Universal Loop-Free Super-Stabilization

We propose an univesal scheme to design loop-free and super-stabilizing protocols for constructing spanning trees optimizing any tree metrics (not only those that are isomorphic to a shortest path tree). Our scheme combines a novel super-stabilizing loop-free BFS with an existing self-stabilizing spanning tree that optimizes a given metric. The composition result preserves the best properties of both worlds: super-stabilization, loop-freedom, and optimization of the original metric without any stabilization time penalty. As case study we apply our composition mechanism to two well known metric-dependent spanning trees: the maximum-flow tree and the minimum degree spanning tree

Optimally Efficient Prefix Search and Multicast in Structured P2P Networks

Searching in P2P networks is fundamental to all overlay networks. P2P networks based on Distributed Hash Tables (DHT) are optimized for single key lookups, whereas unstructured networks offer more complex queries at the cost of increased traffic and uncertain success rates. Our Distributed Tree Construction (DTC) approach enables structured P2P networks to perform prefix search, range queries, and multicast in an optimal way. It achieves this by creating a spanning tree over the peers in the search area, using only information available locally on each peer. Because DTC creates a spanning tree, it can query all the peers in the search area with a minimal number of messages. Furthermore, we show that the tree depth has the same upper bound as a regular DHT lookup which in turn guarantees fast and responsive runtime behavior. By placing objects with a region quadtree, we can perform a prefix search or a range query in a freely selectable area of the DHT. Our DTC algorithm is DHT-agnostic and works with most existing DHTs. We evaluate the performance of DTC over several DHTs by comparing the performance to existing application-level multicast solutions, we show that DTC sends 30-250% fewer messages than common solutions

Spanning Trees with Many Leaves in Graphs without Diamonds and Blossoms

It is known that graphs on n vertices with minimum degree at least 3 have spanning trees with at least n/4+2 leaves and that this can be improved to (n+4)/3 for cubic graphs without the diamond K_4-e as a subgraph. We generalize the second result by proving that every graph with minimum degree at least 3, without diamonds and certain subgraphs called blossoms, has a spanning tree with at least (n+4)/3 leaves, and generalize this further by allowing vertices of lower degree. We show that it is necessary to exclude blossoms in order to obtain a bound of the form n/3+c. We use the new bound to obtain a simple FPT algorithm, which decides in O(m)+O^*(6.75^k) time whether a graph of size m has a spanning tree with at least k leaves. This improves the best known time complexity for MAX LEAF SPANNING TREE.Comment: 25 pages, 27 Figure