Alphabetic Binary Tree Research Articles

Alphabetic coding with exponential costs

This note considers an alphabetic binary tree formulation in a family of nonlinear problems. An application of this family occurs when a random outcome needs to be determined via alphabetically ordered search within a stochastic time window. Rather than finding a decision tree minimizing ∑ i = 1 n w ( i ) l ( i ) , this variant involves minimizing log a ∑ i = 1 n w ( i ) a l ( i ) for a given a ∈ ( 0 , 1 ) . Herein a dynamic programming algorithm finds the optimal solution in O ( n 3 ) time and O ( n 2 ) space; methods traditionally used to improve the speed of optimizations in related problems, such as the Hu–Tucker procedure, fail for this problem. This note thus also introduces two algorithms which can find a suboptimal solution in linear time (for one) or O ( n log n ) time (for the other), with associated redundancy bounds guaranteeing their coding efficiency.

The Optimal Alphabetic Tree Problem Revisited

TheOptimal Alphabetic Binary Tree(OABT) problem is equivalent to the Optimal Binary Search Tree problem where the weights are associated only with the leaves. The problem can be solved inO(nlogn) time, while the best known lower bound is Ω(n).In this paper we relate the complexity of the problem to the complexity of priority queue operations and the complexity of sorting. We giveO(nlogP(k))-time algorithm for the general OABT problem an-time algorithm for the integer OABT problem wherekis a number at most at large as the number of local minima,P(k) is the time complexity of priority queue insert/delete_min operation, andS(n) is the complexity of sorting in the given domain of weights. Our algorithms also give rise to linear time algorithms for some special cases.

A Parallel Algorithm for Optimum Height-Limited Alphabetic Binary Trees

In this paper, anO(Llogn)-timen-processor parallel EREW PRAM algorithm is presented for construction of an optimal alphabetic binary tree with height restricted toL. The technique used isparallel package merge.

Upper and Lower Bounds on Constructing Alphabetic Binary Trees

This paper studies the long-standing open question of whether optimal alphabetic binary trees can be constructed in $o( n\lg n )$ time. We show that a class of techniques for finding optimal alphabetic trees which includes all current methods yielding $O( n\lg n )$-time algorithms are at least as hard as sorting in whatever model of computation is used. We also give $O( n )$-time algorithms for the case where all the input weights are within a constant factor of one another and when they are exponentially separated.

A Fast Algorithm For Optimum Height-Limited Alphabetic Binary Trees

In this paper, an $O(nL\log n)$-time algorithm is presented for construction of an optimal alphabetic binary tree with height restricted to $L$. This algorithm is an alphabetic version of the Package Merge algorithm, and yields an $O(nL\log n)$-time algorithm for the alphabetic Huffman coding problem. The Alphabetic Package Merge algorithm is quite simple to describe, but appears hard to prove correct. Garey [SIAM J. Comput., 3 (1974), 101--110] gives an $O(n^3\log n)$-time algorithm for the height-limited alphabetic binary tree problem. Itai [SIAM J. Comput., 5 (1976), 9--18] and Wessner [Inform. Process. Lett., 4 (1976), pp. 90--94] independently reduce this time to $O(n^2L)$ for the alphabetic problem. In [SIAM J. Comput., 16 (1987), pp. 1115--1123], a rather complex $O(n^{3/2}L\log ^{1/2}n)$-time hybrid algorithm is given for length-limited Huffman coding. The Package Merge algorithm, discussed in this paper, first appeared in [Tech. Report, 88-01, ICS Dept. Univ. of California, Irvine, CA], but without proof of correctness.

An extended result of Kleitman and Saks concerning binary trees

In a recent paper, D.J. Kleitman and M.E. Saks gave a proof of Huang's conjecture on alphabetic binary trees. Given a set E = {e i}, i = 0, 1, 2, ..., m and assigned positive weights to its elements and supposing the elements are indexed such that w(e 0) ≤ w(e 1) ≤ ... ≤w (e m) , where w(e i) is the weight of e i , we call the following sequence E * a ‘saw-tooth’ sequence E *=( e 0, e m , e 1,..., e j , e m− j ,...). Huang's conjecture is: E * is the most expensive sequence for alphabetic binary trees. This paper shows that this property is true for the L-restricted alphabetic binary trees, where L is the maximum length of the leaves and ⌈log 2( m + 1)⌉ ≤ L≤ m.

Set Orderings Requiring Costliest Alphabetic Binary Trees

It is shown that an ordering of a set with weighted elements which requires the most expensive alphabetic binary tree is a “sawtooth order.” For the set $\{ e_0 ,e_1 , \cdots ,e_t \}$, with the elements indexed from least to greatest weight, this order is $e_0 ,e_t ,e_1 ,e_{t - 1} , \cdots ,e_j ,e_{t - 1} , \cdots $. This result was conjectured by Hwang and leads to an upper bound on the cost of alphabetic binary trees.

Isolating a Single Defective Using Group Testing

Abstract Given a finite set of n items, containing at least one defective item, and, for each item, the probability that it is defective, it is desired to determine a group-testing procedure which isolates a single defective with a minimum expected number of tests. We prove that such an optimal procedure can be found by constructing an optimal “alphabetic binary tree” for a derived set of weights obtained directly from the given probabilities. The algorithm for constructing the optimal procedure requires a number of operations proportional only to n log n.

A New Proof of the T-C Algorithm

An algorithm for constructing an alphabetic binary tree of minimum weighted path length was suggested by Hu and Tucker. The algorithm called the T-C algorithm needs $O( {n\log n} )$ operations and $O( n )$ storage locations, where n is the number of terminal nodes in the tree. Although the algorithm is simple to state, the associated proof was extremely complicated and long. Here a more revealing and comparatively shorter proof is given.

Path Length of Binary Search Trees

An algorithm is given for constructing a binary tree of minimum weighted path length and with restricted maximum path length. (Huffman’s tree is a binary tree of minimum weighted path length with no restriction on maximum path length.) When an alphabetical ordering of the terminal nodes is introduced, the binary tree is called an alphabetic binary tree. We show that Huffman’s algorithm for optimal binary trees coincides with the T-C algorithm for optimal alphabetic binary trees when the terminal nodes have monotonically increasing weights from left to right.

Optimal Computer Search Trees and Variable-Length Alphabetical Codes

An algorithm is given for constructing an alphabetic binary tree of minimum weighted path length (for short, an optimal alphabetic tree). The algorithm needs $4n^2 + 2n$ operations and $4n$ storage locations, where n is the number of terminal nodes in the tree. A given binary tree corresponds to a computer search procedure, where the given files or letters (represented by terminal nodes) are partitioned into two parts successively until a particular file or letter is finally identified. If the files or letters are listed alphabetically, such as a dictionary, then the binary tree must have, from left to right, the terminal nodes consecutively. Since different letters have different frequencies (weights) of occurring, an alphabetic tree of minimum weighted path length corresponds to a computer search tree with minimum-mean search time. A binary tree also represents a (variable-length) binary code. In an alphabetic binary code, the numerical binary order of the code words corresponds to the alphabetical orde...