A tournament is an orientation of a complete graph. A vertex that can reach every other vertex within two steps is called a king. We study the complexity of finding k kings in a tournament graph. We show that the randomized query complexity of finding k ≤ 3 kings is O(n), and for the deterministic case it takes the same amount of queries (up to a constant) as finding a single king (the best known deterministic algorithm makes O(n^{3/2}) queries). On the other hand, we show that finding k ≥ 4 kings requires Ω(n²) queries, even in the randomized case. We consider the RAM model for k ≥ 4. We show an algorithm that finds k kings in time O(kn²), which is optimal for constant values of k. Alternatively, one can also find k ≥ 4 kings in time n^{ω} (the time for matrix multiplication). We provide evidence that this is optimal for large k by suggesting a fine-grained reduction from a variant of the triangle detection problem.

We propose DSON, a space efficient δ-based CRDT approach for distributed JSON document stores, enabling high availability at a global scale, while providing strong eventual consistency guarantees. We define the semantics of our CRDT based approach formally, and prove its correctness and convergence. Previous approaches optimize for collaborative document editing and store metadata proportional to the number of updates to a document, which is not acceptable for long lived document management. The metadata stored with our approach is bounded by O ( k 2 D + n log n ), where n is the number of replicas, D is the number of document elements, and k ≤ n is the number of concurrent document updates. We also implement our approach[37] and demonstrate its space efficiency empirically. Experimental analysis shows that the metadata stored is typically significantly less than the worst case. This provides the basis for robust highly available distributed document stores with well defined semantics and safety guarantees, relieving application developers from the burden of conflict resolution.