Counting authorised paths in constrained control-flow graphs Nikola - - PowerPoint PPT Presentation

Counting authorised paths in constrained control-flow graphs

Nikola K. Blanchard1, Siargey Kachanovich2

1Digitrust, Loria, Universit´

e de Lorraine

2Inria Sophia-Antipolis

5th Bordeaux Graph Workshop October 29th, 2019

Why this question?

Model Bounds Counting in practice Conclusion 1/12

Model

We are given:

• A directed graph G
• A vector of dimension k initialised at 0: V = (0, ...0)
• A start node
• For each edge e, a vector V e of dimension k
• An integer B to bound V
• Constraints for each edge e

Model Bounds Counting in practice Conclusion 2/12

Avoiding infinite loops

(1) (−1)

No infinite path ⇐ ⇒ no cycle leaving v unchanged

Model Bounds Counting in practice Conclusion 3/12

Avoiding infinite loops

(1) (−1)

No infinite path ⇐ ⇒ no cycle leaving v unchanged

Model Bounds Counting in practice Conclusion 3/12

Objectives

We look at two indicators:

• Λ : length of longest authorised path
• Π : number of distinct authorised paths

Two goals:

• Give general upper and lower bounds for Λ and Π on graphs of order n
• Compute Λ(G) and Π(G) for any given G

Model Bounds Counting in practice Conclusion 4/12

Objectives

We look at two indicators:

• Λ : length of longest authorised path
• Π : number of distinct authorised paths

Two goals:

• Give general upper and lower bounds for Λ and Π on graphs of order n
• Compute Λ(G) and Π(G) for any given G

Model Bounds Counting in practice Conclusion 4/12

Simple example

(1, 0) (−1, 3) (1, −1)

Λ = 10, Π = 14

Model Bounds Counting in practice Conclusion 5/12

Simple example

(1, 0) (−1, 3) (1, −1)

Λ = 10, Π = 14

Model Bounds Counting in practice Conclusion 5/12

Upper bounds

General case: Λ ≤ n × (B + 1)k Corollary: Π ≤ nn×(B+1)k When m < k: Λ ≤ n × 2mkm/2 k m

• Bm

Thanks to Lˆ e Th` anh D¨ ung Nguyˆ en: Λ ≤ n × (B + 1)m

Model Bounds Counting in practice Conclusion 6/12

Upper bounds

General case: Λ ≤ n × (B + 1)k Corollary: Π ≤ nn×(B+1)k When m < k: Λ ≤ n × 2mkm/2 k m

• Bm

Thanks to Lˆ e Th` anh D¨ ung Nguyˆ en: Λ ≤ n × (B + 1)m

Model Bounds Counting in practice Conclusion 6/12

Upper bounds

General case: Λ ≤ n × (B + 1)k Corollary: Π ≤ nn×(B+1)k When m < k: Λ ≤ n × 2mkm/2 k m

• Bm

Thanks to Lˆ e Th` anh D¨ ung Nguyˆ en: Λ ≤ n × (B + 1)m

Model Bounds Counting in practice Conclusion 6/12

Lower bound: multigraph case

(1, 0, . . . , 0) (−B, 1, 0, . . . , 0) . . . (−B, . . . , −B, 1)

Λ = n × (B + 1)k, Π = 1 + n × (B + 1)k

Model Bounds Counting in practice Conclusion 7/12

Lower bound: multigraph case

(1, 0, . . . , 0) (−B, 1, 0, . . . , 0) . . . (−B, . . . , −B, 1)

Λ = n × (B + 1)k, Π = 1 + n × (B + 1)k

Model Bounds Counting in practice Conclusion 7/12

Lower bound: (n − 1)2 < k case

Kn (1, 0, . . . , 0) (0, . . . , 0) Λ = (2B + 1)B(n−1)2−1, Π = (n − 1)(B−1)B(n−1)2−1

Model Bounds Counting in practice Conclusion 8/12

Lower bound: (n − 1)2 < k case

Kn (1, 0, . . . , 0) (0, . . . , 0) Λ = (2B + 1)B(n−1)2−1, Π = (n − 1)(B−1)B(n−1)2−1

Model Bounds Counting in practice Conclusion 8/12

How to find Π(G)

Central idea:

• Assume we have a Markov chain that allows us to sample paths uniformly at random
• Sample from it repeatedly
• Stop when you get the same path twice
• Output the square of the number of samples multiplied by 4/π2

Model Bounds Counting in practice Conclusion 9/12

Problems with the Markov chain method

Two main problems:

• How do we find a good Markov chain?
• Is there a divergence between the computed Π(G) and the real Π(G)?

Model Bounds Counting in practice Conclusion 10/12

Markov chain example

(1)

Divergence: Π′ =

Π log(Π) Model Bounds Counting in practice Conclusion 11/12

Markov chain example

(1)

Divergence: Π′ =

Π log(Π) Model Bounds Counting in practice Conclusion 11/12

Open problems

A few central questions:

• How fast can we get the Markov chain to converge?
• Can we find general bounds on the divergence?
• Is there any general method that could approximate Π(G) in Π(G)o(1)
• Can we compute Λ(G) in o(Λ(G))

Model Bounds Counting in practice Conclusion 12/12

Thank you for your attention

Model Bounds Counting in practice Conclusion 12/12