Add the tool used to generate the constrained tokenset.

The code that generates the raw distribution is based on a MATLAB
program by Debargha Mukherjee, and the algorithm used to quantize the
distribution comes from the ANS Toolkit by Jarek Duda.

Change-Id: Ic273f7d9e43e3ecd999e9e7e04cde57e8559375a

tools/gen_constrained_tokenset.py

+#!/usr/bin/python
+##
+## Copyright (c) 2016, Alliance for Open Media. All rights reserved
+##
+## This source code is subject to the terms of the BSD 2 Clause License and
+## the Alliance for Open Media Patent License 1.0. If the BSD 2 Clause License
+## was not distributed with this source code in the LICENSE file, you can
+## obtain it at www.aomedia.org/license/software. If the Alliance for Open
+## Media Patent License 1.0 was not distributed with this source code in the
+## PATENTS file, you can obtain it at www.aomedia.org/license/patent.
+##
+"""Generate the probability model for the constrained token set.
+
+Model obtained from a 2-sided zero-centered distribution derived
+from a Pareto distribution. The cdf of the distribution is:
+cdf(x) = 0.5 + 0.5 * sgn(x) * [1 - {alpha/(alpha + |x|)} ^ beta]
+
+For a given beta and a given probability of the 1-node, the alpha
+is first solved, and then the {alpha, beta} pair is used to generate
+the probabilities for the rest of the nodes.
+"""
+
+import heapq
+import sys
+import numpy as np
+import scipy.optimize
+import scipy.stats
+
+
+def cdf_spareto(x, xm, beta):
+  p = 1 - (xm / (np.abs(x) + xm))**beta
+  p = 0.5 + 0.5 * np.sign(x) * p
+  return p
+
+
+def get_spareto(p, beta):
+  cdf = cdf_spareto
+
+  def func(x):
+    return ((cdf(1.5, x, beta) - cdf(0.5, x, beta)) /
+            (1 - cdf(0.5, x, beta)) - p)**2
+
+  alpha = scipy.optimize.fminbound(func, 1e-12, 10000, xtol=1e-12)
+  parray = np.zeros(11)
+  parray[0] = 2 * (cdf(0.5, alpha, beta) - 0.5)
+  parray[1] = (2 * (cdf(1.5, alpha, beta) - cdf(0.5, alpha, beta)))
+  parray[2] = (2 * (cdf(2.5, alpha, beta) - cdf(1.5, alpha, beta)))
+  parray[3] = (2 * (cdf(3.5, alpha, beta) - cdf(2.5, alpha, beta)))
+  parray[4] = (2 * (cdf(4.5, alpha, beta) - cdf(3.5, alpha, beta)))
+  parray[5] = (2 * (cdf(6.5, alpha, beta) - cdf(4.5, alpha, beta)))
+  parray[6] = (2 * (cdf(10.5, alpha, beta) - cdf(6.5, alpha, beta)))
+  parray[7] = (2 * (cdf(18.5, alpha, beta) - cdf(10.5, alpha, beta)))
+  parray[8] = (2 * (cdf(34.5, alpha, beta) - cdf(18.5, alpha, beta)))
+  parray[9] = (2 * (cdf(66.5, alpha, beta) - cdf(34.5, alpha, beta)))
+  parray[10] = 2 * (1. - cdf(66.5, alpha, beta))
+  return parray
+
+
+def quantize_probs(p, save_first_bin, bits):
+  """Quantize probability precisely.
+
+  Quantize probabilities minimizing dH (Kullback-Leibler divergence)
+  approximated by: sum (p_i-q_i)^2/p_i.
+  References:
+  https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence
+  https://github.com/JarekDuda/AsymmetricNumeralSystemsToolkit
+  """
+  num_sym = p.size
+  p = np.clip(p, 1e-16, 1)
+  L = 2**bits
+  pL = p * L
+  ip = 1. / p  # inverse probability
+  q = np.clip(np.round(pL), 1, L + 1 - num_sym)
+  quant_err = (pL - q)**2 * ip
+  sgn = np.sign(L - q.sum())  # direction of correction
+  if sgn != 0:  # correction is needed
+    v = []  # heap of adjustment results (adjustment err, index) of each symbol
+    for i in range(1 if save_first_bin else 0, num_sym):
+      q_adj = q[i] + sgn
+      if q_adj > 0 and q_adj < L:
+        adj_err = (pL[i] - q_adj)**2 * ip[i] - quant_err[i]
+        heapq.heappush(v, (adj_err, i))
+    while q.sum() != L:
+      # apply lowest error adjustment
+      (adj_err, i) = heapq.heappop(v)
+      quant_err[i] += adj_err
+      q[i] += sgn
+      # calculate the cost of adjusting this symbol again
+      q_adj = q[i] + sgn
+      if q_adj > 0 and q_adj < L:
+        adj_err = (pL[i] - q_adj)**2 * ip[i] - quant_err[i]
+        heapq.heappush(v, (adj_err, i))
+  return q
+
+
+def get_quantized_spareto(p, beta, bits):
+  parray = get_spareto(p, beta)
+  parray = parray[1:] / (1 - parray[0])
+  qarray = quantize_probs(parray, True, bits)
+  return qarray.astype(np.int)
+
+
+def main(bits=8):
+  beta = 8
+  for q in range(1, 256):
+    parray = get_quantized_spareto(q / 256., beta, bits)
+    assert parray.sum() == 2**bits
+    print '{', ', '.join('%d' % i for i in parray), '},'
+
+
+if __name__ == '__main__':
+  if len(sys.argv) > 1:
+    main(int(sys.argv[1]))
+  else:
+    main()