TD 6 : Compression

𝑢, 𝑣 ∈Σ∗𝑢−𝑣={𝑢𝑙+1 . . . 𝑢𝑘}𝑢=𝑢1. . . 𝑢𝑘
𝑣=𝑢1. . . 𝑢𝑙𝑙≤𝑘 𝑢 −𝑣=∅𝑎𝑏𝑐 −𝑎={𝑏𝑐}𝑎𝑏 −𝑏𝑎 =∅
𝑆⊆Σ∗
𝑇0=∪
𝑢,𝑣∈𝑆
𝑢∕=𝑣
𝑢−𝑣
𝑇 𝑇0
∪
𝑠∈𝑆,𝑣∈𝑇
((𝑠−𝑣)∪(𝑣−𝑠)) ⊆𝑇
𝑆 𝑇
𝑆
𝑆0={0,10,11}𝑆1={0,01,11}𝑆2={0,01,10}𝑆3={0,01}
𝑆4={00,01,10,11}𝑆5={110,11,10}𝑆6={110,11,100,00,10}
𝑆
𝑝 𝑙𝑜𝑔(1/𝑝(𝑥))
𝑙(𝑥) = 𝑙𝑜𝑔(1/𝑝(𝑥))
𝑙′(𝑥)
𝑃 𝑟[𝑙(𝑥)< 𝑙′(𝑥)] ≥𝑃 𝑟[𝑙(𝑥)> 𝑙′(𝑥)]
∀𝑥:𝑙′(𝑥) = 𝑙(𝑥)𝑠𝑔𝑛(𝑙(𝑥)−𝑙′(𝑥)) ≤2𝑙(𝑥)−𝑙′(𝑥)−1
𝑥
𝑛
𝑘
𝑛

12
3
𝑛
𝑖 𝑎+
𝑖+𝑎− 𝑖−𝑎=
𝑖
=
𝑖 𝑏+
𝑖+𝑏− 𝑖−𝑏=
𝑖=
𝜋 𝑠𝑔𝑛(𝑎𝑖) = 𝑠𝑔𝑛(𝑏𝜋(𝑖))𝑠𝑔𝑛(𝑥)∈
{−1,0,1}𝑥
𝑀[𝑖1, 𝑖2, . . . , 𝑖8]
𝑀[𝑖1, 𝑖2, . . . , 𝑖8]
𝑖1+−
=
𝑖2+−
=
𝑖8