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Single Family Algebra Operation on BDDs and ZDDs Leads To Exponential Blow-Up

Kengo Nakamura, Masaaki Nishino, Shuhei Denzumi

TL;DR

The paper proves that most Knuth-style family algebra operations on ZDDs/BDDs admit no polynomial-time bound in the worst case, producing exponential growth in the output size even when input sizes are polynomial and the element order is favourable. It constructs specific polynomial-size families, such as the hidden weighted bit function $\mathcal{H}_m$ and the permutation function $\mathcal{P}_m$, to force exponential blow-ups for joins, quotients, and related operations, and shows this persists under arbitrary element ordering. The authors provide detailed lower-bound constructions for join, quotient, restrict, and other operations, and extend the results to both ZDDs and BDDs, resolving open questions about the complexity of the latter. The findings have practical implications for the use of BDD/ZDD-based family algebra in applications, indicating that worst-case blow-up cannot generally be avoided by reordering or dynamic reordering strategies in standard packages. Overall, the work delineates which family-algebra operations can be performed in polynomial time and which inherently incur exponential time in the worst case.

Abstract

Binary decision diagram (BDD) and zero-suppressed binary decision diagram (ZDD) are data structures to represent a family of (sub)sets compactly, and it can be used as succinct indexes for a family of sets. To build BDD/ZDD representing a desired family of sets, there are many transformation operations that take BDDs/ZDDs as inputs and output BDD/ZDD representing the resultant family after performing operations such as set union and intersection. However, except for some basic operations, the worst-time complexity of taking such transformation on BDDs/ZDDs has not been extensively studied, and some contradictory statements about it have arisen in the literature. In this paper, we show that many transformation operations on BDDs/ZDDs, including all operations for families of sets that appear in Knuth's book, cannot be performed in worst-case polynomial time in the size of input BDDs/ZDDs. This refutes some of the folklore circulated in past literature and resolves an open problem raised by Knuth. Our results are stronger in that such blow-up of computational time occurs even when the ordering, which has a significant impact on the efficiency of treating BDDs/ZDDs, is chosen arbitrarily.

Single Family Algebra Operation on BDDs and ZDDs Leads To Exponential Blow-Up

TL;DR

The paper proves that most Knuth-style family algebra operations on ZDDs/BDDs admit no polynomial-time bound in the worst case, producing exponential growth in the output size even when input sizes are polynomial and the element order is favourable. It constructs specific polynomial-size families, such as the hidden weighted bit function and the permutation function , to force exponential blow-ups for joins, quotients, and related operations, and shows this persists under arbitrary element ordering. The authors provide detailed lower-bound constructions for join, quotient, restrict, and other operations, and extend the results to both ZDDs and BDDs, resolving open questions about the complexity of the latter. The findings have practical implications for the use of BDD/ZDD-based family algebra in applications, indicating that worst-case blow-up cannot generally be avoided by reordering or dynamic reordering strategies in standard packages. Overall, the work delineates which family-algebra operations can be performed in polynomial time and which inherently incur exponential time in the worst case.

Abstract

Binary decision diagram (BDD) and zero-suppressed binary decision diagram (ZDD) are data structures to represent a family of (sub)sets compactly, and it can be used as succinct indexes for a family of sets. To build BDD/ZDD representing a desired family of sets, there are many transformation operations that take BDDs/ZDDs as inputs and output BDD/ZDD representing the resultant family after performing operations such as set union and intersection. However, except for some basic operations, the worst-time complexity of taking such transformation on BDDs/ZDDs has not been extensively studied, and some contradictory statements about it have arisen in the literature. In this paper, we show that many transformation operations on BDDs/ZDDs, including all operations for families of sets that appear in Knuth's book, cannot be performed in worst-case polynomial time in the size of input BDDs/ZDDs. This refutes some of the folklore circulated in past literature and resolves an open problem raised by Knuth. Our results are stronger in that such blow-up of computational time occurs even when the ordering, which has a significant impact on the efficiency of treating BDDs/ZDDs, is chosen arbitrarily.
Paper Structure (15 sections, 10 theorems, 6 equations, 3 figures, 1 table)

This paper contains 15 sections, 10 theorems, 6 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our Contribution
  4. Preliminaries
  5. Zero-suppressed Binary Decision Diagram
  6. Family Algebra Operations on ZDDs
  7. Blow-Up Operations
  8. High-Level Idea
  9. Proofs with Specific Element Order
  10. Join, Disjoint Join, Joint Join, Meet, and Delta
  11. Quotient and Remainder
  12. Restrict, Permit, Nonsuperset, and Nonsubset
  13. Maximal and Minimal
  14. Minimal Hitting Set and Closure
  15. Consideration for Element Order

Key Result

Lemma 2

For any family $\mathcal{F}$ of subsets of a set of $n$ elements and any order $<$ of elements, $B_<(\mathcal{F})=O( nZ_<(\mathcal{F}) )$ and $Z_<(\mathcal{F})=O( nB_<(\mathcal{F}) )$.

Figures (3)

  • Figure 1: (a) Example of a ZDD representing the family of subsets of $\{x_1,\ldots,x_5\}$ such that the cardinality is less than $3$. (b) Schematic of node sharing. (c) Schematic of zero suppression.
  • Figure 2: Example of blow-up for join (left) and quotient (right) operations. Blue triangles mean that the ZDD size representing this family is polynomial in $m$, while red triangle means that its size is exponential in $m$. Arcs going to $\bot$ terminal are omitted.
  • Figure 3: Example of blow-up for permit (left) and maximal (right) operations.

Theorems & Definitions (14)

  • Definition 1
  • Lemma 2: knuth11
  • Definition 3
  • Theorem 4: bryant91
  • Definition 5
  • Theorem 6: knuth11
  • Theorem 7
  • Lemma 8
  • Theorem 9
  • Theorem 10
  • ...and 4 more