Tight Timing Estimation With theNewton-Gregory Formulae

Robert van Engelen

Kyle Gallivan

Burt Walsh

Florida State University

CPC 2003 – p.1/30

Introduction• Worst-case execution time (WCET) estimation

has important applications in scheduling andreal-time embedded systems.

0

5

10

15

20

25

30

0 20 40 60 80 100

(ms)

TaskWCET

The WCET bound is obtained by static (compiler)analysis of a task and is used to determine if thetask will finish within a given time frame.

CPC 2003 – p.2/30

Parameterized WCET• Parametric timing analysis produces formulae,

where the unknown values affecting theexecution are parameterized.

0

5

10

15

20

0 20 40 60 80 100

(ms)

Input Size

TaskWCET

WCET = 5.0 + 0.1 N

where N is the task input size.

CPC 2003 – p.3/30

Related Work• Presburger formulas [Pugh, 1994]• Summations over τ functions [Haghighat, 1995]• Ehrhart polynomials [Clauss, 1996]• Guarded sums [Sakellariou, 1997]• Parametric timing analysis [Vivancos et al., 2001]

CPC 2003 – p.4/30

Experiences• We incorporated Sakellariou’s guarded sums into

a WCET estimation framework which requiredlinking a small computer algebra system with theVPO compiler [Healy, 2000]

• The timing estimation can be very loose for loopswith slightly more complicated control flow

for i = 0 to 9 do

if a[i] > 0 then

for j = 1 to i do

b[j] = b[i] + a[i]

else

b[i] = a[i]

CPC 2003 – p.5/30

Motivation• WCET analysis focus on loops, because most of

the execution time of a task is typically spent inloops.

• Loops may have symbolic bounds and exhibitcomplicated control flow.

for i = 0 to N do

if . . . then

for j = 0 to i do

. . .

else

for j = 0 to i do

for k = 0 to j do

. . .CPC 2003 – p.6/30

Motivation (cont’d)Existing methods. . .

• are computationally expensive and requireextensive (symbolic) manipulation.

• can lead to an over estimation of the WCET ofloops with complicated control flow due to thedecoupling of the WCET of the loop body andthe iteration count

WCET = WCET (loopHeader)

+ WCET (loopBody) ∗ iterCount

CPC 2003 – p.7/30

Loop Execution Time• Loops are assumed to have been normalized

for i = a to b step s do

S⇒

for i = 0 to b b−a

sc do

S′

• The unknown loop execution time functions are

ω(H(i)) = Loop header execution time

ω(S ′(i)) = Loop body execution time

• The (unknown) real loop execution time ω is

ω = ω(H(0)) +

b b−a

sc

∑

i=0

ω(S ′(i)) + ω(H(i + 1))

CPC 2003 – p.8/30

Loop Execution Time Bound• First we recursively determine (parameterized)

WCET bounds on the unknowns

ω(H(0)) ≤ c0

ω(S ′(i)) + ω(H(i + 1)) ≤ p(i)

• Then, the WCET bound on ω is

ω ≤ WCET = c0 +n−1∑

i=0

p(i)

provided that the number of loop iterationsn = bb−a

sc+ 1 > 0.

CPC 2003 – p.9/30

Newton-Gregory• Let p(i) = p0 + p1 i + · · ·+ pk ik be a polynomial.

There exist Newton series coefficients

Φ(i) = 〈φ0, φ1, . . . , φk〉i

such that

p(i) =k

∑

j=0

φj

(

i

j

)

CPC 2003 – p.10/30

Newton-Gregory WCETLemma.

• To determine the WCET of a loop, we canreplace the WCET iteration summation with aconstant summation

WCET = c0 +n−1∑

i=0

p(i) =k

∑

j=0

φj

(

n

j + 1

)

for all (symbolic) n > 0.

CPC 2003 – p.11/30

Newton Series Conversions• Given coefficients [p0, p1, . . . , pk]i of polynomial

p(i) = p0 + p1 i + · · · + pk ik, then the Newtonseries is

Φ(i) = Nkp(i)

where Nk is the constant integer(k + 1) × (k + 1) “Newton triangle matrix”.

• Given Newton series coefficientsΦ(i) = 〈φ0, φ1, . . . , φk〉i, then

p(i) = N−1k Φ(i)

where N−1k is the constant rational

(k + 1) × (k + 1) “inverse Newton trianglematrix”. CPC 2003 – p.12/30

WCET of a Loop• Let Φ(i) = 〈φ0, . . . , φk〉i denote the Newton

series of a polynomial in i over i = 0, . . . , n− 1,n ≥ 0. We define

σ(Φ(i), n)def=

k∑

j=0

φj

(

n

j + 1

)

• The WCET of a (possibly zero trip) loopi = a, . . . , b, stride s, and iteration WCETpolynomial p(i) with Newton series Φ(i) is

WCET = c0 + σ(Φ(i), max(0, b−as

+ 1))

CPC 2003 – p.13/30

A Useful PropertyLemma.

• The coefficients s0, . . . , sk+1 of the polynomials(n) = σ(Φ(i), n) are given by the matrix-vectorproduct

s(n) = [s0, . . . , sk+1]n = N−1k+1〈0, φ0, . . . , φk〉i

where Φ(i) = 〈φ0, φ1, . . . , φk〉i.

CPC 2003 – p.14/30

Application• Hence, the WCET of a loop is

WCET = c0 + σ(Φ(i), max(0, b−as

+ 1)))

= c0 + s(max(0, b−as

+ 1)))

where s(n) = N−1k+1〈0, φ0, . . . , φk〉i.

CPC 2003 – p.15/30

An Example

for I = 1 to N do /* header H1 */

for J = I to I ∗ I − 2 step 2 do /* header H2 */

S

The normalized iteration space is

i = 0, . . . , N − 1

j = 0, . . . , b i+i2

2 c − 1

The (recursively determined) execution time bounds

ω(H1(0)) ≤ c0

ω(H ′2(i, 0)) ≤ c1

ω(S ′(i, j)) + ω(H ′2(i, j + 1)) ≤ c2

CPC 2003 – p.16/30

An Example (cont’d)

WCET 1

= c0 + σ(N1(c1 + σ(N0c2, max(0, i+i2

2 ))), max(0, N))

(i + i2 ≥ 0, so replace max(0, i+i2

2 ) by i+i2

2 )= c0 + σ(N1(c1 + σ(N0c2,

i+i2

2 )), max(0, N))

= c0 + σ(N1(c1 + σ(c2,i+i2

2 )), max(0, N))

= c0 + σ(N1(c1 + 12c2(i + i2)), max(0, N))

= c0 + σ(〈c1, c2, c2〉i, max(0, N))

= c0 + (c1 −16c2) max(0, N) + 1

6c2 max(0, N)3

CPC 2003 – p.17/30

An Example (cont’d)For c0 = 1, c1 = 1, and c2 = 2 we have

0100200300400500600700800900

1000

0 2 4 6 8 10

WCET 2WCET 1

where the conventional WCET estimation of theexample fragment gives the over estimation

WCET 2 = c0 + c1 N + c2 NbN2−N2 c

CPC 2003 – p.18/30

Loop Bounds Analysis• If the loop iteration space size is non-negative

(n = bb−asc+ 1 ≥ 0), the max operation can be

eliminated from the WCET expressionWCET = c0 + σ(Φ(i), max(0, b−a

s+ 1)).

• Otherwise, splitting with guards is used.• Use symbolic range propagation [Blume, 1995].• Or we can use Newton series representations

directly.

CPC 2003 – p.19/30

Monotonicity• If the values of a (symbolic) expression are

monotonic in a loop, its extreme values can beeasily determined (symbolically).

• Most array index and loop bound expressions aremonotonic (in one loop dimension).

• But how can we determine monotonicity withouttoo much effort?

CPC 2003 – p.20/30

Monotonic Polynomials• Let Φ(i) = 〈φ0, φ1, . . . , φk〉i denote the Newton

series of a polynomial p(i) in i.

• If h(i) = N−1k−1〈φ1, . . . , φk〉i ≥ 0 for all

i = 0, . . . , n− 2, n ≥ 0, then p(i) ismonotonically increasing on the interval[0, n − 1].

• Similarly, if h(i) ≤ 0 for all i = 0, . . . , n− 2,then p(i) is monotonically decreasing on theinterval [0, n − 1].

CPC 2003 – p.21/30

Monotonicity Example• p(i) = i2−i

2 with coefficients [0,−12 ,

12 ]i has

Newton series Φ(i) = N2[0,−12 ,

12 ]i = 〈0, 0, 1〉i.

• Because h(i) = N−11 〈0, 1〉i = i ≥ 0 for all i ≥ 0,

p(i) is monotonically increasing with i.• The value range of p(i) on i = 0, . . . , N is

[p(0), p(N)] = [0, N2−N2 ].

• While conventional value range analysis gives12[0, N ]2 − 1

2 [0, N ] = [−12N, 1

2N2].

CPC 2003 – p.22/30

Value Range Analysis

Value range of expression E = [L(E),U(E)]

Rewrite rules:L(E1 + E2) = L(E1) + L(E2)

L(Φ(i)) =

L(φ0) if L(〈φ1, . . . , φk〉i) ≥ 0

L(N−1

kΦ(n− 1)) if U(〈φ1, . . . , φk〉i) ≤ 0

L(N−1

kΦ(i)) otherwise

U(Φ(i)) =

U(φ0) if U(〈φ1, . . . , φk〉i) ≤ 0

U(N−1

kΦ(n− 1)) if L(〈φ1, . . . , φk〉i) ≥ 0

U(N−1

kΦ(i)) otherwise

CPC 2003 – p.23/30

Finding Critical PathsThe objective is to find the critical path

for i = 0 to n do

if C then

S1 /* path with parametric WCET cost p(i) */

else

S2 /* path with parametric WCET cost q(i) */

CPC 2003 – p.24/30

Comparing WCET Polynomials• Let Φ(i) and Ψ(i) be Newton series of

polynomials p(i) and q(i).• If L(Φ(i) −Ψ(i)) ≥ 0 then p(i) ≥ q(i).• If U(Φ(i) −Ψ(i)) ≤ 0 then p(i) ≤ q(i).

020406080

100120140160

0 2 4 6 8 10

(ms)

i

p(i)q(i)

CPC 2003 – p.25/30

Example

for i = 0 to N do

if a[i] > 0 then

for j = 0 to i do

S4

else

for j = 0 to i do

for k = 0 to j do

S8

ω(H1(0))≤ 1

ω(C2(i))≤ 1

ω(H3(i, 0))≤ 1

ω(S4(i, j)) + ω(H3(i, j + 1))≤ 2

ω(H6(i, 0))≤ 1

ω(H7(i, 0))≤ 1

ω(S8(i, j)) + ω(H7(i, j + 1))≤ 2

CPC 2003 – p.26/30

Example (cont’d)

WCET 1 = 7 + 61

6N + 21

2N 2 + 1

3N 3

WCET 2 = 1 + (N + 1) max(5 + 2N, 6 + 4N + N 2)

WCET 3 = 1 + (N + 1) max(5 + 2N, 6 + 5N + 2N 2)

0

500

1000

1500

2000

2500

3000

0 2 4 6 8 10

WCET 3WCET 2WCET 1

CPC 2003 – p.27/30

Bounding WCET Polynomials• When one of the WCET polynomials does not

dominate, we need to split the iteration space(root finding!) or derive an overall boundingpolynomial.

020406080

100120140160

0 2 4 6 8 10

p(i)q(i)

bound

CPC 2003 – p.28/30

Iteration ReversalWhen one of the WCET polynomials does notdominate and the paths are loop invariant, the iterationdirection can be reversed.

020406080

100120140160

0 2 4 6 8 10

(ms)

i

p(i)q(i)

020406080

100120140160

0 2 4 6 8 10

(ms)

i

p(i)q(i)

CPC 2003 – p.29/30

Conclusions• Parametric WCET has important applications.• Timing of loops with complicated control flow is

difficult.• Relatively simple numeric/symbolic method.• Approach also contributes to value range

analysis.• Implementation is work in progress.

CPC 2003 – p.30/30