10089 - Repackaging

All about problems in Volume 100. If there is a thread about your problem, please use it. If not, create one with its number in the subject.

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liux0229
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I got WA...

Post by liux0229 » Tue May 03, 2005 3:50 pm

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
Thx in advance!

arif_pasha
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Post by arif_pasha » Tue Sep 06, 2005 8:01 pm

can anybody post some test data? thnx in advance...

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Abednego
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Post by Abednego » Mon Oct 24, 2005 10:12 pm

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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Post by Per » Tue Oct 25, 2005 4:09 pm

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.

(So warning, spoiler ahead.)



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?

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Abednego
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Post by Abednego » Tue Oct 25, 2005 4:23 pm

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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Post by Per » Tue Oct 25, 2005 4:39 pm

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
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Post by TrePe » Wed Nov 02, 2005 2:02 am

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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Hmmm... quite interesting

Post by icesoft85 » Sun Sep 10, 2006 7:16 am

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. :D

Erik
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Post by Erik » Mon Sep 11, 2006 9:27 am

Hi,

could anyone please post some test data?

Cu, Erik :)

poixp
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WA

Post by poixp » Wed Nov 19, 2008 6:17 pm

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
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Post by poixp » Fri Nov 21, 2008 7:22 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
3
16277 11507 4034
9738 10437 2768
13616 22265 6031
3
2539 25774 5201
1362 13182 25249
12466 23233 27583
3
26488 5026 28376
17669 13884 17873
29206 3629 360
3
4769 6630 8252
4604 25317 8286
12101 25805 13545
3
27691 8846 12831
21940 29900 14318
4981 16265 14079
3
30329 15272 16820
19833 13408 30007
21891 474 13330
3
23262 17554 27104
7335 32362 1099
25819 20863 29131
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27310 3425 27942
26219 8270 29506
25467 22206 29596
3
12734 3858 28187
29032 13814 11734
8906 26277 18849
3
4339 13918 12141
13824 1011 4164
30533 4884 12070
3
9885 20129 5902
31048 26953 14763
31239 11358 25412
3
7232 16881 25057
1590 10043 2766
31262 21573 10999
3
7968 8925 23598
29455 29856 12609
15252 3944 24882
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21231 7064 11890
7193 23154 23465
27818 24423 10975
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24221 15716 15800
1716 11777 20679
32 17926 26542
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6937 8290 14300
10653 24893 13155
13014 28669 28281
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22494 6418 6530
22397 32088 21327
1535 28321 5975
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1875 823 19823
4303 3467 31729
5985 5085 27122
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7498 23019 31159
19507 26191 11708
30436 23314 21963
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32066 22440 20232
10846 5228 24771
8237 26831 23987
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10331 19292 7331
28763 31542 29329
1775 21494 8748
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23701 12138 7467
7132 6110 3014
18357 7839 17584
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12609 29102 30394
19081 17527 23566
5336 31932 7034
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21615 24759 29276
17956 27667 9249
5467 28844 8231
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8734 11507 21624
22345 11880 26437
11058 22601 22815
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9793 22558 7724
702 5061 15275
11639 27726 16649
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25928 20138 6105
21818 6008 21002
3362 3941 21033
3
16547 22163 8087
22338 7822 19063
6979 23931 8816
3
29801 4323 25969
1603 13068 32482
25850 21849 24061
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12730 15541 23453
2054 19802 11238
6685 12693 1482
3
13953 25797 32620
19217 12677 3437
18841 27735 27019
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16407 5801 30279
6047 29809 22297
20663 6222 6306
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29965 24459 6400
22850 3140 10966
12169 10922 24140
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21059 4090 30575
22258 20404 21693
6185 13565 19752
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4578 1202 3402
3356 19144 31885
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86 23902 12809
16882 13548 20105
23042 24283 2545
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3452 8035 21324
32038 26503 1677
3372 5032 8410
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2111 4691 11197
3901 4466 4324
7117 19714 14977
3
18909 17071 6200
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17319 11049 27031
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30 29952 8995
4049 19508 6324
21356 4873 21397
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26551 28215 1395
9525 9340 7733
20687 1638 14176
3
16741 21994 9292
27205 17966 9575
31121 1917 24235
3
13759 15451 24555
32326 14983 26747
31844 21896 242
3
14824 32347 16431
30102 11115 29630
22913 14195 19867
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8731 10977 23601
2572 3335 10400
18857 9924 15296
3
28949 18599 7662
529 496 13550
7635 20598 22475
3
28285 17669 20353
11266 28689 28381
19068 17605 12011
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10195 1054 6566
23817 15709 14740
3
22347 18756 19307
20594 9177 13155
5107 32614 9620
3
2473 26883 14810
1426 1761 14338
7716 29983 12858
3
23096 5117 8799
4437 20113 12085
24629 16541 16541
3
10917 1380 26071
220 3940 18164
12537 21049 23614
3
4568 24384 12500
16737 10189 29821
30212 17498 901
3
7856 32157 24640
18437 30679 12398
30255 16204 6664
3
6158 8696 11317
14782 24537 18199
29598 14961 26421
3
27110 24778 29033
18714 10512 2308
2771 31953 27463
3
4741 17685 29691
5434 25240 30303
22998 23489 30460
3
4226 22460 1627
28298 5894 748
13787 1303 15222
3
22413 862 1692
5119 25840 2944
32149 8250 3862
3
14800 8358 4165
2418 26753 17284
536 28399 10338
3
20243 10518 6965
17680 1228 5091
10536 22889 4450
3
1142 15026 2117
12965 3695 28099
11617 29862 28586
3
8854 6812 18972
22292 30517 13670
1030 3517 7231
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2165 27794 13871
26345 7609 2366
5631 17475 13459
3
18493 7585 29117
10306 7452 8832
7913 6814 20645
3
32590 23682 14088
2052 3737 31918
8609 18216 13216
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18560 26298 29292
25956 11969 32396
4983 28674 20464
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14633 25952 24639
2951 9907 27670
19813 29312 20428
3
6912 30030 17176
21069 20192 24254
20947 13883 7933
3
30198 960 3831
18817 22405 15870
15540 16160 15714
3
28287 25134 4602
16952 4620 25242
19395 28417 23433
3
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17669 28359 30908
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31828 13574 31215
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11972 5605 17273
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28011 21202 30827
9252 12701 25781
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18997 21818 4093
17844 17979 9361
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13082 15463 19109
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3621 22564 26862
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4211 25704 2318
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31545 14808 13654
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16825 1847 23906
3
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18544 3993 9093
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3
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14406 30684 20
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3
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21414 32190 21550
3
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21836 513 27687
9174 1430 17502
3
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29326 4619 8201
18912 17750 19128
3
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4227 8004 28360
14164 30999 24457
3
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19332 6867 14672
5969 29910 18913
3
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27590 5238 25184
10916 24063 30952
3
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12454 2853 13526
15463 18228 29691
3
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26851 25632 28538
7077 7213 31853
3
9042 20521 15298
24204 20893 395
17426 31866 11898
3
13632 32589 10231
29623 28845 1238
25644 31720 14314
3
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27078 11468 22286
30074 7435 20108
3
28513 8971 25347
31208 14783 4563
4922 16586 30398
3
23849 1158 25669
21106 29403 22341
1484 8406 28169
3
13542 30881 14702
5178 10411 18319
13867 10409 30124
3
17116 6543 2528
3616 10874 8209
19848 4056 18557
3
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4050 8498 24876
4186 4909 142
3
8720 24332 26691
24075 4765 5274
19258 24198 30794
3
1135 5725 13676
12775 26352 25109
5336 30510 27323
3
24852 26875 13980
27436 19668 25281
1407 28710 30272
3
5553 22095 23312
20145 11542 26945
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3
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28348 30273 193
14534 6684 27161
3
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19597 9572 15044
4326 10733 12808
3
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29968 9597 24579
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32445 27794 1321
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23042 4070 11696
8094 2785 23845
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30499 5491 15976
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15866 15734 6150
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3895 3923 31992
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19257 21623 27192
3
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17066 19240 22479
3
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31102 31847 318
11258 6707 30127
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23670 14688 14351
24615 3617 6496
24655 6384 12402
3
10416 15065 11811
18344 3807 29716
29017 6792 1266
3
30973 4977 28427
28496 14712 11661
32596 20187 25827
3
23390 15076 19808
21745 19106 10257
26929 6838 25578
3
31156 1040 25086
28833 11555 22863
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3
7990 14751 7945
28814 22437 11422
30565 16555 5114
3
18648 10572 10364
12341 5213 5207
29636 18779 30513
3
848 31163 29464
10843 12018 9403
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3
8764 29943 17686
17976 20226 11782
2814 4984 13205
3
8081 19409 14049
9458 4336 7831
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3
21779 24539 8773
23730 27291 706
18522 4700 21799
3
5829 11932 8430
27815 28899 32677
19438 23706 27764
3
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11182 27782 25088
26303 16750 27976
3
4601 22366 10294
24648 20277 14382
10865 8399 6191
3
7382 8191 11991
5702 19777 9453
6849 10922 12017
3
11652 22854 17020
22678 7046 5040
13873 991 14516
3
18506 14392 20880
14995 20245 5852
5077 13168 5820
3
21867 4821 5282
11722 24728 32651
9743 27675 25118
3
2257 5874 20624
30630 18110 3035
26375 20715 28722
3
19959 19083 31004
3759 32657 6087
20354 9323 31193
3
25537 29538 30883
14308 2757 10021
32305 29429 27550
3
28342 2145 21499
25357 27948 16897
24154 7027 17833
3
6839 30702 1233
897 7832 12877
2290 19871 27637
3
14669 20455 26917
15224 2759 10420
20307 11888 23450
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27209 23210 22378
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17079 30455 8367
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3
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27268 5977 4849
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3
11493 32598 9847
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3
30578 25393 30482
26500 15770 521
27099 14200 4626
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32738 9676 28014
5148 27206 92
20174 25097 23185
3
4226 11652 429
3226 31435 21435
11536 18581 17323
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23784 22374 9272
7184 8203 32321
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15927 4841 29037
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6916 32266 26632
6536 31887 23321
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12128 9938 23368
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29903 19153 5583
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11319 10122 19422
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12812 31786 27132
3
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30332 29434 11253
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122 16352 27293
3
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30528 10186 11215
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12122 31221 18459
3
503 7399 4667
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3
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30102 12166 17209
3
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3
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3
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26451 4159 116
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337 6183 21118
20512 5560 31786
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10040 219 16515
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19420 30641 9766
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26430 28400 6379
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11259 30279 29638
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23035 26929 16831
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32100 18770 167
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11066 5118 20907
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23237 18497 13734
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11224 13059 25297
3
11240 3535 7307
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228 1566 17330
3
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13419 13909 32131
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3
26770 27782 31460
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22715 30370 25484
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32121 28678 28518
28513 11149 24411
29933 9078 908
3
13286 13811 10158
5963 13494 10139
32170 32440 916
3
26589 406 17177
27445 8932 18946
3993 23988 26548
3
14829 19200 14143
12304 31011 32450
6416 8091 18122
3
12091 28830 15933
23095 2361 32533
12744 17926 2235
3
29956 15068 4457
31992 30294 30984
31680 3491 6788
3
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26061 12821 28631
3
29423 2044 13189
24524 16766 20291
6675 30522 6449
3
28793 14026 13388
27540 4807 24766
27675 27244 19886
3
2809 22738 27978
18507 30392 21264
21856 24026 21976
3
14989 25492 27514
29067 20993 13523
11055 10575 4673
3
22696 16429 24097
14507 13837 22612
7661 5317 12929
3
1783 9034 22817
20314 15585 32100
29572 12637 410
3
29753 12358 18060
20966 32009 11516
32319 18319 21255
3
24149 18294 28260
2133 11586 18798
27266 9007 28506
3
2674 21458 7015
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19473 29622 2780
3
2714 11647 18789
26400 31767 27504
26782 19861 3737
3
11140 26061 12549
6548 6131 28828
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1064 2411 19966
3
14812 25966 353
19714 22888 15690
12692 851 5871
3
17283 16829 19614
9980 17355 21205
31172 15232 31393
3
30832 13532 26564
14710 23628 26231
162 5647 16426
3
39 19701 10538
20956 20546 6734
8292 1440 17123
3
2537 10074 27308
13362 10719 27663
11786 14309 11932
3
17038 16294 31239
24724 11216 19878
7416 4658 17375
3
2553 14518 12817
23356 28211 24410
19823 16254 29080
3
19174 5855 30525
25974 6205 17725
29998 27577 13171
3
24295 1679 15289
22855 17706 198
10582 20974 5115
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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No

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

Re:

Post by HIT_Nessie » Thu Nov 27, 2008 2:42 pm

Thx to your testdata, I finally got AC. :D
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

Post by wallace » Tue Mar 31, 2009 2:16 am

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

User avatar
tryit1
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Posts: 119
Joined: Sun Aug 24, 2008 9:09 pm

Re:

Post by tryit1 » Sun Aug 23, 2009 7:13 pm

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 :wink:
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
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Re: 10089 - Repackaging

Post by mf » Sun Aug 23, 2009 10:26 pm

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.

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