the list is considered as the tallest node.However, our finger could be associated with a ...
Read MoreRandomized Finger Search Trees in Data Structure
Two randomized alternatives to deterministic search trees are the randomized binary search trees, treaps and the skip lists. Both treaps and skip lists are defined as elegant data structures, where the randomization facilitates simple and efficient update operations.In this section we explain how both treaps and skip lists can be implemented as efficient finger search trees without changing the data structures. Both data structures support finger searches by consuming expected O(log d) time, where the expectations are taken over the random choices created by the algorithm during the construction of the data structure.Skip listsIn a skip list, one can finger ...
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