Alon Itai

On the Complexity of Timetable and Multicommodity Flow Problems

A very primitive version of Gotlieb’s timetable problem is shown to be NP-complete, and therefore all the common timetable problems are NP-complete. A polynomial time algorithm, in case all teachers are binary, is shown. The theorem that a meeting function always exists if all teachers and classes have no time constraints is proved. The multicommodity integral flow problem is shown to be NP-complete even if the number of commodities is two. This is true both in the directed and undirected cases.

A fast and simple randomized parallel algorithm for the maximal independent set problem

Abstract A simple parallel randomized algorithm to find a maximal independent set in a graph G = ( V , E ) on n vertices is presented. Its expected running time on a concurrent-read concurrent-write PRAM with O (| E | d max ) processors is O (log n ), where d max denotes the maximum degree. On an exclusive-read exclusive-write PRAM with O (| E |) processors the algorithm runs in O (log 2 n ). Previously, an O (log 4 n ) deterministic algorithm was given by Karp and Wigderson for the EREW-PRAM model. This was recently (independently of our work) improved to O (log 2 n ) by M. Luby. In both cases randomized algorithms depending on pairwise independent choices were turned into deterministic algorithms. We comment on how randomized combinatorial algorithms whose analysis only depends on d -wise rather than fully independent random choices (for some constant d ) can be converted into deterministic algorithms. We apply a technique due to A. Joffe (1974) and obtain deterministic construction in fast parallel time of various combinatorial objects whose existence follows from probabilistic arguments.

On the time-complexity of broadcast in multi-hop radio networks: an exponential gap between determinism and randomization

The time-complexity of deterministic and randomized protocols for achieving broadcast (distributing a message from a source to all other nodes) in arbitrary multi-hop radio networks is investigated. In many such networks, communication takes place in synchronous time-slots. A processor receives a message at a certain time-slot if exactly one of its neighbors transmits at that time-slot. We assume no collision-detection mechanism; i.e., it is not always possible to distinguish the case where no neighbor transmits from the case where several neighbors transmit simultaneously. We present a randomized protocol that achieves broadcast in time which is optimal up to a logarithmic factor. In particular, with probability 1 --E, the protocol achieves broadcast within O((D + log n/s) ‘log n) time-slots, where n is the number of processors in the network and D its diameter. On the other hand, we prove a linear lower bound on the deterministic time-complexity of broadcast in this model. Namely, we show that any deterministic broadcast protocol requires 8(n) time-slots, even if the network has diameter 3, and n is known to all processors. These two results demonstrate an exponential gap in complexity between randomization and determinism.

Finding a Minimum Circuit in a Graph

Finding minimum circuits in graphs and digraphs is discussed. An almost minimum circuit is a circuit which may have only one edge more than the minimum. To find an almost minimum circuit an

Hamilton Paths in Grid Graphs

O(n^2 )

On the complexity of time table and multi-commodity flow problems

algorithm is presented. A direct algorithm for finding a minimum circuit has an

The multi-tree approach to reliability in distributed networks

O(ne)

On the time-complexity of broadcast in radio networks: an exponential gap between determinism randomization

behavior. It is refined to yield an

Symmetry breaking in distributed networks

O(n^2 )

Two Languages Are More Informative Than One

average time algorithm. An alternative method is to reduce the problem of finding a minimum circuit to that of finding a triangle in an auxiliary graph. Three methods for finding a triangle in a graph are given. The first has an