C. Monma and J. Sidney. Sequencing with series-parallel precedence constraints. Math. Oper. Res., 4:215--224, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... include the case where the components are identical, i.e. the cost of inspecting any of the components is the same c and the probability that a component is working is the same p for all components (see, e.g. 1,5] or the case when precedence constraints are prescribed on the tests (see, e.g. [11,21,17]) We also mention as an essentially unsolved special case the sequential testing of Series Parallel systems (see, e.g. 2,4] Another interesting open problem is the optimal sequential testing of general threshold systems. ....

C.L. Monma and J.B. Sidney, Sequencing with series--parallel precedence constraints, Technical Report, Cornell University (1976).

....Stack distinct any poly time Thrm 3.5 Queue, Priority queue, Stack arbitrary interval orders NP complete Thrm 4.1 Queue, Priority queue, Stack arbitrary chains NP complete Thrm 4.2 Counter arbitrary interval orders O(n) time Obs 5. 1 Counter arbitrary chains, series parallel O(n log n) time [MS79] Counter arbitrary any NP complete [AW76] Counter distinct any NP complete Cor 5.4 Fetch Add counter arbitrary interval orders, chains NP complete [GK97] Counter with reads arbitrary interval orders NP complete Thrm 5.8 Counter with reads arbitrary chains NP complete Thrm 5.7 Arbitrary ....

.... counters (Algorithm F in the full paper) Then, we discuss how an O(n log n) time algorithm for testing parallel executions of non linearizable counters, where the partial order is either the union of chains or a series parallel order, can be obtained as a corollary of known results [AWK78, MS79] on sequencing to minimize maximum cumulative profit. On the other hand, we observe that it is an immediate consequence of previous work [AW76] that testing parallel executions of non linearizable counters with an arbitrary partial order is NP complete. Moreover, this hardness result can be ....

C. L. Monma and J. B. Sidney. Sequencing with series-parallel precedence constraints. Mathematics of Operations Research, 4(3):215--224, 1979.

....proved that their formulation completely describes the scheduling polyhedron in the case of series parallel constraints; a by product of our proof of correctness of Lawler s algorithm is an alternate proof of this fact. Other proofs of the correctness of Lawler s algorithm have been given (see [4, 10, 9], for example) but to the best of our knowledge ours is the rst duality based proof. In the course of our proof it is helpful to use what might be called two dimensional Gantt charts. Although 2D Gantt charts were introduced in a 1964 paper by Eastman, Even, and Isaacs [2] they seem to have ....

C. L. Monma and J. B. Sidney. Sequencing with series-parallel precedence constraints. Mathematics of Operations Research, 4:215-224, 1979.

....position relative to the join nodes. The algorithm assumes that join costs are linear in the sizes of the operands. This allows them to assign a rank for each of the join predicate in addition to assigning ranks for expensive predicates. The notion of rank has been studied previously in [Monma and Sidney 1979; Krishnamurthy et al. 1986] Having assigned ranks, the algorithm iterates over each stream, where a stream is a path from a leaf to a root in the execution tree. Every iteration potentially rearranges the placement of the expensive selections. The iteration continues over the streams until the ....

....time with our assumed cost model for user defined predicates. Furthermore, the ordering of the selections does not depend on the size of the relation on which they apply. The problem of selection ordering was addressed in [Hellerstein and Stonebraker 1993] cf. Krishnamurthy et al. 1986; Monma and Sidney 1979; Whang and Krishnamurthy 1990] It utilizes the notion of a rank. The rank of a predicate is the ratio c= 1 Gamma s) where c is its cost per tuple and s is its selectivity. Theorem 4.1. Consider the query oe e (R) where e = e 1 : e n . The optimal ordering of the predicates in e is in the ....

Monma, C. and Sidney, J. 1979. Sequencing with series-parallel precedence constraints. Mathematics of Operations Research 4, 215--224.

....6= return pt; Figure 3: Algorithm Minimum Selectivity is joined with the (so far) intermediate result and moved from R remaining to R used . Figure 3 shows the complete algorithm for left deep processing trees. 4.1. 3 Krishnamurthy Boral Zaniolo Algorithm On the foundation of [Law78] and [MS79] Ibaraki and Kameda showed in [IK84] that it is possible to compute the optimal nesting order in polynomial time, provided the query graph forms a tree (i.e. no cycles) and the cost function is a member of a certain class. Based on this result, Krishnamurthy, Boral and Zaniolo developed in ....

C. Monma and J. Sidney. Sequencing with series-parallel precedence constraints. Mathematics of Operations Research, 4:215--224, 1979.

....for a special block wise nested loop join [2] Independent of any cost function is the dynamic programming approach to join ordering. The number of alternatives this approach generates is known as well [5] Further, it is known that for some cost functions those fulfilling the ASI property [4] the problem can be solved in polynomial time for acyclic query graph, i.e. tree queries [2, 3] Unfortunately, some cost functions like sort merge could not be treated so far. We do so by a slight detour showing that this cost function (and others too) are optimized if and only if the sum of ....

....programming approach is independent of the chosen cost function. Nevertheless, the question arises, whether one can do better with specialized algorithms, if something is known about the cost functions. The answer is yes, if the cost function has the ASI (Adjacent Sequence Interchange) property [4]. For these cost functions, there exist polynomial time algorithms (the fastest is O(n 2 ) producing optimal left deep trees for tree queries [2, 3] Let us shortly review this approach. From a query graph, a precedence tree is constructed by arbitrarily choosing one relation as a root and ....

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Monma, C., Sidney, J.: Sequencing with Series-Parallel Precedence Constraints. Math. Oper. Res,. 4 (1979), 215---224

....max i=1 f i (C i ) For problems without release dates and monotone objective functions it can be shown, that only schedules without idle times have to be considered. Therefore it is sufficient to determine an optimal permutation of the jobs and, thus, the problem becomes a sequencing problem. Monma Sidney [1979] have shown that a large class of polynomially solvable sequencing problems can be solved by a general concept. If a transitive relation exists with the adjacent pair interchange (API) property, the elements only have to be sorted with respect to this relation to obtain an optimal solution. Monma ....

....[1979] have shown that a large class of polynomially solvable sequencing problems can be solved by a general concept. If a transitive relation exists with the adjacent pair interchange (API) property, the elements only have to be sorted with respect to this relation to obtain an optimal solution. Monma Sidney [1979] and Lawler [1992] also developed a general algorithm for these problems with series parallel precedence constraints. It relies on a relation for subsequences with the adjacent sequence interchange (ASI) property. In the first part of this paper we derive sufficient conditions for the existence of ....

[Article contains additional citation context not shown here]

Monma, C.L., Sidney, J.B. [1979] Sequencing with series-parallel precedence constraints, Mathematics of Oper. Research 4, 215-224.

....proved that their formulation completely describes the scheduling polyhedron in the case of series parallel constraints; a by product of our proof of correctness of Lawler s algorithm is an alternate proof of this fact. Other proofs of the correctness of Lawler s algorithm have been given (see [4, 10, 9], for example) but to the best of our knowledge ours is the first duality based proof. In the course of our proof it is helpful to use what might be called two dimensional Gantt charts. Although the 2D Gantt charts we will use are not a new concept see the 1964 paper by Eastman, Even, and ....

C. L. Monma and J. B. Sidney. Sequencing with series-parallel precedence constraints. Mathematics of Operations Research, 4:215--224, 1979.

....Therefore, the Top Down heuristic selects the relation from the set that can be joined with lowest cost with the remaining relations. This strategy is applied repeatedly until no relations are left (Figure 4) 4.1. 4 Krishnamurthy Boral Zaniolo Algorithm On the foundation of [Law78] and [MS79] Ibaraki and Kameda showed in [IK84] that it is possible to compute the optimal nesting order in polynomial time, provided the query graph forms a tree (i.e. no cycles) and the cost function is a member of a certain class. Based on this result, Krishnamurthy, Boral and Zaniolo developed in ....

C. Monma and J. Sidney. Sequencing with series-parallel precedence constraints. Math. Oper. Res., 4:215--224, 1979.

....Garey considers the problem of sequencing with precedence constraints [Garey73] He shows that a series of interchange rules may be applied that allow task pairs to be merged or the precedence constraints to be relaxed. His rules are sufficient to provide optimal schedules for opposing forests. In [Monma79] two interchange rules are proposed that allow global improvement in the cost function. These rules are known as the adjacent sequence interchange rule (ASI) and the series 93 network decomposition (SND) rule. They make it possible to find optimal schedules for parallel chains and for ....

....the set of sequencing problems that can be solved optimally by repeated pair wise interchange of the tasks in a schedule [Sidney81] He also provided four rules for sequencing tasks with general precedence constraints: ASI, SND, consistency and monotonicity. These rules generalize the work of [Monma79] and also provide an optimal solution to the 2 machine flow shop problem where the objective is to minimize maximum flow time. Several other applications are mentioned in [Sidney81] The work in [Monma87] recasts the algorithms of [Monma79] to handle job modules. Any directed graph that is not ....

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Monma, Clyde L. and Jeffrey B. Sidney. Sequencing with Series-Parallel Precedence Constraints. Mathematics of Operations Research (August 1979) vol. 4, no. 3, pp. 215-224.

....Query 4. but with the costly selection pulled to the top. 4.4 Predicate Migration The details of the Predicate Migration algorithm are presented in [Hel92] and we only review them here. The Predicate Migration algorithm repeatedly applies the Series Parallel Algorithm using Parallel Chains [MS79] to each root to leaf path in the plan tree until no progress is made. In essence, Predicate Migration augments PullRank by also considering the possibility that two primary join nodes in a plan tree may be out of rank order, e.g. join node J 2 may appear just above node J 1 in a plan tree, with ....

C. L. Monma and J.B. Sidney. Sequencing with Series-Parallel Precedence Constraints. Mathematics of Operations Research, 4:215-- 224, 1979.

....and the IK KBZ algorithm of Krishnamurthy, Boral and Zaniolo [10] for the join ordering problem. The KBZ algorithm is an improved version of the IK algorithm of Ibaraki and Kameda [6] who were the first to recognize the applicability of results for sequencing problems with ASI cost functions [14] to the area of optimizing join orders. As mentioned earlier, every left deep tree corresponds to a permutation indicating the order in which the base relations are joined with the intermediate result relation. We will henceforth speak of permutations or sequences instead of left deep processing ....

....of these cost functions: C(ffl) 0 C(R j ) 0 if R j is the starting relation C(R j ) g j (jR j j) else C(s 1 s 2 ) C(s 1 ) T (s 1 ) C(s 2 ) with T (ffl) 1 T (s) n Y i=1 f i s i Here, s 1 ; s 2 and s denote sequences of relations. We now define the ASI property 1 [14] of a cost function. Definition 3.1 (ASI property) A cost function C has the ASI property, if there exists a rank function r(s) for sequences s, such that for all sequences a and b and all non empty sequences u and v the following holds: C(auvb) C(avub) r(u) r(v) For a cost function of the ....

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C. Monma and J. Sidney. Sequencing with series-parallel precedence constraints. Math. Oper. Res., 4:215--224, 1979.

....and, in extended relational and object oriented systems additionally in the form of user defined functions. HS93] s work is based on ordering the conditions in a sequence according to their relative selectivity and evaluation cost adapting a technique developed in operations research [MS79] This approach yields the optimal evaluation sequence for conjunctive selection predicates. It is striking that in all these works the optimization of disjunctive query predicates tends to be neglected. Bry89] and [Mur88] are the only works to the authors knowledge who dealt with ....

C. Monma and J. Sidney. Sequencing with series-parallel precedence constraints. Mathematics of Operations Research, 4:215--224, 1979.

....position relative to the join nodes. The algorithm assumes that join costs are linear in the sizes of the operands. This allows them to assign a rank for each of the join predicate in addition to assigning ranks for expensive predicates. The notion of rank has been studied previously in [MS79, KBZ86] Having assigned ranks, the algorithm iterates over each stream, where a stream is a path from a leaf to a root in the execution tree. Every iteration potentially rearranges the placement of the expensive selections. The iteration continues over the streams until the modified operator ....

....selection ordering problem can be solved in polynomial time with our assumed cost model for user defined predicates. Furthermore, the ordering of the selections does not depend on the size of the relation on which they apply. The problem of selection ordering was addressed in [HS93] cf. KBZ86, MS79, WK90] It utilizes the notion of a rank. The rank of a predicate is the ratio c= 1 Gamma s) where c is its cost per tuple and s is its selectivity. Theorem 4.1: Consider the query oe e (R) where e = e 1 : e n . The optimal ordering of the predicates in e is in the order of ascending ranks ....

C.L. Monma and J.B. Sidney. Sequencing with series-parallel precedence constraints. Mathematics of Operations Research, 4:215--224, 1979.

....in the general case are easy in the special case of series parallel networks. Bein, Brucker and Tamir [4] for example, show that the minimum cost flow problem is solved by the greedy algorithm if and only if the graph is series parallel. Other examples include scheduling and sequencing problems [1, 2, 26, 27], location problems [17] as well as many combinatorial problems [21, 35, 34] All these approaches rely on the decomposition tree (see [6] for a general formulation of this idea) or on Duffin s characterization. This paper introduces definitions of st dag complexity, measures that describe how ....

C. Monma and J. Sidney, Sequencing with series-parallel precedence constraints, Mathematics of Operations Research, 4 (1979), pp. 215 -- 224.

....position relative to the join nodes. The algorithm assumes that join costs are linear in the sizes of the operands. This allows them to assign a rank for each of the join predicate in addition to assigning ranks for expensive predicates. The notion of rank has been studied previously in [MS79, KBZ86] Having assigned ranks, the algorithm iterates over each stream, where a stream is a path from a leaf to a root in the execution tree. Every iteration potentially rearranges the placement of the expensive selections. The iteration continues over the streams until the modified operator ....

....the latter problem is NP hard. On the other hand, the selection ordering problem can be solved in polynomial time. Furthermore, the ordering of the selections does not depend on the size of the relation on which they apply. The problem of selection ordering was addressed in [HS93] cf. KBZ86, MS79, WK90] It utilizes the notion of a rank. The rank of a predicate is the ratio c= 1 Gamma s) where c is its cost per tuple and s is its selectivity. Theorem 4.1: Consider the query oe e (R) where e = e 1 : e n . The optimal ordering of the predicates in e is in the order of ascending ....

C.L. Monma and J.B. Sidney. Sequencing with series-parallel precedence constraints. Mathematics of Operations Research, 4:215--224, 1979.

....containing t tuples is e 1 = e p1 t s p1 e p2 t Delta Delta Delta s p1 s p2 Delta Delta Delta s pn Gamma1 e pn t: The following lemma demonstrates that this cost can be minimized by a simple sort on the predicates. It is analogous to the Least Cost Fault Detection problem solved in [MS79] Lemma 1 The cost of applying expensive restriction predicates to a set of tuples is minimized by applying the predicates in ascending order of the metric rank = selectivity Gamma 1 cost per tuple Thus we see that for single table queries, predicates can be optimally ordered by simply ....

....Predicate Migration 0.36 sec 0.57 sec 0 min 3.46 sec 0 min 6.75 sec Table 4: Performance of Plans for Example 2 even though rank(p 1 ) rank(p 2 ) In such situations, we will need to find the optimal ordering of predicates in the stream subject to the precedence constraints. Monma and Sidney [MS79] have shown that finding the optimal ordering under arbitrary precedence constraints can be done fairly simply. Their analysis is based on two key results: 1. A stream can be broken down into modules, where a module is defined as a set of nodes that have the same constraint relationship with all ....

[Article contains additional citation context not shown here]

C. L. Monma and J.B. Sidney. Sequencing with Series-Parallel Precedence Constraints. Mathematics of Operations Research, 4:215--224, 1979.

....t tuples is e = e p1 t s p1 e p2 t : s p1 s p2 Delta Delta Delta s pn Gamma1 e pn t: The following lemma demonstrates that this cost can be minimized by a simple sort on the predicates. It is analogous to the Least Cost Fault Detection problem addressed by Monma and Sidney [Monma and Sidney 1979]. Lemma 1. The cost of applying expensive selection predicates to a set of tuples is minimized by applying the predicates in ascending order of the metric rank = selectivity Gamma 1 differential cost inexpensive predicates on an unindexed nested loop join may be considered primary join ....

....simply ordering a stream by ascending rank, since a predicate p 1 may be constrained to precede a predicate p 2 , even though rank(p 1 ) rank(p 2 ) In such situations, we will need to find the optimal ordering of predicates in the stream subject to the precedence constraints. Monma and Sidney [Monma and Sidney 1979] have shown that finding the optimal ordering for a single stream under these kinds of precedence constraints can be done fairly simply. Their analysis is based on two key results: 1) A set S of plan nodes can be grouped into job modules, where a job module is defined as a subset of nodes S 0 ....

[Article contains additional citation context not shown here]

Monma, C. L. and Sidney, J. 1979. Sequencing with Series-Parallel Precedence Constraints. Mathematics of Operations Research 4, 215--224.

....and object oriented systems additionally in the form of user defined functions. HS93] s work is based on ordering the conditions in a sequence according to their relative selectivity and evaluation cost. This approach yields the optimal evaluation sequence for conjunctive selection predicates [MS79] However, it is striking that in all these works the optimization of disjunctive query predicates tends to be neglected. The traditional approaches transform a query predicate (i.e. either selection or join predicate) into a normal form (namely, conjunctive or disjunctive normal form) thus ....

....subset of the set of bypass plans. DNF For the DNF approach, the entire selection predicate is transformed into the Disjunctive Normal Form (DNF) and each conjunct of this normal form is regarded as a Boolean factor. Then, each Boolean factor is independently optimized by ordering selections [MS79] ordering joins [KBZ86] and ordering selections into join orderings [HS93] However, the derived evaluation plans contain nondisjoint tuple streams that must be united by union operators that eliminate duplicates (unlike the specialcase merge union operators for disjoint operands that can be ....

C. Monma and J. Sidney. Sequencing with series-parallel precedence constraints. Math. Oper. Res., 4:215--224, 1979.

....above, a possible strategy to a solution of Problem 2 would be to obtain topological sorts in which vertices with negative s values are pushed as close as possible to the start. 4 The Solution Problems of this nature, involving precedence constraints, have been studied by Monma and Sidney in [4], in the more general framework of series parallel graphs. A sequencing problem on a set of jobs J with a cost function f , is to find a permutation of J , contained in a set of feasible permutations F , which minimizes f , i.e Minimize f( Pi) Pi 2 F If the set of feasible permutations is the ....

....Max (p 2 ; p 1 ; t 2 p 1 ) Max (p 1 ; p 2 t 1 ) 2. As t 1 0, Max (p 1 ; t 1 p 2 ) Max (p 1 ; p 2 ) Since p 1 t 2 p 1 p 2 , Max (p 1 ; p 2 ) Max (p 2 ; t 2 p 1 ) p 2 . 3. Max (p 1 ; t 1 p 2 ) Max (p 1 ; p 2 ) Max (t 2 p 1 ; p 2 ) In the terminology of Monma and Sidney [4], the function Peak and the order have the adjacent pairwise interchange (API) property . Theorem 4.1 Given a set of blocks B and functions T ; P : B Gamma Z, any permutation Pi, in which the blocks are sorted by the order, is optimal. Proof. Consider any arbitrary optimal permutation Pi. ....

Clyde L. Monma and Jeffrey B. Sidney. Sequencing with series-parallel precedence constraints. Mathematics of Operations Research, 4:215--224, 1979.

....subset of the set of bypass plans. DNF For the DNF approach, the entire selection predicate is transformed into the Disjunctive Normal Form (DNF) and each conjunct of this normal form is regarded as a Boolean factor. Then, each Boolean factor is independently optimized by ordering selections [MS79] ordering joins [KBZ86] and ordering selections into join orderings [HS93] However, the computed evaluation plans contain non disjoint tuple streams that must be united by union operators that eliminate duplicates (unlike the special case merge union operators for disjoint operands that can ....

C. Monma and J. Sidney. Sequencing with series-parallel precedence constraints. Math. Oper. Res., 4:215--224, 1979.

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C. Monma and J. Sidney. Sequencing with series-parallel precedence constraints. Math. Oper. Res., 4:215--224, 1979.

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C. L. Monma and J. B. Sidney. Sequencing with series-parallel precedence constraints. Mathematics of Operations Research, 4:215--224, 1979.

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Monma, C.L. and Sidney, J.B. (1979). Sequencing with series-parallel precedence constraints. Mathematics of Operations Research, 4, 215-234.

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Monma, C.L., Sidney, J.B.: Sequencing with series-parallel precedence constraints. Math. Oper. Res. (1979) 4, N 3, 215-224.

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