Randomized Work-Competitive Scheduling for Cooperative Computing on $k$-partite Task Graphs

A fundamental problem in distributed computing is the task of cooperatively executing a given set of $t$ tasks by $p$ processors where the communication medium is dynamic and subject to failures. The dynamics of the communication medium lead to groups of processors being disconnected and possibly reconnected during the entire course of the computation furthermore tasks can have dependencies among them. In this paper, we present a randomized algorithm whose competitive ratio is dependent on the dynamics of the communication medium and also on the nature of the dependencies among the tasks.

💡 Research Summary

The paper addresses the fundamental problem of cooperatively executing a set of t tasks on p processors when the communication medium is dynamic and subject to failures, causing processors to be partitioned into groups that may later reconnect. Unlike prior work that assumes tasks are independent, this study considers tasks that have a k‑partite directed‑acyclic graph (k‑partite DAG) dependency structure: tasks are divided into k levels, where every task in level ℓ_{i+1} depends on all tasks in level ℓ_i. Each processor group knows the tasks completed by its members, but the adversary controls the timing of group reconfigurations and the amount of work (quota) each group may perform before the next reconfiguration, without dictating which specific tasks are chosen.

The authors formalize the notion of computation width (cw), defined as the maximum width of the poset formed by the successor sets of vertices in the adversarial computation DAG that captures the sequence of reconfigurations and work quotas. Intuitively, cw measures the degree of parallelism forced by the communication pattern: a larger cw indicates more simultaneous independent “branches” of work that must be pursued without coordination.

Building on the earlier Random Select (RS) algorithm, which achieved a competitive ratio of (1 + cw/e) for independent tasks, the paper introduces Modified‑RS (m‑RS), a simple randomized scheduling rule adapted to the leveled dependency structure. When a processor group knows a set K of completed tasks, it selects uniformly at random a task τ from the set of unfinished tasks that has the smallest level among all unfinished tasks (i.e., τ is a minimal element of the remaining poset). This rule respects dependencies while preserving the randomness that underlies the RS analysis.

The main technical contributions are twofold: (1) a lower‑bound on the expected work any online algorithm must perform for 2‑level task graphs, and (2) an upper‑bound showing that m‑RS attains exactly this bound, thereby being optimal (tight) in the competitive sense.

For the lower bound, the adversary first creates w singleton groups, each allowed to execute α t w tasks from level ℓ₁ (α∈(0,1] denotes the fraction of tasks in the first level). After these groups merge, they are split again, each now allowed to execute (1‑α) t w tasks from level ℓ₂. By analyzing the random selection of tasks and applying Azuma‑Hoeffding concentration, the authors show that any algorithm must, in expectation, perform at least

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