The Degree of Piecewise Monotone Interpolation

Let 0 = x₀ < x₁ < ⋯ < xₖ = 1 and let y₀,y₁, ⋯ ,yₖ be real numbers such that yj - 1 ≠ yⱼ,j = 1,2, ⋯ ,k. Estimates are obtained on the degree of an algebraic polynomial $p(x)$ that interpolates the given data piecewise monotonely; i.e., such that (i) p(xⱼ) = yⱼ,j = 0,1, ⋯ ,k, and such that (ii) $p(x)$ is increasing on Iⱼ = (xj - 1,xⱼ) if yⱼ < yj - 1, and decreasing on Iⱼ. if yⱼ < yj - 1,j = 1,2, ⋯ ,k. The problem is seen to be related to the problem of monotone approximation.