Constructing a binary tree from its traversals by reversible recursion and iteration

We present a new algorithm that combines the hash value techniques and suffix array techniques of previous work. The term differential compression refers to encoding a file (a version file) as a set of changes with respect to another file (a reference file). Previous algorithms can be shown empirically to run in linear time, but they have certain drawbacks; namely, they do not find the best matches for every offset of the version file. Our algorithm, hsadelta (hash suffix array delta), finds the best matches for every offset of the version file, with respect to a certain granularity and above a certain length threshold. The algorithm has two variations depending on how we choose the block size. We show that if the block size is kept fixed, the performance of the algorithm is similar to that of the greedy algorithm, without the associated expensive space and time requirements. If the block size is varied linearly with the reference file size, the algorithm can run in linear time and constant space. We also show empirically that the algorithm performs better than other state-of-the-art algorithms in terms of and is comparable in speed.

We design an efficient sublinear time parallel construction of optimal binary search trees. The efficiency of the parallel algorithm corresponds to its total work (the product time × processors). Our algorithm works in O(n1−ɛ log n) time with the total work O(n2−2ɛ), for an arbitrarily small constant 0 < e ≤ 1/2. This is optimal within a factor n2ɛ with respect to the best known sequential algorithm given by Knuth, which needs only O(n2) time due to a monotonicity property of optimal binary search trees, see [6]). It is unknown how to explore this property in an efficient NC construction of binary search trees. Here we show that it can be effectively used in sublinear time parallel computation. Our improvement also relies on the use (in independently processed small subcomputations) of the parallelism present in Knuth's algorithm. The best known sublinear time algorithms for the construction of binary search trees (as an instance of a more general problem) have O(n3) work for time larger than n3/4, see [3] and [7]. For time √n these algorithms need n4 work, while our algorithm needs for this time only n3 work, thus improving the known algorithms by a linear factor. Also if time is O(n1−ɛ) and e is very small our improvement is close to O(n). Such improvement is similar to the one implied by the monotonicity property in sequential computations (from n3 sequential time for a more general dynamic programming problem to n2 time for the special case of optimal binary search trees).

In this paper, a new representation of a binary tree is introduced, called the Catalan Cipher Vector, which is a vector of elements with certain properties. It can be ranked using a special form of the Catalan Triangle designed for this purpose. It is shown that the vector coincides with the level-order traversal of the binary tree and how it can be used to generate a binary tree from it. Streamlined algorithms for directly obtaining the rank from a binary tree and vice versa, using the Catalan Cipher Vector during the processes, are given. The algorithms are analyzed for time and space complexity and shown to be linear for both.The Catalan Cipher Vector enables a straightforward determination of the position and linking for every node of the binary tree, since it contains information for both every node’s ancestor and the direction of linking from the ancestor to that node. Thus, it is especially well suited for binary tree generation. Using another structure, called a canonical state-space tableau, the relationship between the Catalan Cipher Vector and the level-order traversal of the binary tree is explained.

There are various ways to measure the shape difference between two n-node rooted binary trees (binary trees for short). A rotation on a binary tree is a local restructuring that changes the tree into another one preserving the in-order sequence. The rotation distance between two binary trees is the minimum number of rotations needed to transform one into another. Till now, no polynomial–time algorithm exists for computing the rotation distance between any two binary trees. Recently, Lucas (Comput. J., 47, 259–269, 2004) presented an O(n2)–time algorithm for finding the rotation distance between two binary trees, where the source tree is a degenerate tree and the destination tree is an angle tree. This paper improves the time-complexity to O(n) under this constraint.

On subtrees of trees

On Rotations and the Generation of Binary Trees

Indexing technology is hardly studied when data access is skew in real time system. On one hand indexing considers the probability of data accessed, on the other hand indexing needs to satisfy data real-time constraints, and meanwhile, the time complexity of arithmetic computing for indexing must be low. The nearly optimal binary index tree NOBIT is based on the idea of static optimal search tree. We introduce real-time weight when it deals with nodes probabilities weight so as to quickly construct a binary index tree with nearly optimal performance. The search process of NOBIT is similar to binary search, and its average time complexity is O(logN). The NOBIT considers the probability of accessed data and data which are accessed frequently are put into the front part of broadcast sequence so that average tuning time is shortened. Meanwhile, the NOBIT takes the real-time constraints of accessed data into account and data with rigid time constraints also are put into the front part of broadcast sequence so that success rate of data requested is improved.

Can we solve finite-horizon Markov decision processes (FHMDPs) while raising low memory requirements? Such models find application in many cases where a decision-making agent needs to act in a probabilistic environment, from resource management to medicine to service provisioning. However, computing optimal policies such an agent should follow by dynamic programming value iteration raises either prohibitive space complexity, or, in reverse, non-scalable time complexity requirements. This scalability question has been largely neglected. In this paper, we propose SIFTER (Space Efficient Finite Horizon MDPs), a suite of algorithms that achieve a golden middle between space and time requirements. Our former algorithm raises space complexity growing with the square root of the horizon's length without a time-complexity overhead, while the latter's space requirements depend only logarithmically in horizon length with a corresponding logarithmic time complexity overhead. A thorough experimental study under diverse settings confirms that SIFTER algorithms achieve the predicted gains, while approximation techniques do not achieve the same combination of time efficiency, space efficiency, and result quality.

We describe algorithms for constructing optimal binary search trees, in which the access cost of a key depends on the k preceding keys, which were reached in the path to it. This problem has applications to searching on secondary memory and robotics. Two kinds of optimal trees are considered, namely optimal worst case trees and weighted average case trees. The time and space complexity of both algorithms are O(n k+2) and O(n k+1), respectively. The algorithms are based on a convenient decomposition and characterizations of sequences of keys, which are paths of special kinds in binary search trees. Finally, using generating functions, the exact number of steps performed by the algorithms has been calculated. The subject will be introduced by a general discussion on the construction of optimal binary search trees.

The running time and memory requirement of a parallel program with dynamic, lightweight threads depends heavily on the underlying thread scheduler. In this paper we present a new asynchronous, space-efficient scheduling algorithm for shared memory machines that combines the low scheduling overheads and good locality of work stealing with the low space requirements of depth-first schedulers. For a nested-parallel program with depth D and serial space requirement S 1 , we show that the expected space requirement is S 1 + O(K⋅ p⋅ D) on p processors. Here, K is a user-adjustable runtime parameter, which provides a tradeoff between running time and space requirement. Our algorithm achieves good locality and low scheduling overheads by automatically increasing the granularity of the work scheduled on each processor. We have implemented the new scheduling algorithm in the context of a native, user-level implementation of Posix standard threads or Pthreads, and evaluated its performance using a set of C-based benchmarks that have dynamic or irregular parallelism. We compare the performance of our scheduler with that of two previous schedulers: the thread library's original scheduler (which uses a FIFO queue), and a provably space-efficient depth-first scheduler. At a fine thread granularity, our scheduler outperforms both these previous schedulers, but requires marginally more memory than the depth-first scheduler. We also present simulation results on synthetic benchmarks to compare our scheduler with space-efficient versions of both a work-stealing scheduler and a depth-first scheduler. The results indicate that unlike these previous approaches, the new algorithm covers a range of scheduling granularities and space requirements, and allows the user to trade the space requirement of a program with the scheduling granularity.

Fingerprint matching algorithm is a key step in fingerprint recognition system. Though there are many existing matching algorithms, there has been inability to match fingerprints in linear time. In this paper we present a novel biometric approach to match fingerprints that run in linear time. We match the minutiae in the fingerprint by constructing a Nearest Neighbor Vector (NNV) considering its k-nearest neighbors. The consolidation of these matched minutiae points is done by incorporating them in binary tree that propagates simultaneously in both fingerprints. This helps our algorithm to run in O(n) time in contrast to many existing algorithms when reference core point is available. We analyze the resulting improvement in computational complexity and present experimental evaluation over FVC2002 database.

SimRank has been proposed to rank web documents based on a graph model on hyperlinks. The existing techniques for conducting SimRank computation adopt an iteration computation paradigm. The most efficient technique has the time complexity O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) with the space requirement O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) in the worst case for each iteration where n is the number of nodes (web documents). In this paper, we propose novel optimization techniques such that each iteration takes the time O(min{n · m,n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sup> }) and requires space O (n + m) where m is the number of edges in a web-graph model and r ≤ log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 7. We also show that our algorithm accelerates the convergence rate of the existing techniques. Moreover, our algorithm not only reduces the time and space complexity of the existing techniques but is also I/O efficient. We conduct extensive experiments on both synthetic and real data sets to demonstrate the efficiency and effectiveness of our iteration techniques.

A simple algorithm is given which takes an arbitrary binary search tree and rebalances it to form another of optimal shape, using time linear in the number of nodes and only a constant amount of space (beyond that used to store the initial tree). This algorithm is therefore optimal in its use of both time and space. Previous algorithms were optimal in at most one of these two measures, or were not applicable to all binary search trees. When the nodes of the tree are stored in an array, a simple addition to this algorithm results in the nodes being stored in sorted order in the initial portion of the array, again using linear time and constant space.

Génération aléatoire uniforme de mots de langages rationnels

Optimal binary search trees with costs depending on the access paths