It is a well-known and useful problem to find a matching in a given graph G whose size is at most a given parameter k and whose weight is maximized (over all matchings of size at most k in G). In this paper, we consider two natural extensions of this problem. One is to find t disjoint matchings in a given graph G whose total size is at most a given parameter k and whose total weight is maximized, where t is a (small) constant integer. Previously, only the special case where t=2 was known to be fixed-parameter tractable. In this paper, we show that the problem is fixed-parameter tractable for any constant t. When t=2, the time complexity of the new algorithm is significantly better than that of the previously known algorithm. The other is to find a set of vertex-disjoint paths each of length 1 or 2 in a given graph whose total length is at most a given parameter k and whose total weight is maximized. As interesting applications, we further use the algorithms to speed up several known approximation algorithms (for related NP-hard problems) whose approximation ratio depends on a fixed parameter 0<ε<1 and whose running time is dominated by the time needed for exactly solving the problems on graphs in which each connected component has at most ε-1 vertices.

In this letter we show that the dynamic optimal PCID allocation problem in LTE systems is NP-complete. Subsequently we provide a near-optimal solution using SON which models the problem using new merge operations and explores the search space using a suitable randomized algorithmic approach. Two feasible options for dynamic auto-configuration of the system are also discussed. Simulation results point out that the approach provides near-optimal auto-configuration of PCIDs in computationally feasible time.

A radio network (RN for short) is a distributed system with no central arbiter, consisting of n radio transceivers, henceforth referred to as stations. We assume that the stations run on batteries and expends power while broadcasting/receiving a data packet. Thus, the most important measure to evaluate protocols on the radio network is the number of awake time slots, in which a station is broadcasting/receiving a data packet. We also assume that the stations are identical and have no unique ID number, and no station knows the number n of the stations. For given n keys one for each station, the ranking problem asks each station to determine the number of keys in the RN smaller than its own key. The main contribution of this paper is to present an optimal randomized ranking protocol on the k-channel RN. Our protocol solves the ranking problem, with high probability, in O(+log n) time slots with every station being awake for at most O(log n) time slots. We also prove that any randomized ranking protocol is required to run in expected Ω(+log n) time slots with at least one station being awake for expected Ω(log n) time slots. Therefore, our ranking protocol is optimal.

In this work we present an energy efficient leader election protocol for anonymous radio network populated with n mobile stations. Previously, Nakano and Olariu have presented a leader election protocol that terminates, with probability exceeding 1- (f ≥ 1), in log log n+o(log log n)+O(log f) time slots [14]. As the above protocol works under the assumption that every station has the ability to transmit and monitor the channel at the same time, it requires every station to be equipped with two transceivers. This assumption, however, is unrealistic for most mobile stations due to constraints in cost, size, and energy dissipation. Our main contribution is to show that it is possible to elect a leader in an anonymous radio network where each station is equipped with a single transceiver. Quite surprisingly, although every station has only one transceiver, our leader election protocol still runs, with probability exceeding 1- (f ≥ 1), in log log n+o(log log n)+O(log f) time slots. Moreover, our leader election protocol needs only expected O(n) total awake time slots, while Nakano and Olariu's protocol needs expected O(nlog log n) total awake time slots. Since every leader election protocol needs at least Ω(n) awake time slots, our leader election protocol is optimal in terms of the expected awake time slots.

A Radio Network (RN, for short) is a distributed system with no central arbiter, consisting of p radio stations each of which is endowed with a radio transceiver. In this work we consider single-hop, single channel RNs, where each station S(i), (1ip), initially stores si items which are tagged with the unique destination they must be routed. Since each item must be transmitted at least once, every routing protocol must take at least n = s1 + s2 + + sp time slots to route each item to its final destination. Similarly, each station S(i), (1ip), must be awake for at least si + di time slots to broadcast si items and to receive di items, where di denotes the number of items destined for S(i). The main contribution of this work is to present a randomized time- and energy-optimal routing protocol on the RN. Let qi, (1ip), be the number of stations that have items destined for S(i), q=q1 +q2 ++ qp, and ri be the number of stations for which S(i) has items. When qi is known to station S(i), our routing protocol runs, with probability exceeding 1 - , (f > 1), in n + O(q + log f) time slots with each station S(i) being awake for at most si + di + O(qi + ri + log f) time slots. Since qidi, risi, and qn always hold, our randomized routing protocol is optimal. We also show that, when the value of di is known to S(i), our routing protocol runs, with probability exceeding 1 - , (f > 1), in O(n + log f) time slots with no station being awake for more than O(si + di + log f) time slots.

A radio network is a distributed system with no central shared resource, consisting of n stations each equipped with a radio transceiver. One of the most important parameters to evaluate protocols in the radio networks is the number of awake time slots in which each individual station sends/receives a data packet. We are interested in devising energy-efficient initialization protocols in the single-hop radio network (RN, for short) that assign unique IDs in the range [1,n] to the n stations using few awake time slots. It is known that the RN can be initialized in O(log log n) awake time slots, with high probability, if every station knows the number n of stations in the RN. Also, it has been shown that the RN can be initialized in O(log n) awake time slots even if no station knows n. However, it has been open whether the initialization can be performed in O(log log n) awake time slots when no station knows n. Our main contribution is to provide the breakthrough: we show that even if no station knows n, the RN can be initialized by our protocol that terminates, with high probability, in O(n) time slots with no station being awake for more than O(log log n) time slots. We then go on to design an initialization protocol for the k-channel RN that terminates, with high probability, in O(n/k + (log n)2) time slots with no station being awake for more than O(log log n) time slots.

A Wireless Sensor Network (WSN, for short) is a distributed system consisting of n sensor nodes and a base station. In this paper, we propose an energy-efficient protocol to initialize the sensor nodes in a WSN, that is, to assign a unique ID to each sensor node. We show that if an upper bound u on the number n of sensor nodes is known beforehand, for any f 1 and any small µ (0<µ<1), a WSN without collision detection capability can be initialized in O((log (1/µ) + log f)u1+µ) time slots, with probability exceeding 1-(1/f), with no sensor node being awake for more than O(log (1/µ)+ log f) time slots.

The main contribution of this work is to propose energy-efficient randomized initialization protocols for ad-hoc radio networks (ARN, for short). First, we show that if the number n of stations is known beforehand, the single-channel ARN can be initialized by a protocol that terminates, with high probability, in O(n) time slots with no station being awake for more than O(log n) time slots. We then go on to address the case where the number n of stations in the ARN is not known beforehand. We begin by discussing, an elegant protocol that provides a tight approximation of n. Interestingly, this protocol terminates, with high probability, in O((log n)2) time slots and no station has to be awake for more than O(log n) time slots. We use this protocol to design an energy-efficient initialization protocol that terminates, with high probability, in O(n) time slots with no station being awake for more than O(log n) time slots. Finally, we design an energy-efficient initialization protocol for the k-channel ARN that terminates, with high probability, in O(n/k+log n) time slots, with no station being awake for more than O(log n) time slots.

A convex hull is one of the most fundamental and interesting geometric constructs in computational geometry. Considerable research effort has focused on developing algorithms, both in serial and in parallel, for computing convex hulls. In particular, there are few problems whose parallel algorithms are so thoroughly studied as convex hull problems. In this paper, we review the convex hull parallel algorithms and their paradigm. We provide a summary of results and introduce several interesting topics including typical techniques, output-size sensitive methods, randomized approaches, and robust algorithms for convex hull problems, with which we may see the highlights of the whole research for parallel algorithms. Most of our discussion uses the PRAM (Parallel Random Access Machine) computational model, but still we give a glance at the results of the other parallel computational models such as mesh, mesh-of-trees, hypercube, recofigurable array, and models of coarse grained multicomputers like BSP and LogP.