195710562
On fundamental tradeoffs between delay bounds and computational complexity in packet scheduling algorithms
In this work, we clarify, extend and solve an open problem concerning the computational complexity for packet scheduling algorithms to achieve tight end-to-end delay bounds. We first focus on the difference between the time a packet finishes service in a scheduling algorithm and its virtual finish time under a GPS (General Processor Sharing) scheduler, called GPS-relative delay. We prove that, under a slightly restrictive but reasonable computational model, the lower bound computational complexity of any scheduling algorithm that guarantees O(1) GPS-relative delay bound is Ω (log 2 n) (widely believed as a "folklore theorem" but never proved). We also discover that, surprisingly, the complexity lower bound remains the same even if the delay bound is relaxed to O(na) for 0‹a⋵1. This implies that the delay-complexity tradeoff curve is "flat" in the "interval" [O(1), O(n)). We later extend both complexity results (for O(1) or O(na) delay) to a much stronger computational model. Finally, we show that the same complexity lower bounds are conditionally applicable to guaranteeing tight end-to-end delay bounds. This is done by untangling the relationship between the GPS-relative delay bound and the end-to-end delay bound.
10184291
Efficient and scalable query routing for unstructured peer-to-peer networks
Searching for content in peer-to-peer networks is an interesting and challenging problem. Queries in Gnutella-like unstructured systems that use flooding or random walk to search must visit O(n) nodes in a network of size n, thus consuming significant amounts of bandwidth. In this paper, we propose a query routing protocol that allows low bandwidth consumption during query forwarding using a low cost mechanism to create and maintain information about nearby objects. To achieve this, our protocol maintains a lightweight probabilistic routing table at each node that suggests the location of each object in the network. Following the corresponding routing table entries, a query can reach the destination in a small number of hops with high probability. However, maintaining routing tables in a large and highly dynamic network requires non-traditional mechanisms. We design a novel data structure called an exponentially decaying bloom filter (EDBF) that encodes such probabilistic routing tables in a highly compressed manner, and allows for efficient aggregation and propagation. The search primitives provided by our system can be used to search for single keys or multiple keywords with equal ease. Analytical modeling of our design predicts significant improvements in search efficiency, verified through extensive simulations in which we observed an order of magnitude reduction in query path length over previous proposals.
16814075
On algebraic construction of Gallager and circulant low-density parity-check codes
This correspondence presents three algebraic methods for constructing low-density parity-check (LDPC) codes. These methods are based on the structural properties of finite geometries. The first method gives a class of Gallager codes and a class of complementary Gallager codes. The second method results in two classes of circulant-LDPC codes, one in cyclic form and the other in quasi-cyclic form. The third method is a two-step hybrid method. Codes in these classes have a wide range of rates and minimum distances, and they perform well with iterative decoding.
47532017
On fundamental tradeoffs between delay bounds and computational complexity in packet scheduling algorithms
We clarify, extend, and solve a long-standing open problem concerning the computational complexity for packet scheduling algorithms to achieve tight end-to-end delay bounds. We first focus on the difference between the time a packet finishes service in a scheduling algorithm and its virtual finish time under a GPS (General Processor Sharing) scheduler, called GPS-relative delay. We prove that, under a slightly restrictive but reasonable computational model, the lower bound computational complexity of any scheduling algorithm that guarantees O(1) GPS-relative delay bound is /spl Omega/(logn). We also discover that, surprisingly, the complexity lower bound remains the same even if the delay bound is relaxed to O(n/sup a/) for 0<a<1. This implies that the delay-complexity tradeoff curve is flat in the "interval" [O(1),O(n)). We later conditionally extend both complexity results (for O(1) or O(n/sup a/) delay) to a much stronger computational model, the linear decision tree. Finally, we show that the same complexity lower bounds are conditionally applicable to guaranteeing tight end-to-end delay bounds, if the delay bounds are provided through the Latency Rate (LR) framework.
905381
On the computational complexity of maintaining GPS clock in packet scheduling
Packet scheduling is an important mechanism to provide QoS guarantees in data networks. A scheduling algorithm generally consists of two functions: one estimates how the GPS (general processor sharing) clock progresses with respect to the real time; the other decides the order of serving packets based on an estimation of their GPS start/finish times. In this work, we answer important open questions concerning the computational complexity of performing the first function. We systematically study the complexity of computing the GPS virtual start/finish times of the packets, which has long been believed to be /spl Omega/(n) per packet but has never been either proved or refuted. We also answer several other related questions such as "whether the complexity can be lower if the only thing that needs to be computed is the relative order of the GPS finish times of the packets rather than their exact values?".
5230665
Finding global icebergs over distributed data sets
Finding icebergs–items whose frequency of occurrence is above a certain threshold–is an important problem with a wide range of applications. Most of the existing work focuses on iceberg queries at a single node. However, in many real-life applications, data sets are distributed across a large number of nodes. Two naïve approaches might be considered. In the first, each node ships its entire data set to a central server, and the central server uses single-node algorithms to find icebergs. But it may incur prohibitive communication overhead. In the second, each node submits local icebergs, and the central server combines local icebergs to find global icebergs. But it may fail because in many important applications, globally frequent items may not be frequent at any node. In this work, we propose two novel schemes that provide accurate and efficient solutions to this problem: a sampling-based scheme and a counting-sketch-based scheme. In particular, the latter scheme incurs a communication cost at least an order of magnitude smaller than the naïve scheme of shipping all data, yet is able to achieve very high accuracy. Through rigorous theoretical and experimental analysis we establish the statistical properties of our proposed algorithms, including their accuracy bounds.
42254231
A novel cache architecture to support layer-four packet classification at memory access speeds
Existing and emerging layer-4 switching technologies require packet classification to be performed on more than one header field, known as layer-4 lookup. Currently, the fastest general layer-8 lookup scheme delivers a throughput of 1 million lookups per second (MLPS), far off from 25/75 MLPS needed to support 50/150 Gbps layer-4 router. We propose the use of route caching to speed up layer-4 lookup, and design and implement a cache architecture for this purpose. We investigated the locality behavior of the Interent traffic (at layer-4) and propose a near-LRU algorithm that can best harness this behavior. In implementation, to best approximate fully-associative near-LRU using relatively inexpensive set-associative hardware, we invented a dynamic set-associative scheme that exploits the nice properties of N-universal hash functions. The cache architecture achieves a high and stable hit ratio above 90 percent and a fast throughput up to 75 MLPS at a reasonable cost.
9436128
Computing Optimal Beams in Two and Three Dimensions
The problem of computing an optimal beam among weighted regions (called the optimal beam problem) arises in several applied areas such as radiation therapy, stereotactic brain surgery, medical surgery, geological exploration, manufacturing, and environmental engineering. In this paper, we present computational geometry techniques that enable us to develop efficient algorithms for solving various optimal beam problems among weighted regions in two and three dimensional spaces. In particular, we consider two types of problems: the covering problems (seeking an optimal beam to contain a specified target region), and the piercing problems (seeking an optimal beam of a fixed shape to pierce the target region). We investigate several versions of these problems, with a variety of beam shapes and target region shapes in 2-D and 3-D. Our algorithms are based on interesting combinations of computational geometry techniques and optimization methods, and transform the optimal beam problems to solving a collection of instances of certain special non-linear optimization problems. Our approach makes use of interesting geometric observations, such as utilizing some new features of Minkowski sums.
2460253
Approximate processing of massive continuous quantile queries over high-speed data streams
Quantile computation has many applications including data mining and financial data analysis. It has been shown that an /spl epsi/-approximate summary can be maintained so that, given a quantile query (/spl phi/,/spl epsi/), the data item at rank /spl lceil//spl phi/N/spl rceil/ may be approximately obtained within the rank error precision /spl epsi/N over all N data items in a data stream or in a sliding window. However, scalable online processing of massive continuous quantile queries with different /spl phi/ and /spl epsi/ poses a new challenge because the summary is continuously updated with new arrivals of data items. In this paper, first we aim to dramatically reduce the number of distinct query results by grouping a set of different queries into a cluster so that they can be processed virtually as a single query while the precision requirements from users can be retained. Second, we aim to minimize the total query processing costs. Efficient algorithms are developed to minimize the total number of times for reprocessing clusters and to produce the minimum number of clusters, respectively. The techniques are extended to maintain near-optimal clustering when queries are registered and removed in an arbitrary fashion against whole data streams or sliding windows. In addition to theoretical analysis, our performance study indicates that the proposed techniques are indeed scalable with respect to the number of input queries as well as the number of items and the item arrival rate in a data stream.
33339608
Almost Optimal Solutions for Bin Coloring Problems
In this paper we study two interesting bin coloring problems: Minimum Bin Coloring Problem (MinBC) and Online Maximum Bin Coloring Problem (OMaxBC), motivated from several applications in networking. For the MinBC problem, we first show that it is NP-complete, and then present two near linear time approximation algorithms to achieve almost optimal solutions, i.e., no more than OPT+2 and OPT+1 respectively, where OPT is the optimal solution. For the OMaxBC problem, we first introduce a deterministic 2-competitive greedy algorithm, and then give lower bounds for any deterministic and randomized (against adaptive offline adversary) online algorithms. The lower bounds show that our deterministic algorithm achieves the best possible competitive ratio.
30458595
Finding an Optimal Path without Growing the Tree
In this paper, we study a class of optimal path problems with the following phenomenon: The space complexity of the algorithms for reporting the lengths of single-source optimal paths for these problems is asymptotically smaller than the space complexity of the "standard" treegrowing algorithms for finding actual optimal paths. We present a general and efficient algorithmic paradigm for finding an actual optimal path for such problems without having to grow a single-source optimal path tree. Our paradigm is based on the "marriage-before-conquer" strategy, the prune-and-search technique, and a data structure called clipped trees. The paradigm enables us to compute an actual path for a number of optimal path problems and dynamic programming problems in computational geometry, graph theory, and combinatorial optimization. Our algorithmic solutions improve the space bounds (in certain cases, the time bounds as well) of the previously best known algorithms, and settle some open problems. Our techniques are likely to be applicable to other problems.