J.L. Bentley, B.W. Weide, and A.C. Yao. Optimal expected-time algorithms for closestpoint problems. In Allerton Conference, Urbana, Illinois, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from the query point to the cell exceeds the distance to the closest point visited. The algorithm was analyzed for data points uniformly distributed on the vertices of the d dimensional hypercube by Rivest [13] and later for general uniformly distributed point sets by Bentley, Weide and Yao [6] and Cleary [8] Cleary s analysis showed that the q Figure 1: Bucketing Algorithm. number of points examined was independent of n, and that the fewest number of points were examined when there was one data point per cell. His analysis also applies to the k d tree algorithm and furnishes more ....

J. L. Bentley, B. W. Weide, and A. C. Yao. Optimal expected-time algorithms for closest point problems. ACM Transactions on Mathematical Software, 6(4):563--580, 1980.

....xa, xw, we have ]i] V i = 1, n,n w 2 2 Proof W 1 W (7i)2 = i)2 = 1 = 2 f = 7 72w i) 1 12il , i = 1, n From the above lemma, each DFT coefficent ranges from to , therefore the DFT feature space R an is a cube of diameter v. We can use a grid structure [4] to report near neighbors efficiently. We will use the first h, h 2n, dimensions of the DFT feature space for indexing. We superimpose an h dimensional orthogonal regular grid on the DFT fea ture space and partition the cube of diameter v into cells with the same size and shape. There are (2 ....

J. L. Bentley, B. W. Weide, and A. C. Yao. Optimal expected-time algorithms for closest point problems. ACM Transactions on Mathematical Software (TOMS), 6(4):563-580, 1980.

....in particular that F is not parallel to any of the coordinate axes. Let S denote a set of data points sampled from a closed convex, sampling region of F according to some probability distribution function. We assume that the distribution function satis es the following bounded density assumption [BWY80] There exist constants 0 c 1 c 2 , such that for any convex open subregion of the sampling region with k dimensional volume V , the probability that a given sampled point lies within this region is in the interval [c 1 V; c 2 V ] This is just a generalization of a uniform distribution but ....

J. L. Bentley, B. W. Weide, and A. C. Yao. Optimal expected-time algorithms for closest-point problems. ACM Trans. Math. Software, 6(4):563{ 580, 1980.

....space and partition the space into cells of diameter e, the correlation threshold. Each stream is mapped to a number of cells (exactly how many depends on the lag time ) in the feature space based on a subset of its DFT coefficients. They use proximity in this feature space to report correlations [7]. Any application of the above technique has one caveat: the constant of linearity increases exponen tially with the dimension of the space [7] A cell in kspace has 3 k 1 neighbor cells, leading to a search volume of (3 k 1)e . Furthermore, if we use the same grid structure to detect ....

....on the lag time ) in the feature space based on a subset of its DFT coefficients. They use proximity in this feature space to report correlations [7] Any application of the above technique has one caveat: the constant of linearity increases exponen tially with the dimension of the space [7]. A cell in kspace has 3 k 1 neighbor cells, leading to a search volume of (3 k 1)e . Furthermore, if we use the same grid structure to detect correlations with threshold larger than e (e.g. 2e, 3e, ne) we need to check distant cells as well. A range query in k space with radius e has ....

J.L. Bentley, B.W. Weide, and A.C. Yao. Optimal expected time algorithms for closest point problems. In ACM Trans. on Math. Software, volume 6, pages 563-580, 1980.

....and it is but one application of the well studied k nearest neighbours (kNN) problem in the field of computational geometry [17,25,59] To speed up photon mapping, the first hurdle is to be able to optimise this kNN step, and perhaps migrate the kNN solution onto hardware. Jensen uses the kd tree [8,9] data structure to find the kNN. However, solving the kNN problem using the kd tree requires a search that traverses the tree. Even if the tree is balanced and stored as a heap, traversal still requires random order memory access and some memory space to store a stack. More importantly, a ....

....used to solve the kNN problem [21,66] An advantage of the kd tree is that if the tree is balanced it can be stored as a heap in a single array, avoiding the use of pointers and memory for a separate index structure. While it has been shown that kd trees have optimal expected time complexity [9], in the worst case finding the k nearest neighbours may require an exhaustive search of the entire data structure. The tree is searched by recursive descent. This requires a stack the same size as the depth of the tree. During the recursion, a choice is made of which subtree to search next based ....

J. L. Bentley, B. W. Weide, and A. C. Chow. Optimal Expected-Time Algorithms for Closest Point Problems. ACM TOMS, 6(4), December 1980. 3, 2.2

....of k often grows with the size of the database. The second approach has also been extensively analyzed and many techniques have been proposed to reduce the cost of the exhaustive search of the prototypes set to nd the k nearest neighbors to a test point. Most of them make use of kd trees [4] [2], 8] or similar data structures. A kd tree is a binary tree where each node represents a region in a k dimensional space. Each internal node also contains a hyper plane (a linear subspace of dimension k 1) dividing the region into two disjoint sub regions, each inherited by one of its sons. Most ....

J.L. Bentley, B.W. Weide, and A.C. Yao. Optimal expected time algorithms for closest point problems. ACM Trans. on Math. Software, 6:563-580, 1980.

....the manifold, the low expansion property will hold (as it does for low dimensional Euclidean spaces) and our near neighbor structure could be applied. Related Work. As mentioned previously, most research on nearest neighbor search has focused on the case of vector spaces and or Euclidean metrics [1, 2]. There has been a growing interest in general metrics, however. In a recent survey [4] Chavez et al. give an overview on the data structures developed for these applications. The most frequent approach is by pivoting [10, 11] i.e. the space is partitioned into two halves by picking a random ....

J. Bentley, B. Weide, and A. Yao. Optimal expected-time algorithms for closest point problems. ACM Transactions of Mathematical Software, 6(4):563--580, 1980.

....of k often grows with the size of the database. The second approach has also been extensively analyzed and many techniques have been proposed to reduce the cost of the exhaustive search of the prototypes set to find the k nearest neighbors to a test point. Most of them make use of kd trees [4] [2], 8] or similar data structures. A kd tree is a binary tree where each node represents a region in a k dimensional space. Each internal node also contains a hyper plane (a linear subspace of dimension k 1) dividing the region into two disjoint sub regions, each inherited by one of its sons. Most ....

J.L. Bentley, B.W. Weide, and A.C. Yao. Optimal expected time algorithms for closest point problems. ACM Trans. on Math. Software, 6:563--580, 1980.

....DFT of a nor malized sequence Xl, x2, x , we have i 1, n,n w 2 Proof w 1 w i i 1 n n 2 lil i 1, n 2 From the above lemma, each DFT coefficent ranges from to , therefore the DFT feature space R 2n is a cube of diameter . We can use a grid structure [4] to report near neighbors efficiently. We will use the first h, h 2n, dimensions of the DFT feature space for indexing. We superimpose an h dimensional orthogonal regular grid on the DFT fea ture space and partition the cube of diameter into cells with the same size and shape. There are (2 ....

J. L. Bentley, B. W. Weide, and A. C. Yao. Optimal expected-time algorithms for closest point problems. A CM Transactions on Mathematical Software (TOMS), 6(4):563-580, 1980.

....bounds are only obtained for rather complicated theoretical algorithms we expect that simple data structures will work in practical situations. For example, if transmitter positions are uniformly distributed, all operations run in constant expected time using a simple grid based data structure [2]. We now explain how the 2 hop algorithm leads to an efficient algorithm for three and four hops. an optimal path with three or four hops between nodes s and t can be found in time O(n log n) using space O(n) Proof. We first explain the method for a three hop path (s; u; v; t) We build a ....

J. Bentley, B. W. Weide, and A. C. Yao. Optimal expected-time algorithms for closest point problems. ACM Transactions on Mathematical Software, 6(4):563-- 580, December 1980.

....limitations, we will now present only an outline of our solutions for Problems 5 8. We will also restrict our presentation to the case d=2. The complete proofs and the generalization to d=O(1) will be presented in the final version of this paper. Our approach is to use a grid method similar to [4]. The non trivial difference is that the results in [4] are expected sequential time complexities, whereas our parallel time complexity results are with high probability. Consider the following rectangular partitioning of S into p subsets R 1 , R p : Partition [0,1] 2 by p 1 vertical lines ....

....our solutions for Problems 5 8. We will also restrict our presentation to the case d=2. The complete proofs and the generalization to d=O(1) will be presented in the final version of this paper. Our approach is to use a grid method similar to [4] The non trivial difference is that the results in [4] are expected sequential time complexities, whereas our parallel time complexity results are with high probability. Consider the following rectangular partitioning of S into p subsets R 1 , R p : Partition [0,1] 2 by p 1 vertical lines into p vertical slabs K 1 , K p such that each slab ....

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J. L. Bentley, Weide, and A. Yao, "Optimal expected time algorithms for closest point problems," ACM Transactions on Mathematical Software, Vol. 6, No. 4, 1980, pp. 563-580.

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J.L. Bentley, B.W. Weide, and A.C. Yao. Optimal expected-time algorithms for closestpoint problems. In Allerton Conference, Urbana, Illinois, 1978.

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J. L. Bentley, B. W. Weide, and A. C. Yao. Optimal expected-time algorithms for closest-point problems. ACM Trans. Math. Softw., 6:563--580, 1980.

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J. L. Bentley, Weide, and A. Yao, "Optimal expected time algorithms for closest point problems," ACM Transactions on Mathematical Software, Vol. 6, No. 4, 1980, pp. 563-580.

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J. L. Bentley, Weide, and A. Yao, "Optimal expected time algorithms for closest point problems," ACM Transactions on Mathematical Software, Vol. 6, No. 4, 1980, pp. 563-580.

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J. L. Bentley, B. W. Weide, and A. C. Yao. Optimal expected time algorithms for closest point problems. ACM Transactions on Mathematical Software, 6:563--580, 1980.

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J. L. Bentley, Weide, and A. Yao, "Optimal expected time algorithms for closest point problems," ACM Transactions on Mathematical Software, Vol. 6, No. 4, 1980, pp. 563-580.

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J. L. Bentle, B. Weide, and A. C. Yao. Optimal Expected-time Algorithms for Closest-point Problems. ACM Trans. Math. Software, 6:563---579, 1980.

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Bentley, J.L., and Weide, B.W.: `Optimal expected-time algorithms for closest point problems', ACM Trans. Math. Soft., 1980, 6, (4), pp. 563--580

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J.L. Bentley, B.W. Weide, and A.C. Yao. Optimal expected time algorithms for closest point problems. ACM Trans. on Math. Software, 6:563580, 1980.

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J. L. Bentley, Weide, and A. Yao, "Optimal expected time algorithms for closest point problems," ACM Transactions on Mathematical Software, Vol. 6, No. 4, 1980, pp. 563-580.

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J.L. Bentley, B.W. Weide, and A.C. Yao. Optimal expected time algorithms for closest point problems. In ACM Trans. on Math. Software, volume 6, pages 563--580, 1980.

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J. L. Bentley, B. W. Weide, and A. C. Yao. Optimal expected-time algorithms for closest point problems. ACM Transactions on Mathematical Software, 6(4):563-- 580, 1980.

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J. L. Bentley, B. W. Weide, and A. C. Yao. Optimal expected-time algorithms for closest point problems. ACM Transactions on Mathematical Software, 6(4):563-580, 1980.

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J. L. Bentley, B. W. Weide, and A. C. Chow. Optimal Expected-Time Algorithms for Closest Point Problems. ACM TOMS, 6(4), December 1980. 1, 2

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