CN101916229A - Efficient analogy method for replacing cache randomly - Google Patents
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Abstract
The invention relates to an efficient analogy method for replacing a cache randomly, comprising the following steps of: firstly, obtaining an estimating formula (1) or (2): E(Xi)is approximate to 1-(1-1/M)E(Zi)(1)E(Xi)is approximate to 1-Pij(1+E(Xj)/(M-1))-1(2) by approximating E(Xi)=1-E((1-1/MZi); and secondly, calculating the hit probability 1-E(Xi) of each point in a memory array according to E(Xi) obtained by the estimating formula (1) or (2), wherein Xi is an index random variable of a depletion event, E(Xi) is a mathematic expectation, M is the degree of association of the cache, Xj belongs to a reusing window of Xi, namely all depletion events during the period from value of ai appearing previously to i. If the value of the point ai in the memory array does not appear before, Zi is infinite; if the value of the point ai appears before, Zi is equal to time of depletion events during the period from value of ai appearing previously to i. The invention can obtain an estimation of the hit probability of each point in the memory array through one analogue round by using the estimating formula of a probability model.
Description
Technical field
The present invention relates to the buffer memory analogue technique of computer realm, relate in particular to a kind of employing efficient analogy method of the buffer memory of replacement policy at random.
Background technology
Be buffered in and be widely used in improving performance in the computer system.Buffer memory arrives value with map addresses.Because the cost of access cache is less than access memory, so the value of often visiting recently is put into the expense that helps reducing access memory in the buffer memory.When buffer memory received request of access to an address, it can search self content, seeks and this address corresponding cache row.If such row exists, then claim hit event has taken place: return the value that is stored in the correspondence in the buffer memory under this incident; Not if so capablely do not exist, then claim the disappearance incident has taken place: the value that needs by certain back mechanism acquisition under this incident, evict buffer memory from and make a vacant line thereby in alternative cache lines, select a row according to replacement policy then, and address and the new value that obtains are deposited new vacant line.Replacement policy is a strategy of selecting the expulsion object in alternative cache lines.Replacement policy is selected the expulsion object at random in alternative cache lines at random.
The buffer memory simulation is the important technology of assessment buffer memory effect.Buffer memory simulation is by using the parameter and the replacement policy of simulator simulated target buffer memory, understand generate behind a certain address sequence of caching process hit the disappearance sequence of events.But for the buffer memory that adopts replacement policy at random, traditional Monte Carlo simulation method one is taken turns operation can only provide one of multiple possibility, so one take turns that simulation can not reflect an address sequence correspondence fully hit the disappearance sequence of events, thereby need many wheel simulations with the influence of assessment buffer memory.Such as hypothesis＜a, e〉represent a content to comprise the fully-associative buffer storage of 2 sizes of a and e, different cache lines represented in different here letters.When in the face of an access sequence d, during a: at first since d not in buffer memory, so the disappearance incident takes place, the difference of this backsight expulsion object, buffer status has two kinds:＜d, e〉or＜a, d 〉.When accepting this moment again, if buffer status is＜d e to the visit of a 〉, the disappearance incident then takes place; If buffer status is＜a d 〉, hit event then takes place.So d, the disappearance sequence of events that hits of a correspondence has " disappearance, disappearance " and " lack, hit " two kinds, and still obvious this can not be taken turns in the simulation one and be observed.So the Monte Carlo simulation method has following problem: once operation can not provide in the memory access sequence every hit probability, and can only ask the method for average probability to obtain every hit probability by many wheels simulations.
Because the set associative buffer memory is equivalent to the array of a fully-associative buffer storage, so following the simulation that fully-associative buffer storage is discussed.
If access queue is a
0, a
1... a
NSubscript is represented logical time.If X
0, X
1... X
nIndex stochastic variable for the disappearance incident.E (X so
i) be exactly a
iThe disappearance probability, 1-E (X
i) be exactly a
iHit probability, wherein E (x) is the mathematical expectation of x.
It is from a that window is reused in definition
iThe value last time memory access subqueue between the i appears.Such as for a, b, c, a, d, a, b
The pairing window of reusing of last b comprises c, a, d, a from a last b.
Definition Z
iAs follows: if a
iZ did not then appear at preceding
i=∞; If a
iAt preceding occurring, Z then
i=from a
iThe value last time disappearance event times between the i appears.
By mathematical analysis, be that the buffer memory of M can obtain following exact formulas for degree of association:
E(X
i)＝1-E((1-1/M)
Z i)；
E (Z
i)=∞; If a
iDo not occur at preceding;
Or E (Z
i)=∑
jE (X
j), X wherein
jBelong to a
iReuse window.
Summary of the invention
A purpose of the present invention is to provide a kind of efficient analogy method of replacing cache randomly, is used for solving the problem of once moving the hit probability that can not provide every of memory access sequence that traditional Monte Carlo simulation method exists.
To achieve these goals, the invention provides a kind of efficient analogy method of replacing cache randomly, it is characterized in that, comprising:
E(X
i)≈1-(1-1/M)
E(Zi) (1)
E(X
i)≈1-∏
j(1+E(X
j)/(M-1))
-1 (2)
Step 2 is according to the E (X that is obtained by estimation equation (1) or estimation equation (2)
i) calculate in the memory access sequence every hit probability;
Wherein, X
iBe the index stochastic variable of disappearance incident, E (X
i) be X
iMathematical expectation, M is the degree of association of buffer memory, X
jBelong to X
iReuse window, for from a
iThe value last time all disappearance incidents between the i appear, if memory access sequence mid point a
iZ did not then appear at preceding
i=∞; If a
iAt preceding occurring, Z then
i=from a
iThe value last time disappearance event times between the i appears.
The efficient analogy method of described replacing cache randomly, wherein,
In the described step 2, further comprise: calculate in the described memory access sequence every hit probability with following formula:
E (the X that every hit probability=1-estimation equation (1) obtains
i);
Or
E (the X that every hit probability=1-estimation equation (2) obtains
i).
The efficient analogy method of described replacing cache randomly, wherein,
In the described step 2, further comprise: comprise h ', h by band skew ", b, two Hash tables of c element calculate in the described memory access sequence every hit probability, are specially:
Step 31, " point to two different empty Hash tables, wherein a variable assignments is given in " ← " expression, selects estimation equation calculating K value for b ← 0, c ← 0, h ' and h;
Step 32 is according to a
iResiding Hash table returns the sum value;
Step 33 calculates E (X according to sum value, selected estimation equation
i);
Step 34 is according to the E (X that calculates
i), selected estimation equation calculates off-set value delta;
Step 35, according to the delta value, carry out following processing:
b←b+delta，c←c+delta，h′[a
i]←-b。
The efficient analogy method of described replacing cache randomly, wherein,
In the described step 31, further comprise:
When selecting estimation equation (1), K=ln (ε)/ln (1-1/M) then; Or
When selecting estimation equation (2), K=-ln (ε)/ln (1+1/ (M-1)) then;
Wherein:
Ln is a natural logarithm, and the ε control accuracy gets 0.01.
The efficient analogy method of described replacing cache randomly, wherein,
In the described step 32, further comprise:
Work as a
iWhen being in the Hash table of h ' sensing, sum=h ' [a
i]+b; Or
Work as a
iBe in h " during the Hash table that points to, sum=h " [a
i]+b; Or
Work as a
iThe Hash table that neither is in h ' sensing is not in h again " during the Hash table that points to, sum=∞;
Wherein:
H ' [a
i] be a
iValue in the Hash table of h ' sensing, h " [a
i] be a
i" the value in the Hash table that points at h.
The efficient analogy method of described replacing cache randomly, wherein,
In the described step 33, further comprise:
When sum=∞, E (X
i)=1; Or
When sum ≠ ∞ and selection estimation equation (1),
E (X
i) ≈ 1-(1-1/M)
E (Zi), E (Z
i)=sum; Or
When sum ≠ ∞ and selection estimation equation (2),
E(X
i)≈1-exp(-sum*ln(1+1/(M-1)))；
Wherein: exp is to be the exponential function at the end with Euler's numbers.
The efficient analogy method of described replacing cache randomly, wherein,
In the described step 34, further comprise:
When selecting estimation equation (1), delta=E (X then
i); Or
When selecting estimation equation (2), delta=ln (1+E (X then
i)/(M-1))/ln (1+1/ (M-1)).
The efficient analogy method of described replacing cache randomly, wherein,
In the described step 35, further comprise:
Whether judge c greater than K, as greater than, then handle: c ← c-K empties h " Hash table that points to, and exchange h ' and the h " Hash table of sensing.
The efficient analogy method of described replacing cache randomly, wherein,
" the big or small sum of the Hash table that points to is O (K), and O is the asymptotic upper bound commonly used in the computer science for h ' and h.
The efficient analogy method of described replacing cache randomly, wherein,
Described M is the integer more than or equal to 2.
Compared with prior art, useful technique effect of the present invention is:
The invention solves in the prior art once operation and can not provide in the memory access sequence problem of every hit probability.By the estimation equation of probability of use model, can take turns the estimation that simulation obtains in the memory access sequence every hit probability by one; Estimation precision is equivalent to 50 and takes turns traditional Monte Carlo simulation; Calculating can be finished in linear session and almost space.
Description of drawings
Fig. 1 is the efficient analogy method process flow diagram of replacing cache randomly of the present invention;
Fig. 2 is an embodiment of the efficient simulation of replacing cache randomly of the present invention;
Fig. 3 is the probability distribution graph of the absolute error of the absolute error of every some hit probability of the present invention and traditional Monte Carlo method;
Fig. 4 is the program runtime of the present invention and traditional Monte Carlo method and the graph of a relation of Input Address sequence length;
Fig. 5 is the graph of a relation of the size of the peak value of program committed memory of the present invention and institute's simulated cache.
Embodiment
Describe the present invention below in conjunction with the drawings and specific embodiments, but not as a limitation of the invention.
As shown in Figure 1, be the efficient analogy method process flow diagram of replacing cache randomly of the present invention.This method flow comprises:
Step 101 is by to E (X
i)=1-E ((1-1/M)
Zi) approximate, obtain estimation equation (1):
E(X
i)≈1-(1-1/M)
E(Zi) (1)
Utilize this estimation, can with E (Z
i) definition form iterative;
This step forms iterative, thereby can estimate E (X
i), and then calculate in the memory access sequence every hit probability;
Step 102 is by to E (X
i)=1-E ((1-1/M)
Zi) approximate, obtain estimation equation (2):
E(X
i)≈1-∏
j(1+E(X
j)/(M-1))
-1 (2)
Utilize this estimation, itself promptly form iterative, thereby can estimate E (X
i), and then calculate in the memory access sequence every hit probability.Than estimation equation (1), the precision of this estimation equation (2) is further enhanced.
Wherein, X
iBe the index stochastic variable of disappearance incident, E (X
i) be X
iMathematical expectation, M is a degree of association, X
jBelong to X
iReuse window, for from a
iThe value last time all disappearance incidents between the i appear, if memory access sequence mid point a
iZ did not then appear at preceding
i=∞; If a
iAt preceding occurring, Z then
i=from a
iThe value last time disappearance event times between the i appears.
Step 103 is according to the E (X that is obtained by estimation equation (1) or estimation equation (2)
i) calculate in the memory access sequence every hit probability.Particularly:
E (the X that every hit probability=1-estimation equation (1) obtains
i);
Or
E (the X that every hit probability=1-estimation equation (2) obtains
i).
Further, in step 103, by every hit probability in two Hash tables calculating memory access sequences of band skew.Comprise h ', h ", b, four elements of c." two windows that are equivalent to moving window method the inside are used for the space that the minimizing program uses for h ' and h.
Adopt the mode of two Hash tables that calculating can be finished in linear session and the almost linear space of relative cache size.
As shown in Figure 2, be an embodiment of the efficient simulation of replacing cache randomly of the present invention, in conjunction with this embodiment the process of the efficient analogy method of replacing cache randomly is described:
When step 201, program initialization, b ← 0, c ← 0, h ' and h " point to two different empty Hash tables.Wherein a variable assignments is given in " ← " expression.Selected estimation equation.As the estimation equation of choosing is formula (1), then K=ln (ε)/ln (1-1/M); As the estimation equation of choosing is formula (2), then K=-ln (ε)/ln (1+1/ (M-1)).Wherein ln is a natural logarithm, ε control accuracy, desirable 0.01.
All a in step 202, the judgement access queue
iWhether processed, as processed, then finish; Otherwise carry out following processing:
At first judge a
iCan in h ', find, if a
iCan in h ', find, then return h ' [a
iThe value of]+b; If can not in h ', find, then search h ", judge a
iCan be at h find in " in find, if can be ", then return h " [a at h
iThe value of]+b; If can not be at h " in find, then return ∞.
Wherein, h ' [a
i] be a
iValue in the Hash table of h ' sensing, h " [a
i] be a
i" the value in the Hash table that points at h.
Step 203, the rreturn value of establishing in the step 202 are sum, calculate E (X according to sum value, selected estimation equation
i);
Do you judge sum=∞? if sum=is ∞, E (X then
i)=1; If sum ≠ ∞, as the estimation equation of choosing is formula (1), then with E (Z
i)=sum substitution estimation equation (1) calculates E (X
i); As the estimation equation of choosing is formula (2), then with E (X
i) ≈ 1-exp (sum*ln (1+1/ (M-1))) calculates E (X
i), wherein exp is to be the exponential function at the end with Euler's numbers.Correspondingly, can obtain under three kinds of situations every hit probability:
Hit probability=1-1=0 of every;
Hit probability=1-of every (1-(1-1/M)
Sum)=(1-1/M)
Sum
Hit probability=1-of every (1-exp (sum*ln (1+1/ (M-1))))
＝exp(-sum*ln(1+1/(M-1)))。
E (the X that step 204, basis calculate
i), selected estimation equation calculates off-set value delta;
If use estimation equation (1), then delta=E (X
i); If use estimation equation (2), then delta=ln (1+E (X
i)/(M-1))/and ln (1+1/ (M-1)), wherein ln is a natural logarithm;
Step 205, according to the delta value, carry out following processing:
b←b+delta，c←c+delta，h′[a
i]←-b。If c〉K, then c ← c-K empties h ", exchange h ' and the h " Hash table that points to.Jumping to step 202 continues.Because to be processed is an address sequence, so constantly circulate up to series processing is finished by jump procedure.
Realize the service data structure by above-mentioned steps 204,205, i.e. two Hash tables of band skew, the correctness of data in guaranteeing to show.
Can prove h ' and h " the big or small sum of the Hash table that points to all is O (K) at any one time, and the algorithm time be linear, wherein O is the asymptotic upper bound of using always in the computer science.Degree of association M is the integer more than or equal to 2.When M is big, no matter estimation equation (1), (2), the used space of algorithm is O (M*ln (1/ ε)), is almost with respect to the buffer memory degree of association promptly.Since each group of set associative buffer memory to hit the disappearance sequence of events independent mutually, so can prove for the set associative buffer memory, the used time of algorithm still is as the criterion linear for linear and used space with respect to the size of buffer memory.
In conjunction with Fig. 3, the complete association replacing cache randomly during relatively for M=4, the precision of every some hit probability of a certain memory access sequence that the multiple averaging of traditional Monte Carlo algorithm and this algorithm are given.Among Fig. 3 with the average do reference of 500 traditional algorithms.X-axis is the absolute error of every some hit probability, and Y-axis is the probability distribution of absolute error.Curve from left to right descends fast more, and then error concentrates on less situation more, and precision is good more.Such as " estimation formula 2; ε=0.01 " curve value when x=0.1 about 0.00001 among the figure, show that when algorithm of the present invention uses estimation equation (2) and get ε=0.01 absolute error of estimation hit probability equals 0.1 situation and just can take place once on average about 100000 times.5 average and 50 average data of asking the average of 5 times and 50 times traditional Monte Carlo algorithms and obtaining that refer to respectively among Fig. 3.Among Fig. 3 as seen, still estimation equation (2) of estimation equation (1) no matter, when ε=0.01, precision all is better than the average of 5 times even 50 times traditional algorithms.Two curves of ε=0.01 of estimation equation among Fig. 3 (1) and ε=0.1 overlap substantially.
Fig. 4 and Fig. 5 show the time and space use of this algorithm.Among Fig. 4, X-axis is the length of Input Address sequence, and Y-axis is a program runtime.Visible ε=0.01 o'clock of adopting among the figure, the working time of estimation equation (1) and estimation equation (2) and the length of Input Address sequence, promptly the scale of problem is linear, and is better than traditional Monte Carlo algorithm.Among Fig. 5, X-axis is the size of institute's simulated cache, and Y-axis is the peak value of program committed memory.Visible ε=0.01 o'clock of adopting among the figure, the internal memory that estimation equation (1) and estimation equation (2) use becomes almost to concern with the size of institute's simulated cache.
The inventive method has solved the following problem of traditional Monte Carlo simulation method: once operation can not provide in the memory access sequence every hit probability.Method of the present invention is by the probability of use model, can one take turns in service finish a memory access sequence in every the estimation of hit probability.
Certainly; the present invention also can have other various embodiments; under the situation that does not deviate from spirit of the present invention and essence thereof; those of ordinary skill in the art work as can make various corresponding changes and distortion according to the present invention, but these corresponding changes and distortion all should belong to the protection domain of the appended claim of the present invention.
Claims (10)
1. the efficient analogy method of a replacing cache randomly is characterized in that, comprising:
Step 1 is by to E (X
i)=1-E ((1-1/M)
Zi) approximate, obtain estimation equation (1) or estimation equation (2):
E(X
i)≈1-(1-1/M)
E(Zi) (1)
E(X
i)≈1-∏
j(1+E(X
j)/(M-1))
-1 (2)
Step 2 is according to the E (X that is obtained by estimation equation (1) or estimation equation (2)
i) calculate in the memory access sequence every hit probability;
Wherein, X
iBe the index stochastic variable of disappearance incident, E (X
i) be X
iMathematical expectation, M is the degree of association of buffer memory, X
jBelong to X
iReuse window, for from a
iThe value last time all disappearance incidents between the i appear, if memory access sequence mid point a
iZ did not then appear at preceding
i=∞; If a
iAt preceding occurring, Z then
i=from a
iThe value last time disappearance event times between the i appears.
2. the efficient analogy method of replacing cache randomly according to claim 1 is characterized in that,
In the described step 2, further comprise: calculate in the described memory access sequence every hit probability with following formula:
E (the X that every hit probability=1-estimation equation (1) obtains
i);
Or
E (the X that every hit probability=1-estimation equation (2) obtains
i).
3. the efficient analogy method of replacing cache randomly according to claim 1 and 2 is characterized in that,
In the described step 2, further comprise: comprise h ', h by band skew ", b, two Hash tables of c element calculate in the described memory access sequence every hit probability, are specially:
Step 31, " point to two different empty Hash tables, wherein a variable assignments is given in " ← " expression, selects estimation equation calculating K value for b ← 0, c ← 0, h ' and h;
Step 32 is according to a
iResiding Hash table returns the sum value;
Step 33 calculates E (X according to sum value, selected estimation equation
i);
Step 34 is according to the E (X that calculates
i), selected estimation equation calculates off-set value delta;
Step 35, according to the delta value, carry out following processing:
b←b+delta，c←c+delta，h′[a
i]←-b。
4. the efficient analogy method of replacing cache randomly according to claim 3 is characterized in that, in the described step 31, further comprises:
When selecting estimation equation (1), K=ln (ε)/ln (1-1/M) then; Or
When selecting estimation equation (2), K=-ln (ε)/ln (1+1/ (M-1)) then;
Wherein:
Ln is a natural logarithm, and the ε control accuracy gets 0.01.
5. the efficient analogy method of replacing cache randomly according to claim 3 is characterized in that, in the described step 32, further comprises:
Work as a
iWhen being in the Hash table of h ' sensing, sum=h ' [a
i]+b; Or
Work as a
iBe in h " during the Hash table that points to, sum=h " [a
i]+b; Or
Work as a
iThe Hash table that neither is in h ' sensing is not in h again " during the Hash table that points to, sum=∞;
Wherein:
H ' [a
i] be a
iValue in the Hash table of h ' sensing, h " [a
i] be a
i" the value in the Hash table that points at h.
6. the efficient analogy method of replacing cache randomly according to claim 3 is characterized in that,
In the described step 33, further comprise:
When sum=∞, E (X
i)=1; Or
When sum ≠ ∞ and selection estimation equation (1),
E (X
i) ≈ 1-(1-1/M)
E (Zi), E (Z
i)=sum; Or
When sum ≠ ∞ and selection estimation equation (2),
E(X
i)≈1-exp(-sum*ln(1+1/(M-1)))；
Wherein: exp is to be the exponential function at the end with Euler's numbers.
7. according to the efficient analogy method of claim 4,5 or 6 described replacing cache randomlies, it is characterized in that,
In the described step 34, further comprise:
When selecting estimation equation (1), delta=E (X then
i); Or
When selecting estimation equation (2), delta=ln (1+E (X then
i)/(M-1))/ln (1+1/ (M-1)).
8. according to the efficient analogy method of claim 4,5 or 6 described replacing cache randomlies, it is characterized in that,
In the described step 35, further comprise:
Whether judge c greater than K, as greater than, then handle: c ← c-K empties h " Hash table that points to, and exchange h ' and the h " Hash table of sensing.
9. according to the efficient analogy method of claim 4,5 or 6 described replacing cache randomlies, it is characterized in that,
" the big or small sum of the Hash table that points to is O (K), and O is the asymptotic upper bound commonly used in the computer science for h ' and h.
10. according to the efficient analogy method of claim 1,4,5 or 6 described replacing cache randomlies, it is characterized in that,
Described M is the integer more than or equal to 2.
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| CN102945184A (en) * | 2012-11-15 | 2013-02-27 | 江苏凌创电气自动化股份有限公司 | Realization method for configuration control on window data |
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| US6952664B1 (en) * | 2001-04-13 | 2005-10-04 | Oracle International Corp. | System and method for predicting cache performance |
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