How many images?
If you limit the maximum page size, and have a value for the minimum picture height, you can calculate the maximum number of images per page. You would need this when evaluating any solution.
I think there were 27 pictures on the link you gave.
The following uses the first_fit algorithm mentioned by Robin Green earlier but then improves on this by greedy swapping.
The swapping routine finds the column that is furthest away from the average column height then systematically looks for a swap between one of its pictures and the first picture in another column that minimizes the maximum deviation from the average.
I used a random sample of 30 pictures with heights in the range five to 50 ‘units’. The convergenge was swift in my case and improved significantly on the first_fit algorithm.
The code (Python 3.2:
def first_fit(items, bincount=3):
items = sorted(items, reverse=1) # New - improves first fit.
bins = [[] for c in range(bincount)]
binsizes = [0] * bincount
for item in items:
minbinindex = binsizes.index(min(binsizes))
bins[minbinindex].append(item)
binsizes[minbinindex] += item
average = sum(binsizes) / float(bincount)
maxdeviation = max(abs(average - bs) for bs in binsizes)
return bins, binsizes, average, maxdeviation
def swap1(columns, colsize, average, margin=0):
'See if you can do a swap to smooth the heights'
colcount = len(columns)
maxdeviation, i_a = max((abs(average - cs), i)
for i,cs in enumerate(colsize))
col_a = columns[i_a]
for pic_a in set(col_a): # use set as if same height then only do once
for i_b, col_b in enumerate(columns):
if i_a != i_b: # Not same column
for pic_b in set(col_b):
if (abs(pic_a - pic_b) > margin): # Not same heights
# new heights if swapped
new_a = colsize[i_a] - pic_a + pic_b
new_b = colsize[i_b] - pic_b + pic_a
if all(abs(average - new) < maxdeviation
for new in (new_a, new_b)):
# Better to swap (in-place)
colsize[i_a] = new_a
colsize[i_b] = new_b
columns[i_a].remove(pic_a)
columns[i_a].append(pic_b)
columns[i_b].remove(pic_b)
columns[i_b].append(pic_a)
maxdeviation = max(abs(average - cs)
for cs in colsize)
return True, maxdeviation
return False, maxdeviation
def printit(columns, colsize, average, maxdeviation):
print('columns')
pp(columns)
print('colsize:', colsize)
print('average, maxdeviation:', average, maxdeviation)
print('deviations:', [abs(average - cs) for cs in colsize])
print()
if __name__ == '__main__':
## Some data
#import random
#heights = [random.randint(5, 50) for i in range(30)]
## Here's some from the above, but 'fixed'.
from pprint import pprint as pp
heights = [45, 7, 46, 34, 12, 12, 34, 19, 17, 41,
28, 9, 37, 32, 30, 44, 17, 16, 44, 7,
23, 30, 36, 5, 40, 20, 28, 42, 8, 38]
columns, colsize, average, maxdeviation = first_fit(heights)
printit(columns, colsize, average, maxdeviation)
while 1:
swapped, maxdeviation = swap1(columns, colsize, average, maxdeviation)
printit(columns, colsize, average, maxdeviation)
if not swapped:
break
#input('Paused: ')
The output:
columns
[[45, 12, 17, 28, 32, 17, 44, 5, 40, 8, 38],
[7, 34, 12, 19, 41, 30, 16, 7, 23, 36, 42],
[46, 34, 9, 37, 44, 30, 20, 28]]
colsize: [286, 267, 248]
average, maxdeviation: 267.0 19.0
deviations: [19.0, 0.0, 19.0]
columns
[[45, 12, 17, 28, 17, 44, 5, 40, 8, 38, 9],
[7, 34, 12, 19, 41, 30, 16, 7, 23, 36, 42],
[46, 34, 37, 44, 30, 20, 28, 32]]
colsize: [263, 267, 271]
average, maxdeviation: 267.0 4.0
deviations: [4.0, 0.0, 4.0]
columns
[[45, 12, 17, 17, 44, 5, 40, 8, 38, 9, 34],
[7, 34, 12, 19, 41, 30, 16, 7, 23, 36, 42],
[46, 37, 44, 30, 20, 28, 32, 28]]
colsize: [269, 267, 265]
average, maxdeviation: 267.0 2.0
deviations: [2.0, 0.0, 2.0]
columns
[[45, 12, 17, 17, 44, 5, 8, 38, 9, 34, 37],
[7, 34, 12, 19, 41, 30, 16, 7, 23, 36, 42],
[46, 44, 30, 20, 28, 32, 28, 40]]
colsize: [266, 267, 268]
average, maxdeviation: 267.0 1.0
deviations: [1.0, 0.0, 1.0]
columns
[[45, 12, 17, 17, 44, 5, 8, 38, 9, 34, 37],
[7, 34, 12, 19, 41, 30, 16, 7, 23, 36, 42],
[46, 44, 30, 20, 28, 32, 28, 40]]
colsize: [266, 267, 268]
average, maxdeviation: 267.0 1.0
deviations: [1.0, 0.0, 1.0]
Nice problem.
Heres the info on reverse-sorting mentioned in my separate comment below.
>>> h = sorted(heights, reverse=1)
>>> h
[46, 45, 44, 44, 42, 41, 40, 38, 37, 36, 34, 34, 32, 30, 30, 28, 28, 23, 20, 19, 17, 17, 16, 12, 12, 9, 8, 7, 7, 5]
>>> columns, colsize, average, maxdeviation = first_fit(h)
>>> printit(columns, colsize, average, maxdeviation)
columns
[[46, 41, 40, 34, 30, 28, 19, 12, 12, 5],
[45, 42, 38, 36, 30, 28, 17, 16, 8, 7],
[44, 44, 37, 34, 32, 23, 20, 17, 9, 7]]
colsize: [267, 267, 267]
average, maxdeviation: 267.0 0.0
deviations: [0.0, 0.0, 0.0]
If you have the reverse-sorting, this extra code appended to the bottom of the above code (in the ‘if name == …), will do extra trials on random data:
for trial in range(2,11):
print('\n## Trial %i' % trial)
heights = [random.randint(5, 50) for i in range(random.randint(5, 50))]
print('Pictures:',len(heights))
columns, colsize, average, maxdeviation = first_fit(heights)
print('average %7.3f' % average, '\nmaxdeviation:')
print('%5.2f%% = %6.3f' % ((maxdeviation * 100. / average), maxdeviation))
swapcount = 0
while maxdeviation:
swapped, maxdeviation = swap1(columns, colsize, average, maxdeviation)
if not swapped:
break
print('%5.2f%% = %6.3f' % ((maxdeviation * 100. / average), maxdeviation))
swapcount += 1
print('swaps:', swapcount)
The extra output shows the effect of the swaps:
## Trial 2
Pictures: 11
average 72.000
maxdeviation:
9.72% = 7.000
swaps: 0
## Trial 3
Pictures: 14
average 118.667
maxdeviation:
6.46% = 7.667
4.78% = 5.667
3.09% = 3.667
0.56% = 0.667
swaps: 3
## Trial 4
Pictures: 46
average 470.333
maxdeviation:
0.57% = 2.667
0.35% = 1.667
0.14% = 0.667
swaps: 2
## Trial 5
Pictures: 40
average 388.667
maxdeviation:
0.43% = 1.667
0.17% = 0.667
swaps: 1
## Trial 6
Pictures: 5
average 44.000
maxdeviation:
4.55% = 2.000
swaps: 0
## Trial 7
Pictures: 30
average 295.000
maxdeviation:
0.34% = 1.000
swaps: 0
## Trial 8
Pictures: 43
average 413.000
maxdeviation:
0.97% = 4.000
0.73% = 3.000
0.48% = 2.000
swaps: 2
## Trial 9
Pictures: 33
average 342.000
maxdeviation:
0.29% = 1.000
swaps: 0
## Trial 10
Pictures: 26
average 233.333
maxdeviation:
2.29% = 5.333
1.86% = 4.333
1.43% = 3.333
1.00% = 2.333
0.57% = 1.333
swaps: 4