Amos Korman

Online computation with advice

We consider a model for online computation in which the online algorithm receives, together with each request, some information regarding the future, referred to as advice. The advice is a function, defined by the online algorithm, of the whole request sequence. The advice provided to the online algorithm may allow an improvement in its performance, compared to the classical model of complete lack of information regarding the future. We are interested in the impact of such advice on the competitive ratio, and in particular, in the relation between the size b of the advice, measured in terms of bits of information per request, and the (improved) competitive ratio. Since b=0 corresponds to the classical online model, and b=@?log|A|@?, where A is the algorithms action space, corresponds to the optimal (offline) one, our model spans a spectrum of settings ranging from classical online algorithms to offline ones. In this paper we propose the above model and illustrate its applicability by considering two of the most extensively studied online problems, namely, metrical task systems (MTS) and the k-server problem. For MTS we establish tight (up to constant factors) upper and lower bounds on the competitive ratio of deterministic and randomized online algorithms with advice for any choice of 1@?b@?@Q(logn), where n is the number of states in the system: we prove that any randomized online algorithm for MTS has competitive ratio @W(log(n)/b) and we present a deterministic online algorithm for MTS with competitive ratio O(log(n)/b). For the k-server problem we construct a deterministic online algorithm for general metric spaces with competitive ratio k^O^(^1^/^b^) for any choice of @Q(1)@?b@?logk.

Label-guided graph exploration by a finite automaton

A finite automaton, simply referred to as a robot, has to explore a graph, that is, visit all the nodes of the graph. The robot has no a priori knowledge of the topology of the graph, nor of its size. It is known that for any k-state robot, there exists a graph of maximum degree 3 that the robot cannot explore. This article considers the effects of allowing the system designer to add short labels to the graph nodes in a preprocessing stage, for helping the exploration by the robot. We describe an exploration algorithm that, given appropriate 2-bit labels (in fact, only 3-valued labels), allows a robot to explore all graphs. Furthermore, we describe a suitable labeling algorithm for generating the required labels in linear time. We also show how to modify our labeling scheme so that a robot can explore all graphs of bounded degree, given appropriate 1-bit labels. In other words, although there is no robot able to explore all graphs of maximum degree 3, there is a robot R, and a way to color in black or white the nodes of any bounded-degree graph G, so that R can explore the colored graph G. Finally, we give impossibility results regarding graph exploration by a robot with no internal memory (i.e., a single-state automaton).

Labeling schemes for vertex connectivity

This article studies labeling schemes for the vertex connectivity function on general graphs. We consider the problem of assigning short labels to the nodes of any <i>n</i>-node graph is such a way that given the labels of any two nodes <i>u</i> and <i>v</i>, one can decide whether <i>u</i> and <i>v</i> are <i>k</i>-vertex connected in <i>G</i>, that is, whether there exist <i>k</i> vertex disjoint paths connecting <i>u</i> and <i>v</i>. This article establishes an upper bound of <i>k</i>²log <i>n</i> on the number of bits used in a label. The best previous upper bound for the label size of such a labeling scheme is 2^<i>k</i>log <i>n</i>.

Improved compact routing schemes for dynamic trees

A classical routing problem consists of assigning a label and distinct port numbers to each node of a graph, such that for every node v, given its own label and the label of any destination vertex u, node v can find which of its incident port numbers leads to the next vertex on a shortest path connecting v and u. In the static (fixed topology) setting, such a routing scheme is evaluated by the label size, i.e., the maximal number of bits stored in a label. Naturally, special attention is given to compact schemes, which are schemes enjoying asymptotically optimal labels. Many routing schemes were proposed for the static setting. However, the more realistic and complex dynamic setting, in which topology changes may occur at arbitrary nodes, has received much less attention. In the dynamic setting, the occurrence of topology changes may force the scheme to occasionally update the (hopefully short) labels, by delivering information from place to place. This raises a natural tradeoff between the size of the labels and the number of messages required for maintaining them. The above dynamic routing problem was proposed by Afek, Gafni, and Ricklin (1989), who also presented an elegant and rather efficient dynamic routing scheme for trees, supporting one type of topology change, namely, the addition of a leaf. Various attempts for improving the tradeoff between the label size and the message complexity as well as for supporting more types of topology changes on trees, were subsequently proposed. Still, the best known compact routing scheme for dynamic trees has very high message complexity, namely, O(nε) amortized messages per topological change. Moreover, previous routing schemes for dynamic trees support at most two kinds of topology changes, namely, the addition and the removal of a leaf node. In this paper, we present two compact routing schemes for dynamic trees that incur extremely low message complexity and can support more types of topology changes than previous schemes. We first present a dynamic compact routing scheme that supports the additions of both leaves and internal nodes and incurs only O(log n) amortized message complexity per node. We then extend that scheme obtaining a dynamic compact routing scheme that supports additions of both leaves and internal nodes as well as deletions of nodes of degree at most 2. The extended scheme incurs O(log2 n) amortized message complexity per topological change.

New bounds for the controller problem

The (M, W)-controller, originally studied by Afek, Awerbuch, Plotkin, and Saks, is a basic distributed tool that provides an abstraction for managing the consumption of a global resource in a distributed dynamic network. The input to the controller arrives online in the form of requests presented at arbitrary nodes. A request presented at node u corresponds to the “desire” of some entity to consume one unit of the global resource at u and the controller should handle this request within finite time either by granting it with a permit or by denying it. Initially, M permits (corresponding to M units of the global resource) are stored at a designated root node. Throughout the execution permits can be transported from place to place along the network’s links so that they can be granted to requests presented at various nodes; when a permit is granted to some request, it is eliminated from the network. The fundamental rule of an (M, W)-controller is that a request should not be denied unless it is certain that at least M − W permits are eventually granted. The most efficient (M, W)-controller known to date has message complexity

Notions of connectivity in overlay networks

{O (Nlog^{2} N log frac{M}{W + 1})}

Parallel exhaustive search without coordination

, where N is the number of nodes that ever existed in the network (the dynamic network may undergo node insertions and deletions). In this paper we establish two new lower bounds on the message complexity of the controller problem. We first prove a simple lower bound stating that any (M, W)-controller must send