Adolphson, D., and Hu, T. (1973): "Optimal Linear Ordering". In: SIAM J. Appl. Math. 25 (3), 403--423.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of articial tasks representing the schedule of 1 tasks and m tasks can be combined into an out tree with an articial 0 processing time task as a root. Thus, it seems that our problem can be reduced to 1jout Gamma treej c j . Problem 1jout Gamma treej c j is solvable in O(n log n) time [1] or cf. e.g. 2] However, the similarity to 1jout Gamma treej is supercial. Let us examine an example. Example 3. fix 1 = fP 1 g; fix 2 =fP 2 g; t 1 = t 2 =3; fix 3 =fP 1 ; P 2 g; t 3 =2. In the schedule for 1 tasks T 1 and T 2 are executed in parallel. Hence, the chain representing this ....

Adolphson, D. and Hu, T.C., Optimal linear ordering, SIAM Journal of Applied Mathematics, 1973, 25, 403-423.

....all the linear orderings of the vertices or the edges of a graph (for a survey, see [21] One of the most known problems of this type is the problem to compute the cutwidth of a graph. It is also known as the MINIMUM CUT LINEAR ARRANGEMENT problem and has several applications such as VLSI design [2, 3, 35, 33], network relia bility [30] automatic graph drawing [44, 38] and information retrieval [15] Cutwidth has been extensively examined [17, 24, 25, 31, 35, 37, 49] and it appears to be closely related with other graph parameters like pathwidth, linear width, bandwidth, and modi fied bandwidth ....

....All, i] A[i 4 2, JAIl. ii) If the sequence contains two elements ai and aj such that j i 2 and Vi k j ai ak aj or Vi j ai ak aj, then set A A(1,i) A(j, A[ We define the compression r(A) of a sequence A 5, as the unique minimum length ele ment of B [ B A . For example, [5,5,6,7,7, 7,4,4, 3,5,4,6, 8, 2, 9,3,4,6,7, 2, 7,5,4,4,6, 4]) 5,7,3,8,2,9,2,7,4] We call a sequence A typical if A ,5 and q (A) A. The following results have been proved in [12] Lemmata 3.3 and 3.5 respectively) LEMMA 2.4. If A 5 and max A k, then r(A) contains at most 2k 1 elements. LEMMA 2.5. The number of different typical sequences ....

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D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM J. Appl. Math., 25(3):403i23, 1973.

....to one another and (b) reducing visual clutter associated with long connecting lines. While these diagram presentation issues provide motivation, this paper focuses only on the LAP. Harper [12] provides a solution to the LAP for the class of n cubes. Solutions are also available for trees [1, 3, 17]. For general graphs, however, the LAP is known to be NP hard [7] There is a need for heuristic algorithms to address this problem in polynomial time. An eigenvalue based algorithm is available for finding linear arrangements of VLSI components (e.g. logic gates) to reduce the total length of ....

D.ADOLPHSONANDT.C.HU, Optimal linear ordering, SIAM J. Appl. Math., 25 (1973), pp. 403-423.

....storage time product minimization problem [91] MinLA has also received some alternative names, as Optimal Linear Ordering, Edge Sum problem or Minimum 1 sum. The Cutwidth problem was first used as a theoretical model for the number of channels in an optimal layout of a circuit in the seventies [3, 72]. In general, the cutwidth of a graph times the order of the graph gives a measure of the area needed to represent the graph in a VLSI layout when nodes are laid out in a row [68] More recent applications of this problem include network reliability [59] automatic graph drawing [79] information ....

....in finding recursively minimal cuts with minimal capacity among all cuts that separate the graph in two components of equal size. The EdgeBis we have presented use this approach. A di#erent approach to solve the placement phase is to use the MinLA in order to minimize the total wire length [3, 51]. Layout problems in graph drawing. Perhaps the most important goal in graph drawing is to produce aestethic representations of graphs. Reducing the number of crossing edges is a way to improve their readability and comprehension. A bipartite drawing (or 2 layer drawing) is a representation of a ....

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D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM Journal on Applied Mathematics, 25(3):403--423, Nov. 1973.

....all the linear orderings of the vertices or the edges of a graph (for a survey, see [20] One of the most known problems of this type is the problem to compute the cutwidth of a graph. It is also known as the Minimum Cut Linear Arrangement problem and has several applications such as VLSI design [2, 3, 33, 31], network reliability [28] automatic graph drawing [42, 36] and information retrieval [14] Cutwidth has been extensively examined [16, 22, 23, 29, 33, 35, 49] and it appears to be closely related with other graph parameters like pathwidth, linear width, bandwidth, and modi ed bandwidth [16, 17, ....

....of typical sequences. We say that A B if 8 1 i r A i B i . For any integer t we set A t = A 1 t; A jAj t] and max(A) max 1 i jAj fmaxA i g. Finally, for any sequence of typical sequences A we set (A) A(1) A(jAj) As an example, 5; 2; 8; 1] 4; 9; 3] [3]; 3; 9; 2; 5; 3] 5; 2; 8; 1; 4; 9; 3; 3; 3; 9; 2; 5; 3] 5; 2; 8; 1; 9; 2; 5; 3] 2.4. Interleavings of sequences Let two equal size sequences A; B of S where A = a 1 ; a r ] B = b 1 ; b r ] We de ne A B = a 1 b 1 ; a r b r ] and we say that A B i ....

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D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM J. Appl. Math., 25(3):403-423, 1973.

....G n,p model with p = c n for some constant c. Our results establish that, in fact, any algorithm that returns a feasible solution will produce such an approximation for graphs with good expansion properties. 1 Introduction Linear arrangement problems play an important role in Computer Science [27, 1, 8]. A linear layout (or linear arrangement or vertex ordering) of a graph G with n nodes is a one to one mapping of the vertices of G to the set 1, n . A layout # on G = V, E) determines in a unique way a nested sequence of vertex subsets containing those vertices placed up to the i th ....

.... problem in which the cut excludes edges touching the last vertex is known as the minimum modified cut arrangement and is also NP complete for planar graphs with maximum degree 3 [26] The minimum linear arrangement problem (also known as the minimum edge sum [19] or the optimal linear ordering [1]) seeks a layout that minimizes the total edge length. This problem is also NP complete [16] For trees the problem is in P [31] and in NC [9] It can be approximated within a O(log 2 n) factor using the approximate max flow min cut theorem [23] A better approximation factor O(log n log log ....

D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM J. on Applied Mathematics, 25(3):403--423, November 1973.

....consists in placing the modules on a board; the routing problem consists in wiring together the terminals on di#erent modules that should be connected. Several approaches to solve the placement phase use the Minimum Linear Arrangement problem (MinLA) in order to minimize the total wire length [16, 2]. MinLA is also connected with graph drawing: A bipartite drawing (or 2 layer drawing) is a graph representation where the nodes of a bipartite graph are placed in two parallel lines and the edges are drawn with straight lines between them. The bipartite crossing number of a bipartite graph is ....

....for the 55 mesh. The black numbers in the nodes show the labeling. The grey numbers represent the incurred costs for each edge, whose sum is 117. Table 1: Survey of classes of graphs optimaly solvable in polynomial time. Class of graph Complexity Ref. Trees O(n 3 ) 13] Rooted trees O(n log n) [2] Trees O(n 2.2 ) 38] Trees O(n log 3 log 2 ) 5] Rectangular meshes O(n) 29] Square meshes O(n) 28] Hypercubes O(n) 15] de Bruijn graph of order 4 O(n) 16] d dimensional c ary cliques O(n) 30] 3 a linear program with an exponential number of constraints using the Ellipsoid method. As ....

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D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM Journal on Applied Mathematics, 25(3):403--423, Nov. 1973.

....graphs, all these problems are NP hard. Moreover, the decisional version of Cutwidth and V ertex Separation problems are NP complete even when restricted to lattice graphs and unit disk graphs [10] All of them have a long history, owing to their practical relevance in di#erent applications [35, 15, 1, 32, 24, 23, 2, 26, 7, 17, 19, 32, 18, 5, 22, 27, 33, 34]. Our layout problems are formally defined as follows. A layout # of a graph G = V, E) is a one to one function # : V # 1, n with n = V . Given a graph G, a layout # of G and an integer i, let us define the sets L(i, #, G) u # V (G) #(u) # i and R(i, #, G) u # V ....

D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM Journal on Applied Mathematics, 25(3):403--423, November 1973.

....that appears as problem number GT42 in [8] and as GT40 in [3] It is motivated as an abstract model of the placement phase of VLSI layout, where nodes of the graph represent modules and edges represent interconnections. In this case, the cost of the arrangement measures the total wire length [1, 10]. This problem was also originally considered in [21] as an over simplified model of some nervous activity in the cortex. More information on MinLA and its complexity can be found in the survey by [4] The MinLA problem is known to be NP hard and its decision version is NP complete [8] However, ....

.... by [4] The MinLA problem is known to be NP hard and its decision version is NP complete [8] However, there exist exact solutions for some particular kinds of graphs in polynomial time: hypercubes [9] DeBruijn graphs of degree four [10] square and rectangular meshes [21, 22] rooted trees [1], unrooted trees [29] see [5] for an NC algorithm) and d dimensional c ary arrays [23] Besides these positive results, the decision problem remains NP complete when the input graph is bipartite [8] On the other hand, other works as [17, 19] have focussed on the search of lower bounds for the ....

D. Adolphson and T.C. Hu. Optimal linear ordering. SIAM J. Appl. Math., 25(3):403--423, November 1973.

.... the Edgesum problem, is relevant in circuit and VLSI layout [72, 35] single machine job scheduling [1, 70] and as a simplified model for nervous system simulation [54] The Minimum Cut Width problem was first used as a theoretical model for the number of channels in an optimal layout of a circuit [53, 2]. More recent applications of the problem include network reliability [42] automatic graph drawing [60] information retrieval [12] and as a subroutine for the cutting plane algorithm to solve the TSP [40] The Minimum Sum Cut problem, also known as the Profile problem, is equivalent to the ....

D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM Journal on Applied Mathematics, 25(3):403--423, November 1973.

....for m n , OLA if is even, and if is odd. K n m mn n m n n m mn n m n m n = 12 3 6 4 12 3 6 1 2 2 2 2 Note that OLA remains hard for arbitrary bipartite graphs [17] 7 OLA has also been solved when G is a tree following algorithms of Adolphson and Hu [1], Goldberg and Klipker [20] Shiloach [34] and Chung [11] Among these, Chung s algorithm has the best time complexity to date; it solves the problem on trees in O n ( l time where l is any real number satisfying l log log . 3 2 1585 . Frederickson and Hambrusch [16] have given an ....

....sizes of certain cut sets in G. The concepts that lead to our formalizations in this section are not new, but have appeared in the literature in various forms for several decades. Harper [23] appears to be the first to present these ideas, but other researchers (for example, see Adolphson and Hu [1], Bezrukov [3] and Liu and Vannelli [27] have made contributions in this area as well. Following, we present our version of the cut set bound concept and proofs of the relevant theorems. First, we provide some notation and definitions. Let c A A ( denote the number of edges in G from ....

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Adolphson, D. and T. C. Hu, "Optimal Linear Ordering," SIAM J. Appl. Math, Vol. 25, No. 3, 403-423 (1973).

....of naturals from 1 to n. Graph layout problems seek for a layout that minimizes a measure associated with each problem. The particular measures that we consider include Bisection, Minimum Linear Arrangement and Minimum Cutwidth, which play an important role in the evaluation of computer networks [1, 14, 13]. The de nition of these problems is given in Section 2. Our results show that random geometric networks can tolerate a constant edge (or node) failure probability maintaining the order of magnitude of the measures considered here. Finally, we consider Hamiltonian cycles in faulty random geometric ....

D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM Journal on Applied Mathematics, 25(3):403-423, Nov. 1973.

.... arises in a manufacturing process or computational process when we want to get everything done in a hurry, and it is more important that some tasks get done quickly than others (thus the weights) Numerous papers have been written on the solution of special cases and approaches to this problem [1, 2, 3, 7, 17, 18] and Lawler [10] showed that it is strongly NP complete. We in fact give an approximation algorithm for a more general problem, in which the goal is to minimize the storage time product for the process. This problem is intended to model a situation arising in manufacturing or computing in which ....

D. Adolphson and T. C. Hu, "Optimal linear ordering," SIAM J. Appl. Math. 25 (1973), pp. 403-423.

.... [25] Dean [8] Kadane [20] Joyce [19] Garey [10] Simon and Kadane [32] Kadane and Simon [21] Monma and Sidney [26] Work related to the MEF problem has been reported by (among others) Bellman [4] Staroverov [33] Greenberg [12] Matula [24] Black [6] Denby [9] Horn [14] Adolphson and Hu [1], Sidney [30] Stone [34] Hall [13] In this paper we concentrate on the MEF problem, when there are no constraints on the admissible testing sequence. A more comprehensive study is presented elsewhere [29] where both the INF and the MEF problems are treated, with and without constraints on the ....

D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM J. Appl. Math., 25:403-- 423, 1973.

....of the problem. Among the others, Grotschel, Junger, Reinelt [7, 19] and, more recently, Leung, Lee [11] Nutov and Penn [15] MSTS has been studied in different particular cases and with different names, reflecting the area from which this optimization problem is derived. Adolphson and Hu [2] call optimal linear ordering the problem we refer to as DLA and proved it is polynomially solvable when the digraph D = N; A) is a tree. Even and Shiloach proved the NPcompleteness of DLA if D is a general acyclic digraph [4] while Ravi, Agrawal and Klein give an approximation algorithm for MSTS ....

....ordering the problem we refer to as DLA and proved it is polynomially solvable when the digraph D = N; A) is a tree. Even and Shiloach proved the NPcompleteness of DLA if D is a general acyclic digraph [4] while Ravi, Agrawal and Klein give an approximation algorithm for MSTS [18] As shown in [2, 18], DLA [MSTS] is equivalent to the single processor scheduling problem, to minimize the sum [the weighted sum] of completion times, subject to precedence constraints. For the 1=prec= P n i=1 C i problem Horn shows in [8] an algorithm equivalent to the one of Adolphson and Hu, and Lawler proves ....

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A. Adolphson and T. C. Hu, Optimal Linear Ordering, SIAM Journal of Applied Mathematics, 25, (3) pp.403--423, 1973.

....problems are representatives of the problems over permutations. In many cases, effective algorithms are based on the use of local optimization techniques as element exchange techniques which effectively lead to a global optimum. We can point out, for example, the following works: Adolphson and Hu [1], Conway et al. 3] Hardy et al. 6] Johnson [7] and Smith [11] This paper describes a general element exchange techniques which is similar to k search approaches. We propose a general multiple level digraph description 1 for the optimization problems over permutations on the basis of ....

D. Adolphson, and T.C. Hu, Optimal Linear Ordering. SIAM J. Appl. Math., Vol. 25, No. 3, pp. 403-423, 1973.

....in VLSI layout design (or generally, optimal linear arrangement problem (OLA) in which a set of pins connected by wires is placed into a set of holes, one pin at each hole. The problem is to put the pins into holes such that the total wire length is minimal. For a history of the problem, see [1, 5]. The problems of both MLO and MCO have been open according to [16] Since there are n possible configurations in the linear ordering of n elements, a straightforward enumeration approach is not feasible. We prove this problem is NP complete. An effective algorithm will be proposed based on the ....

D. Adolphson and T. C. Hu. "Optimal Linear Ordering,". SIAM Journal on Applied Mathematics, 25:403--423, 1973.

....the jobs in non increasing order of the ratios w(s j ) p(s j ) Conway, Maxwell and Miller [3] developed a procedure for solving the problem when G is in the form of vertex disjoint paths and all w(s j ) 1. Horn [10] proposed algorithms for the case when G is a rooted tree. Adolphson and Hu [2] showed that such an algorithm can be implemented in O(n log n) time. Sidney [14] contributed a number of interesting theorems which apply to the problem with arbitrary precedence constraints. Lawler [12] gave an O(n log n) algorithm for the case when G is a transitive series parallel graph. He ....

D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM Journal Of Applied Math., 25:403--423, 1973.

....costs. NP optimization problems of P. Crescenzi and V. Kann [4] It is motivated as an abstract model of the placement phase of VLSI layout, where nodes of the graph represent modules and edges represent interconnections. In this case, the cost of the arrangement measures the total wire length [11, 1]. This problem was also originally considered by G. Mitchison and R. Durbin [22] as an oversimplified model of some nervous activity in the cortex. The problem can be easily generalized and is related to some other graph layout problems. More information on MinLA and its complexity can be found in ....

....in polynomial time. L.H. Harper shows how to label optimally hypercubes [10] and DeBruijn graphs of degree four [11] The optimal linear arrangement for square and rectangular meshes is characterised by G. Mitshison and R. Durbin [22] and D.O. Muradyan and T.E. Piliposjan [23] Adolphson and Hu [1] solved the problem for rooted trees by an O(n log n) algorithm and Y. Shiloach [27] presented an O(n 2:2 ) algorithm to find an optimal linear arrangement on unrooted trees. Moreover, J. D iaz et al. 5] presented an NC algorithm for trees. The case of d dimensional c ary arrays was studied by ....

D. Adolphson and T.C. Hu. Optimal linear ordering. SIAM J. Appl. Math., 25(3):403--423, November 1973.

.... as follows: given a graph G = V; E) find a layout of G that minimizes the cost c = X uv2E j (u) Gamma (v)j: MinLA is an interesting and appealing problem that appears as problem number GT42 in [8] and as GT40 in [2] with several different applications in Computer Science and Biology [1, 9, 22]. The MinLA problem is known to be NP hard and its decision version is NP comple te [8] However, there exist exact solutions for some particular kinds of graphs in polynomial time [9, 22, 1, 27, 4, 24] The lack of efficient exact algorithms for general graphs has given rise to the possibility ....

.... GT42 in [8] and as GT40 in [2] with several different applications in Computer Science and Biology [1, 9, 22] The MinLA problem is known to be NP hard and its decision version is NP comple te [8] However, there exist exact solutions for some particular kinds of graphs in polynomial time [9, 22, 1, 27, 4, 24]. The lack of efficient exact algorithms for general graphs has given rise to the possibility of finding approximation algorithms. An approximation of MinLA within a 1 ffl factor in time n O(1=ffl) for dense graphs is presented in [7] and an O(log n log log n) approximation for general graphs ....

D. Adolphson and T.C. Hu. Optimal linear ordering. SIAM J. Appl. Math., 25(3):403-- 423, November 1973.

....is interested in only two items. Given a graph G = V; E) with vertex set V and 7 edges E, find a one to one function f : V f1; 2; jV jg such that X (u;v) 2 E jf(u) Gamma f(v)j is minimized. This problem is identical to the known NP complete problem Optimal Linear Arrangement [1]. 2 Data layout problem 2. Suppose the airdisk has an index which specifies the sequence of data items in each rotation, and for simplicity, assume that all the data items are of the same length. A client then accesses the data items of interest by first reading the index and then reading only ....

D. Adolphson and T. C. Hu. Optimal linear ordering. SIMA J. Appl. Math., 25:403--423, 1973.

....is interested in only two items. Given a graph G = V; E) with vertex set V and edges E, find a one to one function f : V f1; 2; jV jg such that s 1 = X (u;v) 2 E jf(u) Gamma f(v)j (1) is minimized. This problem is identical to the known NP complete problem Optimal Linear Arrangement [4]. 2 The Non indexed data layout problem has been studied extensively in other contexts (e.g. see [7] and optimal as well as heuristic algorithms have been designed for it. We will not consider it further in this paper. Instead, we will consider the situation where the airdisk contains an index, ....

D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM J. Appl. Math., 25:403--423, 1973.

....G n;p model with p = c=n for some constant c. Our results establish that, in fact, any algorithm that returns a feasible solution will produce such an approximation for graphs with good expansion properties. 1 Introduction Linear arrangement problems play an important role in Computer Science [27, 1, 8]. A linear layout (or linear arrangement or vertex ordering) of a graph G with n nodes is a one to one mapping of the vertices of G to the set f1; ng. A layout on G = V; E) determines in a unique way a nested sequence of vertex subsets containing those vertices placed up to the i th ....

.... problem in which the cut excludes edges touching the last vertex is known as the minimum modified cut arrangement and is also NP complete for planar graphs with maximum degree 3 [26] The minimum linear arrangement problem (also known as the minimum edge sum [19] or the optimal linear ordering [1]) seeks a layout that minimizes the total edge length. This problem is also NP complete [16] For trees the problem is in P [31] and in NC [9] It can be approximated within a O(log 2 n) factor using the approximate max flow min cut theorem [23] A better approximation factor O(log n log log n) ....

D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM J. on Applied Mathematics, 25(3):403--423, November 1973.

....line graph even though it is guaranteed it will always hit these optima. Recall that Hill Climbing corresponds to SA at a zero temperature. Relation to previous work. Even though the MinLA problem is NP complete [8] there exist exact solutions for hypercubes, meshes and trees in polynomial time [9, 18, 1, 25, 3]. The lack of efficient exact algorithms for general graphs has given rise to the possibility of finding approximation algorithms. A polynomial approximation scheme for MinLA on dense graphs is presented in [7] and a O(log n log log n) approximation using spreading metrics for general graphs is ....

.... this section we present two classes of random graphs that are useful to generate MinLA instances for which a good estimate of the optimum is known by construction but is hidden from the heuristics (whp) Previously, the only known families of graphs that satisfied this property were deterministic [9, 18, 1, 25, 3, 19]. Let L n be the class of random linear graphs on vertex set V = f1; ng where each possible edge fi; jg appears with probability 1=jj Gamma i 1j. It is easy to see that the expected number of edges in a L n graph is concentrated around Theta(n log n) The canonical layout of a L n ....

D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM J. Appl. Math., 25(3):403-- 423, November 1973.

....trying to obtain polynomial time algorithms for particular types of graphs. For instance, Even and Shiloach proved that the problem remains NP complete for bipartite graphs [5] Adolph and Hu gave an O(n log n) algorithm for the case that the graph is a rooted tree, where n is the size of the tree [2]. Finally, Shiloach solved the problem for undirected trees by an O(n 2:2 ) algorithm [12] The second problem that we shall consider is the minimum cut linear arrangement (MINCUT) problem. Given a graph G = V; E) find the layout that minimizes the cutwidth fl( G) An important special ....

D. Adolphson and T.C. Hu. "Optimal linear ordering". SIAM J. on Applied Mathematics, 25(3):403--423, Nov. 1973.

....that appears as problem number GT42 in [15] and as GT40 in [6] It is motivated as an abstract model of the placement phase of VLSI layout, where nodes of the graph represent modules and edges represent interconnections. In this case, the cost of the arrangement measures the total wire length [18, 1]. This problem was also originally considered in [29] as an over simplified model of some nervous activity in the cortex. As other related problems, MinLA modeles the notion that adjacent vertices are close to one another. The MinLA problem is known to be NP hard and its decision version is ....

....in the cortex. As other related problems, MinLA modeles the notion that adjacent vertices are close to one another. The MinLA problem is known to be NP hard and its decision version is NP complete [15] However, there exist exact solutions for some particular kinds of graphs in polynomial time [17, 18, 29, 30, 1, 33, 9, 31]. The lack of efficient exact algorithms for general graphs has given rise to the possibility of finding approximation algorithms. An approximation of MinLA within a 1 ffl factor in time n O(1=ffl) for dense graphs is presented by [14] and [12] find an O(log n log log n) approximation factor ....

D. Adolphson and T.C. Hu. Optimal linear ordering. SIAM J. Appl. Math., 25(3):403-- 423, November 1973.

....that appears as problem number GT42 in [8] and as GT40 in [3] It is motivated as an abstract model of the placement phase of VLSI layout, where nodes of the graph represent modules and edges represent interconnections. In this case, the cost of the arrangement measures the total wire length [1, 10]. This problem was also originally considered in [21] as an over simplified model of some nervous activity in the cortex. More information on MinLA and its complexity can be found in the survey by [4] The MinLA problem is known to be NP hard and its decision version is NP complete [8] However, ....

.... by [4] The MinLA problem is known to be NP hard and its decision version is NP complete [8] However, there exist exact solutions for some particular kinds of graphs in polynomial time: hypercubes [9] DeBruijn graphs of degree four [10] square and rectangular meshes [21, 22] rooted trees [1], unrooted trees [29] see [5] for an NC algorithm) and d dimensional c ary arrays [23] Besides these positive results, the decision problem remains NP complete when the input graph is This work has been supported by ESPRIT LTR Project no. 20244 ALCOM IT. Approximation Heuristics and ....

D. Adolphson and T.C. Hu. Optimal linear ordering. SIAM J. Appl. Math., 25(3):403--423, November 1973.

....meaning that vertex (i) is placed in the i th position in a straight line. The cost of a permutation is m X i=1 w i max k;l2S i fj 1 (k) 1 (l)jg; 20) and it is asked to nd a permutation with the minimum cost. The applications of OLAP are abundant in VLSI design and other areas [1, 2, 13, 24]. A special case of OLAP in which the hypergraph H is a graph (each S i contains exactly two vertices) is referred to as the Graph Optimal Linear Arrangement problem (abbreviated as GOLAP) It is known that GOLAP with edge weights equal to 1 is already NP hard [6] 4.1 Genetic DP Algorithm In this ....

....were performed on SUN SPARC station IPX using C language for SMP and OLAP, and using FORTRAN 77 for TSP. The tested problem instances are generated as follows. SMP: For each n, coe cients p i ; h i ; w i for i 2 V ( f1; ng) are generated by randomly selecting integers from interval [1, 10]. It has been observed in the literature (e.g. 20] that problem hardness is related to two parameters RDD and LF , called the relative range of due dates and the average lateness factor, respectively. In our experiment, RDD = 0:2; 0:4; 0:6; 0:8; 1:0; LF = 0:2; 0:4; are used. Corresponding to ....

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Adolphson, D., and Hu, T.C., \Optimal Linear Ordering," SIAM Journal on Applied Mathematics 25 (1973) 403-423.

....in cell based designs is also determined by the number of wiring tracks used, i.e. the sum of channel widths is also an important objective function. Although minimum wirelength placement and minimum channel width placement are not perfectly correlated, we note that results of Adolphson and Hu [1] may be used to establish a probabilistic relationship between the two metrics. Indeed, our experimental results below show that when we use our new methods to improve wirelength, the channel width usually also improves. In view of these arguments, we choose to minimize a sum of wirelengths ....

D. Adolphson and T.C. Hu, "Optimal Linear Ordering", SIAM J. Appl. Math. 25 (1973), pp. 403--423.

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Adolphson, D., and Hu, T. (1973): "Optimal Linear Ordering". In: SIAM J. Appl. Math. 25 (3), 403--423.

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D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM Journal on Applied Mathematics, 25(3):403-423, Nov. 1973.

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D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM Journal on Applied Mathematics, 25(3):403--423, November 1973.

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D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM J. Appl. Math., 25(3):403--423, 1973.

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Adolphson, D.L., Hu, T.C.: Optimal linear ordering. SIAM J. Appl. Math. (1973) 25, N 3, 403-423.

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D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM J. Appl. Math., 25:403--423, 1973.

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D. Adolphson and T. C. Hu, "Optimal linear ordering," SIAM J. Appl. Math., 25 (1973), pp. 403-423.

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D.Adolphson and T. Hu, "Optimal Linear ordering," SIAM J. Appl. Math., vol.25, pp. 403-423, 1973.

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D.Adolphson and T. Hu, "Optimal Linear ordering," SIAM J. Appl. Math., vol.25, pp. 403-423, 1973.

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D. Adolphson and T.C. Hu, Optimal linear ordering, SIAM J. Appl. Math. 25 (1973), 403--423.

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