C. E. Leiserson and ,1. B. Saxe. A mixed-integer programming problem which is efficiently solvable..1ourmf of Algorithms, 9:114-128, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... w(e) for all u e v 2 E R(v) Gamma r(v) 0 for all v 2 V R(u) Gamma R(v) w(e) Gamma 2 d(v) for all u e v 2 E r(v) Gamma R(v) 2 Gamma 2 (d(v) ffl) for all v 2 V : 15) The constraints defined in Theorem 6 form a mixed integer linear program of the special form MI from [10]. Problem MI can be solved in O(V E V 2 lg V ) time by applying Algorithm MILP from [10] which gives us an O(V E V 2 lg V ) algorithm for retiming when the clocking scheme utilized by the circuit is symmetric. General two phase clocking schemes If the clocking scheme utilized by a ....

.... 2 d(v) for all u e v 2 E r(v) Gamma R(v) 2 Gamma 2 (d(v) ffl) for all v 2 V : 15) The constraints defined in Theorem 6 form a mixed integer linear program of the special form MI from [10] Problem MI can be solved in O(V E V 2 lg V ) time by applying Algorithm MILP from [10], which gives us an O(V E V 2 lg V ) algorithm for retiming when the clocking scheme utilized by the circuit is symmetric. General two phase clocking schemes If the clocking scheme utilized by a circuit has a general form, i.e. is not symmetric, the retiming problem can still be cast as a ....

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C. E. Leiserson and J. B. Saxe. A mixed-integer programming problem which is efficiently solvable. Journal of Algorithms, 9:114--128, 1988.

....optimization problem, that we call restricted mixed integer dual of an uncapacitated minimum cost flow. The polynomial running time is achieved by introducing a set of additional, appropriately chosen constraints. The same idea was used for the solution of a similar mixed integer problem in [22, 16], which did not involve, however, an objective to be optimized. The technique of introducing additional constraints, or cuts as they are known in the literature, in order to solve mixed integer optimization problems, is known in general to require an exponential number of steps [21, 23, 3, 18] ....

C. E. Leiserson and ,1. B. Saxe. A mixed-integer programming problem which is efficiently solvable..1ourmf of Algorithms, 9:114-128, 1988.

....number. Find a vector x = x 1 ; x 2 ; x n ) satisfying the constraints that x i Gamma x j a(i; j) for all (i; j) 2 E, and that x i 2 Z for all i 2 V I , or determine that no feasible vector exists. Problem MI can be solved in O(V E V 2 lg V ) time by applying Algorithm MILP from [14]. Thus, we obtain the following theorem. Theorem 15 The retiming problem with symmetric schemes can be solved for a two phase, level clocked circuit G = hV; E; d; w; i and a symmetric clocking scheme = hOE; fl; OE; fli in O(V E V 2 lg V ) time. Proof. Algorithm RwSCS in Figure 6 solves the ....

....symmetric schemes can be solved for a two phase, level clocked circuit G = hV; E; d; w; i and a symmetric clocking scheme = hOE; fl; OE; fli in O(V E V 2 lg V ) time. Proof. Algorithm RwSCS in Figure 6 solves the retiming problem with symmetric phases. It simply applies Algorithm MILP from [14] to the constraints in Lemma 14. Since jV H j = jV j 1 and jEH j = O(V E) the running time of RwSCS is O(V E V 2 lg V ) 6 Retiming with general clocking schemes In this section, we study the retiming problem: Given a circuit G = hV; E; d; w; i and a clocking scheme = hOE 0 ; fl 0 ; OE 1 ....

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C. E. Leiserson and J. B. Saxe. A mixed-integer programming problem which is efficiently solvable. Journal of Algorithms, 9:114--128, 1988.

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