## 10089 - Repackaging

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liux0229
New poster
Posts: 8
Joined: Thu May 20, 2004 1:30 pm
Location: China

### I got WA...

I first convert (s1,s2,s3) to (s1-s3,s2-s3), then I find the convex hull and determine wether (0,0) is within this convex hull
But my code is rejected as WA. My code for finding convex hull && checking Point in Polygon has been tested on other problems so they should be ok
Can anyone give me a hint about the tricky case, or some test data, plz?
I have already considered points forming a line && identical points

arif_pasha
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Posts: 42
Joined: Fri Jun 13, 2003 3:47 pm
Contact:
can anybody post some test data? thnx in advance...

Abednego
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Posts: 281
Joined: Tue Sep 10, 2002 5:14 am
Location: Mountain View, CA, USA
Contact:
Per, could you explain why 4 different vectors are always enough? I think 3 are always enough, but I don't trust my proof of that. ;-)
If only I had as much free time as I did in college...

Per
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Posts: 429
Joined: Fri Nov 29, 2002 11:27 pm
Location: Sweden
My proof that 4 vectors are enough was a bit messy. But when thinking about using just three vectors, I came up with a quite nice characterization of when the problem is solvable.

Let v_1, ..., v_n be the vectors sorted by angle, and let t_i be the angle from v_i to v_{i+1} (where v_{n+1} = v_1). Then, the problem is solvable if and only if max(t_i) is at most pi.

If some t_i is greater than pi, there is a line through (0, 0) such that every vector lies strictly in the halfplane defined by the line, which means that every move takes you further away from the line, and hence that the problem is not solvable.

If all t_i are at most pi, it is possible to throw away all but three of the v_i:s and still have that condition hold, i.e. there are three vectors w_1, w_2, w_3 so that the angle from w_1 to w_2, the angle from w_2 to w_3, and the angle from w_3 to w_1, all are at most pi. Now, if you think about it a while, you see that one of the w_i:s (wlog, say w_3) must satisfy that -w_3 lies between w_1 and w_2. Then there exists a positive linear combination of w_1 and w_2 which is a multiple of -w_3 (seeing this is just a matter of writing things down and using that -w_3 lies between w_1 and w_2), which means the vectors w_1, w_2 and w_3 can be used to form a solution.

I am too lazy to write down all the details, but it shouldn't be hard to work them out.

I guess maybe that's the proof you had as well?

Abednego
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Posts: 281
Joined: Tue Sep 10, 2002 5:14 am
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Yes, that's what I thought. Thanks. I did not believe that it could work because Adrian had a 3D multi-pyramid, you had an n^2 algorithm, and everyone else was doing convex hulls. How could this simple little algorithm possibly be correct?! ;-)
If only I had as much free time as I did in college...

Per
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Posts: 429
Joined: Fri Nov 29, 2002 11:27 pm
Location: Sweden
I remember spending way too much time thinking about this problem, back when, so I guess I was just happy when I finally solved it.

TrePe
New poster
Posts: 3
Joined: Sat Mar 15, 2003 10:27 pm
Very nice solution, guys. Two thumbs up.
And it's O(n), because you do not need to compare each pair and even do not need to sort, in order to find max(t_i). Cute!

icesoft85
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Joined: Sun Mar 26, 2006 5:51 am
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### Hmmm... quite interesting

It took me a while to figure out why what you guys where saying was true (and I'm not doubting about Abednego or Per statements ), but, hey, it is a really good solution.

Erik
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Location: Germany
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Hi,

could anyone please post some test data?

Cu, Erik

poixp
New poster
Posts: 20
Joined: Mon Apr 07, 2008 11:05 am

### WA

Hi I'm trying to solve that problem but absolutely stuck. Here what I do:
1. Convert package (s1,s2,s3) to 2D vector [s2-s1,s3-s1]
2. Sort it by angle
3. If angle of 2 neighbor vectors is exceed pi than there is no solution, otherwise it has solution
I prepare some test cases, what is correct results?

Code: Select all

3
100 110 102
100 90 98
100 90 90
3
100 110 102
100 90 98
100 110 110
3
100 110 102
100 90 98
200 180 196
3
100 110 101
100 90 98
100 90 90
5
5 10 5
5 8 8
5 6 10
5 8 2
5 5 1
3
1 2 3
2 4 6
3 6 9
0

My program returns:

Code: Select all

Yes
Yes
Yes
No
No
No


poixp
New poster
Posts: 20
Joined: Mon Apr 07, 2008 11:05 am
More test cases:

Code: Select all

3
7162 16586 13541
16799 28225 1564
25273 10247 12189
3
6028 5145 12303
23369 120 28610
19219 11396 8503
3
24187 24908 17135
20977 29338 11079
26167 10163 26825
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16277 11507 4034
9738 10437 2768
13616 22265 6031
3
2539 25774 5201
1362 13182 25249
12466 23233 27583
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26488 5026 28376
17669 13884 17873
29206 3629 360
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4769 6630 8252
4604 25317 8286
12101 25805 13545
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27691 8846 12831
21940 29900 14318
4981 16265 14079
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30329 15272 16820
19833 13408 30007
21891 474 13330
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23262 17554 27104
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25467 22206 29596
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12734 3858 28187
29032 13814 11734
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4339 13918 12141
13824 1011 4164
30533 4884 12070
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9885 20129 5902
31048 26953 14763
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1590 10043 2766
31262 21573 10999
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1716 11777 20679
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3
29253 8775 18707
10728 23216 5551
26395 27726 11842
3
31217 4013 7115
10349 25798 26994
19702 12736 27021
3
4889 22862 23822
12770 10655 23665
16333 1137 973
3
4495 10684 11788
18178 9501 32713
252 14642 20174
3
15921 24213 29743
18367 26089 32526
24103 28913 22845
3
27535 2926 15869
27575 24202 28030
14339 10562 32335
3
29501 11492 6287
9691 32382 24241
31928 12273 21193
3
18275 23075 30829
24927 17612 17066
2004 5753 16589
3
11512 7471 3025
10698 23350 2586
17507 20315 5352
3
5400 21877 30532
14353 22057 8405
16040 26014 16799
3
1122 7033 19713
21805 8909 4659
23325 6977 10657
3
29642 20578 30867
14465 14304 569
30718 19889 18236
3
19362 2192 23513
19064 28955 25053
27909 20729 31685
3
664 30447 25637
20418 32100 17537
23247 481 2023
3
6294 246 12413
32 28161 26544
25506 23547 30928
3
29025 14450 14115
19666 28208 23495
29464 19798 23740
3
6170 7553 18998
8970 31396 4291
28180 21443 21442
3
722 25060 21404
5790 30280 18858
29461 23564 16820
3
32389 20582 3250
8500 10733 5558
30316 1606 9785
3
26511 20719 6619
18532 6031 30124
9389 30480 28512
3
20215 20947 13003
10218 17000 32658
29586 24774 6785
3
4727 32659 11254
13202 32418 7991
468 28634 31589
3
21088 12617 14063
5390 27376 8255
1120 4352 12426
3
28761 18840 22537
29084 23521 9989
23279 9629 24322
3
16506 9548 1171
311 12498 4764
14297 21251 24791
3
22933 2079 29667
17762 25015 11442
13441 18627 5923
3
8190 10538 25442
12464 7505 16514
10151 7228 11233
3
12918 17688 1344
23313 23411 18113
10663 5104 22485
3
4313 28897 9218
29603 24994 31027
25924 17017 22929
3
3921 10090 29637
20143 17454 15979
14134 28479 5395
3
23103 17778 30625
23023 3047 24840
32630 28142 7087
3
25509 8709 17996
5296 11146 1502
27500 10648 32061
3
1980 2084 22190
11647 3248 859
25872 10111 16252
3
22487 13119 9037
15677 7760 26334
8337 30030 32201
3
27575 31101 16488
30240 25599 8704
29454 14563 28123
3
26429 28456 1125
25123 17211 30958
13585 28773 6366
3
21255 255 15405
27670 26098 4549
24451 27222 30374
3
28816 30390 22344
12405 30835 6488
12735 1590 17150
3
25738 9453 11939
17171 31354 6614
16384 501 6953
3
32004 12030 12962
14962 3196 13534
3939 14152 16882
3
24793 13729 6207
11577 22720 29690
29005 21245 3296
3
16020 22226 25524
25384 15793 471
24274 7566 10884
3
2451 30268 32686
28228 22068 17920
30644 14206 12508
3
13379 20093 31151
5353 3451 8953
2834 9849 12510
3
13524 21687 29229
24857 16981 18645
23086 3398 32134
3
4961 21614 9752
23701 14559 24113
21337 18810 6009
3
18635 7575 29914
32334 17323 24602
20523 20236 18328
3
5913 29466 24001
18235 24109 28713
750 18654 13061
3
23709 5086 19827
20572 3991 13621
10456 13608 23681
3
15308 12121 11284
19627 4028 23671
26417 7222 17849
3
22247 1000 32306
25250 30174 2868
27651 32085 11148
3
16910 1033 13000
30950 24104 15232
8278 20137 29405
3
26532 22614 32668
10856 25113 1266
10261 8021 19595
3
11575 11966 27190
14226 20584 17049
14231 12504 19352
3
31298 31284 31869
15735 30837 3433
32186 23661 14799
3
23529 4805 29413
28633 24243 26613
30188 3394 2114
3
1575 27130 26476
28676 8288 2575
4395 21006 25780
3
25858 21193 14094
21606 25878 25421
22657 26448 30159
3
18172 5414 32389
15364 30497 2482
27190 5172 14090
3
8645 1746 19230
14931 13597 13681
17547 103 1290
3
13171 1135 2371
11809 5476 20016
18301 25141 26915
3
20038 29573 32387
17220 516 9260
21037 15282 20917
3
20578 10474 7577
4331 7748 28155
20099 22804 9046
3
31585 12481 29262
31244 26957 1292
15997 20624 29144
3
3948 23194 21695
15604 857 27214
8564 10402 31520
3
16831 4362 24934
32393 25002 6428
24312 23735 31154
3
5007 6461 23887
25847 6355 2421
3293 21739 20325
3
19382 10355 7777
21657 29849 27795
23783 8854 28467
3
19278 24662 6050
30535 19886 6311
672 953 9298
3
29704 22742 13638
11096 25391 30964
7539 25984 30703
3
8018 29123 12941
21001 28538 32557
17647 20036 14054
3
29133 6816 11200
19723 17626 32076
7047 13462 28376
3
19892 28261 25077
7695 7764 21754
19225 6459 24899
3
26125 9338 6277
11612 24355 28508
10930 24219 21328
3
15697 31387 5577
15438 19712 2269
23388 8652 25636
3
11729 27033 19208
2776 30655 25083
11936 3089 14670
3
29144 5286 24428
23615 10818 7922
24618 19547 3884
3
11196 30072 3754
8653 18026 4084
22758 27210 6822
3
8302 15663 17551
7604 27627 15872
29821 25090 24816
3
24162 17739 18689
20989 608 14374
1844 14679 13242
3
1563 11911 27146
7902 31031 13025
21039 23571 24813
3
7420 12907 22776
9655 24437 28344
30253 16384 543
3
23656 11600 5698
6045 12647 21291
1644 12592 30912
3
18987 7814 17613
12386 14612 18583
14707 29733 6644
3
28103 3515 23048
13654 21697 19633
25532 23991 16219
3
17634 10599 32574
2937 13750 8610
22439 23342 26368
3
10947 27265 2085
24276 14690 1433
26176 10649 20502
3
27062 6526 26718
9344 27526 802
6514 5515 8409
3
2578 20270 22257
28173 17021 18710
30210 1993 14528
3
8993 7073 15599
25483 3924 12175
7371 319 19861
3
16620 19857 25755
21951 22522 9990
29848 26610 5413
3
25981 25489 30509
16704 2309 24999
8734 13074 13537
3
20068 11832 11793
9016 11679 14599
7789 270 25503
3
5056 22096 18035
31875 32029 12389
24314 736 11623
3
15255 13933 10669
26836 21531 32191
19804 3092 19888
3
15426 4170 18215
4901 282 5721
31345 24348 18674
3
9179 27152 29199
27630 9139 15213
27024 10208 22710
3
28945 7381 6846
15960 22715 8819
16724 23684 8111
3
25450 20022 4811
336 24418 27897
7873 17393 1588
3
17826 24996 30697
19116 21599 10559
11493 21134 26997
3
14 4483 26582
25978 18595 12572
3908 1476 10312
3
17376 26648 21793
1254 17705 14619
5320 6694 17417
3
20190 12688 27093
8741 3910 6496
2957 13511 26184
3
1255 8082 31790
28658 2445 26874
25897 939 1961
3
21265 18660 1211
18676 26107 7937
7182 25406 28315
3
5610 16180 13151
23988 1382 10413
1137 8782 28523
3
9572 14140 860
820 23417 32242
15515 22162 28227
3
29 25277 18823
9110 31445 21191
18449 28225 12598
3
12472 24132 475
16657 14699 21147
21440 12258 27582
3
14992 11606 17410
19700 15722 13205
32286 19544 31063
3
1990 18152 2595
16058 25994 9484
1440 5574 25119
3
3942 30350 16972
12511 12146 21958
14801 4361 10244
3
22814 16356 18628
13369 26252 7343
20833 12973 17039
3
8651 6369 28910
32202 29017 31927
19249 26098 15182
3
25039 29764 32235
25870 3534 31172
16686 20285 9628
3
22672 13934 11729
29984 13693 5499
9836 25674 3660
3
1126 10474 15703
30425 12262 13877
10320 13972 17146
3
2401 32724 23093
464 22261 25368
3500 3046 5766
3
10540 21079 2415
2699 6528 25976
18424 31340 3494
3
27366 2635 21212
24665 12291 3988
8297 19903 6893
3
5570 6225 10475
6649 23344 9112
2726 10077 25477
3
8279 23919 13612
985 19799 27328
5512 31257 5086
3
19859 13075 5036
9161 28197 11686
10070 5595 27386
3
12102 26205 2133
4897 3411 18667
27079 21945 2563
3
6580 26149 19124
18959 25816 10781
14555 21164 19979
3
28840 15634 9149
4595 3808 30430
16967 14033 29201
3
24100 18213 12388
17426 9780 17917
31499 10710 20997
3
8306 29225 20542
24002 6555 26061
15153 8582 15796
3
13203 2799 20837
29204 26309 11170
16043 16735 9158
3
9015 15976 14333
25265 29035 22828
25803 15677 13865
3
10255 21008 21476
4627 756 2738
3652 30047 25615
3
32682 15045 20708
9381 25013 20860
23924 30483 17228
3
24143 28013 16914
5847 22106 31883
16697 23838 23698
3
29184 3780 6144
14760 18631 31379
21863 2865 4661
3
21110 26235 28163
1745 32679 5910
10210 6501 7576
3
31096 5212 16469
28143 6717 22769
26928 19919 12428
3
9335 19077 4186
7824 32189 2040
25204 22060 25967
3
23700 7113 30186
20281 16227 8465
12853 13871 13884
3
15251 24265 20589
19165 2535 11367
9722 22074 28161
3
28365 3827 5652
7275 32485 23781
16588 12498 3389
0

My output:

Code: Select all

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HIT_Nessie
New poster
Posts: 6
Joined: Wed Nov 19, 2008 8:08 am

### Re:

Thx to your testdata, I finally got AC.
Here's my output on your testdata:

Code: Select all

No
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wallace
New poster
Posts: 10
Joined: Thu Feb 26, 2009 4:03 pm

### Re: 10089 - Repackaging

I didn't understand that problem!
can anyone me help?

tryit1
Experienced poster
Posts: 119
Joined: Sun Aug 24, 2008 9:09 pm

### Re:

cyfra wrote:Hi!

I have an other algorithm that is correct, and got AC.

My algorithm goes like this :

We treat is as the vectors on the plane. if the package is (s1,s2,s3) it is count as an vector [s2-s1,s3-s1]. And our problem is to find whether starting from point (0,0) you can return there.

And that's all..

Good Luck
How can you reduce like this ? I don't know what this theorem is called. Can someone explain me what is the significance of this reduction and how can you do something like this.

http://online-judge.uva.es/p/v100/10089.html
Last edited by tryit1 on Tue Aug 25, 2009 8:20 am, edited 1 time in total.

mf
Guru
Posts: 1244
Joined: Mon Feb 28, 2005 4:51 am
Location: Zürich, Switzerland
Contact:

### Re: 10089 - Repackaging

tryit1 wrote:
cyfra wrote:We treat is as the vectors on the plane. if the package is (s1,s2,s3) it is count as an vector [s2-s1,s3-s1]. And our problem is to find whether starting from point (0,0) you can return there.
How can you reduce like this ? I don't know what this theorem is called. Can someone explain me what is the significance of this reduction and how can you do something like this.
Well, isn't it obvious? A linear combinations of vectors (s2 - s1, s3 - s1) is zero if and only if the same linear combination of vectors (s1, s2, s3) is a vector with three equal components, namely, all equal to this linear combination of s1's. It's just simple algebra and manipulations of sums. It doesn't deserve to be called a theorem, or even lemma. The proper term for this is "left as an exercise to the reader".

Now, to answer why it's useful. To check that you can make a zero vector, you basically need to check that your collection of vectors spans an angle of 180 degrees or more, i.e. there doesn't exist a hyperplane w*x=0 through the origin ("*" is the dot product of vectors here), such that all the given vectors lie strictly to one side of it.

Necessity: if there exists a vector w such that w*x_i > 0 for all i, then for any non-negative coefficients c (not all zeros), the vector s = \sum c_i x_i satisfies: w*s = w * \sum (c_i x_i) = \sum (c_i w*x_i) = \sum (c_i * <something strictly positive>) > 0, and therefore s can't be a zero vector.

Sufficiency: if hypothesis is satistisfied, then you can always find a linear combination with rational non-negative coefficients (not all zeroes) of given vectors, which will sum to zero vector. I've tried writing a proof for this fact, but it was getting too long for a single post... You better try to draw some pictures and get a feel of it for yourself, instead. :)

And once you got a combination with rational coefficients, you can always turn it into integers by multiplying all coefficients by LCM of their denominators.