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Right Skew Pointers

Right Skew Pointers. 2. 2. -. -. 1 3 1 7 0 1. 1 3 1 7 0 1. -. -. 4 6 4 5 3 0. 4 6 4 5 3 0. -. -. 2. 2. -. -. 4. 4. -. -. 8. 8. 1 2 3 4 5 6.

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Right Skew Pointers

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  1. Right Skew Pointers 2 2 - - 1 3 1 7 0 1 1 3 1 7 0 1 - - 4 6 4 5 3 0 4 6 4 5 3 0 - - 2 2 - - 4 4 - - 8 8 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 7 8 9 10 11 12 13 14 15 16 13 14 15 16 Right half Left half As a whole

  2. Finding Right Skew Pointers S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 2 2 - - 1 3 1 7 0 1 1 3 1 7 0 1 - - 4 6 4 5 3 0 4 6 4 5 3 0 - - 2 2 - - 4 4 - - 8 8 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 7 8 9 10 11 12 13 14 15 16 13 14 15 16 0.5 4.5 i=1 Prefix sum: 2, 1, 4, 5, 12, 12, 13, 9, 15, 19, 24, 27, 27, 25, 21, 13 μ(S1)=2.4, μ(S1S2)=13/7, μ(S1S2S3)=9/8, μ(S1…S4)=15/9, μ(S1…S5)=24/11, μ(S1…S6)=27/12, μ(S1…S7)=27/13, μ(S1…S8)=25/14, μ(S1…S9)=21/15, μ(S1…S10)=15/16, So the first DRSP(1,16) starting from i=1 is determined (at index 5).

  3. Finding Right Skew Pointers S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 2 2 - - 1 3 1 7 0 1 1 3 1 7 0 1 - - 4 6 4 5 3 0 4 6 4 5 3 0 - - 2 2 - - 4 4 - - 8 8 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 7 8 9 10 11 12 13 14 15 16 13 14 15 16 i=2 0.5 4.5 Prefix sum: -1, 2, 3, 10, 10, 11, 7, 13, 17, 22, 25, 25, 23, 19, 11 μ(S1)=2.5, μ(S1S2)=11/6, μ(S1S2S3)=7/7, μ(S1…S4)=13/8, μ(S1…S5)=22/10, μ(S1…S6)=25/11, μ(S1…S7)=27/13, μ(S1…S8)=23/13, μ(S1…S9)=19/14, μ(S1…S10)=11/15 So the first DRSP(2,16) starting from i=2 is determined (at index 5).

  4. Finding Right Skew Pointers S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 2 2 - - 1 3 1 7 0 1 1 3 1 7 0 1 - - 4 6 4 5 3 0 4 6 4 5 3 0 - - 2 2 - - 4 4 - - 8 8 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 7 8 9 10 11 12 13 14 15 16 13 14 15 16 i=3 0.5 4.5 Prefix sum: 3, 4, 11, 11, 12, 8, 14, 18, 23, 26, 26, 24, 20, 12 μ(S1)=11/3, μ(S1S2)=12/5, μ(S1…S3)=8/6, μ(S1…S4)=14/7, μ(S1…S5)=23/9, μ(S1…S6)=26/10, μ(S1…S7)=26/11, μ(S1…S8)=24/12, μ(S1…S9)=20/13, μ(S1…S10)=12/14 So the first DRSP(3,16) starting from i=3 is determined (at index 5).

  5. Finding Right Skew Pointers S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 2 2 - - 1 3 1 7 0 1 1 3 1 7 0 1 - - 4 6 4 5 3 0 4 6 4 5 3 0 - - 2 2 - - 4 4 - - 8 8 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 7 8 9 10 11 12 13 14 15 16 13 14 15 16 i=4 0.5 4.5 Prefix sum: 1, 8, 8, 9, 5, 11, 15, 20, 23, 23, 21, 17, 9 μ(S1)=8/2, μ(S1S2)=9/4, … obviously μ(S1) is the largest. So the first DRSP(4,16) starting from i=4 is determined (at index 5).

  6. Finding Right Skew Pointers S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 2 2 - - 1 3 1 7 0 1 1 3 1 7 0 1 - - 4 6 4 5 3 0 4 6 4 5 3 0 - - 2 2 - - 4 4 - - 8 8 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 7 8 9 10 11 12 13 14 15 16 13 14 15 16 i=5 0.5 4.5 Prefix sum: 7, 7, 8, 4, 10, 14, 19, 22, 22, 20, 16, 8 μ(S1)=8/2,…, … obviously μ(S1) is the largest. So the first DRSP(5,16) starting from i=5 is determined (at index 5).

  7. Finding Right Skew Pointers S2 S3 S4 S5 S6 S7 S8 S9 S10 2 2 - - 1 3 1 7 0 1 1 3 1 7 0 1 - - 4 6 4 5 3 0 4 6 4 5 3 0 - - 2 2 - - 4 4 - - 8 8 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 7 8 9 10 11 12 13 14 15 16 13 14 15 16 i=6 Prefix sum: 0, 1, -3, 3, 7, 12, 15, 15, 13, 9, 1 μ(S2)=1/2, μ(S2S3)=-3/3, μ(S2…S4)=3/4, μ(S2…S5)=12/6, μ(S2…S6)=15/7, μ(S2…S7)=15/8, μ(S2…S8)=13/9, μ(S2…S9)=9/10, μ(S2…S10)=1/11 So the first DRSP(6,16) starting from i=6 is determined (at index 12).

  8. Finding Right Skew Pointers S2 S3 S4 S5 S6 S7 S8 S9 S10 2 2 - - 1 3 1 7 0 1 1 3 1 7 0 1 - - 4 6 4 5 3 0 4 6 4 5 3 0 - - 2 2 - - 4 4 - - 8 8 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 7 8 9 10 11 12 13 14 15 16 13 14 15 16 i=7 Prefix sum: 1, -3, 3, 7, 12, 15, 15, 13, 9, 1 μ(S2)=1, μ(S2S3)=-3/2, μ(S2…S4)=3/3, μ(S2…S5)=12/5, μ(S2…S6)=15/6, μ(S2…S7)=15/7, μ(S2…S8)=13/8, μ(S2…S9)=9/9, μ(S2…S10)=1/10 So the first DRSP(7,16) starting from i=7 is determined (at index 12).

  9. Finding Right Skew Pointers S2 S3 S4 S5 S6 S7 S8 S9 S10 2 2 - - 1 3 1 7 0 1 1 3 1 7 0 1 - - 4 6 4 5 3 0 4 6 4 5 3 0 - - 2 2 - - 4 4 - - 8 8 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 7 8 9 10 11 12 13 14 15 16 13 14 15 16 i=8 Prefix sum: -4, 2, 6, 11, 14, 14, 12, 8, 0 μ(S3)=-4, μ(S3S4)=-2/2, μ(S3…S5)=11/4, μ(S3…S6)=14/5, μ(S3…S7)=14/6, μ(S3…S8)=12/7, μ(S3…S9)=8/8, μ(S3…S10)=0 So the first DRSP(8,16) starting from i=8 is determined (at index 12).

  10. Finding Right Skew Pointers S2 S3 S4 S5 S6 S7 S8 S9 S10 2 2 - - 1 3 1 7 0 1 1 3 1 7 0 1 - - 4 6 4 5 3 0 4 6 4 5 3 0 - - 2 2 - - 4 4 - - 8 8 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 7 8 9 10 11 12 13 14 15 16 13 14 15 16 i=8 Note that we have correctly find the right-skew pointers of the whole sequence. Prefix sum: -4, 2, 6, 11, 14, 14, 12, 8, 0 μ(S3)=-4, μ(S3S4)=-2/2, μ(S3…S5)=11/4, μ(S3…S6)=14/5, μ(S3…S7)=14/6, μ(S3…S8)=12/7, μ(S3…S9)=8/8, μ(S3…S10)=0 So the first DRSP(8,16) starting from i=8 is determined (at index 12).

  11. Using Pointer-Jumping Table to Find the Maximum (Take i = 0 as an example) 2 2 - - 1 3 1 7 0 1 1 3 1 7 0 1 - - 4 6 4 5 3 0 4 6 4 5 3 0 - - 2 2 - - 4 4 - - 8 8 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 7 8 9 10 11 12 13 14 15 16 13 14 15 16 Pointer -Jumping table i=1 TPJ(9,0)=9, TPJ(9,1)=11, TPJ(9,2)=13, TPJ(9,3)=16, TPJ(9,4) = 16 TPJ(10,0)=11, TPJ(10,1)=12, TPJ(10,2)=14, TPJ(10,3)=16, TPJ(10,4) = 16 TPJ(11,0)=11, TPJ(11,1)=12, TPJ(11,2)=14, TPJ(11,3)=16, TPJ(11,4) = 16 TPJ(12,0)=12, TPJ(12,1)=13, TPJ(12,2)=15, TPJ(12,3)=16, TPJ(12,4) = 16 TPJ(13,0)=13, TPJ(13,1)=14, TPJ(13,2)=16, TPJ(13,3)=16, TPJ(13,4) = 16 TPJ(14,0)=14, TPJ(14,1)=15, TPJ(14,2)=16, TPJ(14,3)=16, TPJ(10,4) = 16 TPJ(15,0)=15, TPJ(15,1)=16, TPJ(15,2)=16, TPJ(15,3)=16, TPJ(15,4) = 16 TPJ(16,0)=16, TPJ(16,1)=16, TPJ(16,2)=16, TPJ(16,3)=16, TPJ(16,4) = 16

  12. μ(S1…S7)=27/13 Step1:μ(SLSR(TPJ(9,2)))=27/13 > μ(SLSR(TPJ(9,3)))=15/16 2 2 - - 1 3 1 7 0 1 1 3 1 7 0 1 - - 4 6 4 5 3 0 4 6 4 5 3 0 - - 2 2 - - 4 4 - - 8 8 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 7 8 9 10 11 12 13 14 15 16 13 14 15 16 i=1 TPJ(9,0)=9, TPJ(9,1)=11, TPJ(9,2)=13, TPJ(9,3)=16 TPJ(10,0)=11, TPJ(10,1)=12, TPJ(10,2)=14, TPJ(10,3)=16 TPJ(11,0)=11, TPJ(11,1)=12, TPJ(11,2)=14, TPJ(11,3)=16 TPJ(12,0)=12, TPJ(12,1)=13, TPJ(12,2)=15, TPJ(12,3)=16 TPJ(13,0)=13, TPJ(13,1)=14, TPJ(13,2)=16, TPJ(13,3)=16 TPJ(14,0)=14, TPJ(14,1)=15, TPJ(14,2)=16, TPJ(14,3)=16 TPJ(15,0)=15, TPJ(15,1)=16, TPJ(15,2)=16, TPJ(15,3)=16 TPJ(16,0)=16, TPJ(16,1)=16, TPJ(16,2)=16, TPJ(16,3)=16 Pointer -Jumping table

  13. Step1:μ(SLSR(TPJ(9,2)))=27/13 > μ(SLSR(P(TPJ(9,2)+1))=25/14 Step2:μ(SLSR(TPJ(9,1)))=24/11 < μ(SLSR(P(TPJ(9,1)+1)))=27/12 2 2 - - 1 3 1 7 0 1 1 3 1 7 0 1 - - 4 6 4 5 3 0 4 6 4 5 3 0 - - 2 2 - - 4 4 - - 8 8 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 7 8 9 10 11 12 13 14 15 16 13 14 15 16 i=1 TPJ(9,0)=9, TPJ(9,1)=11, TPJ(9,2)=13, TPJ(9,3)=16 TPJ(10,0)=11, TPJ(10,1)=12, TPJ(10,2)=14, TPJ(10,3)=16 TPJ(11,0)=11, TPJ(11,1)=12, TPJ(11,2)=14, TPJ(11,3)=16 TPJ(12,0)=12, TPJ(12,1)=13, TPJ(12,2)=15, TPJ(12,3)=16 TPJ(13,0)=13, TPJ(13,1)=14, TPJ(13,2)=16, TPJ(13,3)=16 TPJ(14,0)=14, TPJ(14,1)=15, TPJ(14,2)=16, TPJ(14,3)=16 TPJ(15,0)=15, TPJ(15,1)=16, TPJ(15,2)=16, TPJ(15,3)=16 TPJ(16,0)=16, TPJ(16,1)=16, TPJ(16,2)=16, TPJ(16,3)=16 Pointer -Jumping table

  14. Step1:μ(SLSR(TPJ(9,2)))=27/13 > μ(SLSR(P(TPJ(9,2)+1))=25/14 Step2:μ(SLSR(TPJ(9,1)))=24/11 < μ(SLSR(P(TPJ(9,1)+1)))=27/12 Step3:μ(SLSR(9,12) > μ(SLSR(P(12+1))) And μ(S1) > μ(SLSR(9,12)) So the maximal is at S1. 2 2 - - 1 3 1 7 0 1 1 3 1 7 0 1 - - 4 6 4 5 3 0 4 6 4 5 3 0 - - 2 2 - - 4 4 - - 8 8 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 7 8 9 10 11 12 13 14 15 16 13 14 15 16 i=1 TPJ(9,0)=9, TPJ(9,1)=11, TPJ(9,2)=13, TPJ(9,3)=16 TPJ(10,0)=11, TPJ(10,1)=12, TPJ(10,2)=14, TPJ(10,3)=16 TPJ(11,0)=11, TPJ(11,1)=12, TPJ(11,2)=14, TPJ(11,3)=16 TPJ(12,0)=12, TPJ(12,1)=13, TPJ(12,2)=15, TPJ(12,3)=16 TPJ(13,0)=13, TPJ(13,1)=14, TPJ(13,2)=16, TPJ(13,3)=16 TPJ(14,0)=14, TPJ(14,1)=15, TPJ(14,2)=16, TPJ(14,3)=16 TPJ(15,0)=15, TPJ(15,1)=16, TPJ(15,2)=16, TPJ(15,3)=16 TPJ(16,0)=16, TPJ(16,1)=16, TPJ(16,2)=16, TPJ(16,3)=16 Pointer -Jumping table

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