Some Interval DP Problems and State Reduction

Revision en23, by lelbaba, 2022-11-08 11:09:46

Hi everyone! Recently I have been solving some classic interval DP problems, and came across some neat problems. Most of these problems have relatively simple recurrence relations, but the straightforward solutions will not pass in complexity, hence requires an observation to reduce the complexity (generally by reducing the number of states by some pre-computation).

Prerequisites : Introductory interval DP such as Longest Increasing Subsequence, Longest Common Subsequence.

1312E - Array Shrinking

You are given an array of $$$a_1,a_2,\dots,a_n$$$ length $$$n \ (n \leq 500, 1 \leq a_i \leq 1000).$$$ You can perform the following operation any number of times:

  • Choose a pair of two neighboring equal elements $$$a_i=a_{i+1}$$$ (if there is at least one such pair).
  • Replace them by one element with value $$$a_i + 1$$$. After each such operation, the length of the array will decrease by one (and elements are renumerated accordingly). What is the minimum possible length of the array a you can get?
Solution
Tags dynamic programming, intervals, tutorial

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en50 English lelbaba 2022-11-10 16:10:07 1209 (published)
en49 English lelbaba 2022-11-10 15:01:42 2223
en48 English lelbaba 2022-11-10 12:54:16 22
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en41 English lelbaba 2022-11-10 10:57:02 1313
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en30 English lelbaba 2022-11-08 14:11:03 188
en29 English lelbaba 2022-11-08 13:58:38 58
en28 English lelbaba 2022-11-08 13:57:19 5 Tiny change: '$f_{i,j,k} denote wh' -> '$f_{i,j,k}$ denote wh'
en27 English lelbaba 2022-11-08 13:56:26 921
en26 English lelbaba 2022-11-08 11:12:27 2 Tiny change: 'elation,\n\n$f_{l,l}' -> 'elation,\n$f_{l,l}'
en25 English lelbaba 2022-11-08 11:11:52 2 Tiny change: 'elation,\n$f_{l,l}' -> 'elation,\n\n$f_{l,l}'
en24 English lelbaba 2022-11-08 11:10:41 4 Tiny change: 'elation,\n\n$f_{l,l} = a_l$\n\n$f_{l,r}' -> 'elation,\n$f_{l,l} = a_l$\n$f_{l,r}'
en23 English lelbaba 2022-11-08 11:09:46 4
en22 English lelbaba 2022-11-08 11:09:08 5
en21 English lelbaba 2022-11-08 11:07:39 4
en20 English lelbaba 2022-11-08 11:05:47 1 Tiny change: 'rations.\n\n<spoiler' -> 'rations.\n<spoiler'
en19 English lelbaba 2022-11-08 11:04:51 2 Tiny change: 'i} + 1)$\nNote: $[' -> 'i} + 1)$\n\nNote: $['
en18 English lelbaba 2022-11-08 11:03:12 22 Tiny change: 'dp_1 = 1.$\n</spoile' -> 'dp_1 = 1.$ $dp_n$ is our answer.\n</spoile'
en17 English lelbaba 2022-11-08 11:01:56 18 Tiny change: '+ f_{j,i}$\n</spoil' -> '+ f_{j,i}$ where $dp_1 = 1.$\n</spoil'
en16 English lelbaba 2022-11-08 10:59:52 2 Tiny change: 'j \leq i, f_{j,i} \' -> 'j \leq i, \ f_{j,i} \'
en15 English lelbaba 2022-11-08 10:59:09 4 Tiny change: 'i} \ne 0} dp_{j-1} ' -> 'i} \ne 0} \ \ dp_{j-1} '
en14 English lelbaba 2022-11-08 10:58:27 237
en13 English lelbaba 2022-11-08 10:41:39 4 Tiny change: 'relation, //\n$f_{l,l}' -> 'relation, \\\n$f_{l,l}'
en12 English lelbaba 2022-11-08 10:41:09 76
en11 English lelbaba 2022-11-08 10:38:21 2 Tiny change: '{l,i} + 1)\n</spoile' -> '{l,i} + 1)$\n</spoile'
en10 English lelbaba 2022-11-08 10:37:57 158
en9 English lelbaba 2022-11-08 10:24:51 5
en8 English lelbaba 2022-11-08 10:22:29 233
en7 English lelbaba 2022-11-08 10:18:02 292
en6 English lelbaba 2022-11-08 09:37:52 9 Tiny change: 'ed to.\n\n#### Claim\nIf the s' -> 'ed to.\n\n**Claim**\nIf the s'
en5 English lelbaba 2022-11-08 09:36:52 305
en4 English lelbaba 2022-11-08 09:08:05 63
en3 English lelbaba 2022-11-08 09:03:16 98
en2 English lelbaba 2022-11-08 08:28:51 65
en1 English lelbaba 2022-11-08 08:19:37 1038 Initial revision (saved to drafts)