We show how to implement divide-and-conquer algorithms without undue overhead on a wide class of ... more We show how to implement divide-and-conquer algorithms without undue overhead on a wide class of networks. We give an optimal generic divide-and-conquer implementation on hypercubes for the class of divide-and-conquer algorithms for which the total size of the subproblems on any level of the recursion does not exceed the parent problem size. For this implementation, appropriately sized subcubes have to be allocated to the subproblems generated by the divide-steps. We take care that these allocation steps do not cause any unbalanced distribution of work, and that, asymptotically, they do not increase the running time. Variants of our generic algorithm also work for the butterfly network and, by a general simulation, for the class of hypercubic networks, including the shuffle-exchange and the cube-connected-cycles network. Our results can also be applied to optimally solve various types of routing problems.
We show how to implement divide- and-conquer algorithms without undue overhead on a wide class of... more We show how to implement divide- and-conquer algorithms without undue overhead on a wide class of networks. We give an optimal generic divide- and-conquer implementation on hypercubes for the class of divide- and-conquer algorithms for which the total size of the subproblems on any level of recursion does not exceed the original problem size. For this implementation, appropriately sized subcubes have to be allocated to the subproblems generated by the divide-step. We take care that these allocation steps do not cause unbalanced distribution of work, and that, asymptotically, they do not increase the running time. Variants of our generic algorithm also work for the butterfly network and, by a general simulation, for the class of hypercubic networks, including the shuffle-exchange and the cube-connected-cycles network. Our results can also be applied to optimally solve various types of routing problems.
An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubi... more An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubic networks, such as the shu e exchange or the butter y network, is presented. The algorithm is based on a recursive approach, uses novel routing techniques, and, for certain small subtrees, simulates optimal PRAM algorithms. For trees of size n, stored on a p processor hypercube in in-order, the running time of the algorithm is O( l n p m log p), which can be shown to be optimal on the hypercube by a corresponding lower bound. Tree contraction can be used to evaluate algebraic expressions consisting of operators +; ?; ; = and rational operands, as well as for the membership problem for certain subclasses of languages in DCFL. The same algorithmic ingredients can also be used to solve the term matching problem, one of the fundamental problems in logic programming.
An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubi... more An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubic networks, such as the shuffle exchange or the butterfly network, is presented. The algorithm is based on novel routing techniques and, for certain small subtrees, simulates optimal PRAM algorithms. For trees of size n, stored on a p processor hypercube in in-order, the running time of the algorithm is O([n/p] log p). The resulting speed-up of O(p/ log p) is optimal due to logarithmic communication overhead, as shown by a corresponding lower bound.
Journal of Parallel and Distributed Computing, 1995
We consider a new class of routing requests, or partial permutations, for which we give optimal o... more We consider a new class of routing requests, or partial permutations, for which we give optimal on-line routing algorithms on the hypercube and shu e-exchange network. For well-formed words of parentheses our algorithm establishes communication between all matching pairs in logarithmic time. It can be applied to the membership problem for Dyck languages and a number of problems dealing with algebraic expressions.
An optimal tree contraction algorithm for the boolean hypercube and the constant-degree hypercubi... more An optimal tree contraction algorithm for the boolean hypercube and the constant-degree hypercubic networks, such as the shuffle exchange or the butterfly network, is presented. The algorithm is based on novel routing techniques and, for certain small subtrees, simulates optimal PRAM algorithms. For trees of size n, stored on a p processor hypercube in in-order, the running time of the algorithm is \(O(\lceil n/p\rceil\log p)\) . The resulting speed-up of \(O(p/\log p)\) is optimal due to logarithmic communication overhead, as shown by a corresponding lower bound. The same algorithmic ingredients can also be used to solve the term matching problem, one of the fundamental problems in logic programming.
We show how to implement divide-and-conquer algorithms without undue overhead on a wide class of ... more We show how to implement divide-and-conquer algorithms without undue overhead on a wide class of networks. We give an optimal generic divide-and-conquer implementation on hypercubes for the class of divide-and-conquer algorithms for which the total size of the subproblems on any level of the recursion does not exceed the parent problem size. For this implementation, appropriately sized subcubes have to be allocated to the subproblems generated by the divide-steps. We take care that these allocation steps do not cause any unbalanced distribution of work, and that, asymptotically, they do not increase the running time. Variants of our generic algorithm also work for the butter y network and, by a general simulation, for the class of hypercubic networks, including the shu e-exchange and the cube-connected-cycles network. Our results can also be applied to optimally solve various types of routing problems.
Circuit size, branching program size, and formula size of Boolean functions, denoted by C(f), BP(... more Circuit size, branching program size, and formula size of Boolean functions, denoted by C(f), BP(f), and L(f), are the most important complexity measures for Boolean functions. Often also the formula size L*(f) over the restricted basis {∨, ∧, ⌝} is considered. It is well-known that C(f) ≤ 3 BP(f), BP(f) ≤ L*(f), L*(f) ≤ L(f) 2, and C(f) ≤ L(f) - 1. These estimates are optimal. But the inequality BP(f) ≤ L(f) 2 can be improved to BP(f) ≤ 1.360 L(f) β, where β = log4(3 + √5) < 1.195.
Abstract In many applications like verification or combinatorial opt imization, OBDDs (ordered bi... more Abstract In many applications like verification or combinatorial opt imization, OBDDs (ordered binary decision diagrams) are used as a representation or data structure for Boolean functions. Efficient algorithms exist for the important operations on O BDDs, and many functions can be represented in reasonable size if a good variable orderin g is chosen. In general, it is NP-hard to compute
We present a randomized on-line algorithm for the list update problem which achieves a competitiv... more We present a randomized on-line algorithm for the list update problem which achieves a competitive factor of 1.6, the best known so far. The algorithm makes an initial random choice between two known algorithms that have different worst-case request sequences. The first is the BIT algorithm that, for each item in the list, alternates between moving it to the front of the list and leaving it at its place after it has been requested. The second is a TIMESTAMP algorithm that moves an item in front of less often requested items within the list.
... In that case the searcher searches a connected subgraph of size k rather than a path of lengt... more ... In that case the searcher searches a connected subgraph of size k rather than a path of length k. The optimal searcher path problem has been investigated in the literature as one-sided search, where a prior hiding location distribution is given [24]. ...
Data structures for Boolean functions build an essential component of design automation tools, es... more Data structures for Boolean functions build an essential component of design automation tools, especially in the area of logic synthesis. The state of the art data structure is the ordered binary decision diagram (OBDD), which results from general binary decision diagrams (BDDs), also called branching programs, by ordering restrictions. In the context of EXOR-based logic synthesis another type of decision diagram (DD), called (ordered) functional decision diagram ((O)FDD) becomes increasingly important. We study the relation between (ordered, free) BDDs and FDDs. Both, BDDs and FDDs, result from DDs by defining the represented function in different ways. If the underlying DD is complete, the relation between both types of interpretation can be described by a Boolean transformation τ. This allows us to relate the FDD-size of f and the BDD-size of τ(f) also in the case that the corresponding DDs are free or ordered, but not (necessarily) complete. We use this property to derive several results on the computational power of OFDDs and OBDDs. Symmetric functions are shown to have efficient representations as OBDDs and OFDDs as well. Classes of functions are given that have exponentially more concise OFDDs than OBDDs, and vice versa. Finally, we determine the complexity of some standard operations if OFDDs are used for the representation of Boolean functions.
Information and Computation/information and Control, 1995
Data structures for Boolean functions build an essential component of design automation tools, es... more Data structures for Boolean functions build an essential component of design automation tools, especially in the area of logic synthesis. The state of the art data structure is the ordered binary decision diagram (OBDD), which results from general binary decision diagrams (BDDs), also called branching programs, by ordering restrictions. In the context of EXORbased logic synthesis another type of decision diagram (DD), called (ordered) functional decision diagram ((O)FDD) becomes increasingly important.
We show how to implement divide-and-conquer algorithms without undue overhead on a wide class of ... more We show how to implement divide-and-conquer algorithms without undue overhead on a wide class of networks. We give an optimal generic divide-and-conquer implementation on hypercubes for the class of divide-and-conquer algorithms for which the total size of the subproblems on any level of the recursion does not exceed the parent problem size. For this implementation, appropriately sized subcubes have to be allocated to the subproblems generated by the divide-steps. We take care that these allocation steps do not cause any unbalanced distribution of work, and that, asymptotically, they do not increase the running time. Variants of our generic algorithm also work for the butterfly network and, by a general simulation, for the class of hypercubic networks, including the shuffle-exchange and the cube-connected-cycles network. Our results can also be applied to optimally solve various types of routing problems.
We show how to implement divide- and-conquer algorithms without undue overhead on a wide class of... more We show how to implement divide- and-conquer algorithms without undue overhead on a wide class of networks. We give an optimal generic divide- and-conquer implementation on hypercubes for the class of divide- and-conquer algorithms for which the total size of the subproblems on any level of recursion does not exceed the original problem size. For this implementation, appropriately sized subcubes have to be allocated to the subproblems generated by the divide-step. We take care that these allocation steps do not cause unbalanced distribution of work, and that, asymptotically, they do not increase the running time. Variants of our generic algorithm also work for the butterfly network and, by a general simulation, for the class of hypercubic networks, including the shuffle-exchange and the cube-connected-cycles network. Our results can also be applied to optimally solve various types of routing problems.
An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubi... more An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubic networks, such as the shu e exchange or the butter y network, is presented. The algorithm is based on a recursive approach, uses novel routing techniques, and, for certain small subtrees, simulates optimal PRAM algorithms. For trees of size n, stored on a p processor hypercube in in-order, the running time of the algorithm is O( l n p m log p), which can be shown to be optimal on the hypercube by a corresponding lower bound. Tree contraction can be used to evaluate algebraic expressions consisting of operators +; ?; ; = and rational operands, as well as for the membership problem for certain subclasses of languages in DCFL. The same algorithmic ingredients can also be used to solve the term matching problem, one of the fundamental problems in logic programming.
An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubi... more An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubic networks, such as the shuffle exchange or the butterfly network, is presented. The algorithm is based on novel routing techniques and, for certain small subtrees, simulates optimal PRAM algorithms. For trees of size n, stored on a p processor hypercube in in-order, the running time of the algorithm is O([n/p] log p). The resulting speed-up of O(p/ log p) is optimal due to logarithmic communication overhead, as shown by a corresponding lower bound.
Journal of Parallel and Distributed Computing, 1995
We consider a new class of routing requests, or partial permutations, for which we give optimal o... more We consider a new class of routing requests, or partial permutations, for which we give optimal on-line routing algorithms on the hypercube and shu e-exchange network. For well-formed words of parentheses our algorithm establishes communication between all matching pairs in logarithmic time. It can be applied to the membership problem for Dyck languages and a number of problems dealing with algebraic expressions.
An optimal tree contraction algorithm for the boolean hypercube and the constant-degree hypercubi... more An optimal tree contraction algorithm for the boolean hypercube and the constant-degree hypercubic networks, such as the shuffle exchange or the butterfly network, is presented. The algorithm is based on novel routing techniques and, for certain small subtrees, simulates optimal PRAM algorithms. For trees of size n, stored on a p processor hypercube in in-order, the running time of the algorithm is \(O(\lceil n/p\rceil\log p)\) . The resulting speed-up of \(O(p/\log p)\) is optimal due to logarithmic communication overhead, as shown by a corresponding lower bound. The same algorithmic ingredients can also be used to solve the term matching problem, one of the fundamental problems in logic programming.
We show how to implement divide-and-conquer algorithms without undue overhead on a wide class of ... more We show how to implement divide-and-conquer algorithms without undue overhead on a wide class of networks. We give an optimal generic divide-and-conquer implementation on hypercubes for the class of divide-and-conquer algorithms for which the total size of the subproblems on any level of the recursion does not exceed the parent problem size. For this implementation, appropriately sized subcubes have to be allocated to the subproblems generated by the divide-steps. We take care that these allocation steps do not cause any unbalanced distribution of work, and that, asymptotically, they do not increase the running time. Variants of our generic algorithm also work for the butter y network and, by a general simulation, for the class of hypercubic networks, including the shu e-exchange and the cube-connected-cycles network. Our results can also be applied to optimally solve various types of routing problems.
Circuit size, branching program size, and formula size of Boolean functions, denoted by C(f), BP(... more Circuit size, branching program size, and formula size of Boolean functions, denoted by C(f), BP(f), and L(f), are the most important complexity measures for Boolean functions. Often also the formula size L*(f) over the restricted basis {∨, ∧, ⌝} is considered. It is well-known that C(f) ≤ 3 BP(f), BP(f) ≤ L*(f), L*(f) ≤ L(f) 2, and C(f) ≤ L(f) - 1. These estimates are optimal. But the inequality BP(f) ≤ L(f) 2 can be improved to BP(f) ≤ 1.360 L(f) β, where β = log4(3 + √5) < 1.195.
Abstract In many applications like verification or combinatorial opt imization, OBDDs (ordered bi... more Abstract In many applications like verification or combinatorial opt imization, OBDDs (ordered binary decision diagrams) are used as a representation or data structure for Boolean functions. Efficient algorithms exist for the important operations on O BDDs, and many functions can be represented in reasonable size if a good variable orderin g is chosen. In general, it is NP-hard to compute
We present a randomized on-line algorithm for the list update problem which achieves a competitiv... more We present a randomized on-line algorithm for the list update problem which achieves a competitive factor of 1.6, the best known so far. The algorithm makes an initial random choice between two known algorithms that have different worst-case request sequences. The first is the BIT algorithm that, for each item in the list, alternates between moving it to the front of the list and leaving it at its place after it has been requested. The second is a TIMESTAMP algorithm that moves an item in front of less often requested items within the list.
... In that case the searcher searches a connected subgraph of size k rather than a path of lengt... more ... In that case the searcher searches a connected subgraph of size k rather than a path of length k. The optimal searcher path problem has been investigated in the literature as one-sided search, where a prior hiding location distribution is given [24]. ...
Data structures for Boolean functions build an essential component of design automation tools, es... more Data structures for Boolean functions build an essential component of design automation tools, especially in the area of logic synthesis. The state of the art data structure is the ordered binary decision diagram (OBDD), which results from general binary decision diagrams (BDDs), also called branching programs, by ordering restrictions. In the context of EXOR-based logic synthesis another type of decision diagram (DD), called (ordered) functional decision diagram ((O)FDD) becomes increasingly important. We study the relation between (ordered, free) BDDs and FDDs. Both, BDDs and FDDs, result from DDs by defining the represented function in different ways. If the underlying DD is complete, the relation between both types of interpretation can be described by a Boolean transformation τ. This allows us to relate the FDD-size of f and the BDD-size of τ(f) also in the case that the corresponding DDs are free or ordered, but not (necessarily) complete. We use this property to derive several results on the computational power of OFDDs and OBDDs. Symmetric functions are shown to have efficient representations as OBDDs and OFDDs as well. Classes of functions are given that have exponentially more concise OFDDs than OBDDs, and vice versa. Finally, we determine the complexity of some standard operations if OFDDs are used for the representation of Boolean functions.
Information and Computation/information and Control, 1995
Data structures for Boolean functions build an essential component of design automation tools, es... more Data structures for Boolean functions build an essential component of design automation tools, especially in the area of logic synthesis. The state of the art data structure is the ordered binary decision diagram (OBDD), which results from general binary decision diagrams (BDDs), also called branching programs, by ordering restrictions. In the context of EXORbased logic synthesis another type of decision diagram (DD), called (ordered) functional decision diagram ((O)FDD) becomes increasingly important.
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Papers by Ralph Werchner