Serap A. Savari

Edge-Cut Bounds on Network Coding Rates

Active networks are network architectures with processors that are capable of executing code carried by the packets passing through them. A critical network management concern is the optimization of such networks and tight bounds on their performance serve as useful design benchmarks. A new bound on communication rates is developed that applies to network coding, which is a promising active network application that has processors transmit packets that are general functions, for example a bit-wise XOR, of selected received packets. The bound generalizes an edge-cut bound on routing rates by progressively removing edges from the network graph and checking whether certain strengthened d-separation conditions are satisfied. The bound improves on the cut-set bound and its efficacy is demonstrated by showing that routing is rate-optimal for some commonly cited examples in the networking literature.

Generalized Tunstall codes for sources with memory

Tunstall codes are variable-to-fixed length codes that maximize the expected number of source letters per dictionary string for discrete, memoryless sources. We analyze a generalization of Tunstall coding to sources with memory and demonstrate that as the dictionary size increases, the number of code letters per source symbol comes arbitrarily close to the minimum among all variable-to-fixed length codes of the same size. We also find the asymptotic relationship between the dictionary size and the average length of a dictionary entry.

Communicating Probability Distributions

A rate distortion problem is solved that is motivated by a quantum data compression problem. The goal is to send information about a source string x so that a receiver can construct a second string y for which the joint empirical probability distribution of x and y is close to some desired distribution. The problem differs from the usual rate distortion problems in that one must consider both remote sources and distortion functions that are not averages of per-letter distortion functions

On optimal permutation codes

Permutation codes are vector quantizers whose codewords are related by permutations and, in one variant, sign changes. Asymptotically, as the vector dimension grows, optimal Variant I permutation code design is identical to optimal entropy-constrained scalar quantizer (ECSQ) design. However, contradicting intuition and previously published assertions, there are finite block length permutation codes that perform better than the best ones with asymptotically large length; thus, there are Variant I permutation codes whose performances cannot be matched by any ECSQ. Along similar lines, a new asymptotic relation between Variant I and Variant II permutation codes is established but again demonstrated to not necessarily predict the performances of short codes. Simple expressions for permutation code performance are found for memoryless uniform and Laplacian sources. The uniform source yields the aforementioned counterexamples.

Cut sets and information flow in networks of two-way channels

A network of two way channels (TWCs) is specified by a graph having an edge between vertex u and vertex v if there is a TWC between these vertices. A new cut-set bound is determined for such networks when network coding is permitted, and some implications of this bound are discussed.

Variable-to-fixed length codes for predictable sources

A number of papers have investigated the compression obtained by some well-known lossless data compression algorithms in the limit as the dictionary size approaches infinity. Using a very detailed analysis of the Tunstall (1967) code as an example, we demonstrate that asymptotic analyses can fail to provide the performance of an algorithm in the important case of a predictable source with a small to moderate dictionary size.

Some notes on Varn coding

A duality between Varn (1971) coding and Tunstall (1968) coding is demonstrated and new bounds on the expected transmission cost per source symbol for Varn codes are established. For binary channels, the superiority of the new upper bound over Krauses (1962) upper bound is proven. >

A probabilistic approach to some asymptotics in noiseless communication

Renewal theory is a powerful tool in the analysis of source codes. We use renewal theory to obtain some asymptotic properties of finite-state noiseless channels. We discuss the relationship between these results and earlier uses of renewal theory to analyze the Lempel-Ziv (1977, 1978) codes and the Tunstall (1967) code. As a new application of our results, we provide the asymptotic performance of two of the Perl, Garey and Even (1975) prefix condition codes.

Precise Asymptotic Analysis of the Tunstall Code

We study the Tunstall code using the machinery from the analysis of algorithms literature. In particular, we propose an algebraic characterization of the Tunstall code which, together with tools like the Mellin transform and the Tauberian theorems, leads to new results on the variance and a central limit theorem for dictionary phrase lengths. This analysis also provides a new argument for obtaining asymptotic results about the mean dictionary phrase length and average redundancy rates

Bounds on the expected cost of one-to-one codes

This paper generalizes one-to-one encodings of a discrete random variable to finite code alphabets with memoryless letter costs and establish some upper and lower bounds on the expected cost per source symbol of the best one-to-one codes. We show that the average cost per symbol of the optimal one-to-one code sometimes is a tighter lower bound to the average cost per symbol of the optimal uniquely decodable code than Shannons original bound.