Some Combinatorial Aspects of Time - Stamp

The aim of this paper is to outline a combinatorial structure appearing in distributed computing, namely a directed graph in which a certain family of subsets with k vertices have a successor. It has been proved that the number of vertices of such a graph is at least 2 k ? 1 and an eeective construction has been given which needs k2 k?1 vertices. This problem is issued from some questions related to the labeling of processes in a system for determining the order in which they were created. By modifying some requirements on the distributed system, we show that there arise other combinatorial structures leading to the construction of solutions whose size becomes a linear function of the input. 0 Introduction Let us rst describe in detail the problem of time-stamping. In a system, we consider two kinds of events, namely the creation and the death of processes. We assume that two such events cannot occur simultaneously. A global "scheduler" assigns a time-stamp to any process at the moment of its creation, according to the actual set of time-stamps assigned to the living processes. Such a time-stamp cannot be modiied during the lifetime of the process. The aim of these time-stamps is the following: any external observer of the system which looks at any two processes must be able, solely by considering their two time-stamps, to determine the order of their creation. Note that a simple solution consists in using the values of a counter as time-stamps, and incrementing it whenever a process is created. Then, the observer can use the natural order on integers for determining which one among any two processes was created rst. Such a solution needs an unbounded set of time-stamps. In 9], Isra eli and Li proved that, when the number of living processes is assumed to be bounded by an integer k, a time-stamp system with a nite number of elements may 1