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The increasing_nvalue Constraint Nicolas Beldiceanu, Fabien - - PowerPoint PPT Presentation
The increasing_nvalue Constraint Nicolas Beldiceanu, Fabien - - PowerPoint PPT Presentation
The increasing_nvalue Constraint Nicolas Beldiceanu, Fabien Hermennier, Xavier Lorca and Thierry Petit LINA CNRS UMR 6241 CPAIOR 2010 Initial Motivation Def: nvalue( N , X) holds iff variable N is equal to the number of distinct values
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Outline
Definition GAC Algorithm Time Complexity and Reformulation Experiments Generalization of increasing_nvalue
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Outline
Definition GAC Algorithm Time Complexity and Reformulation Experiments Generalization of increasing_nvalue
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Definition
Def: increasing_nvalue(N, X) holds iff : Variable N is equal to the number of distinct values assigned to the variables in X x0 ≤ x1 ≤ … xn
Example :
(3, <6,6,8,8,1,8>) violates increasing_nvalue (3, <1,6,6,8,8,8>) satisfies increasing_nvalue
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Def [Pesant CP01]: Given X = [x0, x1, … xn] a sequence of variables, a stretch on X is a maximum length subsequence S of X such that any consecutive pair of variables in S are equal.
Example: <6,6,8,8,1,8,8,8 > contains 4 stretches:
<6,6>, <8,8>, <1>, and <8,8,8>
Definition
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increasing_nvalue: Since x0 ≤ x1 ≤ … ≤ xn the number of distinct values is equal to the number of stretches: Example: <1,6,6,8,8,9,9,9> contains 4 distinct values: <1>, <6,6>, <8,8>, and <9,9,9>
Definition
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increasing_nvalue: Since x0 ≤ x1 ≤ … ≤ xn-1 the number of distinct values is equal to the number of stretches: Example: <1,6,6,8,8,9,9,9> contains 4 distinct values: <1>, <6,6>, <8,8>, and <9,9,9>
GAC algorithm : basically, estimate the minimum (resp. maximum) number of stretches if value v is assigned to xi
Definition
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Outline
Definition GAC Algorithm Time Complexity and Reformulation Experiments Generalization of increasing_nvalue
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GAC Algorithm
Compute the min. number of stretches of values
X0 X1 X2 X3 X4 X5 X6 X7 1 4 5 6 9
Value 4 for x2: smin[x2…x7] = 3
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GAC Algorithm
Compute the max. number of stretches of values
X0 X1 X2 X3 X4 X5 X6 X7 1 4 5 6 9
Value 4 for x2: smax[x2…x7] = 4
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GAC Algorithm Filtering N from variables in X
- 1. Remove values violating necessarily constraints
x0 ≤ x1, …, xn-2 ≤ xn-1 : keep only well-ordered values
- 2. Compute min. and max. number of stretches of x0
- 3. Compare with D(N)
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GAC Algorithm
X0 X1 X2 X3 X4 X5 X6 X7 1 4 5 6 9
(1) Remove all the values that are not well-ordered
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GAC Algorithm
X0 X1 X2 X3 X4 X5 X6 X7 1 4 5 6 9
(1) Remove all the values that are not well-ordered (2) Variable x0 :
- value 4:
smin(4)[x0…x7] = 3 smax(4)[x0…x7] = 4
- value 6:
smin(6)[x0…x7] = 2 smax(6)[x0…x7] = 2
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GAC Algorithm
(1) Remove all the values that are not well-ordered ⇒ SMIN = 2, SMAX = 4 ⇒ Compare [SMIN,SMAX] with D(N)
X0 X1 X2 X3 X4 X5 X6 X7 1 4 5 6 9
(2) Variable x0 :
- value 4:
smin(4)[x0…x7] = 3 smax(4)[x0…x7] = 4
- value 6:
smin(6)[x0…x7] = 2 smax(6)[x0…x7] = 2
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GAC Algorithm
Compare [SMIN, SMAX] with D(N): Obvious: remove from D(N) values not in [SMIN, SMAX] An issue remains: If D(N) ⊂ [SMIN, SMAX] ?
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GAC Algorithm Main theoretical contribution of the paper:
To any k within [SMIN, SMAX] corresponds an assignment of values to variables in X As a consequence,
increasing_nvalue has a solution ⇔ D(N) ∩ [SMIN, SMAX] ≠ ∅
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Intuition of the Result
Properties on the max. and min. number of stretches ⇒ Given two values v1 and v2 in D(xi) [smin(v1), smax(v1)] ∪ [smin(v2), smax(v2)] never has holes
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Intuition of the Result
⇒ Given two values v1 and v2 in D(xi) [smin(v1), smax(v1)] ∪ [smin(v2), smax(v2)] never has holes ⇒Extreme case :
X0 X1 X2 X3 X4 X5 X6 X7 1 4 5 6 9
- value 9:
smin(9)[x0…x7] = 1 smax(9)[x0…x7] = 1
- value 1:
smin(1)[x0…x7] = 2 smax(1)[x0…x7] = 5
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GAC Algorithm Filtering variables in X from N
- 1. Compute for each well-ordered value in D(xi) the
minimum and maximum number of stretches in the prefix [ x0, …, xi ] and in the suffix [ xi, …, xn-1 ] sequences.
- 2. Aggregate information and compare with the
current domain of N
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GAC Algorithm
X0 X1 X2 X3 X4 X5 X6 X7 1 4 5 6 9
(2) Value 5 for x4 : pmin[x0…x4] = 2 pmax[x0…x4] = 2 smin[x4…x7] = 2 smax[x4…x7] = 3
?
prefix suffix (1) Remove values that are not well-ordered
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GAC Algorithm
X0 X1 X2 X3 X4 X5 X6 X7 1 4 5 6 9
(2) Value 5 for x4 : pmin[x0…x4] = 2 pmax[x0…x4] = 2 smin[x4…x7] = 2 smax[x4…x7] = 3 (1) Remove values that are not well-ordered (3) Aggregate prefix and suffix information: smin[x0…x7] = 3 (i.e., 2+2-1) smax[x0…x7] = 4 (i.e., 2+3-1) Compare [smin, smax] with D(N)
?
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GAC Algorithm
From properties on min. and max. v in D(xi) is GAC ⇔ D(N) ∩ [smin, smax] ≠ ∅
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Outline
Definition GAC Algorithm Time Complexity and Reformulation Experiments Generalization of Increasing NValue
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Time Complexity
For each value : compute pmin, pmax, smin, smax Compute values from previous (resp. next) variable i+1 (resp. i-1)
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Time Complexity
For each value : compute pmin, pmax, smin, smax Compute values from previous (resp. next) variable i+1 (resp. i-1) Use of sparse matrix:
Iterate by scanning D(xi) in increasing or decreasing order Values not in domain = default value (0 or n) Cumulated min on consecutive pmin, pmax, …= O(1) for each
⇒ Overall time complexity in O(ΣD(xi))
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Reformulations of Increasing NValue
Automaton*: O(n(∪D(xi))3) (space O((∪D(xi))3) transitions) N=2 D(xi) = [6..8] * Described in (pdf version): http://www.emn.fr/x-info/sdemasse/gccat/
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Outline
Definition GAC Algorithm Time Complexity and Reformulation Experiments Generalization of increasing_nvalue
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Experiments: unary tests Unary tests in CHOCO: enforcing GAC
(200 variables, 200 values, 20% holes in domains) With increasing_nvalue: 100% instances OK 0,1s medium time per instance With an automaton (regular constraint) : 10% intances OK (90% memory overflow) 2,8s medium time per solved instance
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Experiment: entropy
Entropy : Optimize allocation of VMs into clusters (i.e., minimize the number of physical machines WNi)
Source: VM4 is waiting Final configuration
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Experiment: entropy Model overview:
One variable per VM Its domain is the number of available physical machines WN nvalue on VM’s modelizes the number of machines used
Use of increasing_nvalue as an implicit constraint
Many WN’s have the same characteristics increasing_nvalue can be set on equivalence classes
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Experiment: entropy
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Outline
Definition GAC Algorithm Time Complexity and Reformulation Experiments Generalization of increasing_nvalue
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Generalization of Increasing NValue Goal: identify a class of sliding constraint for which GAC can be performed with a time complexity which depends only on the sum
- f domain sizes
Generalization of the notion of stretch
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Generalization of Increasing Nvalue (2)
Def: Given X = [x0, x1, … xn] a sequence of variables and C a binary constraint, a C-stretch on X is a maximum length subsequence S of X such that: Either |S| = 1 (one single variable) and
Either the interpretation C(xi,xi) is true (universal constraint) or false (complementary of the universal constraint) Or C is neither true nor false and C(xi,xi) holds
Or |S| > 1 and C holds on any consecutive pair of variables in S.
Example: <6,6,8,8,1,10,8,8,8,1 > contains 3 ≤-stretches:
<6,6,8,8>, <1,10>, <8,8,8>, <1>
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Generalization of Increasing Nvalue (3)
Def: In SeqBin(N,X,C,B), N is a domain variable, X is a sequence of variables [x0,x1,…,xn-1] C is a binary constraint, B is a binary constraint. The constraint SeqBin* holds iff: (1) For all i in [0,n-2]: B(xi ,xi+1) holds, (2) N is the number of C-stretches in X.
* For more details see: Beldiceanu, Lorca, Petit, “A GAC algorithm for a class of counting constraints”, TR10/1/INFO, École des Mines de Nantes, 2010.
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