## Data Structures for Traveling Salesmen (1995)

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Citations: | 29 - 2 self |

### BibTeX

@MISC{Fredman95datastructures,

author = {M. L. Fredman and D. S. Johnson and L. A. Mcgeoch and G. Ostheimer},

title = {Data Structures for Traveling Salesmen},

year = {1995}

}

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### Abstract

The choice of data structure for tour representation plays a critical role in the efficiency of local improvement heuristics for the Traveling Salesman Problem. The tour data structure must permit queries about the relative order of cities in the current tour and must allow sections of the tour to be reversed. The traditional array-based representation of a tour permits the relative order of cities to be determined in small constant time, but requires worst-case W(N) time (where N is the number of cities) to implement a reversal, which renders it impractical for large instances. This paper considers alternative tour data structures, examining them from both a theoretical and experimental point of view. The first alternative we consider is a data structure based on splay trees, where all queries and updates take amortized time O(logN). We show that this is close to the best possible, because in the cell probe model of computation any data structure must take worst-case amortized time W(...

### Citations

2453 |
The Design and Analysis of Computer Algorithms
- Aho, Hopcroft, et al.
- 1974
(Show Context)
Citation Context ... given tree T, Φ(T) = Σ rank(v) . v∈T It is easy to see that if T is an N-vertex balanced binary tree, then Φ(T) = Θ(N) under this definition. (The argument is essentially the same as the one used in =-=[1]-=- to show that a heap can be constructed in linear time.) Furthermore, since no vertex can have rank exceeding log N and no surgery on the tree can affect the rank of any vertex below thes- 11 - point ... |

609 | Tabu Search
- Glover
- 1990
(Show Context)
Citation Context ...inations, they give rise to the famous Lin-Kernighan algorithm [23] and provide the basic engines for most applications of simulated annealing [5,17,19], genetic algorithms [4,25,26], and tabu search =-=[11]-=- to the TSP. Two of the authors of the current paper have been involved in an extended study [3,14] of the 2-Opt, 3-Opt, and Lin-Kernighan algorithms [22,23] and how they can be adapted to very large ... |

605 | Data Structures and Network Algorithms - Tarjan - 1983 |

481 | Tabu search: part I - Glover - 1989 |

372 |
Self-adjusting binary search trees
- Sleator, Tarjan
- 1985
(Show Context)
Citation Context ...ce three new Tour representations, all designed to improve on the Array representation, and we analyze their worst-case behavior. Asymptotically, the best of the three is the one based on splay trees =-=[28]-=-, which has an amortized worst-case running time of O( logN) per operation. The constants of proportionality are high, however, leaving a surprisingly large window of opportunity for the other two rep... |

321 |
Thermodynamical Approach to the Traveling Salesman Problem: An Efficient Simulation Algorithm
- Cerny
- 1985
(Show Context)
Citation Context ...nown 2-Opt and 3-Opt algorithms. In more complicated combinations, they give rise to the famous Lin-Kernighan algorithm [23] and provide the basic engines for most applications of simulated annealing =-=[5,17,19]-=-, genetic algorithms [4,25,26], and tabu search [11] to the TSP. Two of the authors of the current paper have been involved in an extended study [3,14] of the 2-Opt, 3-Opt, and Lin-Kernighan algorithm... |

314 |
The Traveling Salesman Problem
- Lawler, Lenstra, et al.
- 1985
(Show Context)
Citation Context ...alesman would make in visiting the cities in the order specified by the permutation, returning at the end to the starting city. The TSP is one of the most widely known NP-hard problems. Lawler et al. =-=[21]-=- give an excellent introduction to the broad range of work on this problem. In this paper, we concentrate on the symmetric TSP, where d(c i ,c j ) = d(c j ,c i ) for 1 ≤ i, j ≤ N. Because the TSP is N... |

275 |
Worst-case analysis of a new heuristic for the traveling salesman problem
- Christofides
- 1976
(Show Context)
Citation Context ...our length. (For comparison purposes, note that the famousse f a b c d a b - 3 - c d a b FIGURE 1.1. A 2-change. d c e f a b (a) (b) d c e f FIGURE 1.2. Two possible 3-changes. Christofides algorithm =-=[6]-=- only gets within 9-10% of the Held-Karp bound on such instances and is significantly slower [3].) 1.1. The Tour Datatype To obtain the performance reported in Table 1.1, we had to deal with a key imp... |

262 |
TSPLIB − A traveling salesman problem library
- Reinelt
- 1991
(Show Context)
Citation Context ...ngth of the shorter segment seems to grow roughly as N .7 [3], and similar behavior has been observed on instances from the TSPLIB database of Euclidean instances derived from real-world applications =-=[27]-=-. Despite these savings, the costs of tour manipulation grow to dominate overall running time as N increases. Table 1.2 illustrates this for our implementation of LinKernighan (which we shall treat mo... |

251 |
Computer solutions of the traveling salesman problem
- Lin
(Show Context)
Citation Context ...genetic algorithms [4,25,26], and tabu search [11] to the TSP. Two of the authors of the current paper have been involved in an extended study [3,14] of the 2-Opt, 3-Opt, and Lin-Kernighan algorithms =-=[22,23]-=- and how they can be adapted to very large instances. (Many applications give rise to instances with between 10,000 and 100,000 cities, and VLSI applications with as many as 1.2 million cities have be... |

151 |
Local optimization and the traveling salesman problem
- Johnson
- 1990
(Show Context)
Citation Context ... for most applications of simulated annealing [5,17,19], genetic algorithms [4,25,26], and tabu search [11] to the TSP. Two of the authors of the current paper have been involved in an extended study =-=[3,14]-=- of the 2-Opt, 3-Opt, and Lin-Kernighan algorithms [22,23] and how they can be adapted to very large instances. (Many applications give rise to instances with between 10,000 and 100,000 cities, and VL... |

137 | Should tables be sorted
- Yao
- 1981
(Show Context)
Citation Context ...t any Tour representation must take amortized time Ω( logN /log logN) per operation in the worst case. We prove our amortized lower bounds in the cell probe model of computation, introduced by A. Yao =-=[30]-=- (see also [8]). In this model there is a single parameter b, the number of bits in single word of shared memory, and the cost of a computation is essentially the number of words accessed. This is a v... |

132 |
Fast algorithms for geometric traveling salesman problems
- Bentley
- 1992
(Show Context)
Citation Context ... next and previous pointers in each internal parent node along the path, (2) making appropriate changes to the pointers in the first and last parent nodes in the path and in their non-path neighbors, =-=(3)-=- flipping the reversal bit of each affected parent node, (4) appropriately adjusting the next and previous pointers for the cities at the ends of the two paths, and (5) updating the sequence numbers o... |

130 |
An effective heuristic algorithm for the traveling-salesman problem
- Lin, Kernighan
- 1973
(Show Context)
Citation Context ...d adding edges as shown. By themselves, these operations give rise to the well-known 2-Opt and 3-Opt algorithms. In more complicated combinations, they give rise to the famous Lin-Kernighan algorithm =-=[23]-=- and provide the basic engines for most applications of simulated annealing [5,17,19], genetic algorithms [4,25,26], and tabu search [11] to the TSP. Two of the authors of the current paper have been ... |

128 | The cell probe complexity of dynamic data structures - Fredman, Saks - 1989 |

118 |
The traveling-salesman problem and minimum spanning trees
- Held, Karp
(Show Context)
Citation Context ... Graphics IRIS™ 4D/250 computer. (MIPS is a trademark of MIPS, Inc., IRIS is a trademark of Silicon Graphics, Inc.) A tour’s quality is measured by the percentage it exceeds the Held-Karp lower bound =-=[12,13,15]-=- on optimal tour length. (For comparison purposes, note that the famousse f a b c d a b - 3 - c d a b FIGURE 1.1. A 2-change. d c e f a b (a) (b) d c e f FIGURE 1.2. Two possible 3-changes. Christofid... |

94 | Large-step Markov chains for the traveling salesman problem
- Martin, Otto, et al.
- 1991
(Show Context)
Citation Context ...s. In more complicated combinations, they give rise to the famous Lin-Kernighan algorithm [23] and provide the basic engines for most applications of simulated annealing [5,17,19], genetic algorithms =-=[4,25,26]-=-, and tabu search [11] to the TSP. Two of the authors of the current paper have been involved in an extended study [3,14] of the 2-Opt, 3-Opt, and Lin-Kernighan algorithms [22,23] and how they can be ... |

83 |
Optimization by simulated annealing: Quantitative studies
- Kirkpatrick
- 1984
(Show Context)
Citation Context ...nown 2-Opt and 3-Opt algorithms. In more complicated combinations, they give rise to the famous Lin-Kernighan algorithm [23] and provide the basic engines for most applications of simulated annealing =-=[5,17,19]-=-, genetic algorithms [4,25,26], and tabu search [11] to the TSP. Two of the authors of the current paper have been involved in an extended study [3,14] of the 2-Opt, 3-Opt, and Lin-Kernighan algorithm... |

51 | Asymptotic experimental analysis for the Held-Karp traveling salesman bound
- Johnson, McGeoch, et al.
- 1996
(Show Context)
Citation Context ... Graphics IRIS™ 4D/250 computer. (MIPS is a trademark of MIPS, Inc., IRIS is a trademark of Silicon Graphics, Inc.) A tour’s quality is measured by the percentage it exceeds the Held-Karp lower bound =-=[12,13,15]-=- on optimal tour length. (For comparison purposes, note that the famousse f a b c d a b - 3 - c d a b FIGURE 1.1. A 2-change. d c e f a b (a) (b) d c e f FIGURE 1.2. Two possible 3-changes. Christofid... |

28 |
Optimization strategies gleaned from biological evolution
- Brady
- 1985
(Show Context)
Citation Context ...s. In more complicated combinations, they give rise to the famous Lin-Kernighan algorithm [23] and provide the basic engines for most applications of simulated annealing [5,17,19], genetic algorithms =-=[4,25,26]-=-, and tabu search [11] to the TSP. Two of the authors of the current paper have been involved in an extended study [3,14] of the 2-Opt, 3-Opt, and Lin-Kernighan algorithms [22,23] and how they can be ... |

17 | Near-optimal solutions to very large traveling salesman problems - JOHNSON, MCGEOCH, et al. |

15 |
Blasting through the information theoretic barrier with fusion trees
- Fredman, Willard
- 1990
(Show Context)
Citation Context ...e word of shared memory, and the cost of a computation is essentially the number of words accessed. This is a very general model, encompassing even such baroque data structures as the fusion trees of =-=[9]-=-. Our proof relies on a rather technical generalization of Theorem 3′ from [8], and we shall state this generalization here without proof, calling it Theorem A. The proof of Theorem A, itself a straig... |

14 |
A data structure useful for finding Hamiltonian cycles, Theoretical Computer Science 71
- Chrobak, Szymacha, et al.
- 1990
(Show Context)
Citation Context ...theoretical gap of a factor of log log N, which we discuss in our concluding Section 5, along with other directions for furthers- 7 - research. We also describe two proposed Tour representations from =-=[7,24]-=- that attain O( logN) time per operation in a strict worst-case rather than an amortized worst-case sense (and explain why they are unlikely to be competitive with splay trees in practice). 2. Three A... |

10 |
personal communication
- Applegate, Johnson
- 1993
(Show Context)
Citation Context ...,F,J,C,G,A,E,B,H,D). A B E D Hs- 8 - reference. (The idea of using splay trees is a natural one, and has been independently proposed by several groups of researchers, for instance, Applegate and Cook =-=[2]-=-.) Splay trees were introduced by Sleator and Tarjan [28]. The key idea is that every time a vertex is accessed, it is brought to the root (splayed) by a sequence of rotations (local alterations of th... |

10 |
personal communication
- RUBENSTEIN, LEIGHTON
(Show Context)
Citation Context ...y Tree representation will be O(log N) per operation. 2.2. The Two-Level Tree Representation The idea of using a tree with just two levels to represent the Tour datatype was suggested by Tom Leighton =-=[20]-=-, who observed that for N ≤ 10 6 , it might well be that c√⎺ N⎺with a small c is better than dlogN where d represents the overhead factor for splay trees. (It also occurred independently to Chrobak, S... |

5 |
Worst-case analysis of algorithms for travelling salesman problems
- Frieze
- 1979
(Show Context)
Citation Context ...urs. In practice, the best such algorithms have all been based on the principle of local optimization: One obtains a starting tour using some tour-construction heuristic (such as the Greedy algorithm =-=[10]-=-) and then repeatedly attempts to improve it using local modifications. The most commonly used such modifications are 2- and 3-changes, as illustrated in Figures 1.1 and 1.2. Here for ease of explanat... |

5 |
Quick Updates for p-opt TSP Heuristics
- Margot
- 1992
(Show Context)
Citation Context ...theoretical gap of a factor of log log N, which we discuss in our concluding Section 5, along with other directions for furthers- 7 - research. We also describe two proposed Tour representations from =-=[7,24]-=- that attain O( logN) time per operation in a strict worst-case rather than an amortized worst-case sense (and explain why they are unlikely to be competitive with splay trees in practice). 2. Three A... |

5 | Tarjan, "Self-adjusting binary search trees - Sleator, E - 1985 |

2 |
Applications of combinatorial optimization,’’ talk at the
- KORTE
- 1988
(Show Context)
Citation Context ...they can be adapted to very large instances. (Many applications give rise to instances with between 10,000 and 100,000 cities, and VLSI applications with as many as 1.2 million cities have been cited =-=[18]-=-.) Table 1.1 shows the current level of performance we have been able to obtain with these algorithms for instances consisting of points uniformly distributed in the unit square under the Euclidean me... |

2 | Applications of combinatorial optimization", in: Mathematical Programming, recent developments and applications, edited by M. Iri and - Korte - 1989 |

1 |
An efficient simulated annealing schedule: imple- 47 - mentation and evaluation," manuscript
- LAM, DELOSME
- 1988
(Show Context)
Citation Context ...nown 2-Opt and 3-Opt algorithms. In more complicated combinations, they give rise to the famous Lin-Kernighan algorithm [23] and provide the basic engines for most applications of simulated annealing =-=[5,17,19]-=-, genetic algorithms [4,25,26], and tabu search [11] to the TSP. Two of the authors of the current paper have been involved in an extended study [3,14] of the 2-Opt, 3-Opt, and Lin-Kernighan algorithm... |

1 | Evolution algorithms in combinatorial optimization - HLENBEIN, KRA - 1988 |