Sparse representations

The sparse tree data structure

Yacas has a sparse tree object for use as a storage for storing (key,value) pairs for which the following properties hold:

  • (key, value1) + (key, value2) = (key, value1+value2)

  • (key1, value1) * (key2, value2) = (key1+key2, value1*value2)

The last is optional. For multivariate polynomials (described elsewhere) both hold, but for matrices, only the addition property holds. The function {MultiplyAddSparseTrees} (described below) should not be used in these cases.

Internal structure

A key is defined to be a list of integer numbers \((n_1,\ldots, n_m)\). Thus for a two-dimensional key, one item in the sparse tree database could be reflected as the (key,value) pair { {{1,2},3} }, which states that element {(1,2)} has value {3}. (Note: this is not the way it is stored in the database!).

The storage is recursive. The sparse tree begins with a list of objects { {n1,tree1} } for values of {n1} for the first item in the key. The {tree1} part then contains a sub-tree for all the items in the database for which the value of the first item in the key is {n1}.

The above single element could be created with:

In> r:=CreateSparseTree({1,2},3)
Out> {{1,{{2,3}}}};

{CreateSparseTree} makes a database with exactly one item. Items can now be obtained from the sparse tree with {SparseTreeGet}.:

In> SparseTreeGet({1,2},r)
Out> 3;
In> SparseTreeGet({1,3},r)
Out> 0;

And values can also be set or changed:

In> SparseTreeSet({1,2},r,Current+5)
Out> 8;
In> r
Out> {{1,{{2,8}}}};
In> SparseTreeSet({1,3},r,Current+5)
Out> 5;
In> r
Out> {{1,{{3,5},{2,8}}}};

The variable {Current} represents the current value, and can be used to determine the new value. {SparseTreeSet} destructively modifies the original, and returns the new value. If the key pair was not found, it is added to the tree.

The sparse tree can be traversed, one element at a time, with {SparseTreeScan}:

In> SparseTreeScan(Hold({{k,v},Echo({k,v})}),2,r)
{1,3} 5
{1,2} 8

An example of the use of this function could be multiplying a sparse matrix with a sparse vector, where the entire matrix can be scanned with {SparseTreeScan}, and each non-zero matrix element \(A_{ij}\) can then be multiplied with a vector element \(v_j\), and the result added to a sparse vector \(w_i\), using the {SparseTreeGet} and {SparseTreeSet} functions. Multiplying two sparse matrices would require two nested calls to {SparseTreeScan} to multiply every item from one matrix with an element from the other, and add it to the appropriate element in the resulting sparse matrix.

When the matrix elements \(A_{ij}\) are defined by a function \(f(i,j\)) (which can be considered a dense representation), and it needs to be multiplied with a sparse vector \(v_j\), it is better to iterate over the sparse vector \(v_j\). Representation defines the most efficient algorithm to use in this case.

The API to sparse trees is:

  • {CreateSparseTree(coefs,fact)} - Create a sparse tree with one monomial, where ‘coefs’ is the key, and ‘fact’ the value. ‘coefs’ should be a list of integers.

  • {SparseTreeMap(op,depth,tree)} - Walk over the sparse tree, one element at a time, and apply the function “op” on the arguments (key,value). The ‘value’ in the tree is replaced by the value returned by the {op} function. ‘depth’ signifies the dimension of the tree (number of indices in the key).

  • {SparseTreeScan(op,depth,tree)} - Same as SparseTreeMap, but without changing elements.

  • {AddSparseTrees(depth,x,y)}, {MultiplyAddSparseTrees(depth,x,y,coefs,fact)} - Add sparse tree ‘y’ to sparse tree ‘x’, destructively. in the {MultiplyAdd} case, the monomials are treated as if they were multiplied by a monomial with coefficients with the (key,value) pair (coefs,fact). ‘depth’ signifies the dimension of the tree (number of indices in the key).

  • {SparseTreeGet(key,tree)} - return value stored for key in the tree.

  • {SparseTreeSet(key,tree,newvalue)} - change the value stored for the key to newvalue. If the key was not found then {newvalue} is stored as a new item. The variable {Current} is set to the old value (or zero if the key didn’t exist in the tree) before evaluating {newvalue}.

Implementation of multivariate polynomials

This section describes the implementation of multivariate polynomials in yacas. Concepts and ideas are taken from the books [DST88] and [vzGG99].


The following definitions define multivariate polynomials, and the functions defined on them that are of interest for using such multivariates.

A term is an object which can be written as

\[cx_1^{n_1}x_2^{n_2}\ldots x_m^{n_m}\]

for \(m\) variables (\(x_1\), …, \(x_m\)). The numbers \(n_m\) are integers. \(c\) is called a coefficient, and \(x_1^{n_1}x_2^{n_2}\ldots x_m^{n_m}\) a monomial.

A multivariate polynomial is taken to be a sum over terms.

We write \(c_ax^a\) for a term, where \(a\) is a list of powers for the monomial, and \(c_a\) the coefficient of the term.

It is useful to define an ordering of monomials, to be able to determine the canonical form of a multivariate. For the currently implemented code the lexicographic order has been chosen:

  • First, the ordering of variables is chosen, (\(x_1\), …, \(x_m\))

  • For the exponents of a monomial, \(a = (a_1,\ldots, a_m)\) the lexicographic order first looks at the first exponent, \(a_1\), to determine which of the two monomials comes first in the multivariate. If the two exponents are the same, the next exponent is considered.

This method is called lexicographic because it is similar to the way words are ordered in a usual dictionary.

For all algorithms (including division) there is some freedom in the ordering of monomials. One interesting advantage of the lexicographic order is that it can be implemented with a recursive data structure, where the first variable, \(x_1\) can be treated as the main variable, thus presenting it as a univariate polynomial in \(x_1\) with all its terms grouped together.

Other orderings can be used, by re-implementing a part of the code dealing with multivariate polynomials, and then selecting the new code to be used as a driver, as will be described later on.

Given the above ordering, the following definitions can be stated:

For a non-zero multivariate polynomial

\[f = \sum_{a=a_{max}}^{a_{min}}c_ax^a\]

with a monomial order:

  1. \(c_ax^a\) is a term of the multivariate.

  2. the multidegree of \(f\) is \(\operatorname{mdeg}(f) := a_{max}\).

  3. the leading coefficient of \(f\) is \(\operatorname{lc}(f):=c_{\operatorname{mdeg}(f)}\), for the first term with non-zero coefficient.

  4. the leading monomial of \(f\) is \(\operatorname{lm}(f):=x^{\operatorname{mdeg}(f)}\).

  5. the leading term of \(f\) is \(\operatorname{lt}(f):=\operatorname{lc}(f)\operatorname{lm}(f)\).

The above define access to the leading monomial, which is used for divisions, gcd calculations and the like. Thus an implementation needs be able to determine \((\operatorname{mdeg}(f),\operatorname{lc}(f)\). Note the similarity with the (key,value) pairs described in the sparse tree section. \(\operatorname{mdeg}(f)\) can be thought of as a key, and \(\operatorname{lc}(f)\) as a value.

The multicontent, \(\operatorname{multicont}(f)\), is defined to be a term that divides all the terms in \(f\), and is the term described by \((\min(a), \gcd(c))\), with \(\gcd(c)\) the GCD of all the coefficients, and :math:min(a)` the lowest exponents for each variable, occurring in \(f\) for which \(c\) is non-zero.

The multiprimitive part is then defined as \(\operatorname{pp}(f):=\frac{f}{\operatorname{multicont}(f)}\).

For a multivariate polynomial, the obvious addition and (distributive) multiplication rules hold

  • \((a+b) + c = a+(b+c)\)

  • \(a(b+c) = ab+ac\)

These are supported in the Yacas system through a multiply-add operation: \(\operatorname{muadd}(f,t,g) := f+tg\). This allows for both adding two polynomials (\(t=1\)), or multiplication of two polynomials by scanning one polynomial, and multiplying each term of the scanned polynomial with the other polynomial, and adding the result to the polynomial that will be returned. Thus there should be an efficient \(\operatorname{muadd}\) operation in the system.


For the representation of polynomials, on computers it is natural to do this in an array: \((a_1, a_2,\ldots, a_n)\) for a univariate polynomial, and the equivalent for multivariates. This is called a dense representation, because all the coefficients are stored, even if they are zero. Computers are efficient at dealing with arrays. However, in the case of multivariate polynomials, arrays can become rather large, requiring a lot of storage and processing power even to add two such polynomials. For instance, \(x^{200}y^{100}z^{300}+1\) could take 6000000 places in an array for the coefficients. Of course variables could be substituted for the single factors, \(p:=x^{200}\) etc., but it requires an additional ad hoc step.

An alternative is to store only the terms for which the coefficients are non-zero. This adds a little overhead to polynomials that could efficiently be stored in a dense representation, but it is still little memory, whereas large sparse polynomials are stored in acceptable memory too. It is of importance to still be able to add, multiply divide and get the leading term of a multivariate polynomial, when the polynomial is stored in a sparse representation.

For the representation, the data structure containing the {(exponents,coefficient)} pair can be viewed as a database holding {(key,value)} pairs, where the list of exponents is the key, and the coefficient of the term is the value stored for that key. Thus, for a variable set {{x,y}} the list {{1,2},3} represents \(3xy^2\).

Yacas stores multivariates internally as MultiNomial(vars, terms), where vars is the ordered list of variables, and terms some object storing all the (key, value) pairs representing the terms. Note we keep the storage vague: the terms placeholder is implemented by other code, as a database of terms. The specific representation can be configured at startup (this is described in more detail below).

For the current version, yacas uses the sparse tree representation, which is a recursive sparse representation. For example, for a variable set {x,y,z}, the terms object contains a list of objects of form {deg,terms}, one for each degree deg for the variable x occurring in the polynomial. The terms part of this object is then a sub-sparse tree for the variables {y,z}.

An explicit example:

In> MM(3*x^2+y)
Out> MultiNomial({x,y},{{2,{{0,3}}},{0,{{1,1}, {0,0}}}});

The first item in the main list is {2,{{0,3}}}, which states that there is a term of the form \(x^2y^03\). The second item states that there are two terms, \(x^0y^11\) and \(x^0y^00 = 0\).

This representation is sparse:

In> r:=MM(x^1000+x)
Out> MultiNomial({x},{{1000,1},{1,1}});

and allows for easy multiplication:

In> r*r
Out> MultiNomial({x},{{2000,1},{1001,2},{2,1},{0,0}});
In> NormalForm(%)
Out> x^2000+2*x^1001+x^2;

Internal code organization

The implementation of multivariates can be divided in three levels.

At the top level are the routines callable by the user or the rest of the system: MultiDegree(), MultiDivide(), MultiGcd(), Groebner(), etc. In general, this is the level implementing the operations actually desired.

The middle level does the book-keeping of the MultiNomial(vars,terms) expressions, using the functionality offered by the lowest level.

For the current system, the middle level is in multivar.rep/sparsenomial.ys, and it uses the sparse tree representation implemented in sparsetree.ys.

The middle level is called the driver, and can be changed, or re-implemented if necessary. For instance, in case calculations need to be done for which dense representations are actually acceptable, one could write C++ implementing above-mentioned database structure, and then write a middle-level driver using the code. The driver can then be selected at startup. In the file yacasinit.ys the default driver is chosen, but this can be overridden in the .yacasrc file or some file that is loaded, or at the command line, as long as it is done before the multivariates module is loaded (which loads the selected driver). Driver selection is as simple as setting a global variable to contain a file name of the file implementing the driver:


where “multivar.rep/sparsenomial.ys” is the file implementing the driver (this is also the default driver, so the above command would not change any thing).

The choice was made for static configuration of the driver before the system starts up because it is expected that there will in general be one best way of doing it, given a certain system with a certain set of libraries installed on the operating system, and for a specific version of Yacas. The best version can then be selected at start up, as a configuration step. The advantage of static selection is that no overhead is imposed: there is no performance penalty for the abstraction layers between the three levels.

Driver interface

The driver should implement the following interface:

  • CreateTerm(vars,{exp,coef}) - create a multivariate polynomial with one term, in the variables defined in var, with the (key,value) pair (coefs,fact)

  • MultiNomialAdd(multi1, multi2) - add two multivars, and (possibly) destructively modify multi1 to contain the result:

    [ multi1 := multi1 + multi2; multi1; ];
  • MultiNomialMultiplyAdd(multi1, multi2,exp,coef) - add two multivars, and (possibly) destructively modify multi1 to contain the result. multi2 is considered multiplied by a term represented by the (key,value) pair (exp,coef):

    [ multi1 := multi1 + term * multi2; multi1; ];
  • MultiNomialNegate(multi) - negate a multivar, returning -multi, and destructively changing the original:

    [ multi := - multi; multi1; ];
  • MultiNomialMultiply(multi1,multi2) - Multiply two multivars, and (possibly) destructively modify multi1 to contain the result, returning the result:

    [ multi1 := multi1 * multi2; multi1; ];
  • NormalForm(multi) - convert MultiNomial to normal form (as would be typed in be the user). This is part of the driver because the driver might be able to do this more efficiently than code above it which can use ScanMultiNomial().

  • MultiLeadingTerm(multi) - return the (key,value) pair (mdeg(f),lc(f)) representing the leading term. This is all the information needed about the leading term, and thus the leading coefficient and multidegree can be extracted from it.

  • MultiDropLeadingZeroes(multi) - remove leading terms with zero factors.

  • MultiTermLess(x,y) - for two (key,value) pairs, return True if \(x<y\), where the operation \(<\) is the one used for the representation, and False otherwise.

  • ScanMultiNomial(op,multi) - traverse all the terms of the multivariate, applying the function op to each (key,value) pair (exp,coef). The monomials are traversed in the ordering defined by MultiTermLess. op should be a function accepting two arguments.

  • MultiZero(multi) - return True if the multivariate is zero (all coefficients are zero), False otherwise.