I have a binary program and one of my variables, x_it is defined on two sets, being I: Set of objects and T: Set of the weeks of the year, thus x_it is a binary variable standing for whether object i is assigned to something on week t. The constraint I failed to implement in AMPL/GNU Mathprog is that if x_it equals to 1 then x_i(t+1) and x_i(t+2) also should take value of 1. Is there a way to implement this constraint in a simple mathematical programming language?
The implication you want to implement is:
x(i,t) = 1 ==> x(i,t+1) = 1, x(i,t+2) = 1
AMPL supports implications (with the ==> operator), so we can write this directly. MathProg does not.
A simple way to implement the implication as straightforward linear inequalities is:
x(i,t+1) >= x(i,t)
x(i,t+2) >= x(i,t)
This can easily be expressed in AMPL, MathProg, or any modeling tool.
This is the pure, naive translation of the question. This means however that once a single x(i,t)=1 all following x(i,t+1),x(i,t+2),x(i,t+3)..=1. That could have been accomplished by just the constraint x(i,t+1) >= x(i,t).
A better interpretation would be: we don't want very short run lengths. I.e. patterns: 010 and 0110 are not allowed. This is sometimes called a minimum up-time in machine scheduling and can be modeled in different ways.
Forbid the patterns 010 and 0110:
(1-x(i,t-1))+x(i,t)+(1-x(i,t+1)) <= 2
(1-x(i,t-1))+x(i,t)+x(i,t+1)+(1-x(i,t+2)) <= 3
The pattern 01 implies 0111:
x(i,t+1)+x(i,t+2) >= 2*(x(i,t)-x(i,t-1))
Both these approaches will prevent patterns 010 and 0110 to occur.
Related
I am doing an algorithmic contest, and I'm trying to optimize my code. Maybe what I want to do is stupid and impossible but I was wondering.
I have these requirements:
An inventory which can contains 4 distinct types of item. This inventory can't contain more than 10 items (all type included). Example of valid inventory: 1 / 1 / 1 / 0. Example of invalid inventories: 11 / 0 / 0 / 0 or 5 / 5 / 5 / 0
I have some recipe which consumes or adds items into my inventory. The recipe can't add or consume more than 10 items since the inventory can't have more than 10 items. Example of valid recipe: -1 / -2 / 3 /
0. Example of invalid recipe: -6 / -6 / +12 / 0
For now, I store the inventory and the recipe into 4 integers. Then I am able to perform some operations like:
ApplyRecepe: Inventory(1/1/1/0).Apply(Recepe(-1/1/0/0)) = Inventory(0/2/1/0)
CanAfford: Iventory(1/1/0/0).CanAfford(Recepe(-2/1/0/0)) = False
I would like to know if it is possible (and if yes, how) to store the 4 values of an inventory/recipe into one single integer and to performs previous operations on it that would be faster than comparing / adding the 4 integers as I'm doing now.
I thought of something like having the inventory like that:
int32: XXXX (number of items of the first type) - YYYY (number of items of the second type) - ZZZ (number of items of the third type) - WWW (number of item of the fourth type)
But I have two problems with that:
I don't know how to handle the possible negative values
It seems to me much slower than just adding the 4 integers since I have to bit shift the inventory and the recipe to get the value I want and then proceed with the addition.
Storing multiple int values into one variable
Here are two alternatives:
An array. The advantage of this is that you may iterate over the elements:
int variable[] {
1,
1,
1,
0,
};
Or a class. The advantage of this is the ability to name the members:
struct {
int X;
int Y;
int Z;
int W;
} variable {
1,
1,
1,
0,
};
Then I am able to perform some operations like:
Those look like SIMD vector operations (Single Instruction Multiple Data). The array is the way to go in this case. Since the number of operands appears to be constant and small in your description, an efficient way to perform them are vector operations on the CPU 1.
There is no standard way to use SIMD operations directly in C++. To give the compiler optimal opportunity to use them, these steps need to be followed:
Make sure that the CPU you use supports the operations that you need. AVX-2 instruction set and its expansions have wide support for integer vector operations.
Make sure that you tell the compiler that the program should be optimised for that architecture.
Make sure to tell the compiler to perform vectorisation optimisations.
Make sure that the integers are sufficiently aligned as required by the operations. This can be achieved with alignas.
Make sure that the number of integers is known at compile time.
If the prospect of relying on the optimiser worries you, then you may instead prefer to use vector extensions that may be provided by your compiler. The use of language extensions would come at the cost of portability to other compilers naturally. Here is an example with GCC:
constexpr int count = 4;
using v4si = int __attribute__ ((vector_size (sizeof(int) * count)));
#include <iostream>
int main()
{
v4si inventory { 1, 1, 1, 0};
v4si recepe {-1, 1, 0, 0};
v4si applied = inventory + recepe;
for (int i = 0; i < count; i++) {
std::cout << applied[i] << ", ";
}
}
1 If the number of operands were large, then specialised vector processor such as a GPU could be faster.
Especially if you're learning, it's not a bad opportunity to try implementing your own helper class for vectorization, and consequently deepen your understanding about data in C++, even if your use case might not warrant the technique.
The insight you want to exploit is that arithmetic operations seem invariant to bitshifts, if one considers the pesky carry-bit and effects of signage (e.g. two's complement). But precisely because of these latter factors, it's much better to use some standardized underlying type like an int8_t[], as #Botje suggests.
To begin, implement the following functions. (My C++ is rusty, consider this pseudocode.)
int8_t* add(int8_t[], int8_t[], size_t);
int8_t* multiply(int8_t[], int8_t[], size_t);
int8_t* zeroes(size_t); // additive identity
int8_t* ones(size_t); // multiplicative identity
Also considering:
How would you like to handle overflows and underflows? Let them be and ask the developer to be cautious? Or throw exceptions?
Maybe you'd like to pin down the size of the array and avoid having to deal with a dynamic size_t?
Maybe you'd like to go as far as overloading operators?
The end result of an exercise like this, but generalized and polished, is something like Armadillo. But you'll understand it on a whole different level by doing the exercise yourself first. Also, if all this makes sense so far, you can now take a look at How to vectorize my loop with g++?—even the compiler can vectorize for you in certain cases.
Bitpacking as #Botje mentions is another step beyond this. You won't even have the safety and convenience of an integer type like int8_t or int4_t. Which additionally means the code you write might stop being platform-independent. I recommend at least finishing the vectorization exercise before delving into this.
This will be something of a non-answer, just intended to show what you're up against if you do bitpacking.
Suppose, for simplicity's sake, that recipes can only remove from inventory, and only contain positive values (you could represent negative numbers using two's complement, but it would take more bits, and add much complexity to working with the bit-packed numbers).
You then have 11 possible values for an item, so you need 4 bits for each item. Four items can then be represented in one uint16.
So, say you have an inventory with 10,4,6,9 items; this would be uint16_t inv = 0b1010'0100'0110'1001.
Then, a recipe with 2,2,2,2 items or uint16_t rec = 0b0010'0010'0010'0010.
inv - rec would give 0b1000'0010'0100'0111 for 8,2,4,7 items.
So far, so good. No need here to shift and mask to get at the individual values before doing the calculation. Yay.
Now, a recipe with 6,6,6,6 items which would be 0b0110'0110'0110'0110, giving inv - rec = 0b0011'1110'0000'0011 for 3,14,0,3 items.
Oops.
The arithmetic will work, but only if you check beforehand that the individual 4-bit results don't go out of bounds; in this example this would mean that you know beforehand that there are enough items in the inventory to fill a recipe.
You could get at, say, the third item in the inventory by doing: (inv >> 4) & 0b1111 or (inv << 8) >> 12 for doing your checks.
For testing, you would then get expressions like:
if ((inv >> 4) & 0b1111 >= (rec >> 4) & 0b1111)
or, comparing the 4 bits "in place":
if (inv & 0b0000000011110000 >= rec & 0b0000000011110000)
for each 4-bit part.
All these things are doable, but do you want to? It probably won't be faster than what is suggested in the other answers after the compiler has done its job, and it certainly won't be more readable.
It becomes even more horrible when you allow negative numbers (two's complement or otherwise) in recipes, especially if you want to bit-shift them.
So, bitpacking is nice for storage, and in some rare cases you can even do math without unpacking the bits, but I wouldn't try to go there (unless you are very performance and memory constrained).
Having said that, it could be fun to try to get it to work; there's always that.
Let's say we are optimizing over 2 variables, each a vector of 6. That is, Y=[y0,y1,...y5], and X=[x0, x1, ..., x5]. How do I formulate a constraint in linear programming so that it forces the following solutions: x0=x1=x2=x3 & x4=x5. Or is it better to penalize the differences (e.g. |x0-x1|) in the objective function? Is so, how?
x0=x1 can be expressed as x0-x1 <= 0 and x0-x1 >= 0. The other equalities in the same way.
Edit: As pointed out in the comments, directly stating x0-x1 = 0 is the better way.
what is the difference between Constraint Programming (CP) and Linear Programming (LP) or Mixed Integer Programming (MIP) ? I know what LP and MIP is but dont understand the difference to CP - or is CP just the same as MIP and LP ? I am a but confused on this ...
This may be a little exhaustive, but I will try to provide all the information to cover a good scope of this topic.
I'll start with an example and the corresponding information will make more sense.
**Example**: Say we need to sequence a set of tasks on a machine. Each task i has a specific fixed processing time pi. Each task can be started after its release date ri , and must be completed before its deadline di. Tasks cannot overlap in time. Time is represented as a discrete set of time points, say {1, 2,…, H} (H stands for horizon)
MIP Model:
Variables: Binary variable xij represents whether task i starts at time period j
Constraints:
Each task starts on exactly one time point
* ∑j xij = 1 for all tasks i
Respect release date and deadline
j*xij = 0 for all tasks i and (j < ri ) or (j > di - pi )
Tasks cannot overlap
Variant 1:
∑i xij ≤ 1 for all time points j we also need to take processing times into account; this becomes messy
Variant 2:
introduce binary variable bi representing whether task i comes before task k must be linked to xij; this becomes messy
MIP models thus consists of linear/quadratic optimization functions, linear/ quadratic optimization constraints and binary/integer variables.
CP model:
Variables:
Let starti represent the starting time of task i takes a value from domain {1,2,…, H} - this immediately ensures that each task starts at exactly one time point
Constraints:
Respect release date and deadline
ri ≤ starti ≤ di - pi
Tasks cannot overlap:
for all tasks i and j (starti + pi < startj) OR (starti + pi < starti)
and that is it!
You could probably say that the structure of the CP models and MIP models are the same: using decision variables, objective function and a set of constraints. Both MIP and CP problems are non-convex and make use of some systematic and exhaustive search algorithms.
However, we see the major difference in modeling capacity. With CP we have n variables and one constraint. In MIP we have nm variables and n+m constraints. This way to map global constraints to MIP constraints using binary variables is quite generic
CP and MIP solves problems in a different way. Both use a divide and conquer approach, where the problem to be solved is recursively split into sub problems by fixing values of one variable at a time. The main difference lies in what happens at each node of the resulting problem tree. In MIP one usually solves a linear relaxation of the problem and uses the result to guide search. This is a branch and bound search. In CP, logical inferences based on the combinatorial nature of each global constraint are performed. This is an implicit enumeration search.
Optimization differences:
A constraint programming engine makes decisions on variables and values and, after each decision, performs a set of logical inferences to reduce the available options for the remaining variables' domains. In contrast, an mathematical programming engine, in the context of discrete optimization, uses a combination of relaxations (strengthened by cutting-planes) and "branch and bound."
A constraint programming engine proves optimality by showing that no better solution than the current one can be found, while an mathematical programming engine uses a lower bound proof provided by cuts and linear relaxation.
A constraint programming engine doesn't make assumptions on the mathematical properties of the solution space (convexity, linearity etc.), while an mathematical programming engine requires that the model falls in a well-defined mathematical category (for instance Mixed Integer Quadratic Programming (MIQP).
In deciding how you should define your problem - as MIP or CP, Google Optimization tools guide suggests: -
If all the constraints for the problem must hold for a solution to be feasible (constraints connected by "and" statements), then MIP is generally faster.
If many of the constraints have the property that just one of them needs to hold for a solution to be feasible (constraints connected by "or" statements), then CP is generally faster.
My 2 cents:
CP and MIP solves problems in a different way. Both use a divide and conquer approach, where the problem to be solved is recursively split into sub problems by fixing values of one variable at a time. The main difference lies in what happens at each node of the resulting problem tree. In MIP one usually solves a linear relaxation of the problem and uses the result to guide search. This is a branch and bound search. In CP, logical inferences based on the combinatorial nature of each global constraint are performed.
There is no one specific answer to which approach would you use to formulate your model and solve the problem. CP would probably work better when the number of variables increase by a lot and the problem is difficult to formulate the constraints using linear equalities. If the MIP relaxation is tight, it can give better results - If you lower bound doesn't move enough while traversing your MIP problem, you might want to take higher degrees of MIP or CP into consideration. CP works well when the problem can be represented by Global constraints.
Some more reading on MIP and CP:
Mixed-Integer Programming problems has some of the decision variables constrained to integers (-n … 0 … n) at the optimal solution. This makes it easier to define the problems in terms of a mathematical program. MP focuses on special class of problems and is useful for solving relaxations or subproblems (vertical structure).
Example of a mathematical model:
Objective: minimize cT x
Constraints: A x = b (linear constraints)
l ≤ x ≤ u (bound constraints)
some or all xj must take integer values (integrality constraints)
Or the model could be define by Quadratic functions or constraints, (MIQP/ MIQCP problems)
Objective: minimize xT Q x + qT x
Constraints: A x = b (linear constraints)
l ≤ x ≤ u (bound constraints)
xT Qi x + qiT x ≤ bi (quadratic constraints)
some or all x must take integer values (integrality constraints)
The most common algorithm used to converge MIP problems is the Branch and Bound approach.
CP:
CP stems from a problems in AI, Operations Research and Computer Science, thus it is closely affiliated to Computer Programming.- Problems in this area assign symbolic values to variables that need to satisfy certain constraints.- These symbolic values have a finite domain and can be labelled with integers.- CP modelling language is more flexible and closer to natural language.
Quoted from one of the IBM docs, constraint Programming is a technology where:
business problems are modeled using a richer modeling language than what is traditionally found in mathematical optimization
problems are solved with a combination of tree search, artificial intelligence and graph theory techniques
The most common constraint(global) is the "alldifferent" constraint, which ensures that the decision variables assume some permutation (non-repeating ordering) of integer values. Ex. If the domain of the problem is 5 decision variables viz. 1,2,3,4,5, they can be ordered in any non-repetitive way.
The answer to this question depends on whether you see MIP and CP as algorithms, as problems, or as scientific fields of study.
E.g., each MIP problem is clearly a CP problem, as the definition of a MIP problem is to find a(n optimal) solution to a set of linear constraints, while the definition of a CP problem is to find a(n optimal) solution to a set of (non-specified) constraints. On the other hand, many important CP problems can straightforwardly be converted to sets of linear constraints, so seeing CP problems through a MIP perspective makes sense as well.
Algorithmically, CP algorithms historically tend to involve more search branching and complex constraint propagation, while MIP algorithms rely heavily on solving the LP relaxation to a problem. There exist hybrid algorithms though (e.g., SCIP, which literally means "Solving Constraint Integer Programs"), and state-of-the-art solvers often borrow techniques from the other side (e.g., no-good learning and backjumps originated in CP, but are now present in MIP solvers as well).
From a scientific field of study point of view, the difference is purely historical: MIP is part of Operations Research, originating at the end of WWII out of a need to optimize large-scale "operations", while CP grew out of logic programming in the field of Artificial Intelligence to model and solve problems declaratively. But there is a good case to be made that both these fields study the same problem. Note that there even is a big shared conference: CPAIOR.
So all in all, I would say MIP and CP are the same in most respects, except on the main techniques used in typical algorithms for each.
LP and MIP are solved using mathematical programming, while there are specific methods to solve constraint programming problems. The following reference is helpful in understanding the differences:
http://ibmdecisionoptimization.github.io/docplex-doc/mp_vs_cp.html
i want binary number that only have 0's at the beginning or end, for instance,
1111111
01111110
001111111
000111000
but no:
01001
0011101
do they have an specific name or property to get them?
I'm looking for something like linear integer optimization conditions, my solutions must have this form, but i can't think of any condition i can add to ensure that
Regards,
This is something which is not nice to formulate within mixed-integer programs. Most problems involving this are more suitable for alternative methods (SAT-solving, SMT-solving, Constraint-programming).
It can be done of course, but the solver will have some work as the formulation is non-trivial and introduces a lot of binary-variables (and the basic approach of MIP-solvers won't work amazingly here; bad integrality-gap).
I won't give you a complete solution, but some basic idea on how to formulate this and i also indicate how hard and cumbersome it is (there are alternative formulations; actually infinite many; but nothing much more simple).
The basic idea here is the following:
your binary number is constructed from N binary-variables
let's call them x
you introduce N auxiliary binary-variables l (left)
l[i] == 1 implicates: every l[j] with i<j is 0
you introduce N auxiliary binary-variables r (right)
r[i] == 1 implicates: every r[j] with i>j is 0
you add the following constraint for each position k:
x[k] == 0 implicates: l[i] == 1 for i < k OR r[i] == 1 for i>k
idea::
if there is a zero somewhere, either all on the left-side or all on the right-side are zeroes (or means: at least one side; but can be both)
To formulate this, you need two more ideas:
A: How to formulate the equality-check?
B: How to formulate the implication?
Remark: a -> b == not a or b (propositional calculus)
(this was wrongly stated earlier and corrected by OP)
These are common in MIP and you will find the solution in many integer-programming books, tutorial and papers. Here is an example (start with indicator-variables).
Another small common formulation:
if a is binary, b is binary:
a OR b is equivalent to: a+b >= 1 (the latter is a linear-expression ready to use for MIP)
Remark: The formulas in my idea-setting above might be wrong in regards to indices (i vs. i-1, vs. i+1) and binary-relations (<vs. <=). You will need to do the actual math yourself and just learn from the idea itself!
Remark 2: This kind of constraint is cumbersome in MIP, but more easily formulated within SAT and CP.
The primary use of CRCs and similar computations (such as Fletcher and Adler) seems to be for the detection of transmission errors. As such, most studies I have seen seem to address the issue of the probability of detecting small-scale differences between two data sets. My needs are slightly different.
What follows is a very approximate description of the problem. Details are much more complicated than this, but the description below illustrates the functionality I am looking for. This little disclaimer is intended to ward of answers such as "Why are you solving your problem this way when you can more easily solve it this other way I propose?" - I need to solve my problem this way for a myriad of reasons that are not germane to this question or post, so please don't post such answers.
I am dealing with collections of data sets (size ~1MB) on a distributed network. Computations are performed on these data sets, and speed/performance is critical. I want a mechanism to allow me to avoid re-transmitting data sets. That is, I need some way to generate a unique identifier (UID) for each data set of a given size. (Then, I transmit data set size and UID from one machine to another, and the receiving machine only needs to request transmission of the data if it does not already have it locally, based on the UID.)
This is similar to the difference between using CRC to check changes to a file, and using a CRC as a digest to detect duplicates among files. I have not seen any discussions of the latter use.
I am not concerned with issues of tampering, i.e. I do not need cryptographic strength hashing.
I am currently using a simple 32-bit CRC of the serialized data, and that has so far served me well. However, I would like to know if anyone can recommend which 32-bit CRC algorithm (i.e. which polynomial?) is best for minimizing the probability of collisions in this situation?
The other question I have is a bit more subtle. In my current implementation, I ignore the structure of my data set, and effectively just CRC the serialized string representing my data. However, for various reasons, I want to change my CRC methodology as follows. Suppose my top-level data set is a collection of some raw data and a few subordinate data sets. My current scheme essentially concatenates the raw data and all the subordinate data sets and then CRC's the result. However, most of the time I already have the CRC's of the subordinate data sets, and I would rather construct my UID of the top-level data set by concatenating the raw data with the CRC's of the subordinate data sets, and then CRC this construction. The question is, how does using this methodology affect the probability of collisions?
To put it in a language what will allow me to discuss my thoughts, I'll define a bit of notation. Call my top-level data set T, and suppose it consists of raw data set R and subordinate data sets Si, i=1..n. I can write this as T = (R, S1, S2, ..., Sn). If & represents concatenation of data sets, my original scheme can be thought of as:
UID_1(T) = CRC(R & S1 & S2 & ... & Sn)
and my new scheme can be thought of as
UID_2(T) = CRC(R & CRC(S1) & CRC(S2) & ... & CRC(Sn))
Then my questions are: (1) if T and T' are very different, what CRC algorithm minimizes prob( UID_1(T)=UID_1(T') ), and what CRC algorithm minimizes prob( UID_2(T)=UID_2(T') ), and how do these two probabilities compare?
My (naive and uninformed) thoughts on the matter are this. Suppose the differences between T and T' are in only one subordinate data set, WLOG say S1!=S1'. If it happens that CRC(S1)=CRC(S1'), then clearly we will have UID_2(T)=UID_2(T'). On the other hand, if CRC(S1)!=CRC(S1'), then the difference between R & CRC(S1) & CRC(S2) & ... & CRC(Sn) and R & CRC(S1') & CRC(S2) & ... & CRC(Sn) is a small difference on 4 bytes only, so the ability of UID_2 to detect differences is effectively the same as a CRC's ability to detect transmission errors, i.e. its ability to detect errors in only a few bits that are not widely separated. Since this is what CRC's are designed to do, I would think that UID_2 is pretty safe, so long as the CRC I am using is good at detecting transmission errors. To put it in terms of our notation,
prob( UID_2(T)=UID_2(T') ) = prob(CRC(S1)=CRC(S1')) + (1-prob(CRC(S1)=CRC(S1'))) * probability of CRC not detecting error a few bits.
Let call the probability of CRC not detecting an error of a few bits P, and the probability of it not detecting large differences on a large size data set Q. The above can be written approximately as
prob( UID_2(T)=UID_2(T') ) ~ Q + (1-Q)*P
Now I will change my UID a bit more as follows. For a "fundamental" piece of data, i.e. a data set T=(R) where R is just a double, integer, char, bool, etc., define UID_3(T)=(R). Then for a data set T consisting of a vector of subordinate data sets T = (S1, S2, ..., Sn), define
UID_3(T) = CRC(ID_3(S1) & ID_3(S2) & ... & ID_3(Sn))
Suppose a particular data set T has subordinate data sets nested m-levels deep, then, in some vague sense, I would think that
prob( UID_3(T)=UID_3(T') ) ~ 1 - (1-Q)(1-P)^m
Given these probabilities are small in any case, this can be approximated as
1 - (1-Q)(1-P)^m = Q + (1-Q)*P*m + (1-Q)*P*P*m*(m-1)/2 + ... ~ Q + m*P
So if I know my maximum nesting level m, and I know P and Q for various CRCs, what I want is to pick the CRC that gives me the minimum value for Q + m*P. If, as I suspect might be the case, P~Q, the above simplifies to this. My probability of error for UID_1 is P. My probability of error for UID_3 is (m+1)P, where m is my maximum nesting (recursion) level.
Does all this seem reasonable?
I want a mechanism to allow me to avoid re-transmitting data sets.
rsync has already solved this problem, using generally the approach you outline.
However, I would like to know if anyone can recommend which 32-bit CRC
algorithm (i.e. which polynomial?) is best for minimizing the
probability of collisions in this situation?
You won't see much difference among well-selected CRC polynomials. Speed may be more important to you, in which case you may want to use a hardware CRC, e.g. the crc32 instruction on modern Intel processors. That one uses the CRC-32C (Castagnoli) polynomial. You can make that really fast by using all three arithmetic units on a single core in parallel by computing the CRC on three buffers in the same loop, and then combining them. See below how to combine CRCs.
However, most of the time I already have the CRC's of the subordinate
data sets, and I would rather construct my UID of the top-level data
set by concatenating the raw data with the CRC's of the subordinate
data sets, and then CRC this construction.
Or you could quickly compute the CRC of the entire set as if you had done a CRC on the whole thing, but using the already calculated CRCs of the pieces. Look at crc32_combine() in zlib. That would be better than taking the CRC of a bunch of CRCs. By combining, you retain all the mathematical goodness of the CRC algorithm.
Mark Adler's answer was bang on. If I'd taken my programmers hat off and put on my mathematicians hat, some of it should have been obvious. He didn't have the time to explain the mathematics, so I will here for those who are interested.
The process of calculating a CRC is essentially the process of doing a polynomial division. The polynomials have coefficients mod 2, i.e. the coefficient of each term is either 0 or 1, hence a polynomial of degree N can be represented by an N-bit number, each bit being the coefficient of a term (and the process of doing a polynomial division amounts to doing a whole bunch of XOR and shift operations). When CRC'ing a data block, we view the "data" as one big polynomial, i.e. a long string of bits, each bit representing the coefficient of a term in the polynomial. Well call our data-block polynomial A. For each CRC "version", there has been chosen the polynomial for the CRC, which we'll call P. For 32-bit CRCs, P is a polynomial with degree 32, so it has 33 terms and 33 coefficients. Because the top coefficient is always 1, it is implicit and we can represent the 32nd-degree polynomial with a 32-bit integer. (Computationally, this is quite convenient actually.) The process of calculating the CRC for a data block A is the process of finding the remainder when A is divided by P. That is, A can always be written
A = Q * P + R
where R is a polynomial of degree less than degree of P, i.e. R has degree 31 or less, so it can be represented by a 32-bit integer. R is essentially the CRC. (Small note: typically one prepends 0xFFFFFFFF to A, but that is unimportant here.) Now, if we concatenate two data blocks A and B, the "polynomial" corresponding to the concatenation of the two blocks is the polynomial for A, "shifted to the left" by the number of bits in B, plus B. Put another way, the polynomial for A&B is A*S+B, where S is the polynomial corresponding to a 1 followed by N zeros, where N is the number of bits in B. (i.e. S = x**N ). Then, what can we say about the CRC for A&B? Suppose we know A=Q*P+R and B=Q'*P+R', i.e. R is the CRC for A and R' is the CRC for B. Suppose we also know S=q*P+r. Then
A * S + B = (Q*P+R)*(q*P+r) + (Q'*P+R')
= Q*(q*P+r)*P + R*q*P + R*r + Q'*P + R'
= (Q*S + R*q + Q') * P + R*r + R'
So to find the remainder when A*S+B is divided by P, we need only find the remainder when R*r+R' is divided by P. Thus, to calculate the CRC of the concatenation of two data streams A and B, we need only know the separate CRC's of the data streams, i.e. R and R', and the length N of the trailing data stream B (so we can compute r). This is also the content of one of Marks other comments: if the lengths of the trailing data streams B are constrained to a few values, we can pre-compute r for each of these lengths, making combination of two CRC's quite trivial. (For an arbitrary length N, computing r is not trivial, but it is much faster (log_2 N) than re-doing the division over the entire B.)
Note: the above is not a precise exposition of CRC. There is some shifting that goes on. To be precise, if L is the polynomial represented by 0xFFFFFFFF, i.e. L=x*31+x*30+...+x+1, and S_n is the "shift left by n bits" polynomial, i.e. S_n = x**n, then the CRC of a data block with polynomial A of N bits, is the remainder when ( L * S_N + A ) * S_32 is divided by P, i.e. when (L&A)*S_32 is divided by P, where & is the "concatenation" operator.
Also, I think I disagree with one of Marks comments, but he can correct me if I'm wrong. If we already know R and R', comparing the time to compute the CRC of A&B using the above methodology, as compared with computing it the straightforward way, does not depend on the ratio of len(A) to len(B) - to compute it the "straight forward" way, one really does not have to re-compute the CRC on the entire concatenated data set. Using our notation above, one only needs to compute the CRC of R*S+B. That is, instead of pre-pending 0xFFFFFFFF to B and computing its CRC, we prepend R to B and compute its CRC. So its a comparison of the time to compute B's CRC over again with the time to compute r, (followed by dividing R*r+R' by P, which is trivial and inconsequential in time likely).
Mark Adler's answer addresses the technical question so that's not what I'll do here. Here I'm going to point out a major potential flaw in the synchronization algorithm proposed in the OP's question and suggest a small improvement.
Checksums and hashes provide a single signature value for some data. However, being of finite length, the number of possible unique values of a checksum/hash is always smaller than the possible combinations of the raw data if the data is longer. For instance, a 4 byte CRC can only ever take on 4 294 967 296 unique values whilst even a 5 byte value which might be the data can take on 8 times as many values. This means for any data longer than the checksum itself, there always exists one or more byte combinations with exactly the same signature.
When used to check integrity, the assumption is that the likelihood of a slightly different stream of data resulting in the same signature is small so that we can assume the data is the same if the signature is the same. It is important to note that we start with some data d and verify that given a checksum, c, calculated using a checksum function, f that f(d) == c.
In the OP's algorithm, however, the different use introduces a subtle, detrimental degradation of confidence. In the OP's algorithm, server A would start with the raw data [d1A,d2A,d3A,d4A] and generate a set of checksums [c1,c2,c3,c4] (where dnA is the n-th data item on server A). Server B would then receive this list of checksums and check its own list of checksums to determine if any are missing. Say Server B has the list [c1,c2,c3,c5]. What should then happen is that it requests d4 from Server A and the synchronization has worked properly in the ideal case.
If we recall the possibilty of collisions, and that it doesn't always take that much data to produce one (e.g. CRC("plumless") == CRC("buckeroo")), then we'll quickly realize that the best guarantee our scheme provides is that server B definitely doesn't have d4A but it cannot guarantee that it has [d1A,d2A,d3A]. This is because it is possible that f(d1A) = c1 and f(d1B) = c1 even though d1A and d1B are distinct and we would like both servers to have both. In this scheme, neither server can ever know about the existence of both d1A and d1B. We can use more and more collision resistant checksums and hashes but this scheme can never guarantee complete synchronization. This becomes more important, the greater the number of files the network must keep track of. I would recommend using a cryptographic hash like SHA1 for which no collisions have been found.
A possible mitigation of the risk of this is to introduce redundant hashes. One way of doing is is to use a completely different algorithm since whilst it is possible crc32(d1) == crc32(d2) it is less likely that adler32(d1) == adler32(d2) simultaneously. This paper suggests you don't gain all that much this way though. To use the OP notation, it is also less likely that crc32('a' & d1) == crc32('a' & d2) and crc32('b' & d1) == crc32('b' & d2) are simultaneously true so you can "salt" to less collision prone combinations. However, I think you may just as well just use a collision resistant hash function like SHA512 which in practice likely won't have that great an impact on your performance.