Optimal Binary Search Trees with Near Minimal Height

Suppose we have n keys, n access probabilities for the keys, and n+1 access probabilities for the gaps between the keys. Let h_min(n) be the minimal height of a binary search tree for n keys. We consider the problem to construct an optimal binary search tree with near minimal height, i.e.\ with height h <= h_min(n) + Delta for some fixed Delta. It is shown, that for any fixed Delta optimal binary search trees with near minimal height can be constructed in time O(n^2). This is as fast as in the unrestricted case. So far, the best known algorithms for the construction of height-restricted optimal binary search trees have running time O(L n^2), whereby L is the maximal permitted height. Compared to these algorithms our algorithm is at least faster by a factor of log n, because L is lower bounded by log n

On the Huffman and Alphabetic Tree Problem with General Cost Functions

We address generalized versions of the Huffman and Alphabetic Tree Problem where the cost caused by each individual leaf i, instead of being linear, depends on its depth in the tree by an arbitrary function. The objective is to minimize either the total cost or the maximum cost among all leaves. We review and extend the known results in this direction and devise a number of new algorithms and hardness proofs. It turns out that the Dynamic Programming approach for the Alphabetic Tree Problem can be extended to arbitrary cost functions, resulting in a time O(n (4)) optimal algorithm using space O(n (3)). We identify classes of cost functions where the well-known trick to reduce the runtime by a factor of n via a "monotonicity" property can be applied. For the generalized Huffman Tree Problem we show that even the k-ary version can be solved by a generalized version of the Coin Collector Algorithm of Larmore and Hirschberg (in Proc. SODA'90, pp. 310-318, 1990) when the cost functions are nondecreasing and convex. Furthermore, we give an O(n (2)logn) algorithm for the worst case minimization variants of both the Huffman and Alphabetic Tree Problem with nondecreasing cost functions. Investigating the limits of computational tractability, we show that the Huffman Tree Problem in its full generality is inapproximable unless P = NP, no matter if the objective function is the sum of leaf costs or their maximum. The alphabetic version becomes NP-hard when the leaf costs are interdependent.ArticleALGORITHMICA. 69(3): 582-604 (2014)journal articl

From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity

The equivalence of finite automata and regular expressions dates back to the seminal paper of Kleene on events in nerve nets and finite automata from 1956. In the present paper we tour a fragment of the literature and summarize results on upper and lower bounds on the conversion of finite automata to regular expressions and vice versa. We also briefly recall the known bounds for the removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free nondeterministic devices. Moreover, we report on recent results on the average case descriptional complexity bounds for the conversion of regular expressions to finite automata and brand new developments on the state elimination algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527

Hu-Tucker alogorithm for building optimal alphabetic binary search trees

The purpose of this thesis is to study the behavior of the Hu- Tucker algorithm for building Optimal Alphabetic Binary Search Trees (OABST), to design an efficient implementation, and to evaluate the performance of the algorithm, and the implementation. The three phases of the algorithm are described and their time complexities evaluated. Two separate implementations for the most expensive phase, Combination, are presented achieving 0(n2) and O(nlogn) time and 0(n) space complexity. The break even point between them is experimentally established and the complexities of the implementations are compared against their theoretical time complexities. The electronic version of The Complete Works of William Shakespeare is compressed using the Hu- Tucker algorithm and other popular compression algorithms to compare the performance of the different techniques. The experiments justified the price that has to be paid to implement the Hu- Tucker algorithm. It is shown that an efficient implementation can process extremely large data sets relatively fast and can achieve optimality close to the Optimal Binary Tree, built using the Huffman algorithm, however the OABST can be used in both encoding and decoding processes, unlike the OBT where an additional mapping mechanism is needed for the decoding phase

Dynamic Trees with Almost-Optimal Access Cost

An optimal binary search tree for an access sequence on elements is a static tree that minimizes the total search cost. Constructing perfectly optimal binary search trees is expensive so the most efficient algorithms construct almost optimal search trees. There exists a long literature of constructing almost optimal search trees dynamically, i.e., when the access pattern is not known in advance. All of these trees, e.g., splay trees and treaps, provide a multiplicative approximation to the optimal search cost. In this paper we show how to maintain an almost optimal weighted binary search tree under access operations and insertions of new elements where the approximation is an additive constant. More technically, we maintain a tree in which the depth of the leaf holding an element e_i does not exceed min(log(W/w_i),log n)+O(1) where w_i is the number of times e_i was accessed and W is the total length of the access sequence. Our techniques can also be used to encode a sequence of m symbols with a dynamic alphabetic code in O(m) time so that the encoding length is bounded by m(H+O(1)), where H is the entropy of the sequence. This is the first efficient algorithm for adaptive alphabetic coding that runs in constant time per symbol

Minimax Trees in Linear Time with Applications

A minimax tree is similar to a Huffman tree except that, instead of minimizing the weighted average of the leaves\u27 depths, it minimizes the maximum of any leaf\u27s weight plus its depth. Golumbic (1976) introduced minimax trees and gave a Huffman-like, $O (n log n)$-time algorithm for building them. Drmota and Szpankowski (2002) gave another $O (n log n)$-time algorithm, which takes linear time when the weights are already sorted by their fractional parts. In this paper we give the first linear-time algorithm for building minimax trees for unsorted real weights

Query Learning with Exponential Query Costs

In query learning, the goal is to identify an unknown object while minimizing the number of "yes" or "no" questions (queries) posed about that object. A well-studied algorithm for query learning is known as generalized binary search (GBS). We show that GBS is a greedy algorithm to optimize the expected number of queries needed to identify the unknown object. We also generalize GBS in two ways. First, we consider the case where the cost of querying grows exponentially in the number of queries and the goal is to minimize the expected exponential cost. Then, we consider the case where the objects are partitioned into groups, and the objective is to identify only the group to which the object belongs. We derive algorithms to address these issues in a common, information-theoretic framework. In particular, we present an exact formula for the objective function in each case involving Shannon or Renyi entropy, and develop a greedy algorithm for minimizing it. Our algorithms are demonstrated on two applications of query learning, active learning and emergency response.Comment: 15 page