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高效前綴約簡的三維Hilbert空間填充曲線編解碼算法

賈連印 范瑤 丁家滿 李曉武 游進(jìn)國

賈連印, 范瑤, 丁家滿, 李曉武, 游進(jìn)國. 高效前綴約簡的三維Hilbert空間填充曲線編解碼算法[J]. 電子與信息學(xué)報(bào), 2024, 46(2): 633-642. doi: 10.11999/JEIT230013
引用本文: 賈連印, 范瑤, 丁家滿, 李曉武, 游進(jìn)國. 高效前綴約簡的三維Hilbert空間填充曲線編解碼算法[J]. 電子與信息學(xué)報(bào), 2024, 46(2): 633-642. doi: 10.11999/JEIT230013
JIA Lianyin, FAN Yao, DING Jiaman, LI Xiaowu, YOU Jinguo. 3D Hilbert Space Filling Curve Encoding and Decoding Algorithms Based on Efficient Prefix Reduction[J]. Journal of Electronics & Information Technology, 2024, 46(2): 633-642. doi: 10.11999/JEIT230013
Citation: JIA Lianyin, FAN Yao, DING Jiaman, LI Xiaowu, YOU Jinguo. 3D Hilbert Space Filling Curve Encoding and Decoding Algorithms Based on Efficient Prefix Reduction[J]. Journal of Electronics & Information Technology, 2024, 46(2): 633-642. doi: 10.11999/JEIT230013

高效前綴約簡的三維Hilbert空間填充曲線編解碼算法

doi: 10.11999/JEIT230013
基金項(xiàng)目: 國家自然科學(xué)基金(62262035, 62262034, 62062046)
詳細(xì)信息
    作者簡介:

    賈連印:男,博士,副教授,研究方向?yàn)閿?shù)據(jù)庫、數(shù)據(jù)挖掘、信息檢索等

    范瑤:男,碩士生,研究方向?yàn)閿?shù)據(jù)庫、信息檢索

    丁家滿:男,碩士,教授,研究方向?yàn)閿?shù)據(jù)挖掘、云計(jì)算等

    李曉武:男,博士,講師,研究方向?yàn)閿?shù)據(jù)庫等

    游進(jìn)國:男,博士,副教授,研究方向?yàn)閿?shù)據(jù)庫、數(shù)據(jù)挖掘等

    通訊作者:

    丁家滿 jiamanding@kust.end.cn

  • 中圖分類號: TN911.2; TP301.6

3D Hilbert Space Filling Curve Encoding and Decoding Algorithms Based on Efficient Prefix Reduction

Funds: The National Natural Science Foundation of China (62262035, 62262034, 62062046)
  • 摘要: 3維Hilbert空間填充曲線(3D HSFC)的編碼和解碼效率對空間查詢處理、圖像處理等領(lǐng)域的應(yīng)用舉足輕重?,F(xiàn)有的3維編解碼算法獨(dú)立編解碼每一個(gè)點(diǎn),忽略了Hilbert曲線的局部保持特性。為了提高編解碼效率,該文設(shè)計(jì)了高效的3D狀態(tài)視圖,并提出一種新的前綴約簡的3D HSFC編碼算法(PR-3HE)和前綴約簡3D HSFC解碼算法(PR-3HD),這兩個(gè)算法通過公共前綴的定義和識別、公共前綴約簡及多種優(yōu)化技術(shù)來最小化需要編碼的階數(shù),從而提高3D HSFC的編解碼效率。理論上證明:當(dāng)編碼或解碼一個(gè)$k$階的窗體(窗體內(nèi)總共含有${2^k} \times {2^k} \times {2^k}$個(gè)點(diǎn))時(shí),PR-3HE平均每個(gè)點(diǎn)的編碼階數(shù)不超過2,PR-3HD平均解碼階數(shù)不超過8/7。相對于傳統(tǒng)的基于迭代的方法,編解碼時(shí)間復(fù)雜度從$O(k)$降低到了$O(1)$。實(shí)驗(yàn)結(jié)果表明,該文算法在模擬數(shù)據(jù)集和真實(shí)數(shù)據(jù)集上的表現(xiàn)顯著優(yōu)于現(xiàn)有算法。
  • 圖  1  24種基礎(chǔ)狀態(tài)

    圖  2  1階Hilbert曲線

    圖  3  2階Hilbert曲線

    圖  4  編碼示例

    圖  6  迭代編碼的次數(shù)(3階)

    圖  5  編碼當(dāng)前點(diǎn)$q$的流程圖

    圖  7  離散數(shù)據(jù)集上編碼效率對比

    圖  8  窗體數(shù)據(jù)集上編碼效率對比

    圖  9  GeoLife數(shù)據(jù)集上編碼效率對比

    圖  10  離散數(shù)據(jù)集上解碼效率對比

    圖  11  窗體數(shù)據(jù)集上解碼效率對比

    圖  12  GeoLife數(shù)據(jù)集上解碼效率對比

    表  1  CHM

    狀態(tài)000001010011100101110111
    001327645
    101763245
    203127465
    307163425
    403741265
    507341625
    621305647
    761705243
    823105467
    967105423
    1043705261
    1147305621
    1221563047
    1361527043
    1423541067
    1567541023
    1643527061
    1747563021
    1825163407
    1965127403
    2025341607
    2165741203
    2245327601
    2345763201
    下載: 導(dǎo)出CSV

    表  2  CSM

    狀態(tài)000001010011100101110111
    051130132250
    1307237131
    241932194162
    3192179133
    422121254104
    501515041155
    66712116201112
    7216197791
    88189108101418
    9153879793
    108232381010411
    116171761151110
    1212131218617176
    1319121313015150
    14142014161516820
    159515131413155
    16142216162214217
    17121117311121716
    1818181912823238
    1913191819221212
    20202014222162214
    217214192021194
    22202216221602023
    23182310110231822
    下載: 導(dǎo)出CSV
    算法1 PR-3HE
     輸入:${H_p}$:$p$的編碼;$|{P_{p,q}}|$:$p$和$q$的公共前綴長度;
        $q$:當(dāng)前點(diǎn);$r$:后鄰點(diǎn);$k$:階數(shù)
     輸出:${H_q}$:當(dāng)前點(diǎn)$q$的Hilbert編碼;$|{P_{p,r} }|$:$q$和$r$的公共前綴
        長度;
     1. ${H_q}$←$ \overrightarrow {H_p^{|{P_{p,q}}|}} $
     2. $|{P_{p,r}}|$←計(jì)算qr的最大公共前綴長度
     3. $s$←$ M[\text{|}{P}_{p,q}|+1] $
     4. FOR $i$←$ \text{|}{P}_{p,q}| $+1 to $k$
     5.   ${H_q} = {H_q} < <3|{ { {\rm{CHM} }[s][x} }_q^i][y_q^i][z_q^i]$
     6.   $s = {\text{CSM}}[s][x_q^i][y_q^i][z_q^i]$
     7.   IF $i <= |{P_{q,r} }|$
     8.     $M[i + 1] = s$
     9.   END IF
     10. END FOR
    下載: 導(dǎo)出CSV

    表  3  HCM

    狀態(tài)01234567
    0000001011010110111101100
    1000001101100110111011010
    2000010011001101111110100
    3000010110100101111011001
    4000100101001011111110010
    5000100110010011111101001
    6011001000010110100101111
    7011001101111110100000010
    8011010000001101100110111
    9011010110111101100000001
    10011111101001000100110010
    11011111110010000100101001
    12101001000100110010011111
    13101001011111110010000100
    14101100000001011010110111
    15101100110111011010000001
    16101111011001000010110100
    17101111110100000010011001
    18110010000100101001011111
    19110010011111101001000100
    20110100000010011001101111
    21110100101111011001000010
    22110111011010000001101100
    23110111101100000001011010
    下載: 導(dǎo)出CSV

    表  4  HSM

    狀態(tài)01234567
    051013502213
    1301731237
    243219421619
    3123913179
    425421241021
    504515051115
    61176121162012
    7967197211
    81098181081418
    9789379153
    108111023810423
    116101117611517
    1217131261712186
    1315121301513190
    14161514201614820
    151314155131595
    16141716221416222
    17121617111217311
    1823191882318128
    1921181922119132
    20222120142220614
    211920214192174
    22202322162022016
    23182223101823110
    下載: 導(dǎo)出CSV
    算法2 PR-3HD
     輸入:${X_p}$, ${Y_p}$,${Z_p}$$p$的坐標(biāo)分量;$|{P_{Hp,Hq} }|$:$Hp$和$Hq$的最大公共前
        綴長度;${H_q}$, ${H_r}$:$q$和$r$的Hilbert編碼;$k$:階數(shù)
     輸出:$q$的坐標(biāo)分量;$|{P_{Hq,Hr} }|$:$Hq$和$Hr$的最大公共前綴長度;
     1. ${X_q}$,${Y_q}$,${Z_q}$←$(\overleftarrow{X_p^{ {\text{|} }{P_{ {H_p},{H_q} } }|} },\overleftarrow{Y_p^{ {\text{|} }{P_{ {H_p},{H_q} } }|}},\overleftarrow{Z_p^{ {\text{|} }{P_{ {H_p},{H_q} } }|}})$
     2. $ {\text{|}}{P_{{H_q},{H_r}}}| $←計(jì)算HqHr的最大公共前綴長度
     3. $s$←$ M{\text{[|}}{P_{{H_p},{H_q}}}| + 1] $
     4. FOR $i$←$ {\text{|}}{P_{{H_p},{H_q}}}| $+1 to $k$
     5.     $x_q^iy_q^iz_q^i = {\text{HCM}}[s][H_q^i]$
     6.     ${X_q} = {X_q} < < 1|x_q^i$
     7.     ${Y_q} = {Y_q} < < 1|y_q^i$
     8.     ${Z_q} = {Z_1} << 1|z_q^i$
     9.     $s = {\text{HSM}}[s][H_q^i]$
     10.     IF $i <= |{P_{ {H_q},{H_r} } }|$
     11.      $M[i + 1] = s$
     12.     END IF
     13. END FOR
    下載: 導(dǎo)出CSV
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  • 收稿日期:  2023-01-12
  • 修回日期:  2023-06-01
  • 網(wǎng)絡(luò)出版日期:  2023-06-20
  • 刊出日期:  2024-02-29

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