We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Artificial neural networks: fundamentals, computing, design, and application

Gene families are growing rapidly, but standard methods for inferring phylogenies do not scale to alignments with over 10,000 sequences. We present FastTree, a method for constructing large phylogenies and for estimating their reliability. Instead of storing a distance matrix, FastTree stores sequence profiles of internal nodes in the tree. FastTree uses these profiles to implement neighbor-joining and uses heuristics to quickly identify candidate joins. FastTree then uses nearest-neighbor interchanges to reduce the length of the tree. For an alignment with N sequences, L sites, and a different characters, a distance matrix requires O(N^2) space and O(N^2 L) time, but FastTree requires just O( NLa + N sqrt(N) ) memory and O( N sqrt(N) log(N) L a ) time. To estimate the tree's reliability, FastTree uses local bootstrapping, which gives another 100-fold speedup over a distance matrix. For example, FastTree computed a tree and support values for 158,022 distinct 16S ribosomal RNAs in 17 hours and 2.4 gigabytes of memory. Just computing pairwise Jukes-Cantor distances and storing them, without inferring a tree or bootstrapping, would require 17 hours and 50 gigabytes of memory. In simulations, FastTree was slightly more accurate than neighbor joining, BIONJ, or FastME; on genuine alignments, FastTree's topologies had higher likelihoods. FastTree is available at http://microbesonline.org/fasttree.

/pdf/fast-tree-computing-large-minimum-evolution-trees-with-4grq0gwfpy.pdf

Fast Tree: Computing Large Minimum-Evolution Trees with Profiles instead of a Distance Matrix

In this paper we introduce the minimum-order approach to frequency assignment and present a theory which relates this approach to the traditional one. This new approach is potentially more desirable than the traditional one. We model assignment problems as both frequency-distance constrained and frequency constrained optimization problems. The frequency constrained approach should be avoided if distance separation is employed to mitigate interference. A restricted class of graphs, called disk graphs, plays a central role in frequency-distance constrained problems. We introduce two generalizations of chromatic number and show that many frequency assignment problems are equivalent to generalized graph coloring problems. Using these equivalences and recent results concerning the complexity of graph coloring, we classify many frequency assignment problems according to the "execution time efficiency" of algorithms that may be devised for their solution. We discuss applications to important real world problems and identify areas for further work.

Frequency assignment: Theory and applications

Objective measurement of test quality is one of the key issues in software testing. It has been a major research focus for the last two decades. Many test criteria have been proposed and studied for this purpose. Various kinds of rationales have been presented in support of one criterion or another. We survey the research work in this area. The notion of adequacy criteria is examined together with its role in software dynamic testing. A review of criteria classification is followed by a summary of the methods for comparison and assessment of criteria.

/pdf/software-unit-test-coverage-and-adequacy-43qk91tobg.pdf

Software unit test coverage and adequacy

Modern distributed systems involving large number of nonstationary clients (mobile hosts, MH) connected via unreliable low-bandwidth communication channels are very prone to frequent disconnections. This disconnection may occur because of different reasons: The clients may voluntarily switch off (to save battery power), or a client may be involuntarily disconnected due to its own movement in a mobile network (hand-off, wireless link failures, etc.). A mobile computing environment is characterized by slow wireless links and relatively underprivileged hosts with limited battery powers. Still, when data at the server changes, the client hosts must be made aware of this fact in order for them to invalidate their cache, otherwise the host would continue to answer queries with the cached values returning incorrect data. The nature of the physical medium coupled with the fact that disconnections from the network are very frequent in mobile computing environments demand a cache invalidation strategy with minimum possible overheads. In this paper, we present a new cache maintenance scheme, called AS. The objective of the proposed scheme is to minimize the overhead for the MHs to validate their cache upon reconnection, to allow stateless servers, and to minimize the bandwidth requirement. The general approach is (1) to use asynchronous invalidation messages and (2) to buffer invalidation messages from servers at the MH's Home Location Cache (HLC) while the MH is disconnected from the network and redeliver these invalidation messages to the MH when it gets reconnected to the network. Use of asynchronous invalidation messages minimizes access latency, buffering of invalidation messages minimizes the overhead of validating MH's cache after each disconnection and use of HLC off-loads the overhead of maintaining state of MH's cache from the servers. The MH can be disconnected from the server either voluntarily or involuntarily. We capture the effects of both by using a single parameter: The percentage of time a mobile host is disconnected from the network. We demonstrate the efficacy of our scheme through simulation and performance modeling. In particular, we show that the average data access latency and the number of uplink requests by a MH decrease by using the proposed strategy at the cost of using buffer space at the HLC. We provide analytical comparison between our proposed scheme and the existing scheme for cache management in a mobile environment. Extensive experimental results are provided to compare the schemes in terms of performance metrics like latency, number of uplink requests, etc., under both a high and a low rate of change of data at servers for various values of the parameters. A mathematical model for the scheme is developed which matches closely with the simulation results.

https://people.cs.vt.edu/%7Eirchen/6204/paper/KAH01-cache-proxy.pdf

A strategy to manage cache consistency in a disconnected distributed environment

In the self-stabilizing algorithmic paradigm for distributed computation each node has only a local view of the system, yet in a finite amount of time, the system converges to a global state satisfying some desired property. In this paper we present polynomial time self-stabilizing algorithms for finding a dominating bipartition, a maximal independent set, and a minimal dominating set in any graph.

https://www.cs.clemson.edu/stabiliz/Papers/cam-2003.pdf

Self-stabilizing algorithms for minimal dominating sets and maximal independent sets

In this paper we propose a new protocol for reliable multicast in a multi-hop mobile radio network. The protocol is reliable, i.e., it guarantees message delivery to all multicast nodes even when the topology of the network changes during multicasting. The proposed protocol uses a core based shared tree. The multicast tree may get fragmented due to node movements. A notion of forwarding region is introduced which is used to glue together fragments of a multicast trees. The gluing process involves flooding of the forwarding region of the nodes which witness topology change due to node movements. Delivery of multicast messages to mobile nodes is expedited through (i) pushing of the message by witness nodes in their forwarding regions and (ii) pulling of messages by a mobile node during the (re)joining process. Hence, the protocol conserves network bandwidth by using a combination of the push-pull approach and limiting network flooding to only minimal parts of the network that is affected by topology change. The protocol adapts to both topology change and the distribution of the multicast group members to minimize the use of system resources.

An adaptive protocol for reliable multicast in mobile multi-hop radio networks

We propose two distributed algorithms to maintain, respectively, a maximal matching and a maximal independent set in a given ad hoc network; our algorithms are fault tolerant (reliable) in the sense that the algorithms can detect occasional link failures and/or new link creations in the network (due to mobility of the hosts) and can readjust the global predicates. We provide time complexity analysis of the algorithms in terms of the number of rounds needed for the algorithm to stabilize after a topology change, where a round is defined as a period of time in which each node in the system receives beacon messages from all its neighbors. In any ad hoc network, the participating nodes periodically transmit beacon messages for message transmission as well as to maintain the knowledge of the local topology at the node; as a result, the nodes get the information about their neighbor nodes synchronously (at specific time intervals). Thus, the paradigm to analyze the complexity of the self-stabilizing algorithms in the context of ad hoc networks is very different from the traditional concept of an adversary daemon used in proving the convergence and correctness of self-stabilizing distributed algorithms in general.

/pdf/self-stabilizing-protocols-for-maximal-matching-and-maximal-fnwmi4ge36.pdf

Self-stabilizing protocols for maximal matching and maximal independent sets for ad hoc networks

https://www.cs.clemson.edu/stabiliz/Papers/coloring-ipl-2003.pdf

Pradip K. Srimani

Papers

A strategy to manage cache consistency in a disconnected distributed environment

Self-stabilizing algorithms for minimal dominating sets and maximal independent sets

An adaptive protocol for reliable multicast in mobile multi-hop radio networks

Self-stabilizing protocols for maximal matching and maximal independent sets for ad hoc networks

Linear time self-stabilizing colorings