Tighter Bounds on the OBDD size of Integer Multiplication

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Tighter Bounds on the OBDD size of Integer Multiplication

KAZUYUKIAMANO∗

GSIS, Tohoku Univ.

Aoba 6-6-05, Sendai, Miyagi, 980-8579 Japan ama@ecei.tohoku.ac.jp

AKIRAMARUOKA

GSIS, Tohoku Univ.

Aoba 6-6-05, Sendai, Miyagi, 980-8579 Japan

maruoka@ecei.tohoku.ac.jp

Abstract: We show that the middle bit of the multiplication of two n-bit integers can be computed by an OBDD of size less than 2.8·2^6n/5. This improves the previously known upper bound of(7/3)·2^4n/3by Woelfel (STACS, 2001). The experimental results suggest that our exponent of 6n/5 is (at least very close to) optimal. A general upper bound of O(2^3n/2)on the OBDD size of each output bit of the multiplication and some conjecture on the structural properties of multipliers inspired by the experimental results are also presented.

Keywords: OBDD, integer multiplication, upper bounds

1 Introduction

Ordered binary decision diagrams (OBDDs), which were first introduced by Bryant [4], are nowadays one of the most well- established computational models for representing and manipulating Boolean functions. OBDDs are widely used in the areas of hardware verification, model checking, and computer aided design (see e.g., [9, 12]).

Definition 1 Let X_n={x₁, . . . ,x_n}be a set of Boolean variables. A variable orderingπ^{on X}nis a permutation on{1, . . . ,n} leading to the ordered list x_π(1), . . . ,x_π(n)of the variables.

Aπ^{-OBDD on X}nis a directed acyclic graph whose sinks are labeled by a constant 0 or 1 and whose inner nodes are labeled by a Boolean variables from X_n. Each inner node has two outgoing edges, one of them labeled by 0, the other by 1.

The edges between inner nodes have to respect the variable orderingπ, i.e., if an edge leads from an xi-nodes to an xj-node, thenπ⁻¹(i)<π⁻¹(j). Each node v represents a Boolean function fv:{0,1}ⁿ→ {0,1}defined in the following way: An assignment(a1, . . . ,an)∈ {0,1}ⁿto variables Xndefines a uniquely determined path from v to one of the sinks. The label of the reached sink gives f_v(a). The size of aπ-OBDD is defined as the number of its nodes. The OBDD size of f , denoted by OBDD(f)is the minimum size of allπ-OBDDs that compute f . Aπ-OBDD for some unspecified variable order is simply called OBDD.

For many practically relevant functions, such as symmetric functions, the corresponding OBDD representations are quite small. However, for several important functions, exponential lower bounds on the size of an OBDD representation are known.

The most “famous” such function is, probably, (the middle bit of) integer multiplication.

Definition 2 For each 0≤k≤2n−1, let MUL_k,n:{0,1}²ⁿ→ {0,1}denote the Boolean function that outputs z_k of the product(z2n−1···z₀)of two n-bit integers(xn−1···x₀)and(yn−1···y₀), where x0, y₀and z₀are the least significant bits.

The middle bit of integer multiplication is denoted byMULn−1,n. Since for any Boolean function on m variables, there exists an OBDD of size(2+ε)2^m/m [8], the trivial upper bound on OBDD(MULn−1,n) is O(2²ⁿ/n). In 1991, Bryant [5]

first proved an exponential lower bound of 2^8/non OBDD(MUL_n₋_1,n). Only recently, Woelfel have succeeded to improve the upper and lower bounds on the size of OBDD forMUL_n₋_1,n[14]. Precisely, he showed that OBDD(MUL_n₋_1,n) is between

2^n/2/61 and(7/3)·2^4n/3. His lower bound rules out the possibility of constructing an OBDD for 64-bit multiplication with a

reasonable size. Nevertheless, there still exists a considerable gap between the upper and lower bounds.

The main objective of this work is to determine the asymptotic behavior of the size of OBDDs forMULn−1,n, or more generally, forMULk,n. Though we started this work by theoretical interest, it may also have an application in the design of multipliers. In addition, we believe that this may help to make progress on the investigations of the complexity of multiplication for more general models of branching programs, which have been extensively studied recently (e.g., [1, 3, 11, 13]).

In the paper, we mainly consider a restricted variant of OBDDs which is called leveled OBDDs or quasi-OBDDs, denoted by QOBDDs. This is because analyzing the size of QOBDDs is easier than that of OBDDs.

∗Corresponding Author. The work was supported in part by Grant-in-Aid for Scientific Research on Priority Areas “New Horizons in Computing” from MEXT of Japan.

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10 KAZUYUKIAMANO, AKIRAMARUOKA

Definition 3 Aπ-QOBDD is aπ-OBDD with the additional property that each edge from an x_π(i)-node for i<n reaches an x_π(i+1)-node. In other words, each path in aπ-QOBDD examines every variable exactly once in the order ofπ^{. Let} π-QOBDD(f)denote the minimum size ofπ-QOBDDs that compute f . The QOBDD size of a Boolean function f , denoted by QOBDD(f)is the minimum size of allπ-QOBDDs that compute f , i.e., QOBDD(f) =min_ππ-QOBDD(f). Aπ^-QOBDD for some unspecified variable order is simply called QOBDD.

Since everyπ-OBDD can be transformed into aπ-QOBDD by inserting dummy nodes on paths from the root to a sink, it is obvious that

OBDD(f)≤QOBDD(f)≤(n+1)OBDD(f)

for every Boolean function f on n variables. Thus, the size of QOBDDs can be considered essentially the same as that of OBDDs, especially for a function having an exponential complexity, such as multiplication. A detailed discussion on the relationship between the OBDD size and the QOBDD size can be found in e.g., [2, 8].

The contributions of the paper are as follows: First, in Section 2, we show thatMULn−1,ncan be computed by a QOBDD of size less than 2.8·2^6n/5, which improves the previously known upper bound of(7/3)·2^4n/3[14]. Second, we obtain the optimal QOBDDs forMULn−1,n for small values of n by an exhaustive search using a computer, and analyze them.

Interestingly, our experimental results suggest that the exponent of 6n/5 in our upper bound is the true exponent of the QOBDD size ofMULn−1,n. This will be described in Section 3. Next, in Section 4, we give a general upper bound on the QOBDD size of each output bit of integer multiplication. Precisely, we show that QOBDD(MUL_k,n) =O(2^c)where c=6k/5 for 0≤k≤5n/4, c=3n/2 for 5n/4<k≤3n/2, and c=3n−k for 3n/2<k≤2n−1. Finally, in Section 5, we describe some conjecture on the structural properties of multipliers inspired by the results of our experiments, which may lead to a better lower bound on the OBDD size ofMULn−1,n.

2 Upper Bounds for MUL

_n₋_1,n

In this section, we show an upper bound of 2.8·2^6n/5on the OBDD size of the middle bit of integer multiplication.

Let X= (xn−1···x₀)be an n-bit binary string. We also use X to denote the integer represented by x_n₋₁···x₀, i.e., X=

∑ⁿ_i=0⁻¹2ⁱxi. For 0≤i≤j≤n−1, let[X]ⁱ_jbe the integer represented by the substring xj···xi. Formally, [X]ⁱ_j = X div 2ⁱmod 2^j⁻ⁱ⁺¹=X mod 2^j+1div 2ⁱ.

Here we use the operators “mod” that gives the integer reminder of division, and “div” that gives the integer result of division.

We abbreviate[X]ⁱ_iby[X]i. For a set S,|S|denotes the cardinality of S.

Theorem 4 There is a QOBDD forMUL_n₋_1,nwhose size is less than 2.8·2^6n/5.

PRO OF: Let X = (xn−1···x₀) and Y= (yn−1···y₀)be the input variables for MULn−1,n. Let π be the variable ordering (x0,y0,x₁,y₁, . . . ,x_n₋₁,y_n₋₁). For the sake of simplicity, we suppose that n is a multiple of 5. (Other cases will be discussed later.) Below we show thatπ-QOBDD(MULn−1,n)≤19/7·2^6n/5<2.72·2^6n/5.

Let n=5k. Let _i,jdenote the set of subfunctions ofMUL_n₋_1,nthat obtained by replacing the variables x₀, . . . ,x_i₋₁and y₀, . . . ,y_j₋₁with constants. It is easy to verify that

π-QOBDD(MULn−1,n) =| 0,0|+| 1,0|+| 1,1|+···+| n,n|. (1) This is because that the number of x_i-nodes (y_i-nodes, resp.) in an optimalπ^{-QOBDD for}^MULn−1,nis shown to be| i,i| (| i+1,i|, resp.). Thus, our goal is to bound the number of different subfunctions in _i,iand in _i+1,i.

We first bound the size of _i,i. Suppose that n/2≤i<n.

Let X_L= (xi−1···x₀), YL = (yi−1···y₀), XH =X\X_L= (xn−1···x_i) and Y_H =Y\Y_L= (yn−1···y_i), which means that X=2ⁱX_H+X_Land Y=2ⁱY_H+YL. We focus on the “middle part” of X·Y , namely,[X·Y]ⁱ_n₋₁of which the most significant bit representsMUL_n₋_1,n(X,Y). We have

X·Y mod 2ⁿ = (2ⁱX_H+X_L)·(2ⁱY_H+YL)mod 2ⁿ

= 2ⁱ(XH·YL+YH·XL) +XL·YLmod 2ⁿ

= 2ⁱ(XH·[YL]⁰_n₋_i₋₁+YH·[XL]⁰_n₋_i₋₁) + [XL·Y_L]⁰_n₋₁mod 2ⁿ. Further, we have

[X·Y]ⁱ_n₋₁= (X·Y mod 2ⁿ)div 2ⁱ= (XH·[YL]⁰_n₋_i₋₁+YH·[XL]⁰_n₋_i₋₁) + [XL·Y_L]ⁱ_n₋_i.

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KAZUYUKIAMANO, AKIRAMARUOKA 11

This implies that the value of[X·Y]ⁱ_n₋₁and henceMUL_n₋_1,n(X,Y)is uniquely determined by XH, YH,[XL]⁰_n₋_i₋₁,[YL]⁰_n₋_i₋₁ and[XL·Y_L]ⁱ_n₋₁. Hence, each subfunction f_X_L_,Y_L∈ i,iis uniquely determined by[XL]⁰_n₋_i₋₁,[YL]⁰_n₋_i₋₁and[XL·Y_L]ⁱ_n₋₁, each of them has length n−i. Therefore, we have| i,i| ≤2³⁽ⁿ⁻ⁱ⁾for n/2≤i<n. Note that this bound is better than the trivial upper bound of| i,i| ≤2²ⁱwhen i≥3n/5. By an analogous argument to the case of| i,i|, we can also show that| i+1,i| ≤2³⁽ⁿ⁻ⁱ⁾⁻¹ for n/2≤i<n, which is better than the trivial bound of| i+1,i| ≤2²ⁱ⁺¹for i≥3n/5.

The upper bound ofπ-QOBDD(MULn−1,n)is now easily derived by plugging these bounds into Eq. (1). Namely, we have

π-QOBDD(MULn−1,n) =

3k−1 i=0

∑

(| i,i|+| i+1,i|) +

5k−1 i=3k

∑

(| i,i|+| i+1,i|) +| n,n|

≤

6k−1 i=0

∑

2ⁱ+

5k−1 i=3k

∑

(2^3(5k⁻ⁱ⁾+2^3(5k⁻ⁱ⁾⁻¹) +2

= 2^6k−1+

1+1

2 2k

i=1

∑

2³ⁱ+2

= 2^6k+3

2·8(2^6k−1)

7 +1<19 7 ·2^6k, which completes the proof for the case n=5k.

The other cases, i.e., n=5k+α^forα∈ {1,2,3,4}, can be shown analogously. The upper bounds are 40/7·2^6k, 96/7·2^6k, 208/7·2^6kand 544/7·2^6kforα=1,2,3 and 4, respectively. By a simple case checking, we can see that all these values are bounded by 2.8·2^6n/5. Note that the largest constant (= (544/7)/2^4.8∼2.7897) is attained whenα=4. 2

Remark that we use the variable orderingπ= (x0,y₀, . . . ,xn−1,y_n₋₁)in the proof of Theorem 4, whereas Woelfel [14]

used the orderingπ = (x0, . . . ,x_n₋₁,y₀, . . . ,y_n₋₁)to show the upper bound of O(2^4n/3). We also remark that Theorem 4 guarantees that the middle bit of a 16-bit multiplication can be computed by a QOBDD with less than 1.5 million nodes. In addition, the proof gives an explicit construction of such a QOBDD. Remarkably, the experimental results suggest that the exponent of 6n/5 in Theorem 4 is (at least very close to) the true exponent of the QOBDD size ofMULn−1,n, which we will describe in the next section.

3 Empirical Results

In this section, we describe the empirical results supporting the conjecture that the exponent of 6n/5 in the upper bound in Theorem 4 is optimal.

We did an exhaustive search by using a computer to find the optimal QOBDDs forMULn−1,nfor small values of n. Note that the best known algorithm for computing an optimal OBDD for a given function has an exponential running time [6, 7].

In addition, we believe that computing an optimal QOBDD is almost as hard as computing an optimal OBDD.

We use a standard dynamic programming approach to compute the size of optimal QOBDDs forMUL_n₋_1,nwhich we briefly describe below.

Let f be a Boolean function over the set of variables X ={x₁, . . . ,x_n}. For I⊆X , let sub(f,I)denote the number of subfunctions of f which we obtain by fixing all variables in X\I to constants. It is easy to verify that

QOBDD(f) = min

={I₀,...,In}

∑

0≤i≤n

sub(f,Ii)

where the minimum ranges over all sequences of sets /0=I0⊂I1⊂ ··· ⊂In=X with|Ii|=i. If we define QOBDD(f,I)for I⊆X by the following recursion,

QOBDD(f,I) = min

x∈I{QOBDD(f,I\{x}) +sub(f,I)}, (2) then QOBDD(f)is given by QOBDD(f,X). It should be noted that if we replace the term sub(f,I)in Eq. (2) by sub_x(f,I), which denotes the number of subfunctions of f obtained by fixing all variables in X\I and that essentially depend on x, then we can obtain OBDD(f)in a similar fashion [6, 7].

Using the above algorithm, we compute the size of optimal QOBDDs forMULn−1,n for n≤11. In addition, we also compute the minimum size ofπ-QOBDDs forMULn−1,nwith the variable orderingπ= (x0,y₀,y₁,y1, . . . ,x_n₋₁,y_n₋₁), which is used in the proof of Theorem 4.

The results are shown in Table 1. Remarkably, Table 1 shows that the QOBDD size ofMUL_n₋_1,nand also the minimum size ofπ-QOBDDs forMULn−1,nare almost proportional to 2^6n/5. This leads to a conjecture that QOBDD(MULn−1,n) =

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Table 1: The QOBDD size of MULn−1,n is shown in the second column, and the minimum size of π-QOBDDs for MULn−1,nwith the variable orderingπ= (x0,y₀, . . . ,x_n₋₁,y_n₋₁)is shown in the fourth column. Note that the computation of QOBDD(MULn−1,n)for n=12 has not completed. We have only confirmed that it is not larger than 22,180.

n QOBDD QOBDD/2^6n/5 π-QOBDD π-QOBDD/2^6n/5

4 39 1.40 56 2.01

5 72 1.13 109 1.70

6 156 1.06 230 1.56

7 348 1.03 490 1.45

8 797 1.03 1,106 1.43

9 1,808 1.01 2,490 1.40

10 4,106 1.00 5,751 1.40

11 9,796 1.04 13,228 1.41

12 ≤22,180 ≤1.03 30,862 1.43

13 71,239 1.43

14 166,981 1.46

Θ(2^6n/5), which means that the upper bound in Theorem 4 is tight up to a constant factor. Table 1 also shows that the optimal QOBDDs forMUL_n₋_1,nare almost 30% smaller than the optimalπ^-QOBDDs.

During the experiments, the optimal variable orderings forMUL_n₋_1,nare also obtained. For example, one of the optimal variable orderings forMUL_7,8is

(x1,x₂,x₃,x4,y₃,y₄,y₂,x₅,y₅,y₁,x6,y₆,x₀,y₇,x₇,y₀), and that forMUL_10,11is

(x3,x4,x₅,x₆,x7,y₄,y₅,y6,y₇,y₃,x₂,y2,x₁,y₁,x₈,y₈,x₉,x₉,x0,y₁₀,x₁₀,y₀).

We remark that the optimal variable ordering is not unique in general. Generalizing these orders may yield an upper bound with a slightly better constant factor than Theorem 4.

4 General Upper Bounds

In this section, we consider the size of a smallest QOBDD for the k-th bit of integer multiplication for general values of k.

The problem of determining the hardest bit of the multiplication and its complexity is interesting and important since the total complexity of the multiplication may essentially depend on the complexity of the hardest bit. It is well known that the middle bit is the “hardest” bit, in the sense that if it can be computed by OBDDs of size s(n), then any other bit can be computed with size at most s(2n)(e.g., [10]). However, this does not assert that the middle bit is exactly the hardest bit. The experimental results suggest that the hardest bit is located higher than the middle. For example, for a 8 bit multiplication, we verified that the 10-th output bit is the hardest for QOBDDs, namely, QOBDD(MUL_k,8) =797,1623,1937,2041,1755,1175 for k=7,8, . . . ,12, respectively. Recall that the 0-th bit is the least significant bit.

It is clear that computingMULk,nis not harder than computingMULk,k+1for every k. This is because if k<n, then MUL_k,n(X,Y) = MUL_k,k+1([X]⁰_k,[Y]⁰_k),

and if k≥n, then

MULk,n(X,Y) = MULk,k+1(0ⁿ⁻^k+1X,0ⁿ⁻^k+1Y).

Hence, the following corollary is a direct consequence of Theorem 4.

Corollary 5 For every k, there exists a QOBDD forMUL_k,nwhose size is O(2^6k/5).

Apparently, the upper bound in the above corollary overestimates the actual size of a smallest QOBDD forMUL_k,nif k is close to 2n. The following theorem asserts that the OBDD size of every single bit of multiplication is bounded by O(2^3n/2).

Theorem 6 For every k, there exists a QOBDD forMUL_k,nwhose size is O(2^3n/2).

For a∈ {0, . . . ,2ⁿ−1}, letMULâ_k,n:{0,1}ⁿ→ {0,1}be the function that outputs the k-th bit of the product of a with an n-bit number, i.e.,MULâ_k,n(X) =MUL_k,n(a,X). Here the 0-th bit is the least significant bit. To prove the theorem, we use the following lemma, which is a generalization of the results of Woelfel [14]. They showed that QOBDD(MULâ_k,n) =O(2^n/2) for k=n−1.

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KAZUYUKIAMANO, AKIRAMARUOKA 13

Lemma 7 For every k and for every a∈ {0, . . . ,2ⁿ−1}, there exists a QOBDD forMUL^a_k,nwhose size is O(2^n/2).

PROOF: If k<n, then the theorem follows immediately from the result of Woelfel described above. We therefore assume k≥n. Let Y = (yn−1···y₀)be the input variables forMUL^a_k,nand letπ be the variable ordering(y0, . . . ,y_n₋₁). Let i be the set of subfunctions ofMUL^a_k,nwhich we obtain by fixing the variables y₀, . . . ,y_i₋₁to constants. We will upper bound the number of subfunctions in _i. Let Y_L= (yi−1···y₀)and Y_H= (yn···y_i). Note that

MUL^a_k,n(Y) = MUL^a_k,n(YH,YL) =aY_H2ⁱ+aY_Lmod 2^k+1div 2^k.

For h∈ {0, . . . ,2ⁿ⁻ⁱ−1}, let z_h=ah2ⁱmod 2^k. Letρ^:{0, . . . ,2ⁿ⁻ⁱ−1} → {0, . . . ,2ⁿ⁻ⁱ−1}be a permutation that satisfies 0≤z_ρ(0)≤z_ρ(1)≤ ··· ≤z_ρ(2n−1)<2^k. For the sake of simplicity, we denote that z_ρ(₋₁₎=0 and z_ρ(2n−i)=2^k.

We claim that for every two distinct integers l,l⁰∈ {0, . . .,2ⁱ−1}such that 2^k−(al mod 2^k+1)and 2^k−(al⁰mod 2^k+1), or 2^k+1−(al mod 2^k+1)and 2^k+1−(al⁰mod 2^k+1)lie in the same interval, precisely,

z_ρ(t₋₁₎<2^k−(al mod 2^k+1)≤z_ρ(t)and z_ρ(t₋₁₎<2^k−(al⁰mod 2^k+1)≤z_ρ(t), (3) or

z_ρ(t₋₁₎<2^k+1−(al mod 2^k+1)≤z_ρ(t)and z_ρ(t₋₁₎<2^k+1−(al⁰mod 2^k+1)≤z_ρ(t), (4) for some 0≤t≤2ⁿ⁻ⁱ, the subfunctions ofMUL^a_k,nobtained by fixing Y_Lto l and to l⁰are identical.

The claim is proved as follows. We assume that l and l⁰satisfy condition (3). (The proof for the case of (4) is analogous to this case.) Since 0≤z_ρ(t₋₁₎+al mod 2^k+1<2^kand 0≤z_ρ(t₋₁₎+al⁰mod 2^k+1<2^k, we have thatMULâ_k,n(h,l) = MULâ_k,n(h,l⁰)for every h=ρ(t⁰) with t⁰∈ {0,1, . . .,t−1}. Similarly, since 2^k ≤z_ρ(t)+al mod 2^k+1<2^k+1 and 2^k≤ z_ρ(t)+al⁰mod 2^k+1<2^k+1, we have thatMULâ_k,n(h,l) =MULâ_k,n(h,l⁰)for every h=ρ(t⁰)with t⁰∈ {t, . . . ,2ⁿ⁻ⁱ−1}. Hence, we can conclude thatMULâ_k,n(h,l) =MULâ_k,n(h,l⁰)for every h∈ {0, . . . ,2ⁿ⁻ⁱ−1}, completing the proof of the claim.

The claim implies that the number of subfunctions in iis bounded by the number of such “intervals”, i.e., 2(2ⁿ⁻ⁱ+1), which is smaller than the trivial upper bound of 2ⁱwhen i>(n+1)/2. Hence, the size of aπ-QOBDD is bounded by

π-QOBDD(MUL^a_k,n) =

∑

n i=0

| i|=

∑

0≤i≤dn/2e

2ⁱ+

∑

dn/2e<i≤n

2(2ⁿ⁻ⁱ+1) =O(2^n/2).

This complete the proof of Lemma 7. 2

Theorem 6 follows immediately from Lemma 7.

PROOF:[of Theorem 6] Let X = (xn−1···x₀)and Y = (yn−1···y₀)be the input variables forMUL_k,n. We first construct a full binary tree T of depth n which examines all the variables in X . Note that each leaf in T corresponds to an n-bit integer.

Then, for each leaf in T that corresponds to an integer a, we connect a QOBDD that computesMUL^a_k,nto the leaf. Lemma 7 guarantees that the resulting QOBDD computesMUL_k,nand whose size is O(2ⁿ2^n/2) =O(2^3n/2). 2

As one might be expected, if the value of k is large enough, then a better upper bound can be obtained.

Theorem 8 For every k≥n, there exists a QOBDD forMULk,nwhose size is O(2³ⁿ⁻^k).

PROOF: Let X = (xn−1···x₀) and Y = (yn−1···y₀) be the input variables forMUL_k,n. Let π be the variable ordering (xn−1,yn−1,x_n₋₂,y_n₋₂, . . . ,x₀,y₀). Let i,j denote the set of subfunctions of MUL_k,n that obtained by fixing the variables x_n₋₁, . . . ,x_n₋_iand y_n₋₁, . . . ,y_n₋_jto constants. Note that the suffixes i and j indicate the number of fixed variables in X and Y , respectively.

In the following, we bound the number of subfunctions in _i,i. Let X_H= (xn−1···x_n₋_i), YH = (yn−1···y_n₋_i), XL= (xn−i−1···x₀)and Y_L= (yn−i−1···y₀). We obtain

X·Y mod 2^k+1 = (2ⁿ⁻ⁱX_H+X_L)·(2ⁿ⁻ⁱY_H+YL)mod 2^k+1

= 2²⁽ⁿ⁻ⁱ⁾X_H·Y_H+2ⁿ⁻ⁱ(XH·Y_L+X_L·Y_H) +X_L·Y_Lmod 2^k+1. (5) Suppose that k≥2n−i+3. Let l=k−(2n−i) +1. For 0≤x<2ⁿ, let x_H and x_L denote two integers that satisfy x=2ⁿ⁻ⁱx_H+x_L where 0≤x_H <2ⁱ and 0≤x_L<2ⁿ⁻ⁱ. For 0≤y<2ⁿ, y_H and y_L are defined analogously. For a pair of integers(xH,yH), let zH and z_Lbe two integers, depending on(xH,yH), such that

2²⁽ⁿ⁻ⁱ⁾x_H·y_Hmod 2^k+1=2²ⁿ⁻ⁱz_H+z_L, (6)

(14)

where 0≤zH<2^land 0≤zL<2²ⁿ⁻ⁱ. From Eqs. (5) and (6),MUL_k,n(x,y)is given by the most significant bit of 2²ⁿ⁻ⁱz_H+z_L+2ⁿ⁻ⁱ(xH·y_L+x_L·y_H) +xL·y_Lmod 2^k+1.

Since

z_L+2ⁿ⁻ⁱ(xH·y_L+x_L·y_H) +x_L·y_L<3·2²ⁿ⁻ⁱ, (7) if the j-th bit of the binary representation of z_H is 0 for some j≥2, then the carry generated by Eq. (7) is not propagated to a higher bit than the j-th bit of z_H. Hence, for every pairs of integers(xH,y_H)that satisfies[zH]j=0 for some j≥2, the subfunction ofMUL_k,n obtained by fixing X_H to x_H and Y_H to y_H is a constant function. Therefore, the number of subfunctions in _i,iis bounded by 2 plus the number of pairs of integers(xH,y_H)that satisfy the condition that[zH]j=1 for every 2≤j≤l−2, equivalently, the output bits i+2 through i+l−2 of the product x_H·y_Hare all 1. The number of such pairs is at most

∑

1≤y<2ⁱ

d2ⁱ⁺² y ed y2ⁱ

2^i+l⁻¹e <

∑

1≤y<2ⁱ

22ⁱ⁺² y ·2 y2ⁱ

2^i+l⁻¹

=

∑

1≤y<2ⁱ

2ⁱ⁻^l+5<2²ⁱ⁻^l+5. Hence, we have

| i,i| ≤ 2+2²ⁱ⁻^l+5<2²ⁱ⁻^k+(2n⁻ⁱ⁾⁺⁷=2²ⁿ⁻^k+i+7, (8)

which is better than the trivial bound of| i,i| ≤2²ⁱwhen i≥2n−k+7. The total size ofπ^{-QOBDD for}MUL_k,nis given by π-QOBDD(MUL_k,n) =

∑

0≤i<n

(| i,i|+| i+1,i|) +| n,n| ≤3

∑

0≤i<n

| i,i|+2. (9)

The last inequality follows from the simple fact that| i+1,i| ≤2| i,i|for every i. By plugging Eq (8) into Eq. (9), we have

π-QOBDD(MULk,n) ≤ 3

∑

0≤i≤2n−k+6

2²ⁱ+

∑

2n−k+7≤i<n

2²ⁿ⁻^k+i+7

! +2

≤ 3(2⁴ⁿ⁻^2k+13+2³ⁿ⁻^k+7) +2=O(2³ⁿ⁻^k).

The last equality follows from the assumption that k≥n. 2

Combining Corollary 5, Theorems 6 and 8, we have the following theorem.

Theorem 9 The QOBDD size ofMUL_k,nis O(2^c)where

c =





6k/5, for 0≤k≤5n/4, 3n/2, for 5n/4<k≤3n/2, 3n−k, for 3n/2<k≤2n−1.

The theorem says that every single bit of the multiplication of two n-bit integers can be computed by a QOBDD (and also by an OBDD) of size O(2^3n/2). Note that the best known lower bound for MUL_k,n is 2^b^(k+1)/2^c/61 for k<n and 2^b⁽²ⁿ⁻^k⁻^1)/2^c/61 for k≥n by Woelfel [14].

5 Concluding Remarks

Although we have not succeeded to improve the lower bound on the size of OBDDs forMULn−1,n, the experiments gave us some insight to improvements. Inspired by our experimental results, we make the following conjecture.

Let s(i), i=0,1, . . . ,be a sequence of integers defined by s(0) =1 and s(i) =





(6s(i−1)−1)/5 if i mod 4 is 0, (5s(i−1)−1)/3 if i mod 4 is 2,

2s(i−1) if i is odd.

(10) The first few numbers of this sequence are: 1,2,3,6,7,14,23,46,55,110,183,366,439,. . . . Note that if i is a multiple of 4, then s(i)has a simple closed formula, i.e., s(4 j) = (6·2^{3 j}+1)/7. This can be verified by observing that s(4(j+1)) =8·s(4 j)−1 from Eq. (10).

Tighter Bounds on the OBDD size of Integer Multiplication

Contents

Tighter Bounds on the OBDD size of Integer Multiplication

1 Introduction

2 Upper Bounds for MUL

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3 Empirical Results

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4 General Upper Bounds

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5 Concluding Remarks