Tight Timing Estimation With the Newton-Gregory Formulae

Transcription

1 CPC 2003 p.1/30 Tight Timing Estimation With the Newton-Gregory Formulae Robert van Engelen Kyle Gallivan Burt Walsh Florida State University

2 CPC 2003 p.2/30 Introduction Worst-case execution time (WCET) estimation has important applications in scheduling and real-time embedded systems (ms) Task WCET The WCET bound is obtained by static (compiler) analysis of a task and is used to determine if the task will finish within a given time frame.

3 CPC 2003 p.3/30 Parameterized WCET Parametric timing analysis produces formulae, where the unknown values affecting the execution are parameterized (ms) Task WCET Input Size W CET = N where N is the task input size.

4 CPC 2003 p.4/30 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]

5 CPC 2003 p.5/30 Experiences We incorporated Sakellariou s guarded sums into a WCET estimation framework which required linking a small computer algebra system with the VPO compiler [Healy, 2000] The timing estimation can be very loose for loops with slightly more complicated control flow for i = 0 to 9 do if a[i] > 0 then for j = 1 to i do else b[j] = b[i] + a[i] b[i] = a[i]

6 CPC 2003 p.6/30 Motivation WCET analysis focus on loops, because most of the execution time of a task is typically spent in loops. Loops may have symbolic bounds and exhibit complicated 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...

7 CPC 2003 p.7/30 Motivation (cont d) Existing methods... are computationally expensive and require extensive (symbolic) manipulation. can lead to an over estimation of the WCET of loops with complicated control flow due to the decoupling of the WCET of the loop body and the iteration count WCET = WCET (loopheader) + WCET (loopbody) itercount

8 CPC 2003 p.8/30 Loop Execution Time Loops are assumed to have been normalized for i = a to b step s do S b a for i = 0 to s 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 a s i=0 ω(s (i)) + ω(h(i + 1))

9 CPC 2003 p.9/30 Loop Execution Time Bound First we recursively determine (parameterized) WCET bounds on the unknowns ω(h(0)) c 0 ω(s (i)) + ω(h(i + 1)) p(i) Then, the WCET bound on ω is ω WCET = c 0 + n 1 i=0 p(i) provided that the number of loop iterations n = b a s + 1 > 0.

10 CPC 2003 p.10/30 Newton-Gregory Let p(i) = p 0 + p 1 i + + p k i k be a polynomial. There exist Newton series coefficients Φ(i) = φ 0, φ 1,..., φ k i such that p(i) = k φ j ( i j ) j=0

11 CPC 2003 p.11/30 Newton-Gregory WCET Lemma. To determine the WCET of a loop, we can replace the WCET iteration summation with a constant summation WCET = c 0 + n 1 i=0 for all (symbolic) n > 0. p(i) = k j=0 ( ) n φ j j + 1

12 Newton Series Conversions Given coefficients [p 0, p 1,..., p k ] i of polynomial p(i) = p 0 + p 1 i + + p k i k, then the Newton series is Φ(i) = N k p(i) where N k is the constant integer (k + 1) (k + 1) Newton triangle matrix. Given Newton series coefficients Φ(i) = φ 0, φ 1,..., φ k i, then p(i) = N 1 k Φ(i) where N 1 k is the constant rational (k + 1) (k + 1) inverse Newton triangle matrix. CPC 2003 p.12/30

13 CPC 2003 p.13/30 WCET of a Loop Let Φ(i) = φ 0,..., φ k 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 ( ) n φ j j + 1 The WCET of a (possibly zero trip) loop i = a,..., b, stride s, and iteration WCET polynomial p(i) with Newton series Φ(i) is WCET = c 0 + σ(φ(i), max(0, b a s + 1))

14 CPC 2003 p.14/30 A Useful Property Lemma. The coefficients s 0,..., s k+1 of the polynomial s(n) = σ(φ(i), n) are given by the matrix-vector product s(n) = [s 0,..., s k+1 ] n = N 1 k+1 0, φ 0,..., φ k i where Φ(i) = φ 0, φ 1,..., φ k i.

15 CPC 2003 p.15/30 Application Hence, the WCET of a loop is WCET = c 0 + σ(φ(i), max(0, b a s + 1))) = c 0 + s(max(0, b a s + 1))) where s(n) = N 1 k+1 0, φ 0,..., φ k i.

16 CPC 2003 p.16/30 An Example for I = 1 to N do /* header H 1 */ for J = I to I I 2 step 2 do /* header H 2 */ S The normalized iteration space is i = 0,..., N 1 j = 0,..., i+i2 2 1 The (recursively determined) execution time bounds ω(h 1 (0)) c 0 ω(h 2 (i, 0)) c 1 ω(s (i, j)) + ω(h 2(i, j + 1)) c 2

17 CPC 2003 p.17/30 An Example (cont d) WCET 1 = c 0 + σ(n 1 (c 1 + σ(n 0 c 2, max(0, i+i2 2 (i + i 2 0, so replace max(0, i+i2 2 = c 0 + σ(n 1 (c 1 + σ(n 0 c 2, i+i2 2 = c 0 + σ(n 1 (c 1 + σ(c 2, i+i2 2 ))), max(0, N)) i+i2 ) by 2 ) )), max(0, N)) )), max(0, N)) = c 0 + σ(n 1 (c c 2(i + i 2 )), max(0, N)) = c 0 + σ( c 1, c 2, c 2 i, max(0, N)) = c 0 + (c c 2) max(0, N) c 2 max(0, N) 3

18 CPC 2003 p.18/30 An Example (cont d) For c 0 = 1, c 1 = 1, and c 2 = 2 we have WCET 2 WCET where the conventional WCET estimation of the example fragment gives the over estimation WCET 2 = c 0 + c 1 N + c 2 N N 2 N 2

19 CPC 2003 p.19/30 Loop Bounds Analysis If the loop iteration space size is non-negative (n = b a s + 1 0), the max operation can be eliminated from the WCET expression W CET = c 0 + 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.

20 CPC 2003 p.20/30 Monotonicity If the values of a (symbolic) expression are monotonic in a loop, its extreme values can be easily determined (symbolically). Most array index and loop bound expressions are monotonic (in one loop dimension). But how can we determine monotonicity without too much effort?

21 CPC 2003 p.21/30 Monotonic Polynomials Let Φ(i) = φ 0, φ 1,..., φ k i denote the Newton series of a polynomial p(i) in i. If h(i) = N 1 k 1 φ 1,..., φ k i 0 for all i = 0,..., n 2, n 0, then p(i) is monotonically 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 the interval [0, n 1].

22 CPC 2003 p.22/30 Monotonicity Example p(i) = i2 i 2 with coefficients [0, 1 2, 1 2 ] i has Newton series Φ(i) = N 2 [0, 1 2, 1 2 ] i = 0, 0, 1 i. Because h(i) = N 1 1 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, N 2 N 2 ]. While conventional value range analysis gives 1 2 [0, N]2 1 2 [0, N] = [ 1 2 N, 1 2 N 2 ].

23 CPC 2003 p.23/30 Value Range Analysis Value range of expression E = [L(E), U(E)] Rewrite rules:l(e 1 + E 2 ) = L(E 1 ) + L(E 2 ) L(Φ(i)) = U(Φ(i)) = L(φ 0 ) if L( φ 1,..., φ k i ) 0 L(N 1 k Φ(n 1)) if U( φ 1,..., φ k i ) 0 L(N 1 k Φ(i)) otherwise U(φ 0 ) if U( φ 1,..., φ k i ) 0 U(N 1 k Φ(n 1)) if L( φ 1,..., φ k i ) 0 U(N 1 k Φ(i)) otherwise

24 CPC 2003 p.24/30 Finding Critical Paths The objective is to find the critical path for i = 0 to n do if C then S 1 /* path with parametric WCET cost p(i) */ else S 2 /* path with parametric WCET cost q(i) */

25 CPC 2003 p.25/30 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) (ms) i p(i) q(i)

26 CPC 2003 p.26/30 Example for i = 0 to N do if a[i] > 0 then for j = 0 to i do S 4 else for j = 0 to i do for k = 0 to j do S 8 ω(h 1 (0)) 1 ω(c 2 (i)) 1 ω(h 3 (i, 0)) 1 ω(s 4 (i, j)) + ω(h 3 (i, j + 1)) 2 ω(h 6 (i, 0)) 1 ω(h 7 (i, 0)) 1 ω(s 8 (i, j)) + ω(h 7 (i, j + 1)) 2

27 CPC 2003 p.27/30 Example (cont d) WCET 1 = N N N 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 ) WCET 3 WCET 2 WCET

28 CPC 2003 p.28/30 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 bounding polynomial p(i) q(i) bound

29 Iteration Reversal When one of the WCET polynomials does not dominate and the paths are loop invariant, the iteration direction can be reversed (ms) (ms) i p(i) q(i) p(i) q(i) CPC 2003 p.29/30

30 CPC 2003 p.30/30 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.