public override string ToString()
{
var s = "[";
var i = 0;
for (; InterpolationCDFRanks[i] == i + 1 && i < InterpolationCDFRanks.Count; i++)
{
}
s += $"1-{i - 1}";
var ranksToShow = 100 + i;
for (; i < ranksToShow && i < InterpolationSize; i++)
{
s += ",";
s += InterpolationCDFRanks[i];
}
if (InterpolationSize > ranksToShow)
{
s += "...";
s += InterpolationCDFRanks.Last();
}
s += "]";
return($"Zipf for N = {N}, α = {Alpha}, K = {K}, C = {C}, ϵ = {Epsilon}. Size = {InterpolationSize}. Ranks = {s}");
}
private void InitInterpolationWithBoundedError(double maxPermittedError)
{
int rank = 1, startRank = 1, midPointRank = 1;
var prevPdf = PDF(rank);
var prevCdf = prevPdf;
var midPointCdf = prevCdf;
InterpolationCDFRanks.Add(1);
InterpolationCDFValues.Add(prevCdf);
// Advance three pointers, one at the start of the segment, one at the midpoint, and one at the end.
// The midpoint advances at half the speed of the end of the segment.
for (rank = 2; rank <= N - 1; rank++)
{
var currPdf = PDF(rank);
var currCdf = Min(1.0, prevCdf + currPdf);
if ((rank - startRank) % 2 == 0)
{
midPointRank++;
midPointCdf += PDF(midPointRank);
}
// No need to interpolate if the segment has no gap that needs interpolating.
if (rank - startRank >= 2)
{
// Add a provisional interpolation point.
InterpolationCDFRanks.Add(rank);
InterpolationCDFValues.Add(currCdf);
// Calculate the interpolation error for the midpoint.
// If it is still small enough, remove the provisional interpolation point.
var endPointPosition = InterpolationSize - 1;
var cdfInterpolator = CDFInterpolator(endPointPosition);
var estimatedMidpointCdf = Min(1.0, cdfInterpolator.Y(midPointRank));
var relativeError = Abs(midPointCdf - estimatedMidpointCdf) / midPointCdf;
InterpolationCDFRanks.RemoveAt(endPointPosition);
InterpolationCDFValues.RemoveAt(endPointPosition);
// It is not guaranteed that the midpoint of the segment is where the worst error would be found.
// Thus permit less error than requested in the hope that this will keep the true maximum error
// below maxPermittedError as well.
if (relativeError >= maxPermittedError / 7)
{
// We decided we need an interpolation point, thus are starting a new segment.
midPointRank = startRank = rank - 1;
midPointCdf = prevCdf;
InterpolationCDFRanks.Add(rank - 1);
InterpolationCDFValues.Add(prevCdf);
}
}
prevPdf = currPdf;
prevCdf = currCdf;
}
// Close out the last segment.
InterpolationCDFRanks.Add(N);
InterpolationCDFValues.Add(1);
}
/// <summary>
/// Perform no interpolation; store CDF values for all ranks, which may consume much memory.
/// </summary>
private void InitInterpolationWithZeroError()
{
var cdf = 0.0;
for (var rank = 1; rank <= N; rank++)
{
cdf += PDF(rank);
InterpolationCDFRanks.Add(rank);
InterpolationCDFValues.Add(cdf);
}
}
/// <summary>
/// Interpolate the CDF using the control points.
/// </summary>
/// <param name="rank">Rank for which the CDF is sought.</param>
/// <returns>The cumulative density value for the given rank, which varies from a small value at rank = 1
/// to one for rank = N.</returns>
public double ApproximateCDF(int rank)
{
// NOTE: This is expensive: execution time proportional to rank/2.
if (rank >= N)
{
return(1.0);
}
if (rank <= 1)
{
return(PDF(1));
}
var position = InterpolationCDFRanks.BinarySearch(rank);
if (position > 0)
{
return(InterpolationCDFValues[position]); // Exact match - no interpolation needed.
}
position = ~position;
Interpolator <double> interpolator = CDFInterpolator(position);
return(Min(1, interpolator.Y(rank)));
}
