Andreas Jakoby

Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth

An algorithmic meta theorem for a logic and a class C of structures states that all problems expressible in this logic can be solved eciently for inputs from C. The prime example is Courcelle’s Theorem, which states that monadic second-order (mso) definable problems are linear-time solvable on graphs of bounded tree width. We contribute new algorithmic meta theorems, which state that mso-definable problems are (a) solvable by uniform constant-depth circuit families (AC 0 for decision problems and TC 0 for counting problems) when restricted to input structures of bounded tree depth and (b) solvable by uniform logarithmic-depth circuit families (NC 1 for decision problems and #NC 1 for counting problems) when a tree decomposition of bounded width in term representation is part of the input. Applications of our theorems include a TC 0 -completeness proof for the unary version of integer linear programming with a fixed number of equations and extensions of a recent result that counting the number of accepting paths of a visible pushdown automaton lies in #NC 1 . Our main technical contributions are a new tree automata model for unordered, unranked, labeled trees; a method for representing the tree automata’s computations algebraically using convolution circuits; and a lemma on computing balanced width-3 tree decompositions of trees in TC 0 , which encapsulates most of the technical diculties surrounding earlier results connecting tree automata and NC 1 . 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes

Private Computation - k-Connected versus 1-Connected Networks

We study the role of connectivity of communication networks in private computations under information theoretic settings. It will be shown that some functions can be computed by private protocols even if the underlying network is 1-connected but not 2-connected. Then we give a complete characterisation of non-degenerate functions that can be computed on non-2-connected networks.Furthermore, a general technique for simulating private protocols on arbitrary networks will be presented. Using this technique every private protocol can be simulated on arbitrary k-connected networks using only a small number of additional random bits.Finally, we give matching lower and upper bounds for the number of random bits needed to compute the parity function on k-connected networks.

The Complexity of Scheduling Problems with Communication Delays for Trees

Scheduling problems with interprocessor communication delays have been shown to be NP-complete ([PY88],[C90],[CP91],[P91]). This paper considers the scheduling problem with the restriction that the underlying DAGs are trees and each task has unit execution time. It is shown that the problem remains NP-complete for binary trees and uniform communication delays. The same holds for complete binary trees, but varying communication delays. On the other hand, by a nontrivial analysis a polynomial time algorithm is obtained that solves the problem for complete k-ary trees and uniform communication delays.

Logspace algorithms for computing shortest and longest paths in series-parallel graphs

For many types of graphs, including directed acyclic graphs, undirected graphs, tournament graphs, and graphs with bounded independence number, the shortest path problem is NL-complete. The longest path problem is even NP-complete for many types of graphs, including undirected K5-minor-free graphs and planar graphs. In the present paper we present logspace algorithms for computing shortest and longest paths in series-parallel graphs where the edges can be directed arbitrarily. The class of series-parallel graphs that we study can be characterized alternatively as the class of K4-minor-free graphs and also as the class of graphs of tree-width 2. It is well-known that for graphs of bounded tree-width many intractable problems can be solved efficiently, but previous work was focused on finding algorithms with low parallel or sequential time complexity. In contrast, our results concern the space complexity of shortest and longest path problems. In particular, our results imply that for directed graphs of tree-width 2 these problems are L-complete.

Space efficient algorithms for directed series-parallel graphs

The subclass of directed series-parallel graphs plays an important role in computer science. Whether a given graph is series-parallel is a well studied problem in algorithmic graph theory, for which fast sequential and parallel algorithms have been developed in a sequence of papers. Also methods are known to solve the reachability and the decomposition problem for series-parallel graphs time efficiently. However, no dedicated results have been obtained for the space complexity of these problems when restricted to series-parallel graphs - the topic of this paper.Deterministic algorithms are presented for the recognition, reachability, decomposition and the path counting problem for series-parallel graphs that use only logarithmic space. Since for arbitrary directed graphs reachability and path counting are believed not to be solvable in Logspace, the main contribution of this work are novel deterministic path finding routines that work correctly in series-parallel graphs, and a characterization of series-parallel graphs by forbidden subgraphs that can be tested space-efficiently. The space bounds are best possible, i.e. the decision problem is shown to be L-complete with respect to AC0-reductions. They have also implications for the parallel time complexity of these problems when restricted to series-parallel graphs.Finally we sketch how these results can be generalized to extension of the series-parallel graph family: to graph with multiple sources or multiple sinks and to the minimal vertex series-parallel graphs.

Circuit complexity: from the worst case to the average case

In contrast to machine models like Turing machines or random access machines, circuits are a rigid computational model. The internal information flow of a computation is fixed in advance, independent of the actual input. Therefore, in complexity theory only worst case complexity measures have been used to analyse this model. Concerning the circuit size this seems to be the best one can do. The delay between feeding the input bits into a circuit and being able to read off the output bits at the output gates is usually measured by the depth of the circuit. One might try to take advantage of favorable cases in which the output values are obtained much earlier. This will be the case when critical paths, e.g. paths between input and output gates of maximal length, have no influence on the final output. Inspired by recent successful attempts to develop a meaningful average case analysis for TM computations [Levi86, Gure91, BCGL92, ReSc93a], we follow the same goal for the circuit model. For this purpose, a new comlexity measure for the internal delay is defined, called time. This may be given implicitly or explicitly, where in the latter case a gate has to signal that it is “ready”, e.g. has computed the desired result. Using a simple coding both models are shown to be equivalent. By doubling the number of gates any circuit with implicit timing can easily be transformed to an equivalent circuit with explicit time signals. Based on the notion of time, two average case measures for the circuit delay are defined. The analysis is not restricted to uniform distributions over the input space, *supported by DFG Research Grant Re 672-2 t Institut fur Theoretische Informatik, AlexanderstraBe 10, 64283 Darmstadt, Germany email: jakoby / reischuk / schindel Ql iti.informatik.thdarmstadt .de Permission to copy without fee all or part of this material is granted provided that the copies are not made or cfiatributecf for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association of Computing Machinery. To copy othervdse, or to republish, requires a fee and/or specific permission. STOC 945/94 Montreal, Quebec, Canada @ 1994 ACM 0-89791 -663-6/94/0005..

The complexity of broadcasting in planar and decomposable graphs

3.50 instead a large class of distributions will be considered. For this purpose, a complexity notion is needed for distributions. We define a measure based on the circuit model by considering the complexity of circuits with uniform distributed random input bits that generate such distributions. Finally, this new approach is applied to concrete examples. We derive matching lower and upper bounds for the average circuit delay for basic functions like the OR, ADDITION, THRESHOLD and PARITY. It will be shown that for PARITY the average delay remains logarithmic. In many cases, however, an exponential speedup compared to the worst case can be achieved. For example, the average delay for n-Bit-ADDITION is of order log log n. The circuit designs to achieve these bounds turn out to be very different from the standard ones for optimal worst case results.

The Average Case Complexity of the Parallel Prefix Problem

Broadcasting in processor networks means disseminating a single piece of information, which is originally known only at some nodes, to all members of the network. The goal is to inform everybody using as few rounds as possible, that is to minimize the broadcasting time.

Communications in unknown networks: Preserving the secret of topology

We analyse the average case complexity of evaluating all prefixes of an input vector over a given semigroup. As computational model circuits over the semigroup are used and a complexity measure for the average delay of such circuits, called time, is introduced. Based on this notion, we then define the average case complexity of a computational problem for arbitrary input distributions.

The Non-Recursive Power of Erroneous Computation

In cryptography we investigate security aspects of data distributed in a network. This kind of security does not protect the secrecy of the network topology against being discovered if some kind of communication has taken place. But there are several scenarios where the network topology has to be a part of the secret. In this paper we study the question of communication within a secret network where the processing nodes of the network have only partial knowledge (e.g. given as routing tables) of the topology. We introduce a model for measuring the loss of security of the topology when far distance communication takes place. A communication protocol preserves the secret of topology if no processing node can deduce additional information about the topology from the communication. We will investigate lower bounds on the knowledge that can be revealed from the communication string and show, for instance, that some knowledge about distances can always be revealed. Then, we consider routing tables. We show that several kinds of routing tables are not sufficient to guarantee the secrecy of topology. On the other hand, if a routing table allows us to specify the direction from which a message is coming, we can run a protocol solving the all-to-all communication problem such that no processing node can gain additional knowledge about the network. Finally, we investigate the problem of whether routing tables can be generated from the local knowledge of the processing nodes without losing the secrecy of the network topology with respect to the resulting knowledge base. It will be shown that this is not possible for static networks and most kinds of dynamic networks.