In this thesis we study the Input/Output (I/O) complexity of large-scale problems arising e.g. in the areas of database systems, geographic information systems, VLSI design systems and computer graphics, and design I/O-efficient algorithms for them. A general theme in our work is to design I/O-efficient algorithms through the design of I/O-efficient data structures. One of our philosophies is to try to isolate all the I/O specific parts of an algorithm in the data structures, that is, to try to design I/O algorithms from internal memory algorithms by exchanging the data structures used in internal memory with their external memory counterparts. The results in the thesis include a technique for transforming an internal memory tree data structure into an external data structure which can be used in a batched dynamic setting, that is, a setting where we for example do not require that the result of a search operation is returned immediately. Using this technique we develop batched dynamic external versions of the (one-dimensional) range-tree and the segment-tree and we develop an external priority queue. Following our general philosophy we show how these structures can be used in standard internal memory sorting algorithms

OBDD's are the state-of-the-art data structure for Boolean function manipulation since basic tasks of Boolean manipulation such as testing equivalence, satisfiability, or tautology, and performing single Boolean synthesis steps can be done efficiently. In the following we show that the efficient manipulation of OBDD's can be extended to a more general data structure, so-called FBDD's. In detail, the advantages of using FBDD's instead of OBDD's are ffl FBDD's are generally more (sometimes even exponentially more) succinct than OBDD's, ffl FBDD's provide, similarly to OBDD's, canonical representations of Boolean functions, and ffl in terms of FBDD's basic tasks of Boolean manipulation can be performed similarly efficient as in terms of OBDD's. The power of the FBDD-concept is demonstrated by showing that the verification of the benchmark circuit design for the hidden weighted bit function HWB proposed by Bryant can be carried out efficiently in terms of FBDD's while, for princip...

Ordered Binary-Decision Diagrams (OBDD) are the state-of-the-art data structure for boolean function manipulation and there exist several software packages for OBDD manipulation. OBDDs have been successfully used to solve problems in e.g. digital-systems design, verification and testing, in mathematical logic, concurrent system design and in artificial intelligence. The OBDDs used in many of these applications quickly get larger than the avaliable main memory and it becomes essential to consider the problem of minimizing the Input/Output (I/O) communication. In this paper we analyze why existing OBDD manipulation algorithms perform poorly in an I/O environment and develop new I/O-efficient algorithms.

Reduced ordered binary decision diagrams (OBDDs) are nowadays the state-of-theart representation scheme for Boolean functions in Boolean manipulation. Recent results have shown that it is possible to use the more general concept of free binary decision diagrams (FBDDs) without giving up most of the useful computational properties of OBDDs, but possibly reducing the space requirements considerably. The amount of space reduction depends essentially on the shape of so-- called FBDD--types the Boolean manipulation in terms of FBDDs is based on. In this paper we start to propose some heuristics for deriving tree-like FBDD--types from given circuit descriptions. The experimental results we obtained demonstrate clearly that the FBDD--approach is not only of theoretical interest but also of practical usefulness even in the case of using merely such simple--structured tree--based FBDD-- types as produced by the investigated heuristics. Keywords--- data structures for switching functions, OBDD,...

We present the concept of TBDD's which considerably enlarges the class of Boolean functions that can be efficiently manipulated in terms of OBDD's. It extends the idea of using domain transformations, which is well-known in many areas of mathematics, physics, and technical sciences, to the context of OBDD--based Boolean function manipulation in CAD: Instead of working with the OBDDrepresentation of a function f , TBDD's allow working with an OBDD-representation of a suited cube transformed version of f . Besides of giving some theoretical insights into the new concept, we investigate in some detail cube transformations which are based on complete types. We ffl show that such TBDD--representations can be derived similarly as OBDD--representations, ffl give evidence of the practical importance of such TBDD's by presenting very small-size TBDDrepresentations of the hidden weighted bit functions HWBn which were proved to have only very large OBDD-representations, and ffl report some pr...

This work is settled in the area of complexity theory for restricted variants of branching programs. Today, branching programs can be considered one of the standard nonuniform models of computation. One reason for their popularity is that they allow to describe computations in an intuitively straightforward way and promise to be easier to analyze than the traditional models. In complexity theory, we are mainly interested in upper and lower bounds on the size of branching programs. Although proving superpolynomial lower bounds on the size of general branching programs still remains a challenging open problem, there has been considerable success in the study of lower bound techniques for various restricted variants, most notably perhaps read-once branching programs and OBDDs (ordered binary decision diagrams). Surprisingly, OBDDs have also turned out to be extremely useful in practical applications as a data structure for Boolean functions. So far, research has concentrated on determinis...

Ordered binary decision diagrams (OBDDs) are an efficient graph representation for Boolean functions, if good variable orderings are used. Variable orderings are computed by heuristic algorithms and improved with the simulated annealing approach. New neighborhoods for simulated annealing algorithms are presented. It is investigated how the OBDD size may change in one iteration step for the different neighborhoods. The hidden weighted bit function HWB is a function whose OBDD size depends heavily on the variable ordering although the OBDD size grows exponentially in any case. Variable orderings for HWB are investigated theoretically and experimentally. All known heuristics work badly for HWB, while the new simulated annealing algorithm yields good results for this and other functions. Supported in part by DFG grant We 1066/7--1. 1 1. INTRODUCTION Ordered binary decision diagrams (OBDDs) are an efficient graph representation (Bryant [3]), if good variable orderings are know...

Multiple-valued logic allows us to formulate problems by using symbolic variables which are often more naturally associated with the problem speci cation than the variables obtained by a binary encoding. In this paper we present a data structure for representation and manipulation of multiple-valued functions - Mod-p Decision Diagrams (Mod-p-DDs). Mod-p-DDs differ from conventional Multiple-Valued Decision Diagrams (MDDs) in that they contain not only branching nodes but also functional nodes, labeled by addition modulo p operation, p - prime. Mod-p-DDs are potentially much more space-ecient than MDDs. However, they are not a canonical representation and thus, the equivalence test of two Mod-p-DDs is more difficult then the test of two MDDs. To overcome this problem, we design a fast probabilistic equivalence test for Mod-p-DDs that requires time linear in the number of nodes.

Abstract. We present a new lower bound technique for a restricted Branching Program model, namely for nondeterministic graph-driven read-once Branching Programs (g.d.-BP1s). The technique is derived by drawing a connection between ω-nondeterministic g.d.-BP1s and ωnondeterministic communication complexity (for the nondeterministic acceptance modes ω ∈ {∨, ∧, ⊕}). We apply the technique in order to prove an exponential lower bound for integer multiplication for ωnondeterministic well-structured g.d.-BP1s. (For ω = ⊕ an exponential lower bound was already obtained in [5] by using a different technique.) Further, we use the lower bound technique to prove for an explicitly defined fnction which can be represented by polynomial size ω-nondeterministic BP1s that it has exponential complexity in the ωnondeterministic well-structured g.d.-BP1 model for ω ∈ {∨, ⊕}. This answers an open question from Brosenne, Homeister, and Waack [7], whether the nondeterministic BP1 model is in fact more powerful than the well-structured graph-driven variant. 1

Graph driven BDDs introduced by Sieling and Wegener (1992) and Gergov and Meinel (1993) are an extension of OBDDs that admit more concise representations of Boolean functions than OBDDs. We investigate the complexity of the operations replacement by constants and functions, quantification, redundancy test and reordering on graph driven BDDs and on a restricted variant of graph driven BDDs called tree driven BDDs. We show that polynomial time algorithms for the replacement operations and quantification on graph driven BDDs are unlikely to exist and we present polynomial time algorithms for replacement by constants, redundancy test and reordering on tree driven BDDs. 1. INTRODUCTION Ordered binary decision diagrams (OBDDs) introduced by Bryant (1986) are the most popular data structure for Boolean functions. They have many applications in hardware design and verification, test pattern generation, logic synthesis and design and analysis of sequential circuits. This is due to the existenc...