Jasmin Smula

The string guessing problem as a method to prove lower bounds on the advice complexity

The advice complexity of an online problem describes the additional information both necessary and sufficient for online algorithms to compute solutions of a certain quality. In this model, an oracle inspects the input before it is processed by an online algorithm. Depending on the input string, the oracle prepares an advice bit string that is accessed sequentially by the algorithm. The number of advice bits that are read to achieve some specific solution quality can then serve as a fine-grained complexity measure. The main contribution of this paper is to study a powerful method for proving lower bounds on the number of advice bits necessary. To this end, we consider the string guessing problem as a generic online problem and show a lower bound on the number of advice bits needed to obtain a good solution. We use special reductions from string guessing to improve the best known lower bound for the online set cover problem and to give a lower bound on the advice complexity of the online maximum clique problem.

The String Guessing Problem as a Method to Prove Lower Bounds on the Advice Complexity

The advice complexity of an online problem describes the additional information both necessary and sufficient for online algorithms to compute solutions of a certain quality. In this model, an oracle inspects the input before it is processed by an online algorithm. Depending on the input string, the oracle prepares an advice bit string that is accessed sequentially by the algorithm. The number of advice bits that are read to achieve some specific solution quality can then serve as a fine-grained complexity measure. The main contribution of this paper is to study a powerful method for proving lower bounds on the number of advice bits necessary. To this end, we consider the string guessing problem as a generic online problem and show a lower bound on the number of advice bits needed to obtain a good solution. We use special reductions from string guessing to improve the best known lower bound for the online set cover problem and to give a lower bound on the advice complexity of the online maximum clique problem.

On the Power of Advice and Randomization for the Disjoint Path Allocation Problem

In the disjoint path allocation problem, we consider a path of L + 1 vertices, representing the nodes in a communication network. Requests for an unbounded-time communication between pairs of vertices arrive in an online fashion and a central authority has to decide which of these calls to admit. The constraint is that each edge in the path can serve only one call and the goal is to admit as many calls as possible.

Time-optimal information exchange on multiple channels

This paper presents an efficient algorithm for detecting and disseminating information in a single-hop multi-channel network: k arbitrary nodes have information they want to share with the entire network. Neither the nodes that have information nor the number k of these nodes are known initially. This communication primitive lies between the two other fundamental primitives regarding information dissemination, broadcasting (one-to-all communication) and gossiping (total information exchange). The time complexity of the information exchange algorithm we present in this paper is linear in the number of information items and thus asymptotically optimal with respect to time. The algorithm does not require collision detection and thanks to using several channels the lower bound of Ω(k+log n) established for single-channel communication can be broken.

Treasure Hunt with Advice

The node searching problem a.k.a. treasure hunt is a fundamental task performed by mobile agents in a network and can be viewed as an online version of the shortest path problem: an agent starts in a vertex of an unknown weighted undirected graph, and its goal is to reach a given vertex. The cost is the overall distance measured by the weights of the traversed edges traversed by the agent. We consider the setting in which the agent sees the identifier of the vertex it is located in, the weights of the incident edges, and also the identifiers of the neighboring vertices. We analyze the problem from the point of view of advice complexity: at the beginning, the agent has a tape with an advice string that gives some a priori information about the input instance. This information has no restricted form; instead, the aim is to study the relationship between the size of this advice and the competitive ratio that can be obtained. We give tight bounds of the form i¾źn/r bits of advice for a competitive ratio r possibly depending on the number of vertices n. In particular, this means that an a priori knowledge of any graph parameter which would be of size Ologn cannot yield a competitive ratio better than Ωn/logn. Moreover, we give a lower bound on the expected competitive ratio of any randomized online algorithm for treasure hunt.

Monitoring churn in wireless networks

Wireless networks often experience a significant amount of churn, the arrival and departure of nodes. In this paper we propose a distributed algorithm for single-hop networks that detects churn and is resilient to a worst-case adversary. The nodes of the network are notified about changes quickly, in asymptotically optimal time up to an additive logarithmic overhead. We establish a trade-off between saving energy and minimizing the delay until notification for single- and multi-channel networks.

Length-Weighted Disjoint Path Allocation

We modify one of the foundational online problems, Disjoint Path Allocation, to include weighted requests. We provide a comprehensive competitive analysis, incorporating the viewpoints of both advice complexity and parametrized complexity. Our bounds feature a consistent parametrization and closely trace the trade-off between advice complexity and competitiveness.

Information dissemination on multiple channels

This article presents an algorithm for detecting and disseminating information in a single-hop multi-channel wireless network: k arbitrary nodes have information they want to share with the entire network. Neither the nodes that have information nor the number k of these nodes are known initially. This communication primitive lies between the two other fundamental primitives regarding information dissemination: broadcasting (one-to-all communication) and gossiping (total information exchange). The time complexity of the algorithm is linear in the number of information items and thus asymptotically optimal with respect to time. The algorithm does not require collision detection and thanks to using several channels the lower bound of Ω(k+log n) established for single-channel communication can be broken.

Brief announcement: self-monitoring in dynamic wireless networks

Wireless networks often experience a significant amount of churn, the arrival and departure of nodes. We propose a distributed algorithm that detects churn and is resilient to a worst-case adversary. The nodes of the network are notified about changes quickly, in asymptotically optimal time up to an additive logarithmic overhead.