TopCoder Statistics - Problem Statement

Problem Statement

Given a sequence of bits (0's and 1's), we want to find an arbitrary monotonically increasing curve that best fits the bits. That is, the i^th bit is b(i), and we want to find some curve, f, such that for x<y, f(x) <= f(y), and the sum over i of (f(i)-b(i))² (the squared error) is minimized.

Given the sequence of bits as a String[] where you concatenate all the elements together, return the minimum possible squared error.

Definition

Class:: MonotoneSEMin
Method:: min
Parameters:: String[]
Returns:: double
Method signature:: double min(String[] bits)
(be sure your method is public)

Notes

Your return must have relative or absolute error less than 1e-9.

Constraints

bits will contain between 1 and 50 elements, inclusive.
Each element of bits will contain between 1 and 50 bits ('0' or '1'), inclusive.

Examples

{"10001110"}

Returns: 1.5

The flat curve f(x) = 0.5 would give a squared error of 0.52*8 = 2. Naturally, we can do better than this though, and it turns out that the best we can do is a squared error of 1.5.
{"00"}

Returns: 0.0
{"11"}

Returns: 0.0
{"1010100101010101001010101001", "0101010100100010010001010101", "1110110101001011010111011011", "1010111101110110111000111100"}

Returns: 26.244842801985662
{"0"}

Returns: 0.0
{"1"}

Returns: 0.0
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Returns: 579.8817573645022
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Returns: 545.0970048193549
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Returns: 600.6762521920788
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Returns: 579.216752450332
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Returns: 330.5078392861091
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Returns: 506.16197478066016
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Returns: 440.1165102225989
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Returns: 277.84487459811254
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Returns: 468.3924471741213
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Returns: 556.1927998586552
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Returns: 150.76628467385166
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Returns: 592.2222753152366
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Returns: 593.5283686590507
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Returns: 588.5836266639033
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Returns: 359.5025180263054
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Returns: 464.46981354463855
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Returns: 530.0922803265539
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Returns: 580.6546792517744
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Returns: 229.90873641347946
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Returns: 446.2785725895594
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Returns: 597.0963981295463
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Returns: 541.3731870599095
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Returns: 600.4010746988947
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Returns: 619.1436995687652
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Returns: 505.7295264131258
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Returns: 386.4547832319554
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Returns: 379.0783738765844
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Returns: 237.55076689027587
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Returns: 399.18181404569407
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Returns: 440.5224808624014
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Returns: 557.0648411715243
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Returns: 492.894952248188
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Returns: 608.9374156522341
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Returns: 455.29825490901356
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Returns: 203.92296359332067
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Returns: 473.7942942587598
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Returns: 590.110111358052
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Returns: 493.00866604693795
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Returns: 583.7044074963262
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Returns: 421.78556040323446
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Returns: 500.9037245693683
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Returns: 430.90732561044234
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Returns: 572.2477714587611
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Returns: 236.20907594716616
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Returns: 251.01867057034377
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Returns: 462.696250009413
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Returns: 444.4694268459882
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Returns: 343.2893778765228
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Returns: 118.48413749960375
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Returns: 547.4745016577905
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Returns: 582.1711298753518
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Returns: 566.980343600062
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Returns: 383.863014069264
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Returns: 609.2989195678272
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Returns: 383.32670103092784
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Returns: 407.0143751261789
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Returns: 615.3900000000001
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Returns: 617.0740296118447
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Returns: 620.1219463356028
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Returns: 395.82046461784364
{"00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000"}

Returns: 0.0
{"11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111"}

Returns: 0.0
{"0000000000000000000000000000000", "000000000000000000000000000000000000000001", "010101010101010000000000000000010101", "00000000000000000000000000000010101010101", "0101010000000000000000000000000000000000000000001", "01010101010101010101010101010101010101010101010101", "010101010000000000000000000000000000000000010101", "01010101010101010101010101010101010101010101010101", "010100000000000000000000000000000000000000101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "111111111111111111111111111111111111111111111101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01111111111111111111111111111111111111111111010101", "01010101010101010101010101010101010101010101010101", "01011111111111111111111111111111111110101", "01111111111111111111111111111111101010101010101", "0101011111111111111111111111111111" }

Returns: 537.2492554299207
{"01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101011101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101000101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010001010101", "01010101010101010101010101010101010100010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101110101110100010101010101010101010101", "11010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101110101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010111010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101110101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010101010101010101010101010101010101010101", "01010101010001010001010111011101010101010101010101" }

Returns: 624.3321668129328
{"10000000100000000000000100000000000000000000000001", "11000000000000000000000000000000000000000000000000", "11100000000000000000000000000000000000000000000000", "11110000000000000000000000000000000000000000000000", "11111000000000000000000000000000000000000000000000", "11111100000000000000000000000000000000000000000000", "11111110000000000000000000000000000000000000000000", "11111111000000000000000000000000000000000000000000", "11111111100010000000000000000000100010000000000001", "11111111110000000000000000000000000000000000000000", "11111111111000000000000000000000000000000000000000", "11111111111100000000000000000000000000000000000000", "11111111111110000000000000000000000000000000000000", "11111111111111000000000000000000000000000000000000", "11111111111111100000000000000000000000000000000000", "11111111111111110000000000000000000000000000000000", "11111111111111111000000000000000000000000000000000", "11111111111111111100000000000000000000000000000000", "11111111111111111110000000000000000000000000000000", "11111111111111111111000000000000000000000000000000", "11111111111111111111100000000000000000000000000000", "11111111111111111111110000000000000000000000000000", "11111111111111111111111000000000000000000000000000", "11111111111111111111111100000000000000000000000000", "11111111101111111111111110000000000011000000000000", "11111111111111111111111111000000000000000000000000", "11111111111111111111111111100000000000000000000000", "11111111111111111111111111110000000000000000000000", "11111111111111111111111111111000000000000000000000", "11111111111111111111111111111100000000000000000000", "11111111111111111111111111111110000000000000000000", "11111111111111111111111111111111000000000000000000", "11110111111111111110111111111111100000000000000000", "11111111111111111111111111111111110000000000000000", "11111111111111111111111111111111111000000001000000", "11111111111111111111111111111111111100000000000000", "11111111111111111111111111111111111110000000000000", "11111111111111111111111111111111111111000000000000", "11111111111111111111111111111111111111100000000000", "11111111101111111111111111111101111111110000000000", "11111111111111111111111111111111111111111000000000", "11111111111111111111111111111111111111111100000000", "11111111111111111111111111111111111111111110000000", "11111111111111111111111111111111111111111111000000", "11111111111111111111111111111111111111111111100000", "11111111111111111111111111111111111111111111110000", "11111111111111111111111111111111111111111111111000", "11111111111111111111111111111111111111111111111100", "11111111111111111111111111111111111111111111111110", "11111111011111111111111111111111111111111111111101" }

Returns: 424.7605367231639
{"11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111", "11101001101010101011101010101010000111110101010111" }

Returns: 608.2295238095226
{"11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "11111111111111111111111111111111111111111111111111", "10000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000", "00000000000000000000000000000000000000000000000000" }

Returns: 399.39959999999996
{"100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010", "100101001010100101001011100101010110001010101010" }

Returns: 595.6999999999989

Statistics

Problem Statement for "MonotoneSEMin"

Problem Statement

Definition

Notes

Constraints

Examples