Udi Manber

DIB—a distributed implementation of backtracking

DIB is a general-purpose package that allows a wide range of applications such as recursive backtrack, branch and bound, and alpha-beta search to be implemented on a multicomputer. It is very easy to use. The application program needs to specify only the root of the recursion tree, the computation to be performed at each node, and how to generate children at each node. In addition, the application program may optionally specify how to synthesize values of tree nodes from their childrens values and how to disseminate information (such as bounds) either globally or locally in the tree. DIB uses a distributed algorithm, transparent to the application programmer, that divides the problem into subproblems and dynamically allocates them to any number of (potentially nonhomogeneous) machines. This algorithm requires only minimal support from the distributed operating system. DIB can recover from failures of machines even if they are not detected. DIB currently runs on the Crystal multicomputer at the University of Wisconsin-Madison. Many applications have been implemented quite easily, including exhaustive traversal (N queens, knights tour, negamax tree evaluation), branch and bound (traveling salesman) and alpha-beta search (the game of NIM). Speedup is excellent for exhaustive traversal and quite good for branch and bound.

Pierce point minimization and optimal torch path determination in flame cutting

Abstract This paper addresses the problem of sequencing a torch for the cutting of a stock sheet nested with regular or irregular parts. The image of the nesting is reduced to an equivalent graph, and the objective is to traverse this graph with a minimum number of pierce points, or blowthroughs. If the graph has 2k vertices of odd degree, then k pierce points are necessary and sufficient to traverse the graph. The torch path problem formulation includes manufacturing cost, efficiency, and distortion considerations. We present three algorithms for the problem. The first algorithm shows how to determine the k torch paths, and is optimal in run time complexity. The second algorithm uses trim margin edges to investigate further reduction in the number of pierce points. The third algorithm guarantees that for the special case where a torch path has no vertices of odd degree, no piece will be dropped that needs further interior cuts. Some possible extensions to this work are also addressed.

Distributed round-robin and first-come first-serve protocols and their applications to multiprocessor bus arbitration

Two new distributed protocols for fair and efficient bus arbitration are presented. The protocols implement round-robin (RR) and first-come first-serve (FCFS) scheduling, respectively. Both protocols use relatively few control lines on the bus, and their logic is simple. The round-robin protocol, which uses statically assigned arbitration numbers to resolve conflict during an arbitration, is more robust and simpler to implement than previous distributed RR protocols that are based on rotating agent priorities. The proposed FCFS protocol uses partly static arbitration numbers, and is the first practical proposal for a FCFS arbiter known to the authors. The proposed protocols thus have a better combination of efficiency, cost, and fairness characteristics than existing multiprocessor bus arbitration algorithms.nThree implementations of our RR protocol, and two implementations of our FCFS protocol, are discussed. Simulation results are presented that address: 1) the practical potential for unfairness in the simpler implementation of the FCFS protocol, 2) the practical implications of the higher waiting time variance in the RR protocol, and 3) the allocation of bus bandwidth among agents with unequal request rates in each protocol. The simulation results indicate that there is very little practical difference in the performance of the two protocols.

Applications of Ramsey's theorem to decision tree complexity

Combinatorial techniques for extending lower bound results for decision trees to general types of queries are presented. Problems that are defined by simple inequalities between inputs, called <italic>order invariant</italic> problems, are considered. A decision tree is called <italic>k-bounded</italic> if each query depends on at most <italic>k</italic> variables. No further assumptions on the type of queries are made. It is proved that one can replace the queries of any <italic>k</italic>-bounded decision tree that solves an order-invariant problem over a large enough input domain with <italic>k</italic>-bounded queries whose outcome depends only on the relative order of the inputs. As a consequence, all existing lower bounds for comparison-based algorithms are valid for general <italic>k</italic>-bounded decision trees, where <italic>k</italic> is a constant.nAn &OHgr;(<italic>n</italic> log <italic>n</italic>) lower bound for the element uniqueness problem and several other problems for any <italic>k</italic>-bounded decision tree, such that <italic>k</italic> = <italic>O</italic>(<italic>n<supscrpt>c</supscrpt></italic>) and <italic>c</italic> < 1/2 is proved. This lower bound is tight since there exist <italic>n</italic><supscrpt>1/2</supscrpt>-bounded decision trees of complexity <italic>O</italic>(<italic>n</italic>) that solve the element-uniqueness problem. All the lower bounds mentioned above are shown to hold for nondeterministic and probabilistic decision trees as well.

The effect of number of Hamiltonian paths on the complexity of a vertex-coloring problem

A generalization of Dobkin and Liptons element uniqueness problem is introduced: for any fixed undirected graph G on vertex set {v1, v2, ..., vn}, the problem is to determine, given n real numbers x1, x2, ..., xn, whether xi ≠ xj for every edge {vi, vj} in G. This problem is shown to have upper and lower bounds of Θ(nlogn) linear comparisons if G is any dense graph. The proof of the lower bound involves showing that any dense graph must contain a subgraph with many Hamiltonian paths, and demonstrating the relevance of these Hamiltonian paths to a geometric argument. In addition, we exhibit relatively sparse graphs for which the same lower bound holds, and relatively dense graphs for which a linear upper bound holds.

Concurrent Maintenance of Binary Search Trees

The problem of providing efficient concurrent access for independent processes to a dynamic search structure is the topic of this paper. We develop concurrent algorithms for search, update, insert, and delete in a simple variation of binary search trees, called external trees. The algorithm for deletion, which is usually the most difficult operation, is relatively easy in this data structure. The advantages of the data structure and the algorithms are that they are simple, flexible, and efficient, so that they can be used as a part in the design of more complicated concurrent algorithms where maintaining a dynamic search structure is necessary. In order to increase the efficiency of the algorithms we introduce maintenance processes that independently reorganize the data structure and relieve the user processes of nonurgent operations. We also discuss questions of transactions in a dynamic environment and replicated copies of the data structure.

On non-intersecting Eulerian circuits

Abstract The following question arises in flame-cutting and similar applications. “Given a graph drawn in the plane, is there an Eulerian circuit in which successive edges always belong to a common face?” We prove that this question and related ones are NP-complete.

Distributed computation via active messages

An extension to the token ring protocol which allows a special type of arithmetic, called shift arithmetic, to be performed directly on the nodes interfaces is proposed. The new protocol is based on an approach in which the communication channel and the interfaces form an environment in which simple commands can be executed. Each command operates on operands located at the interfaces, and places the result at the interface which initiated the command. The commands utilize the mandatory 1-bit delay of the token ring protocol to implement arithmetic and logical operations on the operands without further delay. The goal of this protocol is to enhance the performance of distributed algorithms on ring networks by performing many simple tasks in the lowest possible level. The authors show that this protocol is especially useful for load sharing in local area networks. Other potential application areas include parallel algorithms, distributed simulation, distributed operating systems, distributed databases and real-time computations.

A probabilistic lower bound for checking disjointness of sets

On prouve, dans cet article, que pour un probleme particulier, determiner si deux ensembles (pas necessairement egaux en taille) sont disjoints montre que lacceleration est limitee a une constante. Une borne inferieure probabiliste Ω(m log n) est prouvee, ou m et n sont les cardinalites densembles et m≥n

Efficient storage of nonadaptive routing tables

An algorithm for improving storage utilization for nonadaptive routing tables in point-to-point networks is presented. It allows storage vs. access time tradeoffs. The algorithm is quite general, and it can be used for other applications to improve storage in tree-like structures. The algorithm was applied to the UUCP routing tables resulting in a threefold storage improvement.