  The pivot algorithm -- we also call it the pivot chain -- is an algorithm for
approximately uniform sampling from $\Omega _{N},$ the set of $N$-step
self-avoiding walks on $\mathbb{Z}^{d}$ ($N,$ $d\geq 1$). Based on this
algorithm and the $G$ method, we construct another algorithm/chain, called the
pivot$^{\text{+}}$ algorithm/chain, for approximately uniform sampling from
$\Omega _{N},$ here, $N\geq 2$. The pivot$^{\text{+}}$ algorithm samples the
pivot from the set $\left\{ 1,2,...,N-1\right\} $ according to the uniform
probability distribution on this set while the pivot algorithm samples the
pivot from the set $\left\{0,1,2,...,N-1\right\} $ according to the uniform
probability distribution on this set, so, on the pivot, the pivot$^{\text{+}}$
algorithm is better than the pivot algorithm. Further, we obtain another
important thing, namely, the pivot$^{\text{+}}$ algorithm/chain enters, at time
$1$, a set $2d$ times smaller than $\Omega _{N},$ and stays forever in this
set, so, at times $1,2,...$ we work with a chain having a state space $2d$
times smaller than $\Omega _{N}$. As to the speed of convergence, we conjecture
that the pivot$^{\text{+}}$ algorithm/chain is faster than the pivot
algorithm/chain.
