## Parity graph-driven read-once branching programs and an exponential lower bound for integer multiplication (2002)

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Venue: | In Proc. of 2nd TCS |

Citations: | 8 - 4 self |

### BibTeX

@INPROCEEDINGS{Bollig02paritygraph-driven,

author = {Beate Bollig and Stephan Waack},

title = {Parity graph-driven read-once branching programs and an exponential lower bound for integer multiplication},

booktitle = {In Proc. of 2nd TCS},

year = {2002},

pages = {83--94}

}

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### Abstract

Abstract Branching programs are a well-established computation model for boolean functions, especially read-once branching programs have been studied intensively. Exponential lower bounds for deterministic and nondeterministic read-once branching programs are known for a long time. On the other hand, the problem of proving superpolynomial lower bounds for parity read-once branching programs is still open. In this paper restricted parity read-once branching programs are considered and an exponential lower bound on the size of well-structured parity graph-driven read-once branching programs for integer multiplication is proven. This is the first strongly exponential lower bound on the size of a nonoblivious parity read-once branching program model for an explicitly defined boolean function. In addition, more insight into the structure of integer multiplication is yielded.

### Citations

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(Show Context)
Citation Context ...et of its internal nodes can be partitioned into disjoint sets Vi, 1 ≤ i ≤ l, such that all nodes from Vi are labeled by si and the edges which leave Vi-nodes reach a sink or a Vj-node, j > i. Bryant =-=[9]-=- has introduced ordered binary decision diagrams (OBDDs) which are up to now the most popular representation for formal circuit verification. OBDDs are oblivious BP1s, where on each path from the sour... |

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Citation Context ... size of a boolean function f is known to be a measure for the space complexity of nonuniform Turing machines and known to lie between the circuit size of f and its {∧, ∨, ¬}-formula size (see, e.g., =-=[19]-=-). Hence, one is interested in exponential lower bounds for more and more general types of BPs (for the latest breakthrough for semantic super-linear length BPs see [1], [3] and [4]). In order to deve... |

252 | On the complexity of VLSI implementations and graph representations of Boolean functions with application to integer multiplication
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Citation Context ...on defined as the ith bit of MULn. The middle bit of multiplication (MULn−1,n) is known to be the hardest bit. Hence, in the following we only consider the function MULn := MULn−1,n. For OBDDs Bryant =-=[10]-=- has presented an exponential lower bound of size 2 n/8 for MULn. Incorporating Ramsey theoretic arguments of Alon and Maass [2] and using the rank method of communication complexity, Gergov [11] has ... |

173 |
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Citation Context ...en marked. Because of Lemma 1 this node is uniquely determined. The resulting graph is a read-once branching program with one sink and at most |G| nodes. Finally, we use the usual algorithm (see also =-=[20]-=-) to insert nodes such that on each path from the source to the sink there exist for each variable xi exactly one node labeled by xi. According to a topological ordering of the nodes, for each node v ... |

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Citation Context ...f a (parity) G0-BP1 G is the number of nodes in G and not in G and G0.) For many restricted (nondeterministic) variants of branching programs exponential lower bounds are known (for a survey see e.g. =-=[15]-=-). Moreover, Thathachar [18] has been able to prove an exponential gap between the size of nondeterministic BPks and deterministic BP(k + 1)s for an explicitly defined boolean function. His results ha... |

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Citation Context ...∨, ¬}-formula size (see, e.g., [19]). Hence, one is interested in exponential lower bounds for more and more general types of BPs (for the latest breakthrough for semantic super-linear length BPs see =-=[1]-=-, [3] and [4]). In order to develop and strengthen lower bound techniques one considers restricted computation models. Definition 2 i) A branching program is called (syntactically) read k times (BPk) ... |

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44 |
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Citation Context ...fortunately, several important and also quite simplesParity Graph-driven Read-Once Branching Programs 3 functions have exponential OBDD size. Therefore, Gergov and Meinel [12] and Sieling and Wegener =-=[17]-=- have generalized independently the concept of variable orderings. Definition 3 A graph ordering is a branching program with a single sink, where on each path from the source to the sink all variables... |

41 |
Efficient analysis and manipulation of OBDDs can be extended to read-once-only branching programs
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Citation Context ...ion π on the variable set. Unfortunately, several important and also quite simplesParity Graph-driven Read-Once Branching Programs 3 functions have exponential OBDD size. Therefore, Gergov and Meinel =-=[12]-=- and Sieling and Wegener [17] have generalized independently the concept of variable orderings. Definition 3 A graph ordering is a branching program with a single sink, where on each path from the sou... |

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Citation Context ...ic linear-length oblivious BPs. Recently, Woelfel [21] has improved Bryant’s lower bound up to Ω(2 n/2 ). The first exponential lower bound on the size of deterministic BP1s has been proven by Ponzio =-=[14]-=-. His lower bound is of order 2 Ω(n1/2 ) and has been improved by Bollig and Woelfel [7] to the first strongly exponential lower bounds4 of size Ω(2 n/4 ) for MULn. Bollig [5] has presented the first ... |

32 | Super-linear time-space tradeoff lower bounds for randomized computation
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Citation Context ...-formula size (see, e.g., [19]). Hence, one is interested in exponential lower bounds for more and more general types of BPs (for the latest breakthrough for semantic super-linear length BPs see [1], =-=[3]-=- and [4]). In order to develop and strengthen lower bound techniques one considers restricted computation models. Definition 2 i) A branching program is called (syntactically) read k times (BPk) if ea... |

27 |
Meanders and their applications in lower bound arguments
- Alon, Maass
- 1988
(Show Context)
Citation Context ...owing we only consider the function MULn := MULn−1,n. For OBDDs Bryant [10] has presented an exponential lower bound of size 2 n/8 for MULn. Incorporating Ramsey theoretic arguments of Alon and Maass =-=[2]-=- and using the rank method of communication complexity, Gergov [11] has extended the lower bound to arbitrary nondeterministic linear-length oblivious BPs. Recently, Woelfel [21] has improved Bryant’s... |

20 | Time-space tradeoffs multiparty communication complexity and nearest neighbor problems
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(Show Context)
Citation Context ... size (see, e.g., [19]). Hence, one is interested in exponential lower bounds for more and more general types of BPs (for the latest breakthrough for semantic super-linear length BPs see [1], [3] and =-=[4]-=-). In order to develop and strengthen lower bound techniques one considers restricted computation models. Definition 2 i) A branching program is called (syntactically) read k times (BPk) if each varia... |

15 |
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- 1994
(Show Context)
Citation Context ...yant [10] has presented an exponential lower bound of size 2 n/8 for MULn. Incorporating Ramsey theoretic arguments of Alon and Maass [2] and using the rank method of communication complexity, Gergov =-=[11]-=- has extended the lower bound to arbitrary nondeterministic linear-length oblivious BPs. Recently, Woelfel [21] has improved Bryant’s lower bound up to Ω(2 n/2 ). The first exponential lower bound on ... |

13 | On separating the read-k-times branching program hierarchy
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- 1998
(Show Context)
Citation Context ... number of nodes in G and not in G and G0.) For many restricted (nondeterministic) variants of branching programs exponential lower bounds are known (for a survey see e.g. [15]). Moreover, Thathachar =-=[18]-=- has been able to prove an exponential gap between the size of nondeterministic BPks and deterministic BP(k + 1)s for an explicitly defined boolean function. His results have demonstrated that the low... |

13 | New bounds on the OBDD-size of integer multiplication via universal hashing
- Woelfel
- 2005
(Show Context)
Citation Context ...ents of Alon and Maass [2] and using the rank method of communication complexity, Gergov [11] has extended the lower bound to arbitrary nondeterministic linear-length oblivious BPs. Recently, Woelfel =-=[21]-=- has improved Bryant’s lower bound up to Ω(2 n/2 ). The first exponential lower bound on the size of deterministic BP1s has been proven by Ponzio [14]. His lower bound is of order 2 Ω(n1/2 ) and has b... |

10 |
A read-once branching program lower bound of Ω(2n/4) for integer multiplication using universal hashing
- Bollig, Woelfel
- 2001
(Show Context)
Citation Context ...up to Ω(2 n/2 ). The first exponential lower bound on the size of deterministic BP1s has been proven by Ponzio [14]. His lower bound is of order 2 Ω(n1/2 ) and has been improved by Bollig and Woelfel =-=[7]-=- to the first strongly exponential lower bounds4 of size Ω(2 n/4 ) for MULn. Bollig [5] has presented the first (not strongly) exponential lower bound on the size of MULn for so-called nondeterministi... |

10 | Linear hash functions - Alon, Dietzfelbinger, et al. - 1999 |

10 | Graph-based algorithms for Boolean manipulation - Bryant - 1986 |

8 | Restricted nondeterministic read-once branching programs and an exponential lower bound for integer multiplication
- Bollig
- 2001
(Show Context)
Citation Context ...s been proven by Ponzio [14]. His lower bound is of order 2 Ω(n1/2 ) and has been improved by Bollig and Woelfel [7] to the first strongly exponential lower bounds4 of size Ω(2 n/4 ) for MULn. Bollig =-=[5]-=- has presented the first (not strongly) exponential lower bound on the size of MULn for so-called nondeterministic tree-driven BP1s. Her result also holds for parity tree-driven BP1s. Until now expone... |

7 | A hierarchy result for read-once branching programs with restricted parity nondeterminism
- Savicky, Sieling
- 2000
(Show Context)
Citation Context ...r parity read-once branching programs is still open. Krause [13] has proved the first exponential lower bounds for oblivious parity branching programs with bounded length. Later, Savick´y and Sieling =-=[16]-=- have presented exponential lower bounds for restricted parity read-once branching programs. In their model only at the top of the read-once branching program parity nodes are allowed. Recently, Brose... |

7 | Efficient strongly universal and optimally universal hashing - Woelfel - 1999 |

5 |
Graph-driven free parity BDDs: algorithms and lower bounds
- Brosenne, Homeister, et al.
- 2001
(Show Context)
Citation Context ...l lower bounds for restricted parity read-once branching programs. In their model only at the top of the read-once branching program parity nodes are allowed. Recently, Brosenne, Homeister, and Waack =-=[8]-=- have proved the first (not strongly) exponential lower bound on the size of restricted parity graph-driven BP1s representing the characteristic function of linear codes. Motivated by applications the... |

3 |
Separating #-L from L, NL, co-NL and AL(= P) for oblivious Turing machines of linear access time
- Krause
- 1992
(Show Context)
Citation Context ... that the lower bound techniques for these models are highly developed. Nevertheless, the problem of proving superpolynomial lower bounds for parity read-once branching programs is still open. Krause =-=[13]-=- has proved the first exponential lower bounds for oblivious parity branching programs with bounded length. Later, Savick´y and Sieling [16] have presented exponential lower bounds for restricted pari... |

2 | Polynomial size Ω-branching programs and their computational power - Meinel - 1990 |