Travelling Santa Problem: Optimization of a Million-Households Tour Within One Hour

Finding the shortest tour visiting all given points at least ones belongs to the most famous optimization problems until today [travelling salesman problem (TSP)]. Optimal solutions exist formany problems up to several ten thousand points. Themajor difficulty in solving larger problems is the required computational complexity. This shifts the research from finding the optimum with no time limitation to approaches that find good but sub-optimal solutions in pre-defined limited time. This paper proposes a new approach for two-dimensional symmetric problems with more than a million coordinates that is able to create good initial tours within few minutes. It is based on a hierarchical clustering strategy and supports parallel processing. In addition, a method is proposed that can correct unfavorable paths with moderate computational complexity. The new approach is superior to state-of-the-artmethods when applied to TSP instances with non-uniformly distributed coordinates

Travelling Santa Problem: Optimization of a Million-Households Tour Within One Hour

Finding the shortest tour visiting all given points at least ones belongs to the most famous optimization problems until today [travelling salesman problem (TSP)]. Optimal solutions exist formany problems up to several ten thousand points. Themajor difficulty in solving larger problems is the required computational complexity. This shifts the research from finding the optimum with no time limitation to approaches that find good but sub-optimal solutions in pre-defined limited time. This paper proposes a new approach for two-dimensional symmetric problems with more than a million coordinates that is able to create good initial tours within few minutes. It is based on a hierarchical clustering strategy and supports parallel processing. In addition, a method is proposed that can correct unfavorable paths with moderate computational complexity. The new approach is superior to state-of-the-artmethods when applied to TSP instances with non-uniformly distributed coordinates

Redesigning the Wheel for Systematic Travelling Salesmen

This paper investigates the systematic and complete usage of k-opt permutations with k=2…6 in application to local optimization of symmetric two-dimensional instances up to 107 points. The proposed method utilizes several techniques for accelerating the processing, such that good tours can be achieved in limited time: candidates selection based on Delaunay triangulation, precomputation of a sparse distance matrix, two-level data structure, and parallel processing based on multithreading. The proposed approach finds good tours (excess of 0.72–8.68% over best-known tour) in a single run within 30 min for instances with more than 105 points and specifically 3.37% for the largest examined tour containing 107 points. The new method proves to be competitive with a state-of-the-art approach based on the Lin–Kernigham–Helsgaun method (LKH) when applied to clustered instances

Travelling Santa Problem: Optimization of a Million-Households Tour Within One Hour

Finding the shortest tour visiting all given points at least ones belongs to the most famous optimization problems until today [travelling salesman problem (TSP)]. Optimal solutions exist formany problems up to several ten thousand points. Themajor difficulty in solving larger problems is the required computational complexity. This shifts the research from finding the optimum with no time limitation to approaches that find good but sub-optimal solutions in pre-defined limited time. This paper proposes a new approach for two-dimensional symmetric problems with more than a million coordinates that is able to create good initial tours within few minutes. It is based on a hierarchical clustering strategy and supports parallel processing. In addition, a method is proposed that can correct unfavorable paths with moderate computational complexity. The new approach is superior to state-of-the-artmethods when applied to TSP instances with non-uniformly distributed coordinates

Run-Length Encodings — Corrected Probabilities for Golomb-Codes

In 1966, Secret Agent 00111 [was] back at the Casino again, playing a game of chance, while the fate of mankind hangs in the balance. Each game consists of sequence of favorable events (probability p1), terminated by the first occurrence of an unfavorable event (Probability p0 = 1 − p1)... the game is roulette, and the unfavorable event is the occurrence of 0... The problem perplexing the [Secret] Service is how to encode the vicissitudes of a wheel... Finally a junior clerk who has been reading up on Information Theory, suggests encoding the run length between successive unfavorable events. In general, the probability of a run length of r is p r 1 · p0, for n = 0, 1, 2, 3,..., which is the familiar geometric distribution... [Gol66]. Now in 2002, another young research assistant has computed the probabilities of the geometric distribution p r 1 · p0 using a computer program. He found quite many rounding errors (caused assumedly by using limited calculation tools as logarithm tables and sliding rule) and probably one typo in the run-length dictionaries of Solomon W. Golomb. Table 1 shows the corrected values for pr = p r 1 · p0

Image Data Compression With Pdf-Adaptive Reconstruction of Wavelet Coefficients

The present contribution proposes a new remarkably efficient image compression algorithm for graylevel images based on dyadic wavelet transformation. In order to achieve perfect reconstruction, orthogonal decomposition is applied. Scalar quantization of wavelet coefficients is combined with run-length coding. Code word assignment is performed by semi-adaptive Huffman coding (determined by validity tables). To improve the reconstruction quality of images a new technique of pdf-adaptive reconstruction of wavelet coefficients (PAR) is used