Optimal Local Buffer Management for Information Gathering with Adversarial Traffic

We consider a problem of routing on directed paths and trees to a single destination, with rate-limited, adversarial traffic. In particular, we focus on local buffer management algorithms that ensure no packet loss, while minimizing the size of the required buffers. While a centralized algorithm for the problem that uses constant-sized buffers has been recently shown [21], there is no known local algorithm that achieves a sub-linear buffer size. In this paper we show tight bounds for the maximum buffer size needed by l-local algorithms for information gathering on directed paths and trees, where an algorithm is called l-local if the decision made by each node v depends only on the sizes of the buffers at most l hops away from v. We show three main results: A lower bound of Ω(c log n/l) for all l-local algorithms on both directed and undirected paths, where c is an upper bound on the link capacity and injection rate. A surprisingly simple 1-local algorithm for directed paths that uses buffers of size O(log n), when c=1. A natural 2-local extension of this algorithm to directed trees, for c=1, with the same asymptotic bound. Our Ω(log n) lower bound is significantly lower than the Ω(n) lower bound for greedy algorithms, and perhaps surprisingly, there is a matching upper bound. The algorithm that achieves it can be summarized in two lines: If the size of your buffer is odd, forward a message if your successor's buffer size is equal or lower. If your buffer size is even, forward a message only if your successor's buffer size is strictly lower. For trees, a simple arbitration between siblings is added.