S. Anily and A. Federgruen. Structured partitioning problems. Operations Research, 13, 130--149, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... partitioning problems have been studied extensively in application areas including databases (e.g. histograms, grid files, index compression) parallel computing (e.g. load balancing) computer graphics (e.g. spatial data structures) and video compression (e.g. block matching) see [17, 22, 2, 4, 9] for some discussion. Following are a few examples that explicitly study partitionings. In databases, several authors [1, 11] have used partitionings to construct histograms in two or more dimensions. In this case, the metric used is often either MAX SUM for V Optimal histograms [30] or ....

S. Anily and A. Federgruen. Structured partitioning problems. Operations Research, 13, 130--149, 1991.

....will write [r j ; r j 1 ] for the sequence foe r j ; oe r j 1 Gamma1 g. From now on we assume that r 0 = 0 and r p = n. The rst reference to the MinMax problem that we are aware of is by Bokhari [2] who presented an O(n p) algorithm using a bottleneck path algorithm. Anily and Federgruen [1] and Hansen and Lih [6] independently presented the same dynamic programming algorithm with time complexity O(n p) The dynamic programming algorithm is as follows: Let g(i; k) be the cost of the most expensive interval in an optimal partition of oe i ; oe i 1 ; oe n Gamma1 into k ....

S. Anily and A. Federgruen, Structured partitioning problems, Operations Research, 13 (1991), pp. 130149.

....Partitioning problems have been studied extensively in various application areas including databases, parallel computing (e.g. load balancing) computational geometry (e.g. clustering) video compression (e.g. block matching) etc. Some related papers from a variety of application areas include [17, 23, 25, 24, 1, 3, 10]. Here we review a selection of related work most relevant to us. Hardness Results. Hardness results exist only for a simple metric function, namely, MAX SUM ID [18] proved it to be NP hard for arbitrary partitions, and [11, 6] proved it to be NP hard for p Theta p partition. Our NP hardness ....

S. Anily and A. Federgruen. Structured partitioning problems. Operations Research, 13, 130--149, 1991.

....optimal algorithms running in log and 3 times, respectively. Here, denotes the sum of the weights in the workload array , i.e. 1 . Iqbal and Bokhari [12] and Nicol and O Hallaron [18] later proposed an log algorithm, and finally Nicol [19] proposed an log 2 algorithm. Anily and Federgruen [1] initiated the DP approach with an 2 algorithm. Hansen and Lih [9] independently proposed an 2 algorithm. Choi and Narahari [6] Manne and Srevik [16] and Olstad and Manne [20] introduced asymptotically faster , log and algorithms, respectively. Theoretical work on optimal 2D partitioning ....

Anily, S., and Federgruen, A. Structured partitioning problems. Operations Research, 13 (1991), 130--149.

....array partitioning problems. Here we briefly describe the application context for each; further details of modeling will be discussed in the journal version. One dimensional case under F . This problem was abstracted for load balancing in pipelined, parallel environments in [B88] and studied in [OM95, AF91, HL92, MS95, M93, CN91, HNC92, N91] etc. Two dimensional case under F . This problem arises in balanced data distribution as implemented in the Superb environment [ZBG86] and HPF2 [HPF] High Performance Fortran) See [M93, CM 95] for more applications to particlein cell computations and sparse matrix computations. Two ....

....of our interest. 1D p partition under F . This problem has been extensively researched. We summarize the previous work and our results in the table below, providing all citations where identical bounds were obtained independently. Reference Bound Bokhari [B88] O(n 3 p) Anily Federgruen [AF91] O(n 2 p) Hansen Liu [HL92] O(n 2 p) Manne Sorevik [MS95] O(np log p) Choi Narahari [CN91] O(np) Olstad Manne [OM95] O(np) Nicol [N91] O(n p 2 log 2 n) Charikar, Chekuri Motwani [CCM96] O(n p 2 log 2 n) Han, Narahari Choi [HNC92] O(n p 1 ffl ) ffl 1 This ....

S. Anily and A. Federgruen. Structured partitioning problems. Operations Research, 13, 130--149, 1991.

....of size n into p intervals such that the maximum number of nonzero elements in any interval is minimized. This gives our 1D p partition problem under F . Further details on the application of the 1D p partition problem under F for load balancing in pipelined, parallel environments can be found in [B88, OM95, AF91, HL92, MS95, M93, MH95]. Two dimensional case under F . For data stored in two dimensional arrays, several high performance computing languages allow the user to specify a partitioning and distribution of data onto a logical set of processors. An example of such a scheme is what is known as the generalized block ....

....in this paper. 1.3 Results We state our results for each of the three problems of our interest. 1D p partition under F . This problem has been extensively researched. We summarize the previous work and our results in the table below. Reference Bound Bokhari [B88] O(n 3 p) Anily Federgren [AF91] O(n 2 p) Hansen Liu [HL92] O(n 2 p) Manne Sorevik [MS95] O(np log p) Olstad Manne [OM95] O(np) This paper O(n p 2 log 2 n) This paper O(n log n) It is not difficult to design an O(n 2 p) time algorithm for this problem using dynamic programming. In [OM95] the authors used ....

S. Anily and A. Federgruen. Structured partitioning problems. Operations Research, 13, 130--149, 1991.

....the interval foe i ; oe i 1 ; oe j g by [i; j] The cost of a partition is the cost of the most expensive interval. The first reference to the MinMax problem that we are aware of is by Bokhari [4] who presented an O(n 3 p) algorithm using a bottleneck path algorithm. Anily and Federgruen [2] and Hansen and Lih [9] independently presented the same dynamic programming algorithm with time complexity O(n 2 p) Manne and S revik [10] then presented an O(p(n Gamma p) log p) algorithm based on iteratively improving a given partition. They also described a bisection method for finding an ....

....obtained by a straightforward calculation. The following boundary conditions apply to g: g(i; 1) f(i; n Gamma 1) 3) g(i; n Gamma i) max ij n f(j; j) 4) Note that for n Gamma i k p we have g(i; k) g(i; n Gamma i) The following recursion, first presented by Anily and Federgruen [2], shows how g(i; k) can be computed for 2 k n Gamma i: g(i; k) min ijn Gammak fmaxff(i; j) g(j 1; k Gamma 1)gg (5) This formula suggests that if one has access to each value of g(j 1; k Gamma 1) i j n Gamma k, then g(i; k) can be computed by looking up n Gamma k Gamma i 1 ....

S. Anily and A. Federgruen, Structured partitioning problems, Operations Research, 13 (1991), pp. 130--149.

....then: MinMax Find a partition R such that the associated cost max p Gamma1 i=0 ff(r i ; r i 1 Gamma 1)g is minimum over all partitions of foe 0 ; oe n Gamma1 g. Bokhari [3] presents the MinMax problem and gives an O(n 3 p) algorithm using a bottleneck path algorithm. Anily and Federgruen [2] and Hansen and Lih [8] independently presented the same dynamic programming algorithm with time complexity O(n 2 p) Manne and S revik [11] then presented an O(p(n Gamma p) log p) algorithm based on iteratively improving a given partition. They also described a bisection method based on a ....

.... each time f(i; j) has exceeded g(0; p) The following boundary conditions apply to g: g(i; 1) f(i; n Gamma 1) 3) g(i; n Gamma i) max ij n f(j; j) 4) Note that for n Gamma i k p we have g(i; k) g(i; n Gamma i) The following recursion, first presented by Anily and Federgruen [2], shows how g(i; k) can be computed for 2 k n Gamma i: g(i; k) min ijn Gammak maxff(i; j) g(j 1; k Gamma 1)g (5) This formula suggests that if one has access to each value of g(j 1; k Gamma 1) i j n Gamma k, then g(i; k) can be computed by looking up n Gamma k Gamma i 1 ....

S. Anily and A. Federgruen, Structured partitioning problems, Operations Research, 13 (1991), pp. 130--149.

....write [r j ; r j 1 ] for the sequence foe r j ; oe r j 1 Gamma1 g. From now on we assume that r 0 = 0 and r p = n. The rst reference to the MinMax problem that we are aware of is by Bokhari [2] who presented an O(n 3 p) algorithm using a bottleneck path algorithm. Anily and Federgruen [1] and Hansen and Lih [6] independently presented the same dynamic programming algorithm with time complexity O(n 2 p) The dynamic programming algorithm is as follows: Let g(i; k) be the cost of the most expensive interval in an optimal partition of oe i ; oe i 1 ; oe n Gamma1 into k ....

S. Anily and A. Federgruen, Structured partitioning problems, Operations Research, 13 (1991), pp. 130149.

....algorithm is based on finding a minimum path on a layered graph. Nicol and O Hallaron [24] reduced the complexity to O(M 2 K) The algorithm paradigms used in the following works can be classified as dynamic programming (DP) iterative refinement , and probe approaches. Anily and Federgruen [1] initiated the DP approach with an O(M 2 K) time algorithm. Hansen and Lih [11] independently proposed an O(M 2 K) time algorithm. Choi and Narahari [6] and Olstad and Manne [27] introduced asymptotically faster O(MK) time, and O( M Gamma K)K) time DP based algorithms, respectively. ....

S. Anily, and A. Federgruen, "Structured Partitioning Problems," Operations Research, Vol. 13, No. 1, 1991, pp.130--149.

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S. Anily and A. Federgruen. Structured partitioning problems. Operations Research, 13, 130--149, 1991.

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