Distributed termination detection with roughly synchronized clocks

Detection of Conjunctive Stable Predicates in Dynamic Systems

“…To detect a solution, control messages used in the computation trees are fewer than or equal to (in the worst case) that of basic messages, which is message-optimal [17], [48], [49]. Control messages used in the logical ring circulate the ring one round at most, which is more efficient than those in previous works for termination detection [13], [19]. Let M be the number of basic messages and M c be the number of control messages to detect a solution.…”

An efficient delay-optimal distributed termination detection algorithm

“…We prove the second result by deriving another linear ordering G (k, n) of the P PEs in which consecutive PEs are (D) distance apart by swapping suitable pairs of PEs in G(k, n). For example, G(6, 2) = 00, 01, 02, 03, 04, 05, 15,14,13,12,11,10,20,21,22,23,24,25,35,34,33,32,31,30,40,41,42,43,44,45, 55, 54, 53, 52, 51, 50 and G(5, 2) = 00, 01, 02, 03, 04, 14,13,12,11,10,20,21,22,23,24,34,33,32,31,30,40,41,42,43,44 , where "swappable" PEs-to be defined shortly-are underlined and PEs at odd-numbered positions that are not swappable are overlined. Note that when k is even, P is even, when k is odd, P is odd, and that there are an even number of PEs at odd-numbered positions (i.e., p i such that i is odd), regardless of whether k is even or odd.…”

“…Similarly, some of the algorithms are either less compute intensive [15] or require less memory [14], but are vulnerable to underflow [15] or overflow [14,3] problems. 2 Although the system model described in Section 1.1 is most commonly used, there are algorithms meant for other system characteristics: those that rely on a common clock [9,25,31], or assume a specific target system topology [11], or require first-in first-out (FIFO) communication between PEs [39], 3 or are designed to tolerate PE or link failures [9,18,22,32,37,38]. A general approach to transform a DTD algorithm meant for a fault-free system into that for a system with PE failures (assuming the non-faulty system graph remains connected) is given in [27].…”

“…Although DTD is a long-standing problem, new algorithms for it with different performance characteristics and varying assumptions regarding the target computing system model appear regularly [2][3][4]9,11,14,15,17,18,22,24,25,28,27,29,31,32,[36][37][38][39]. Some of these algorithms are efficient in terms of the number of secondary messages used [4,17,28], but may take a long time to detect termination [4].…”

A general model for detecting distributed termination in dynamic systems

“…These include solutions for static systems [14,13,30,31,25,16,15,18,22] and dynamic systems [11,29,6,19,9,7]. Some algorithms take advantage of a global time base in the system [30,27,8]. While most of the solutions are for non-faulty systems, some of them are fault-tolerant [1,33,21,32,8].…”