Krzysztof Diks

Improved deterministic parallel integer sorting

Abstract We consider the problem of deterministic sorting of integers on a parallel RAM (PRAM). The best previous result ( T. Hagerup, 1987 , Inform. and Comput.75, 39–51) states that n integers of size polynomial in n can be sorted in time O(log n) on a Priority CRCW PRAM with O( n log log n log n ) processors. We prove that n integers drawn from a set {0, …, m−1} can be sorted on an Arbitrary CRCW PRAM in time O( log n log log n + log log m) with a time-processor product of O(n log log m). In particular, if m = n(log n)O(1), the time and number of processors used are O( log n log log n ) and O( n( log log n) 2 log n ) , respectively. This improves the previous result in several respects: The new algorithm is faster, it works on a weaker PRAM model, and it is closer to optimality for input numbers of superpolynomial size. If log log m = O( log n log log n ) , the new algorithm is optimally fast, for any polynomial number of processors, and if log log m = (1 + Ω(1)) log log n and log log m = 0 ( log n ), it has optimal speedup relative to the fastest known sequential algorithm. The space needed is O(nme), for arbitrary but fixed e > 0. The sorting algorithm derives its speed from a fast solution to a special list ranking problem of possible independent interest, the monotonic list ranking problem. In monotonic list ranking, each list element has an associated key, and the keys are known to increase monotonically along the list. We show that monotonic list ranking problems of size n can be solved optimally in time O( log n log log n ) . We also discuss and attempt to solve some of the problems arising in the precise description and implementation of parallel recursive algorithms. As part of this effort, we introduce a new PRAM variant, the allocated PRAM.

The Impact of Knowledge on Broadcasting Time in Radio Networks

We consider the problem of distributed deterministic broadcasting in radio networks. Nodes send messages in synchronous time-slots. Each node v has a given transmission range. All nodes located within this range can receive messages from v. However, a node situated in the range of two or more nodes that send messages simultaneously, cannot receive these messages and hears only noise. Each node knows only its own position and range, as well as the maximum of all ranges. Broadcasting is adaptive: Nodes can decide on the action to take on the basis of previously received messages, silence or noise.We prove a lower bound on broadcasting time in this model and construct a broadcasting protocol whose performance matches this bound for the simplest case when nodes are situated on a line and the network has constant depth.We also show that if nodes do not even know their own range, every broadcasting protocol must be hopelessly slow. While distributed randomized broadcasting algorithms, and, on the other hand, deterministic off-line broadcasting algorithms assuming full knowledge of the radio network, have been extensively studied in the literature, ours are the first results concerning broadcasting algorithms that are distributed and deterministic at the same time.We show that in this case the amount of knowledge available to nodes influences the efficiency of broadcasting in a significant way.

The impact of information on broadcasting time in linear radio networks

We consider the problem of distributed deterministic broadcasting in radio networks whose nodes are located on a line. Nodes send messages in synchronous time-slots. Each node υ has a given transmission range. All nodes located within this range can receive messages from υ. However, a node situated in the range of two or more nodes that send messages simultaneously, cannot receive these messages and hears only noise. Each node knows only its own position and range, as well as the maximum of all ranges. Broadcasting is adaptive: nodes can decide on the action to take on the basis of previously received messages, silence or noise. We prove lower bounds on broadcasting time in this model and construct broadcasting protocols whose performance nearly matches these bounds for the simplest case when nodes are situated on a line. We also show that if nodes do not even know their own range, every broadcasting protocol must be hopelessly slow. While distributed randomized broadcasting algorithms, and, on the other hand, deterministic off-line broadcasting algorithms assuming full knowledge of the radio network, have been extensively studied in the literature, ours are the first results concerning broadcasting algorithms that are distributed and deterministic at the same time. We show that in this case, information available to nodes influences the efficiency of broadcasting in a significant way.

Efficient Simulations Between Concurrent-Read Concurrent-Write PRAM Models

We give several simple and efficient algorithms for simulations of stronger CRCW PRAMs on weaker ones. The models that we consider are the well-known PRIORITY, ARBITRARY and COMMON PRAMs, and COLLISION and COLLISION+, defined by the property that a special collision symbol is stored in each memory cell into which more than one processor attempts to write, or more than one value is attempted to be written, respectively, in a given step. Our results are the following, where n denotes the number of processors of the simulated PRAM: 1) A O(1)-time simulation between any pair of models, provided that the simulating machine has O(n log n) processors; 2) Two n-processor simulations: of PRIORITY on ARBITRARY with O(loglog n) slowdown, and of PRIORITY on COLLISION+ with O((loglog n)2) slowdown.

Anonymous wireless rings

We introduce anonymous wireless rings: a new computational model for ring networks. In the well-known hardware ring each processor has two buffers, one corresponding to each of its neighbors. In the wireless ring each processor has a single buffer and cannot distinguish which neighbor the arriving bit comes from. This feature substantially increases anonymity of the ring. A priori it is not clear whether any nontrivial computation can be performed on wireless rings. Nevertheless we show that wireless rings are computationally equivalent to hardware rings.

A proposal for an IOI Syllabus

The International Olympiad in Informatics (IOI) is the premier competition in computing science for secondary education. The competition problems are algorithmic in nature, but the IOI Regulations do not clearly define the scope of the competition. The international olympiads in physics, chemistry, and biology do have an official syllabus, whereas the International Mathematical Olympiad has made the deliberate decision not to have an official syllabus. We argue that the benefits of having an official IOI Syllabus outweigh the disadvantages. Guided by a set of general principles we present a proposal for an IOI Syllabus, divided into four main areas: mathematics, computing science, software engineering, and computer literacy.

Efficient gossiping by packets in networks with random faults

Every node of a communication network has a constant size value which should be made known to all other nodes. Nodes and links fail independently with constant probabilities

Broadcasting in synchronous networks with dynamic faults

p 0

Almost safe gossiping in bounded degree networks

we present an algorithm to exchange values between all fault-free nodes of an n-node network in time

New Simulations between CRCW PRAMs

O(\frac{n}{b(n)}) + \log n