Computing a Hamiltonian Path of Minimum Euclidean Length Inside a Simple Polygon
TL;DR: This paper reformulates the above shortest-path problems in terms of a dynamic programming scheme involving falling staircase anti-Monge weight-arrays, and provides an O(nlogn) time and Θ(n) space algorithm to solve the following one-dimensional dynamic programming recurrence.
Abstract: Given an n-vertex convex polygon, we show that a shortest Hamiltonian path visiting all vertices without imposing any restriction on the starting and ending vertices of the path can be found in O(nlogn) time and Θ(n) space. The time complexity increases to O(nlog2
n) for computing this path inside an n-vertex simple polygon. The previous best algorithms for these problems are quadratic in time and space. For our purposes, we reformulate the above shortest-path problems in terms of a dynamic programming scheme involving falling staircase anti-Monge weight-arrays, and, in addition, we provide an O(nlogn) time and Θ(n) space algorithm to solve the following one-dimensional dynamic programming recurrence $$E[i] = \min _{1\le j\le k}\min _{k\le i} \{V[k-1] + b(i,j) + c(j,k)\},\quad i=1, \dots,n,$$
where V[0] is known, V[k], for k=1,…,n, can be computed from E[k] in constant time, and B={b(i,j)} and C={c(j,k)} are known falling staircase anti-Monge weight-arrays of size n×n.
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TL;DR: A Two-Level solution approach for solving the Clustered Capacitated Vehicle Routing Problem and a cluster generation to define problem instances is presented, indicating the high performance of the approach over a wide range of real-size problem instances.
Abstract: We address the Clustered Capacitated Vehicle Routing Problem.We propose a Two-Level solution approach for solving the Clustered Capacitated Vehicle Routing Problem.Our heuristic algorithm allows to find high-quality solutions in short times. The Clustered Capacitated Vehicle Routing Problem (CCVRP) aims to determine the routes of a fleet of capacitated vehicles to fulfill the service demands of customers organized into clusters. The optimization criterion is to minimize the total travel cost of the routes. The main constraint is that all the customers belonging to the same cluster have to be served by the same vehicle consecutively. In this work, a mathematical formulation for the CCVRP is developed. Since the problem is NP -hard, an approximate Two-Level solution approach is proposed. It is based upon breaking the problem down into two routing problems. The first one aims to determine routes targeted at visiting the clusters. A metaheuristic algorithm is considered to address this sub-problem. Furthermore, the second one is aimed at finding a visiting order of the customers belonging to each cluster. Exact and approximate methods are proposed to tackle this sub-problem. The computational results indicate the high performance of our approach over a wide range of real-size problem instances. In this regard, a cluster generation to define problem instances is presented.
56 citations
01 Jan 1998
TL;DR: This paper provides an O(nmax(?,β)) time algorithm for finding a minimal cost prefix-free code in which the encoding alphabet consists of unequal cost (length) letters, with lengths ? and β.
Abstract: In this paper we discuss a variation of the classical Huffman coding problem: finding optimal prefix-free codes for unequal letter costs. Our problem consists of finding a minimal cost prefix-free code in which the encoding alphabet consists of unequal cost (length) letters, with lengths α and β. The most efficient algorithm known previously required O(n2+max(αβ)) time to construct such a minimal-cost set of n codewords. In this paper we provide an O(nmax(αβ)) time algorithm. Our improvement comes from the use of a more sophisticated modeling of the problem combined with the observation that the problem possesses a "Monge property" and that the SMAWK algorithm on monotone matrices can therefore be applied.
22 citations
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01 Jan 2000TL;DR: The Davenport-Schinzel sequences and their geometric applications, as well as randomizedalgorithms in computaional geometry, are described.
Abstract: reface. List of contrinbutors. 1. Davenport-Schinzel sequences and their geometric applications (P.K. Agarwal and M. Sharir). 2. Arrangements and their applications (P.K. Agarwal and M. Sharir). 3. Discrete geometric shapes: Matching, interpolation, and approximation (H. Alt and L.J. Guibas). 4. Deterministic parallel computational geometry (M.J. Attalah and D.Z. Chen). 5. Voronoi diagrams (F. Aurenhammer and R. Klein). 6. Mesh generation (M. Bern and P. Plassmann). 7. Applications of computational geometry to geographic information systems (L. de Floriani, P. Magillo and E. Puppo). 8. Making geometry visible: An introduction to the animation of geometric algorithms (A. Hausner and D.P. Dobkin). 9. Spanning trees and spanners (D. Eppstein). 10. Geometric data structures (M.T. Goodrich and K. Ramaiyer). 11. Polygon decomposition (J.M. Keil). 12. Link distance problems (A. Maheshwari, J.-R. Sack and H. N. Djidjev). 13. Derandomization in computational geometry (J. Matousek). 14. Robustness and precision issues in geometric computation (S. Schirra). 15. Geometric shortest paths and network optimization (J.S.B. Mitchell). 16. Randomizedalgorithms in computaional geometry (K. Mulmuley).
688 citations
TL;DR: The Θ(m) bound on finding the maxima of wide totally monotone matrices is used to speed up several geometric algorithms by a factor of logn.
Abstract: LetA be a matrix with real entries and letj(i) be the index of the leftmost column containing the maximum value in rowi ofA.A is said to bemonotone ifi
1 >i
2 implies thatj(i
1) ≥J(i
2).A istotally monotone if all of its submatrices are monotone. We show that finding the maximum entry in each row of an arbitraryn xm monotone matrix requires Θ(m logn) time, whereas if the matrix is totally monotone the time is Θ(m) whenm≥n and is Θ(m(1 + log(n/m))) whenm
506 citations
01 Jan 2000
407 citations
"Computing a Hamiltonian Path of Min..." refers background in this paper
...In this regard, we refer the reader to Guibas and Hershberger [19] and Hershberger [21] for full details (cf. also Mitchell [24] for an excellent and comprehensive review concerning shortest paths in computational geometry)....
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...also Mitchell [24] for an excellent and comprehensive review concerning shortest paths in computational geometry)....
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TL;DR: This paper shows that for many of these concave cost economic lot size problems, the dynamic programming formulation of the problem gives rise to a special kind of array, called a Monge array, and shows how the structure of Monge arrays can be exploited to obtain significantly faster algorithms for these economic lots size problems.
Abstract: Many problems in inventory control, production planning, and capacity planning can be formulated in terms of a simple economic lot size model proposed independently by A. S. Manne (1958) and by H. M. Wagner and T. M. Whitin (1958). The Manne-Wagner-Whitin model and its variants have been studied widely in the operations research and management science communities, and a large number of algorithms have been proposed for solving various problems expressed in terms of this model, most of which assume concave costs and rely on dynamic programming. In this paper, we show that for many of these concave cost economic lot size problems, the dynamic programming formulation of the problem gives rise to a special kind of array, called a Monge array. We then show how the structure of Monge arrays can be exploited to obtain significantly faster algorithms for these economic lot size problems. We focus on uncapacitated problems, i.e., problems without bounds on production, inventory, or backlogging; capacitated problem...
349 citations
"Computing a Hamiltonian Path of Min..." refers background or methods in this paper
...tions of scheme (9) when W satisfies monotonicity properties can be found in Galil and Giancarlo [14], Aggarwal and Park [2], García and Tejel [16], García et al....
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...Perhaps the best-known is the one developed by Larmore and Schieber [23], which we call the LARSCH algorithm (see, for instance, Galil and Park [15] for another algorithm and also Bar-Noy et al. [7] for other types of on-line and off-line versions)....
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...where E[2] = d(p1,p2), E[k] is computed from E[k] and L(k) according to (4), for k = 3, ....
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...It is also interesting to mention that the notions of two-dimensional totally monotone arrays, as well as Monge arrays, were generalized by Aggarwal and Park [1] to multidimensional arrays (cf. also Park [26])....
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...Applications of scheme (9) when W satisfies monotonicity properties can be found in Galil and Giancarlo [14], Aggarwal and Park [2], García and Tejel [16], García et al. [17], among others, together with the aforementioned papers [3, 15, 22, 23]....
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TL;DR: This paper presents a survey on Monge matrices and related Monge properties and their role in combinatorial optimization, and deals with the following three main topics: fundamental combinatorsial properties of Monge structures, applications of MonGE properties to optimization problems and recognition ofMonge properties.
Abstract: An m × n matrix C is called Monge matrix if c ij + c rs ⩽ c is + c rj for all 1 ⩽ i r ⩽ m , 1 ⩽ j s ⩽ n . In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) fundamental combinatorial properties of Monge structures, (ii) applications of Monge properties to optimization problems and (iii) recognition of Monge properties.
321 citations