A Streaming Algorithm for the Undirected Longest Path Problem

Recently, there has been a growing focus in solving approximate pattern matching problems in the streaming model. Of particular interest are the pattern matching with k-mismatches (KMM) problem and the pattern matching with w-wildcards (PMWC) problem. Motivated by reductions from these problems in the streaming model to the dictionary matching problem, this paper focuses on designing algorithms for the dictionary matching problem in the multi-stream model where there are several independent streams of data (as opposed to just one in the streaming model), and the memory complexity of an algorithm is expressed using two quantities: (1) a read-only shared memory storage area which is shared among all the streams, and (2) local stream memory that each stream stores separately. In the dictionary matching problem in the multi-stream model the goal is to preprocess a dictionary D={P_1,P_2,...,P_d} of d=|D| patterns (strings with maximum length m over alphabet Sigma) into a data structure stored in shared memory, so that given multiple independent streaming texts (where characters arrive one at a time) the algorithm reports occurrences of patterns from D in each one of the texts as soon as they appear. We design two efficient algorithms for the dictionary matching problem in the multi-stream model. The first algorithm works when all the patterns in D have the same length m and costs O(d log m) words in shared memory, O(log m log d) words in stream memory, and O(log m) time per character. The second algorithm works for general D, but the time cost per character becomes O(log m+log d log log d). We also demonstrate the usefulness of our first algorithm in solving both the KMM problem and PMWC problem in the streaming model. In particular, we obtain the first almost optimal (up to poly-log factors) algorithm for the PMWC problem in the streaming model. We also design a new algorithm for the KMM problem in the streaming model that, up to poly-log factors, has the same bounds as the most recent results that use different techniques. Moreover, for most inputs, our algorithm for KMM is significantly faster on average.

Estimating the size of the maximum matching is a canonical problem in graph analysis, and one that has attracted extensive study over a range of different computational models. We present improved streaming algorithms for approximating the size of maximum matching with sparse (bounded arboricity) graphs. 
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\n* (Insert-Only Streams) We present a one-pass algorithm that takes O(alpha log n) space and approximates the size of the maximum matching in graphs with arboricity alpha within a factor of O(alpha). This improves significantly upon the state-of-the-art tilde{O}(alpha n^{2/3})-space streaming algorithms, and is the first poly-logarithmic space algorithm for this problem. 
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\n* (Dynamic Streams) Given a dynamic graph stream (i.e., inserts and deletes) of edges of an underlying alpha-bounded arboricity graph, we present an one-pass algorithm that uses space tilde{O}(alpha^{10/3}n^{2/3}) and returns an O(alpha)-estimator for the size of the maximum matching on the condition that the number edge deletions in the stream is bounded by O(alpha n). For this class of inputs, our algorithm improves the state-of-the-art tilde{O}(\\alpha n^{4/5})-space algorithms, where the \\tilde{O}(.) notation hides logarithmic in n dependencies.
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\nIn contrast to prior work, our results take more advantage of the streaming access to the input and characterize the matching size based on the ordering of the edges in the stream in addition to the degree distributions and structural properties of the sparse graphs.

Bidirectional heuristic search has the potential to decrease search time in combinatorial search problems amenable to backward search. To date, bidirectional search has been limited to minimization or shortest path problems. This paper extends the notion of bidirectional heuristic search to (constrained) longest path problems, which turns out to be non-trivial due to the path necessarily being part of the state and the inapplicability of standard bidirectional heuristic search techniques such as meet-in-the-middle (MM) and BAE*. We present a basic bidirectional heuristic search for longest simple path (LSP) in undirected graphs, and prove its correctness. We then suggest several refinements and optimizations, as well as a generalization to other types of longest path problems Coil-in-a-box (CIB). Empirical evaluation shows that, as with many forms of bidirectional search, sometimes unidirectional search wins, but for a sizable chunk of problem instance types, bidirectional search performs better by expanding fewer nodes and achieves a shorter runtime despite the increased overhead per expansion.

In this chapter we will consider the hamiltonian path and cycle problems for digraphs as well as some related problems such as the longest path and cycle problems and the minimum path factor problem. We describe and prove a number of results in the area as well as formulate several open questions.

A graph is called Hamiltonian connected if it contains a Hamiltonian path between any two distinct vertices. In the past, we proved the Hamiltonian path and cycle problems for general supergrid graphs to be NP-complete. However, they are still open for solid supergrid graphs. In this paper, first we will verify the Hamiltonian cycle property of C-shaped supergrid graphs, which are a special case of solid supergrid graphs. Next, we show that C-shaped supergrid graphs are Hamiltonian connected except in a few conditions. For these excluding conditions of Hamiltonian connectivity, we compute their longest paths. Then, we design a linear-time algorithm to solve the longest path problem in these graphs. The Hamiltonian connectivity of C-shaped supergrid graphs can be applied to compute the optimal stitching trace of computer embroidery machines, and construct the minimum printing trace of 3D printers with a C-like component being printed.

We consider computing a longest palindrome in the streaming model, where the symbols arrive one-by-one and we do not have random access to the input. While computing the answer exactly using sublinear space is not possible in such a setting, one can still hope for a good approximation guarantee. Our contribution is twofold. First, we provide lower bounds on the space requirements for randomized approximation algorithms processing inputs of length n. We rule out Las Vegas algorithms, as they cannot achieve sublinear space complexity. For Monte Carlo algorithms, we prove a lower bounds of Omega(M log min {|Sigma|, M}) bits of memory; here M=n/E for approximating the answer with additive error E, and M= log n / log (1 + epsilon) for approximating the answer with multiplicative error (1 + epsilon). Second, we design three real-time algorithms for this problem. Our Monte Carlo approximation algorithms for both additive and multiplicative versions of the problem use O(M) words of memory. Thus the obtained lower bounds are asymptotically tight up to a logarithmic factor. The third algorithm is deterministic and finds a longest palindrome exactly if it is short. This algorithm can be run in parallel with a Monte Carlo algorithm to obtain better results in practice. Overall, both the time and space complexity of finding a longest palindrome in a stream are essentially settled.

In this paper, we study streaming and online algorithms in the context of randomness in the input. For several problems, a random order of the input sequence - as opposed to the worst-case order - appears to be a necessary evil in order to prove satisfying guarantees. However, algorithmic techniques that work under this assumption tend to be vulnerable to even small changes in the distribution. For this reason, we propose a new adversarial injections model, in which the input is ordered randomly, but an adversary may inject misleading elements at arbitrary positions. We believe that studying algorithms under this much weaker assumption can lead to new insights and, in particular, more robust algorithms. We investigate two classical combinatorial-optimization problems in this model: Maximum matching and cardinality constrained monotone submodular function maximization. Our main technical contribution is a novel streaming algorithm for the latter that computes a 0.55-approximation. While the algorithm itself is clean and simple, an involved analysis shows that it emulates a subdivision of the input stream which can be used to greatly limit the power of the adversary.

Live Road Surveillance video streaming is a new type of safety data that can be transmitted in Vehicular Ad Hoc Networks (VANETs). Video streaming is a high-bandwidth-demanded service and causes congestion if there are many vehicles that stream video in the wireless network at the same time. Clustering is an efficient technique to relieve network congestion because it can reduce the number of vehicles to transmit data. Based on the effective-vision of the camera in vehicles, an Effective-Vision-Area-based Clustering (EVAC) algorithm for LRS service is proposed in this paper. EVAC can construct clusters such that a driver X can join a suitable cluster to get the LRS service from the appropriate vehicle ahead of X to have additional vision. In this paper, we compare the proposed EVAC algorithm for LRS service with other LRS service provider selection schemes that do not adopt the clustering mechanism in term of different performance metrics to show the effectiveness of the proposed work.

The H.264 video coding standard has been widely applied to multimedia data transmission. To prevent the contents of video from being recognized by someone unauthorized, providing sufficient security to H.264 has become more and more important. In this thesis, an improved H.264/AVC video encryption method is proposed. As compared with the Shahid’s method, more data are encrypted. And this makes the outcome more desirable. In the proposed scheme, selective encryption (SE) is performed in the motion vector difference (MVD), delta quantization parameter (delta QP), and the context-based adaptive variable length coding (CAVLC) modules. Owing that the encrypted stream should be decoded correctly before decryption, a format-compliant method is presented. Furthermore, in order to keep the bitrate unchanged, keeping the same stream length is another important characteristic of our proposed scheme. Since there is no escalation in bitrate, the proposed encryption algorithm is suitable for real-time multimedia streaming over heterogeneous networks. The encrypting key is generated by using the advanced encryption standard (AES) algorithm, and every frame has its own unique key. Four benchmark video sequences with different combinations of motion, texture and objects were used for experimental evaluation of the proposed scheme. The structural similarity (SSIM) index, a reference value to image quality, of the proposed scheme is 0.202 which is lower than 0.233 of the Shahid’s scheme. The key security is shown reliable under the test of only 1 bit change. The extra computation needed by our scheme is less than 0.5%, so it is feasible for real-time transmission. The above performance on security and sensitivity demonstrates that our proposed scheme is a good candidate for practical applications.

In this paper, we address the problem of QoS (Quality of Service) provisioning for multi-user video streaming over multiple heterogeneous wireless networks based on the distributed, cross-layer design framework. By jointly considering the rate allocation and the Joint Source-Channel Coding (JSCC), our proposal aims at maximizing the QoS provisioning under the given resource constraint. At first, we develop and evaluate a framework for optimal video rate allocation over multiple networks based on the observed Available Bit Rate (ABR) and the Round Trip Time (RTT) over each access network, as well as the video rate-distortion characteristics. The rate allocation is formulated as a convex optimization problem that minimizes the sum of all video streams expected distortion. Then, we propose an analytical JSCC scheme for error-resilient scalable encoded video, and integrate the JSCC with the specific rate allocation algorithm to improve the constructed video quality by optimally applying the appropriate channel coding rate given the constraints imposed by the transmission rate and the prevailing channel conditions. Objective and subjective simulation results are provided which demonstrate the effectiveness of our proposed joint scheme.

The longest path problem asks for a path with the largest number of vertices in a given graph. The first polynomial time algorithm (with running time O(n4)) has been recently developed for interval graphs. Even though interval and circular-arc graphs look superficially similar, they differ substantially, as circular-arc graphs are not perfect. In this paper, we prove that for every path P of a circular-arc graph G, we can appropriately “cut” the circle, such that the obtained (not induced) interval subgraph G′ of G admits a path P′ on the same vertices as P. This non-trivial result is of independent interest, as it suggests a generic reduction of a number of path problems on circular-arc graphs to the case of interval graphs with a multiplicative linear time overhead of O(n). As an application of this reduction, we present the first polynomial algorithm for the longest path problem on circular-arc graphs, which turns out to have the same running time O(n4) with the one on interval graphs, as we manage to get rid of the linear overhead of the reduction. This algorithm computes in the same time an n-approximation of the number of different vertex sets that provide a longest path; in the case where G is an interval graph, we compute the exact number. Moreover, our algorithm can be directly extended with the same running time to the case where every vertex has an arbitrary positive weight.

Peer-2-Peer (P2P) technologies are rapidly advancing and have already proven successful in the form of various bandwidth demanding applications such as file-transfer and video streaming. The major advantage of the P2P technologies is its built-in scalability feature, relative to traditional client- server architecture. Therefore, keeping in view the large number of users involved in live video-streaming, P2P technology can be considered as a valuable tool aiming to broadcast live-video contents to a large number of users over the internet. Although, there is lot of research going on in analyzing the performance of P2P video streaming in heterogeneous networking scenarios, but to the best of our knowledge no work is done in analyzing the impact of Forward Error Correction (FEC) on the performance of P2P live-video streaming. In this paper we analyze the performance and characteristics of P2P live-video streaming on end-user quality of experience while employing FEC on the coded video packets. Additionally, in-order to carry out the performance analysis with reference to non-error protected benchmarker system; we utilized a realistic realtime French distributed Grid facility known as Grid'5000. More specifically, the performance of the system is evaluated while considering various realistic P2P video streaming constraints, such as limited playout delay, limited upload bandwidth and video rates. We concluded that the FEC will result in noticeable performance improvement in a scenario where each peer is connected with limited number of other peers that have the required contents for download. In such a situation the portion of the stream corrupted due to channel errors may not be recovered with the help of other accessible peers. However, if each peer has more connected peers with various alternative options to download the corrupted contents, then the employment of FEC may not have very noticeable advantage on the performance of the P2P video streaming system.

Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs. In particular, we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time O(1.7159^n), of split graphs in time O(1.3803^n), and of AT-free, strongly chordal, and distance-hereditary graphs in time O^*(3^{n/3}), where n is the number of vertices of the input graph. Our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes.

We provide a simple algorithm for constructing Hamiltonian graph cycles (visiting every vertex exactly once) on a set of arbitrarily large finite subgraphs of aperiodic two-dimensional Ammann-Beenker (AB) tilings. Using this result, and the discrete scale symmetry of AB tilings, we find exact solutions to a range of other problems which lie in the complexity class NP-complete for general graphs. These include the equal-weight traveling salesperson problem, providing, for example, the most efficient route a scanning tunneling microscope tip could take to image the atoms of physical quasicrystals with AB symmetries; the longest path problem, whose solution demonstrates that collections of flexible molecules of any length can adsorb onto AB quasicrystal surfaces at density one, with possible applications to catalysis; and the three-coloring problem, giving ground states for the q-state Potts model (q≥3) of magnetic interactions defined on the planar dual to AB, which may provide useful models for protein folding. Published by the American Physical Society 2024

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