DIMACS Center, CoRE Building (Room 431), Rutgers University, Busch Campus, Piscataway, NJ

**Mordecai Golin**, Hong Kong University of Science and Technology**Julia Abrahams**, Rutgers University, DIMACS

**Co-sponsored by DIMACS and the IEEE Information Theory Society.**

1. Julia Abrahams, Rutgers University Tutorial - Huffman Code Trees and Problem Variants: Some Open Questions Huffman code trees will be reviewed as the solution to the minimum-average-codeword-length coding problem, and also as the solution to the minimum-average-number-of-tests sequential diagnosis or search problem. Variants of the underlying "word problem" which impose costs or constraints on the paths through the tree tend to arise in coding problems while costs or constraints on the interior nodes of the tree tend to arise in testing problems, although some variants are plausible problems of interest in both contexts. Several examples of problem variants will be categorized as instances of constrained Huffman problems or as examples of Huffman problems with an imposed cost structure. In order for a Huffman variant problem to be considered resolved, we would like to have a parsimonious characterization of the set of candidate trees, an entropy-type bound on the cost of the optimal tree, and an efficient algorithm for the construction of the optimal tree. Particular problems which are open in this sense include the case that partial order constraints are imposed on the leaf nodes of the tree, the case that the edges in the paths through the tree occur according to a finite state cost model, and the case in which costs are associated with setting up distinct tests. These three examples will be presented in some detail as research challenges. 2. Joy Thomas, IBM Huffman Algebras In this talk we define a class of two operator algebras for which the Huffman algorithm is optimal, based on a rearrangement inequality by Hardy, Littlewood and Polya. These algebras, called Huffman algebras, include both real numbers and independent random variables. Many examples of such algebras are given. For the case with random weights of the leaves, we prove the optimality of the tree constructed by the power of two rule, i.e., the Huffman algorithm assuming identical weights, when the weights of the leaves are independent and identically distributed. As a natural extension of our algebras, we consider tree extensions, and derive an equivalent of entropy for some of the algebras considered in the talk. Applications of these algebras include not only the source coding problem, but also circuit design and scheduling in parallel computers. (Joint work with C.S. Chang) 3. Kingo Kobayashi, University of Electro Communications Coding of Trees: From the Information-Theoretic Point of View The tree structure has often played central roles in both computer science and information theory. Various expressions for several classes of trees have been devised in computer science. We can compare these expressions by the efficiency of "coding" which is an important concept of information theory. Here, from the coding point of view, we give some results on the expression of trees. 4. Lawrence L. Larmore, University of Nevada - Las Vegas Optimal Trees and the Monge Property In several cases, it is possible to reduce the problem of finding an optimal tree to a seemingly unrelated problem in which the Monge property applies. The first and simplest example is the reduction of the Huffman coding problem to the concave least weight subsequence problem. Although in this one simplest case no speedup is achieved, it is nevertheless instructive. When the technique is extended to more complex problems, there is typically a speedup of an order of magnitude. 5. Rudolf Fleischer, Max-Planck Institut Saarbrucken Self-Organizing Data Structures and Compression: A Survey Self-organizing data structures like linear lists (with the Move-to-Front rule, for example) or Splay Trees have been proposed to be used in efficient implementations of encoders in compression algorithms. But they can also be used directly to encode data yielding good compression rates, in particular if used together with the Burrows-Wheeler transform. In this talk, I will give a short survey on self-organizing lists and Splay Trees, and how these data structures can be used for data compression. 6. Mike Fredman, Rutgers University (with David M. Cohen, IDA) Weighted Binary Trees for Concurrent Searching A traditional cost measure for binary search trees is given by weighted path length, which measures the expected cost of a single random search. In this paper we investigate a generalization, the k-cost, which is suitable for applications involving independent parallel processors each utilizing a common search tree. The k-cost is defined to be the expected value of the maximum length of the search paths encountered during the course of k independently performed random searches. For example, if k = 1, then the k-cost represents the traditional weighted path length. Our results include efficient algorithms for constructing binary search trees having near optimal k-cost. An interesting feature of one of our algorithms concerns the fact that it constructs a search tree that simultaneously has near optimal k-cost for all k; in fact, the structure of this tree is determined only by the order permutation of the sequence of probability weights that are to be assigned to the tree nodes. This stands in contrast to an example which shows that a tree which is precisely optimal relative to weighted path length (1-cost) can have k-cost which is exponentially greater than the k-cost of the optimal tree. 7. Bob Sedgewick, Princeton University Is Quicksort Optimal? This talk describes some relatively bold theoretical results and some relatively new practical implementations that imply tight relationships between the number of comparisons used by a properly implemented Quicksort and the entropy of the set of keys being sorted. 9. Veronique Bruyere, University of Mons-Hainaut Le Pentagone Tutorial - Codes: A Language Theoretic Approach The topic of this tutorial is the theory of variable-length codes. This theory was initiated by Schutzenberger in the fifties. It takes its origin in the coding and information theory developed by Shannon. In the first part of the tutorial, we survey the definitions and basic results about codes, with an emphasis on the interconnections with finite automata. In the second part, we discuss some of the results recently obtained in this direction together with some open problems, that might be interesting for the information theory community and the algorithms community. When the proofs are simple, we expose them in a way that the audience has a flavor on the tools used in this domain. 9. Dominique Perrin, Universite de Marne-la-Vallee Bifix Codes Bifix codes are codes which have the prefix property in both directions. Many interesting -and not widely known - results hold for this particular class of prefix codes introduced long ago by Gilbert and Moore and by Schutzenberger. In this talk, I will present several constructions allowing to embed a bifix code into a maximal one. The results are due to Zhang and Shen (1995) and to Veronique Bruyere and myself. 10. Ian Munro, Waterloo (with Venatesh Raman) Succinct Tree Representations for Fast Navigation We consider applications of large, mostly static, tree structures and the implementation of abstract data types for the objects: binary tree, rooted ordered tree and balanced parenthesis expression. We review a number of representations, and present some new ones, that use an amount of space within a lower order term of the information theoretic minimum and support, in constant time, a rich set of navigational operations. In the case of binary trees, for instance, we can move from a node to its left or right child or to the parent in constant time while retaining knowledge of the size of the subtree at which we are positioned. 11. Erik D. Demaine, Waterloo (with Ian Munro and Alex Lopez-Ortiz, University of New Brunswick) Adaptive Computation of the Intersection of a Collection of Sets This talk discusses the problem of computing the intersection of a collection of sets, which is motived by the goal of efficiently evaluating boolean queries in text retrieval systems. An optimal adaptive algorithm is proposed to solve this problem. Information theory is an important tool for analyzing this algorithm. In particular, we study sets of comparisons that prove correctness of an answer, and study optimal binary encodings of such proofs. The lengths of these encodings form an information theoretic lower bound. We present matching upper and lower bounds on the worst-case running time with respect to the information theoretic lower bound. 12. Stott Parker, UCLA The Construction of Huffman Codes is a Submodular ('Convex') Problem Over a Lattice of Binary Trees We show that the space of all binary Huffman codes for a finite alphabet defines a lattice, ordered by the imbalance of the code trees. Representing code trees as path-length sequences, we show that the imbalance ordering is closely related to a majorization ordering on real-valued sequences that correspond to discrete probability density functions. Furthermore, this tree imbalance is a partial ordering that is consistent with the total orderings given by either the external path length (sum of tree path lengths), or the entropy determined by the tree structure. On the imbalance lattice, we show the weighted path-length of a tree (the usual objective function for Huffman coding) is a submodular function, as is the corresponding function on the majorization lattice. Submodular functions are discrete analogues of convex functions. These results give perspective on Huffman coding, and suggest new approaches to coding as optimization over a lattice. 13. Mihai Sipitca, Georgia Tech (with David Gillman) Lossless Encoding of DCT Coefficients for Video Compression The last stage of many image or video coders is to losslessly encode an orthogonal transformation of an image or a residual image, after the transformed image has been quantized, or reduced in numerical precision. Traditionally, image and video coders use a fixed table for entropy coding the tokens that represent the quantized transform coefficients. However, the characteristics of images are highly non-stationary, i.e., different portions of images and different images have different statistics which would require different entropy coding tables for a more efficient encoding. In our presentation, we will investigate the improvement obtained when using multiple entropy tables instead of one. Issues addressed include: 1) choosing the variance as the characteristic of a block, 2) table compaction - keeping the memory requirements reasonable at a minimum coding efficiency cost, 3) presenting the trade-off between memory requirements and the coding efficiency improvement. The results presented were obtained for resolutions between SIF (similar to VHS) and CCIR-601 (broadcast TV). Future work is going to target the low bit rate range. 14. Sajal Das, U. North Texas (with Amiya Bhattacharya) Predictive Location Management Using Lossless Compression A user in a cellular or PCS system, while enjoying the freedom to be mobile, creates an uncertainty about the exact location of the registered mobile terminal being carried. The system has to search for the mobile against this uncertainty, to ensure a successful call delivery within a time constarint. The collector network deployed is usually organized in the form of a three-tier tree like hierarchy, viz. with a mobile switching exchange at the root, a number of base station controllers at the second tier, and several base stations as leaves at the third tier. Each base station is dedicated to handle the air-link within one cell. On a call arrival, the search for the target mobile is inititaed by the system by paging or polling the cells where the user can possibly be found. Paging messages originate from mobile switching exchange and when it reaches a base station, it is broadcast so that the target mobile can respond if present. To limit the level of uncertainty, the mobile can report its position from time to time. This mobile-initiated reporting is called a location update or registration. Both paging and update contribute to the location management cost, and the location management problem aims at minimizing this total cost. Any more message exchange than necessary would lead to wastage of valuable resources such as wireless bandwidth and mobile equipment power, in effect increasing the operational cost that the user has to bear. In this paper, we propose a predictive location management technique that makes use of path updates as opposed to the widely used position updates. To enhance predictability, the mobility profile for each mobile user must be built in the system database. For this, the path traced by the user in the past needs to be made completely known to the system. The cost minimization can come out of achieving this information exchange in its information theoretic limit. The technique evolves out of the mapping of the user's trail onto a string, as reducing the location management cost translates to lossless compression of this string. The compressibility of the fixed-to-variable length encoding of the acclaimed Lempel-Ziv family of algorithms reduce the update cost, whereas their built-in predictive power can be effectively used to reduce paging cost. What we end up with is a family of adaptive on-line algorithms, which are asymptotically optimal if the user mobility can be modeled as an ergodic source. We discuss the underlying data structures used for building the profile on the fly, and how good paging strategies can be designed based on the profile. We take into consideration the limit on the memory available on a mobile terminal for executing the proposed class of algorithms. 15. Yuriy Reznik, RealNetworks On Compact Level Partitioning of Digital Trees A modification of digital tree structure is proposed that supports non-uniform distribution of digits between levels (i.e. each level may have nodes of different size). We call this distribution of digits - level partitioning, and the resulting data structure - LP-tree. We study general properties of LP-trees, and develop an algorithm that for any given binary tree constructs an adequate LP-tree that uses minimum amount of memory. We call the resulting trees - compact LP-trees, and perform detailed analysis of their properties. It turns out that, on average, the compact LP-tree uses approximately 20.65% less memory than a binary tree, and even more importantly, the average complexity of the successful search in such a tree is O(1) (vs. O(log N) for the standard binary trie). 16. Mamoru Hoshi, University of Electro Communications Homophonic Coding by the Algorithm: Preliminary Considerations We shall show that the idea of the successive refinement of interval partitions, which plays the key role in the interval algorithm for random number generation [Han and Hoshi], is directly applicable also to the homophonic coding. We propose an efficient and very simple algorithm for homophonic coding which produces an i.i.d. sequence. The lower and upper bounds for the expected length of the code generated by the algorithm are given. In a special case, the expected length of the code generated by the algorithm is bounded by H(X)/log M + 3, where H(X) is the entropy of a source X. Homophonic coding with cost is also discussed. 17. Andrej Brodnik, Lulea Technical University and IMFM, and Svante Carlsson, Lulea Technical University Sub-Linear Huffman Decoding: Some Theoretical and Practical Results In the first part of our presentation we describe a succinct data structure storing the Huffman encoding that permits sub-linear decoding in the number of transmitted bits ([BroCar98]). The size of the extra storage except for the storage of the symbols in the alphabet for the new data structure is O(l log N) bits, where l is the longest Huffman code and N is the number of symbols in the alphabet. Similar solution was independently presented by Moffat and Turpin ([MofTur97]). We use the presented data structure to describe two different decoding algorithms -- one running in time O(l') and the other in O(log l'), where l' is a number of distinct lengths of Huffman codes. We proceed by showing that our algorithm decodes Huffman codes on average in time O(log log N) for any probability distribution. In the second part we present practical results obtained from the implementation of the described algorithms. We show that with such an efficient decoding algorithm, handling of the input stream of bits (Huffman codes) becomes as expensive as the decoding itself. To improve the latter we developed a more detail machine model which better captures the memory hierarchy. The model was influenced by the work on the external memory data structures (eg. [Arge96,ArgVit96]). We conclude the presentation with some comments and open questions related to the dynamic Huffman encoding ([Vitter87]). The presented algorithms will be used in the fall release of the ``Phone Book of Slovenia'' ([PhBook]). References ---------- [Arge96] L. Arge. Efficient External-Memory Data Structures and Applications. PhD thesis, University of {\AA}rhus, 1996. [ArgVit96] L. Arge and J.S. Vitter. Optimal dynamic interval management in external memory. In 37\Th FOCS, 1996. [BroCar98] A. Brodnik and S. Carlsson. Sub-linear Decoding of Huffman Codes Almost In-Place IMFM, Technical Report IMFM-(1998)-PS-600. (ftp://www.ijp.si/pub/preprints/ps/98/pp600.ps) [MofTur97] A. Moffat and A. Turpin. On implementation of minimum redundancy prefix codes. IEEE Transactions on Communications, 45(10):1200--1207, 1997. [Vitter87] J.S. Vitter. Design and analysis of dynamic Huffman codes. JACM, 34(4):825--845, October 1987. [PhBook] Telekom Slovenije & ADACTA. Telefonski imenik Slovenije. URL: http://tis.telekom.si.Other Workshops

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Document last modified on October 2, 1998.