In the second part of the series papers, we set out to study the algorithmic efficiency of sparsity-constrained sensing (sparse sensing). Stemmed from co-prime sampling/array, we propose a generalized framework, termed Diophantine sensing, which utilizes generic <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Diophantine equation</i> theory and higher-order <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sparse ruler</i> to strengthen sampling performance with respect to the sampling time (delay), the degree of freedom (DoF), and the sampling sparsity, simultaneously. With a careful design of sampling spacings either in the temporal or spatial domain, co-prime sensing can reconstruct the autocorrelation of a sequence with significantly denser and more lags based on Bézout theorem. However, Bézout theorem also puts two practical constraints in the co-prime sensing framework. For frequency estimation, co-prime sampling needs <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Theta ((M_{1}+M_{2})L_{c})$</tex-math></inline-formula> samples, which results in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Theta (M_{1}M_{2}L_{c})$</tex-math></inline-formula> sampling time, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M_{1}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M_{2}$</tex-math></inline-formula> are co-prime down-sampling rates and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{c}$</tex-math></inline-formula> is the number of least snapshots needed for each lag. As for Direction-of-arrival (DoA) estimation, the sensors cannot be arbitrarily sparse in co-prime arrays, where the least inter spacing has to be less than a half of wavelength. Resorting to higher-moment statistics, the proposed Diophantine framework presents two fundamental improvements. First, on frequency estimation, we prove that given arbitrarily large down-sampling rates, there exist sampling schemes where the number of samples needed is only proportional to the sum of DoF and the least number of snapshots required for each lag, which implies a linear sampling time. Second, on DoA estimation, we propose two generic array constructions such that given <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> sensors, the minimal spacing among sensors can be as large as a polynomial of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Theta (N^{q})$</tex-math></inline-formula> , which indicates that an arbitrarily sparse array (with arbitrarily small mutual coupling) exists given sufficiently many sensors. In addition, the proposed array configurations produce the best known asymptotic DoF bound compared to existing coarray designs.