# performance tuning – How can I more quickly get the interchangeable variables in a function?

Let’s say I have an expression like `a b + c d e`. This is equal to: `a b + c d e`, `a b + c e d`, `a b + d c e`, `a b + d e c`, `a b + e c d`, `a b + e d c`, `b a + c d e`, `b a + c e d`, `b a + d c e`, `b a + d e c`, `b a + e c d`, and `b a + e d c`

As you can see, swapping `a` and `b` will always result in an expression equal to the original expression and swapping `c`, `d`, and `e` will always result in an expression equal to the original expression.

The following code is too slow for more complicated expressions. Is there a speedier way to get Mathematica to give me all equivalence classes of swappable variables such that `f(a b + c d e) == {{a, b}, {c, d, e}}` for some `f`?

``````SwapVariables(expr_, variable1_, variable2_) := expr /. variable1 -> replacedInSwapVariablesFunction /. variable2 -> variable1 /. replacedInSwapVariablesFunction -> variable2;

VariablesIn(expr_) := Integrate`getAllVariables({expr}, {});

SwappableVariablesIn(expr_) := (
vars = VariablesIn(expr);
originalVars = vars;
results = {};
While(Length(vars) > 0, (
var = First(vars);
vars = Rest(vars);
swappable = Map(TrueQ(ForAll(originalVars, SwapVariables(expr, var, #) == expr)) &, vars);
results = Append(results, Prepend(Pick(vars, swappable), var));
vars = Pick(vars, swappable, False);
));
results
);

SwappableVariablesIn(a b + c d e)

(* {{a,b},{c,d,e}} *)
``````

A function which can be used for a complicated expression to test timing:

``````DetNByN(n_) := Det(Table(Table(Indexed(x, {i, j}), {j, 1, n}), {i, 1, n}));

First(Timing(SwappableVariablesIn(DetNByN(6))))

(* 10.7118 *)
``````

Posted on Categories Articles