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* Huffman encoding: minimum weighted path length of a binary prefix code.
*
* Repeatedly merging the two smallest live weights constructs a Huffman tree.
* The sum of all merge weights is that tree's weighted path length and is
* minimum among all full binary prefix-code trees over the input frequencies.
*
* This verification-oriented implementation uses a fresh selection scan for
* each of the two minima. Removing a selected item is done by moving the last
* live item into its slot, so only the prefix work[0..active) is live.
*/
/*@ Extern Coq
(Forall : {A} -> (A -> Prop) -> list A -> Prop)
(Z::le : Z -> Z -> Prop)
(Z::ge : Z -> Z -> Prop)
(HuffmanOptimalCost : list Z -> Z -> Prop)
(HuffmanProgress : list Z -> list Z -> Z -> Z -> Prop)
(HuffmanMinScan : list Z -> Z -> Z -> Prop)
(HuffmanFirstHeld : list Z -> list Z -> Z -> Z -> Z -> Prop)
(HuffmanPairReady : list Z -> list Z -> Z -> Z -> Z -> Z -> Prop)
*/
/*@ Import Coq Require Import PVbench.Algorithms.huffman_encoding.rocq.spec_lib */
/*@ Import Coq Require Import PVbench.Algorithms.huffman_encoding.rocq.helper_lib */
int huffman_cost(int *weights, int n)
/*@ With (weights_l : list Z)
Require
1 <= n && n <= 8 &&
Zlength(weights_l) == n &&
Forall(Z::le(1), weights_l) && Forall(Z::ge(1000), weights_l) &&
IntArray::full(weights, n, weights_l)
Ensure
HuffmanOptimalCost(weights_l, __return) &&
IntArray::full(weights, n, weights_l)
*/
{
int work[8];
/*@ Inv Assert
exists copied remaining,
weights == weights@pre && n == n@pre &&
1 <= n@pre && n@pre <= 8 &&
Zlength(weights_l) == n@pre &&
Forall(Z::le(1), weights_l) && Forall(Z::ge(1000), weights_l) &&
weights_l == app(copied, remaining) &&
Zlength(copied) == i &&
0 <= i && i <= n@pre &&
IntArray::full(weights, n@pre, weights_l) *
IntArray::seg(work, 0, i, copied) *
IntArray::undef_seg(work, i, n@pre) *
IntArray::undef_seg(work, n@pre, 8)
*/
for (int i = 0; i < n; ++i) {
work[i] = weights[i];
}
int active = n;
int total = 0;
/*@ Inv Assert
exists work_l,
weights == weights@pre && n == n@pre &&
1 <= n@pre && n@pre <= 8 &&
Zlength(weights_l) == n@pre &&
Zlength(work_l) == n@pre &&
Forall(Z::le(1), weights_l) && Forall(Z::ge(1000), weights_l) &&
1 <= active && active <= n@pre &&
0 <= total && total <= 56000 &&
Forall(Z::le(1), sublist(0, active, work_l)) &&
Forall(Z::ge(8000), sublist(0, active, work_l)) &&
HuffmanProgress(weights_l, work_l, active, total) &&
IntArray::full(weights, n@pre, weights_l) *
IntArray::full(work, n@pre, work_l) *
IntArray::undef_seg(work, n@pre, 8)
*/
while (active > 1) {
int first = 0;
/*@ Inv Assert
exists work_l,
weights == weights@pre && n == n@pre &&
1 <= n@pre && n@pre <= 8 &&
Zlength(weights_l) == n@pre &&
Zlength(work_l) == n@pre &&
Forall(Z::le(1), weights_l) && Forall(Z::ge(1000), weights_l) &&
2 <= active && active <= n@pre &&
1 <= i && i <= active && i <= n@pre &&
0 <= first && first < i && first < n@pre &&
0 <= total && total <= 56000 &&
Forall(Z::le(1), sublist(0, active, work_l)) &&
Forall(Z::ge(8000), sublist(0, active, work_l)) &&
HuffmanProgress(weights_l, work_l, active, total) &&
HuffmanMinScan(work_l, i, first) &&
IntArray::full(weights, n@pre, weights_l) *
IntArray::full(work, n@pre, work_l) *
IntArray::undef_seg(work, n@pre, 8)
*/
for (int i = 1; i < active; ++i) {
/*@ 0 <= i && i < n@pre */
if (work[i] < work[first]) {
first = i;
}
}
int x = work[first];
--active;
work[first] = work[active];
int second = 0;
/*@ Inv Assert
exists work_l,
weights == weights@pre && n == n@pre &&
1 <= n@pre && n@pre <= 8 &&
Zlength(weights_l) == n@pre &&
Zlength(work_l) == n@pre &&
Forall(Z::le(1), weights_l) && Forall(Z::ge(1000), weights_l) &&
1 <= active && active < n@pre &&
1 <= i && i <= active && i < n@pre &&
0 <= first && first < n@pre &&
0 <= second && second < i && second < n@pre &&
1 <= x && x <= 8000 &&
0 <= total && total <= 56000 &&
Forall(Z::le(1), sublist(0, active, work_l)) &&
Forall(Z::ge(8000), sublist(0, active, work_l)) &&
HuffmanFirstHeld(weights_l, work_l, active, x, total) &&
HuffmanMinScan(work_l, i, second) &&
IntArray::full(weights, n@pre, weights_l) *
IntArray::full(work, n@pre, work_l) *
IntArray::undef_seg(work, n@pre, 8)
*/
for (int i = 1; i < active; ++i) {
if (work[i] < work[second]) {
second = i;
}
}
int y = work[second];
--active;
work[second] = work[active];
int merged = x + y;
total += merged;
work[active] = merged;
++active;
}
/*@ Assert
weights == weights@pre && n == n@pre &&
active == 1 &&
HuffmanOptimalCost(weights_l, total) &&
IntArray::full(weights, n@pre, weights_l) *
IntArray::undef_full(work, 8)
*/
return total;
}
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