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/*-------------------------------------------------------------------------
*
* knapsack.c
* Knapsack problem solver
*
* Given input vectors of integral item weights (must be >= 0) and values
* (double >= 0), compute the set of items which produces the greatest total
* value without exceeding a specified total weight; each item is included at
* most once (this is the 0/1 knapsack problem). Weight 0 items will always be
* included.
*
* The performance of this algorithm is pseudo-polynomial, O(nW) where W is the
* weight limit. To use with non-integral weights or approximate solutions,
* the caller should pre-scale the input weights to a suitable range. This
* allows approximate solutions in polynomial time (the general case of the
* exact problem is NP-hard).
*
* Copyright (c) 2017-2023, PostgreSQL Global Development Group
*
* IDENTIFICATION
* src/backend/lib/knapsack.c
*
*-------------------------------------------------------------------------
*/
#include "postgres.h"
#include <math.h>
#include <limits.h>
#include "lib/knapsack.h"
#include "miscadmin.h"
#include "nodes/bitmapset.h"
#include "utils/builtins.h"
#include "utils/memutils.h"
/*
* DiscreteKnapsack
*
* The item_values input is optional; if omitted, all the items are assumed to
* have value 1.
*
* Returns a Bitmapset of the 0..(n-1) indexes of the items chosen for
* inclusion in the solution.
*
* This uses the usual dynamic-programming algorithm, adapted to reuse the
* memory on each pass (by working from larger weights to smaller). At the
* start of pass number i, the values[w] array contains the largest value
* computed with total weight <= w, using only items with indices < i; and
* sets[w] contains the bitmap of items actually used for that value. (The
* bitmapsets are all pre-initialized with an unused high bit so that memory
* allocation is done only once.)
*/
Bitmapset *
DiscreteKnapsack(int max_weight, int num_items,
int *item_weights, double *item_values)
{
MemoryContext local_ctx = AllocSetContextCreate(CurrentMemoryContext,
"Knapsack",
ALLOCSET_SMALL_SIZES);
MemoryContext oldctx = MemoryContextSwitchTo(local_ctx);
double *values;
Bitmapset **sets;
Bitmapset *result;
int i,
j;
Assert(max_weight >= 0);
Assert(num_items > 0 && item_weights);
values = palloc((1 + max_weight) * sizeof(double));
sets = palloc((1 + max_weight) * sizeof(Bitmapset *));
for (i = 0; i <= max_weight; ++i)
{
values[i] = 0;
sets[i] = bms_make_singleton(num_items);
}
for (i = 0; i < num_items; ++i)
{
int iw = item_weights[i];
double iv = item_values ? item_values[i] : 1;
for (j = max_weight; j >= iw; --j)
{
int ow = j - iw;
if (values[j] <= values[ow] + iv)
{
/* copy sets[ow] to sets[j] without realloc */
if (j != ow)
{
sets[j] = bms_del_members(sets[j], sets[j]);
sets[j] = bms_add_members(sets[j], sets[ow]);
}
sets[j] = bms_add_member(sets[j], i);
values[j] = values[ow] + iv;
}
}
}
MemoryContextSwitchTo(oldctx);
result = bms_del_member(bms_copy(sets[max_weight]), num_items);
MemoryContextDelete(local_ctx);
return result;
}
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