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Approximating Inclusion-based Points-to Analysis

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Department of Computer Science and Automation,

Indian Institute of Science, Bangalore, India

MSPC 2011

June 05, 2011

Approximating Inclusion-based Points-to AnalysisPlacement of Pointer Analysis

Improved runtime.

Parallelizing compiler.

Lock synchronizer.

Memory leak detector.

Secure code.

Pointer Analysis.

Data flow analyzer.

String vulnerability finder.

Better compile time.

Affine expression analyzer.

Type analyzer.

Program slicer.

Better debugging.

Inclusion-based Points-to Analysis

p

p

q

q

- p = &q address-of
- p = q copy
- p = *q load
- *p = q store

p

q

p

q

p

q

p

q

p

q

p

q

Inclusion-based Points-to Analysis

Points-to Analysis

...

a = &x

c = b

d = *b

*b = a

...

Program

Points-to Sets

main ( ) {

if (...) {

...

}

}

...

a → {x,y}

b → {a,z}

c → {a,z,x}

...

Optimizations

- Online cycle elimination (Fahndrich et al., 1998)
- Offline variable substitution(Rountev and Chandra, 2000)
- Pointer and location equivalence (Hardekopf and Lin, 2007)

These optimizations preserve the precision of the underlying analysis.

Pointer Equivalence (PE)

x1

x1

x2

x2

x3

x3

P1

P2

P1, P2

x4

x4

x5

x5

x6

x6

x7

x7

Original points-to sets

Modified points-to sets

Location Equivalence (LE)

P1

P1

P2

P2

P3

P3

x1

x2

X1, X2

P4

P4

P5

P5

P6

P6

P7

P7

Original points-to sets

Modified points-to sets

Issues and Learnings

- Cubic time complexity.
- High absolute running times.
- Approximations are inevitable for scalability.

Basic Idea

Approximate Pointer Equivalence (APE)

x1

x1

x2

x2

x3

x3

P1

P2

P1, P2

x4

x4

x5

x5

x6

x6

x7

x7

Original points-to sets

Approximate points-to sets

Basic Idea

Approximate Location Equivalence (ALE)

P1

P1

P2

P2

P3

P3

x1

x2

X1, X2

P4

P4

P5

P5

P6

P6

P7

P7

Original points-to sets

Approximate points-to sets

Our Contributions

- Approximate pointer and location equivalence
- Sound algorithm to compute APE and ALE online
- Optimizations:
- Proximity merge
- Eager/lazy merging
- Merge order
- Equivalence identification frequency

- Extensive empirical evaluation

APE and ALE

Pointers P1 and P2 are approximately pointer equivalent with similarity αif sim(ptsto(P1), ptsto(P2)) ≥α.

Objects x1 and x2 are approximately location equivalent with similarity βif sim(ptdby(x1), ptdby(x2)) ≥ β.

s1 Ո s2

sim(s1, s2) =

s1 Ս s2

Examples

ptsto(p1) = {x,y,z} ptdby(x) = {p1, p3}

ptsto(p2) = {y,z,w} ptdby(y) = {p1, p2}

ptsto(p3) = {x,w} ptdby(z) = {p1, p2}

ptdby(w) = {p2, p3}

α= 0.5

p1 and p2 are APE with similarity 2/4 = 0.5

p1 and p3 are not APE with similarity 1/4 = 0.25

β= 0.7

y and z are ALE with similarity 2/2 = 1.0

x and w are not ALE with similarity 1/3 = 0.33

Approximate Points-to Analysis

Input: set of constraints, α, β.

Process address-of constraints

Add edges to constraint graph G using copy edges

repeat

Propagate points-to information in G

for each variable pair (x,y) do

simα = sim(ptsto(x), ptsto(y))

simβ = sim(ptdby(x), ptdby(y))

if simα ≥ αor simβ ≥ βthen

merge(x,y)

end if

end for

Add edges to G using load and store constraints

until fixed point

Example

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

α = 0.5, β = 0.7

Example

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

a {u,v,w,x}

0: Processing address-of constraints

p {a}

b {y}

q {b}

c {z}

d { }

r {a}

v {u}

u {v}

Constraint graph

Example

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

a {u,v,w,x}

0: Processing address-of constraints

0: Processing copy constraints

p {a}

b {y}

q {b}

c {z}

d { }

r {a}

v {u}

u {v}

Constraint graph

Example

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

a {u,v,w,x}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

p {a}

b {y}

q {b}

c {z}

d {u,v,w,x}

r {a}

v {u}

u {v}

Constraint graph

Example

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

α = 0.5, β = 0.7

[a,d] {u,v,w,x}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

p {a}

b {y}

q {b}

c {z}

r {a}

v {u}

u {v}

Constraint graph

Example

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

α = 0.5, β = 0.7

[a,d] {u,v,w,x}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

[p,r] {a}

b {y}

q {b}

c {z}

v {u}

u {v}

Constraint graph

Example

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

α = 0.5, β = 0.7

[a,d] {u,v,[w,x]}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

1: merge(w, x)

[p,r] {a}

b {y}

q {b}

c {z}

v {u}

u {v}

Constraint graph

Example

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

[a,d] {u,v,[w,x]}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

1: merge(w, x)

1: Processing load/store constraints

[p,r] {a}

b {y}

q {b}

c {z}

v {u}

u {v}

Constraint graph

Example

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

[a,d] {u,v,[w,x]}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

1: merge(w, x)

1: Processing load/store constraints

[p,r] {a}

b {y}

q {b}

c {z}

v {u}

u {v}

Constraint graph

Example

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

[a,d] {u,v,[w,x],z}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

1: merge(w, x)

1: Processing load/store constraints

2: Propagate points-to information

[p,r] {a}

b {y,z}

q {b}

c {z}

v {u}

u {v}

Constraint graph

Example

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

α = 0.5, β = 0.7

[a,d] {u,v,[w,x],z}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

1: merge(w, x)

1: Processing load/store constraints

2: Propagate points-to information

2: merge(b, c)

[p,r] {a}

q {b}

[b,c] {y,z}

v {u}

u {v}

Constraint graph

Example

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

[a,d] {u,v,[w,x],y,z}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

1: merge(w, x)

1: Processing load/store constraints

2: Propagate points-to information

2: merge(b, c)

3: Propagate points-to information

[p,r] {a}

q {b}

[b,c] {y,z}

v {u}

u {v}

Constraint graph

Example

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

[a,d] {u,v,[w,x],y,z}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

1: merge(w, x)

1: Processing load/store constraints

2: Propagate points-to information

2: merge(b, c)

3: Propagate points-to information

Fixed point

[p,r] {a}

q {b}

[b,c] {y,z}

v {u}

u {v}

Constraint graph

Example

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

[a,d] {u,v,[w,x],y,z}

Exact analysis: 18 points-to pairs

Approximate analysis: 19 points-to pairs

[p,r] {a}

q {b}

[b,c] {y,z}

v {u}

u {v}

Constraint graph

Proximity Merge

Similarity of a node is checked against another that is at most k-reachable.

0

1

1

2

2

2

3

3

A proximity of k=10 gives a 38% improvement in analysis time and the precision loss is only 2.6%.

Lazy Merging

When to merge matters.

Example:

α = 0.5

ptsto(p1) = {a,b}, ptsto(p2) = {b,c}, ptsto(p3) = {c,d},

ptsto(p4) = {d,e}, ptsto(p5) = {e,f}, ptsto(p6) = {f,g}

Eager merging: [p1,p2], [p3,p4], [p5,p6]

Lazy merging: [p1,p2,p3,p4,p5,p6]

Lazy merging improves analysis time over eager merging by 8%, but reduces precision.

Merge Order

Merge order matters.

Example:

α = 0.5

ptsto(a) = {x,y,z}, ptsto(b) = {w,y,z}, ptsto(c) = {w,x}

Order (a,c), (b,c), (a,b) merges a and b.

Order (a,b), (a,c), (b,c) merges all a, b and c.

We arrange nodes in non-increasing and non-decreasing similarities and find that the former reduces analysis time by 10% while the latter requires 8% more time than not ordering the variables.

Effect of α

Precision improves non-linearly with α.

Effect of β

Precision improves almost linearly with β.

Equivalence Identification Frequency

A client provides MAXMEM and the analysis updates α, β to choose the maximum amount of precision possible.

Note: observations are averaged across benchmarks.

Conclusions

- APE is more important that ALE
- Proximity merging helps the analysis scale.
- Lazy vs eager merging, order of merging offer nice trade-offs.
- Client can specify a MAXMEM to make its judicious use.

Department of Computer Science and Automation,

Indian Institute of Science, Bangalore, India

MSPC 2011

June 05, 2011

Approximating Inclusion-based Points-to Analysis
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