The Traveling Salesman Problem (TSP) is among the most famous NP-hard optimization problems. The special case of TSP in bounded-dimensional Euclidean spaces has been a particular focus of research: The celebrated resu...
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(纸本)9780769551357
The Traveling Salesman Problem (TSP) is among the most famous NP-hard optimization problems. The special case of TSP in bounded-dimensional Euclidean spaces has been a particular focus of research: The celebrated results of Arora [Aro98] and Mitchell [Mit99] - along with subsequent improvements of Rao and Smith [RS98] - demonstrated a polynomial time approximation scheme for this problem, ultimately achieving a runtime of O-d,O-e(n log n). In this paper, we present a linear time approximation scheme for Euclidean TSP, with runtime O-d,O-e(n). This improvement resolves a 15 year old conjecture of Rao and Smith, and matches for Euclidean spaces the bound known for a broad class of planar graphs [Kle08].
Let A be a sequence of n real numbers, L-1 and L-2 be two integers such that L-1 <= L-2, and let R-1 and R-2 be two real numbers such that R-1 <= R-2. An interval of A is feasible if its length is between L-1 an...
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Let A be a sequence of n real numbers, L-1 and L-2 be two integers such that L-1 <= L-2, and let R-1 and R-2 be two real numbers such that R-1 <= R-2. An interval of A is feasible if its length is between L-1 and L-2, and its average is between R-1 and R-2. In this paper, we study the following problems: finding all feasible intervals of A, counting all feasible intervals of A, finding a maximum cardinality set of nonoverlapping feasible intervals of A, locating a longest feasible interval of A, and locating a shortest feasible interval of A. The problems are motivated from the problem of locating CpG islands in biomolecular sequences. In this paper, we first show that all the problems have an Omega(n log n)-time lower bound in the comparison model. Then, we use geometric approaches to design optimal algorithms for the problems. All the presented algorithms run in an online manner and use O(n) space.
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