Go1.19で採用された Pattern-defeating Quicksort の紹介

by po3rin

Slide 1

Slide 1 text

(PͰ࠾༻͞Εͨ 1BUUFSOEFGFBUJOH2VJDLTPSUͷ঺հ )JSPNV/BLBNVSB !QPSJO 1BUUFSOEFGFBUJOH2VJDLTPSUJO(P

Slide 2

Slide 2 text

தଜ߂෢ ΤϜεϦʔͷ"*ɾػցֶशνʔϜͷιϑτ΢ΣΞ ΤϯδχΞͰओʹݕࡧج൫पΓΛ୲౰͍ͯ͠Δɻ ֤छΞΧ΢ϯτQPSJOͰ΍ͬͯ·͢ɻ )*SPNV/BLBNVSB The Gopher was designed by Renée French

Slide 3

Slide 3 text

*OUSPEVDUJPO

Slide 4

Slide 4 text

ϦϦʔεϊʔυݟΔͱTPSUύοέʔδͷ಺෦ΞϧΰϦζϜ͕Կ΍Βมߋ͞ΕͨͬΆ͍ (PͷTPSUύοέʔδͷมߋ

Slide 5

Slide 5 text

࣮ࡍʹTPSU[TPSUGVODHPͱTPSU[TPSUJOUFSGBDFHPʹͦΕͧΕQERTPSU@GVODɺQERTPSUͱ͍͏ ؔ਺͕௥Ճ͞Ε͍ͯΔɻݺͼग़͍ͯ͠Δͷ͸TPSU4PSUͱTPSU4MJDF (PͷTPSUύοέʔδͷมߋ

Slide 6

Slide 6 text

ࠓճ͸1BUUFSOEFGFBUJOH2VJDLTPSU<> ͷ࿦จΛಡΜͰ͍͖ɺ(PͰͲͷΑ͏ʹ࣮ ૷͞Ε͍ͯΔ͔؆୯ʹݟ͍ͯ͘ 1BUUFSOEFGFBUJOH2VJDLTPSU [1] https://arxiv.org/abs/2106.05123

Slide 7

Slide 7 text

1BUUFSOEFGFBUJOH2VJDLTPSUͷͬ͘͟Γͱͨ͠ཧղ ূ໌ͳͲࡉ͔͍࿩͸εΩοϓͯ͠ͳΜͱͳ͘ԿΛ΍ͬͯΔͷ͔ͷཧղΛ໨ࢦ͢ ͜ͷൃදͰಘΕΔ΋ͷ

Slide 8

Slide 8 text

1BUUFSOEFGFBUJOH2VJDLTPSU

Slide 9

Slide 9 text

*OUSPTPSUͷ෮श 1BUUFSOEFGFBUJOH2VJDLTPSU͸Πϯτϩιʔτͷվྑ൛ͳͷͰɺ؆୯ʹΠϯτϩιʔτΛ෮ श͓͖ͯ͠·͢ɻ Πϯτϩιʔτ͸ۙ೥࠷΋࢖༻͞Ε͍ͯΔϋΠϒϦουιʔτΞϧΰϦζϜͰ͋Γɺૠೖιʔ τɺώʔϓιʔτɺΫΠοΫιʔτΛ૊Έ߹ΘͤͨϋΠϒϦουΞϧΰϦζϜͰ͢ɻ ώϡʔϦεςΟοΫͳํ๏Ͱɺ͍ͭͲͷઓུΛ࠾༻͢Δ͔Λܾఆ͠·͢ɻྫ͑͹ɺ࠶ؼͷਂ͞ ͕ਂ͘ͳΓ͗͢Δͱώʔϓιʔτʹ੾Γସ͑ͨΓɺύʔςΟγϣϯͷαΠζ͕খ͘͞ͳΓ͗͢ Δͱૠೖιʔτʹ੾Γସ͑ͨΓ͠·͢ɻ

Slide 10

Slide 10 text

1BUUFSOEFGFBUJOH2VJDLTPSU͸ΠϯτϩιʔτΛ֦ு͓Αͼվྑͨ͠΋ͷͰ͢ɻ ޙͰৄ͘͠આ໌͍͖ͯ͠·͕͢ɺ1BUUFSOEFGFBUJOH2VJDLTPSUͷ؊͸؆୯ʹݴ͏ͱ ύʔςΟγϣϯॲཧΛ͍͍ײ͡ʹͨ͠ ࠷ѱέʔεΛ্ख͘ݕ஌ͯ͠ϑΥʔϧόοΫ͢ΔΑ͏ʹͨ͠ ͷͭͰ͢ɻ 1BUUFSOEFGFBUJOH2VJDLTPSU֓ཁ

Slide 11

Slide 11 text

ॏෳΛؚΉ໿ཁૉͷγʔέϯεΛιʔτ͢Δ1BUUFSOEFGFBUJOH2VJDLTPSUͷࢹ֮Խɻ 1BUUFSOEFGFBUJOH2VJDLTPSU֓ཁ

Slide 12

Slide 12 text

1BSUJUJPOJOH

Slide 13

Slide 13 text

1BUUFSOEFGFBUJOH2VJDLTPSU͸Φϥϯμࠃض໰୊ͷߟ͑ํ͕ϕʔεʹͳ͍ͬͯΔͷͰɺ·ͣ͸Φϥϯμ ࠃض໰୊Λ෮श͠·͢ɻ Φϥϯμࠃض໰୊͸ɺɺͷΈΛؚΉγʔέϯεΛಉ͡ཁૉͰύʔςΟγϣϯ͢Δ໰୊Ͱ͢ɻྫ͑͹Լ هͷJOΛPVUʹม׵͢Δ໰୊Ͱ͢ɻ JO PVU 1BSUJUJPOJOH

Slide 14

Slide 14 text

1BSUJUJPOJOH ΞϧΰϦζϜ͸୯७Ͱઢܗ࣌ؒͰղ͚·͢ɻNJEͱMPXΛ഑ྻͷ࠷ॳͷཁૉͷJOEFYʹηοτ͠ɺIJHIΛ࠷ ޙͷཁૉͷJOEFYʹηοτ͠·͢ɻ w ΋͠NJEͳΒMPXͱNJEͷཁૉΛSFQMBDFͯ͠IJHIͰσΫϦϝϯτ w ΋͠NJEͳΒNJEΛΠϯΫϦϝϯτ w ΋͠NJEͳΒIJHIͱNJEͷཁૉΛSFQMBDFͯ͠IJHIΛσΫϦϝϯτ

Slide 15

Slide 15 text

MPX NJE IJHI

Slide 16

Slide 16 text

MPX NJE IJHI MPX NJE IJHI NJEͳͷͰɺIJHIͱSFQMBDFͯ͠IJHI

Slide 17

Slide 17 text

MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI NJEͳͷͰɺIJHIͱSFQMBDFͯ͠IJHI NJEͳͷͰɺNJE

Slide 18

Slide 18 text

MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI NJEͳͷͰɺIJHIͱSFQMBDFͯ͠IJHI NJEͳͷͰɺNJE NJEͳͷͰɺMPXͱSFQMBDFͯ͠NJE MPX

Slide 19

Slide 19 text

MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI NJEͳͷͰɺIJHIͱSFQMBDFͯ͠IJHI NJEͳͷͰɺNJE NJEͳͷͰɺMPXͱSFQMBDFͯ͠NJE MPX NJEͳͷͰɺMPXͱSFQMBDFͯ͠NJE MPX

Slide 20

Slide 20 text

MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI NJEͳͷͰɺIJHIͱSFQMBDFͯ͠IJHI NJEͳͷͰɺNJE NJEͳͷͰɺMPXͱSFQMBDFͯ͠NJE MPX NJEͳͷͰɺMPXͱSFQMBDFͯ͠NJE MPX NJEͳͷͰɺNJE

Slide 21

Slide 21 text

MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI NJEͳͷͰɺIJHIͱSFQMBDFͯ͠IJHI NJEͳͷͰɺNJE NJEͳͷͰɺMPXͱSFQMBDFͯ͠NJE MPX NJEͳͷͰɺMPXͱSFQMBDFͯ͠NJE MPX NJEͳͷͰɺNJE NJEͳͷͰɺIJHIͱSFQMBDFͯ͠IJHI NJE͕IJHIΛ௒͑ͨͷͰऴྃ

Slide 22

Slide 22 text

1BSUJUJPOJOHJO1BUUFSOEFGFBUJOH2VJDLTPSU ୯७ͳΫΠοΫιʔτͷ࣮૷Ͱ͸ɺ౳͍͠ൺֱཁૉΛಉ͡ύʔςΟγϣϯʹ഑ஔ͢Δ͜ͱʹ ΑΓϺ O? ͷ࠷ѱͷέʔε͕Ҿ͖ى͜͞Ε·͢ɻ ౳͍͠ཁૉΛޮ཰తʹॲཧ͢Δʹ͸ɺ౳͍͠ཁૉΛूΊͯɺͦΕΑΓେ͖͍ཁૉɺখ͍͞ཁ ૉͷ෼ׂʹͰ͖Ε͹ɺͦͷޙͷιʔτͰ౳͍͠ཁૉΛؚΉύʔςΟγϣϯͷιʔτΛε ΩοϓͰ͖·͢ɻ·͞ʹΦϥϯμࠃض໰୊ͱಉ౳ͷࡾ෼ׂͷߟ͑ํ͕࢖͑ͦ͏Ͱ͢ɻ ͜ͷΑ͏ʹύʔςΟγϣϯΛ෼ׂͰ͖Δͱخ͍͠

Slide 23

Slide 23 text

1BSUJUJPOJOHJO1BUUFSOEFGFBUJOH2VJDLTPSU ͜Ε·Ͱʹઆ໌ͨ͠Φϥϯμͷࠃض໰୊ͷղ๏Λ୯७ʹద༻͢Δͱɺ͢΂ͯͷཁૉΛεϫο ϓ͢ΔલʹϐϘοτͱ౳͍͔͠Ͳ͏͔Λ໌ࣔతʹνΣοΫ͢Δඞཁ͕͋Γ·͢ɻ͜Ε͸ɺ౳ ͍͠ཁૉ͕ଟ਺ଘࡏ͢Δ͔Ͳ͏͔ʹؔ܎ͳ͘ൃੜ͠ɺฏۉతͳέʔεͰ͸ύϑΥʔϚϯε͕ ௿Լ͠·͢ɻ ͜ͷ࿦จͰ͸͜ͷ໰୊ʹରॲ͢Δ৽͍͠ղܾࡦΛٞ࿦͓ͯ͠ΓɺLݸͷҟͳΔཁૉΛ࣋ͭೖ ྗʹରͯ͠0 OL ͷ࠷ѱέʔεʹऩ·Δख๏ΛఏҊ͍ͯ͠·͢ɻ1BUUFSOEFGFBUJOH 2VJDLTPSUͰ͸͜ͷख๏ΛϕʔεʹΠϯτϩιʔτΛվྑ͍ͯ͠·͢ɻ

Slide 24

Slide 24 text

1BSUJUJPO-FGU3JHIU 1 1 QBSUJUJPO@SJHIU QBSUJUJPO@MFGU 1BUUFSOEFGFBUJOH2VJDLTPSU͸ͭͷύʔςΟγϣϯ༻ؔ਺Λ૊Έ߹ΘͤͯઌఔͷύʔςΟγϣϯ Λୡ੒͠·͢ɻͭ͸ϐϘοτQʹ౳͍͠ཁૉΛࠨͷύʔςΟγϣϯʹάϧʔϓԽ QBSUJUJPOMFGU ͠ɺ΋͏ͭ͸ϐϘοτʹ౳͍͠ཁૉΛӈͷύʔςΟγϣϯ QBSUJUJPOSJHIU ʹάϧʔϓԽ͠·͢ɻ ͜ͷύʔςΟγϣϯͷ૊Έํͷར఺͸ʮཁૉ͕ϐϘοτͷ஋Ҏ্͔Ͳ͏͔ʯͱ͍͏৚݅ҰݸͰ൑ఆͰ ͖Δ͜ͱͰ͢ɻ

Slide 25

Slide 25 text

1BUJUJPOJO(P (PͰ͸TPSUύοέʔδ಺ͷQBUJUJPOؔ਺͕ύʔςΟγϣϯΛ୲͏ɻ࣮ࡍʹؔ਺ΛFYQPSUͯ͠Ұճͩ ͚ಈ͔ͯ͠ΈΔͱ͔֬ʹQBUJUJPO@MFGUͷ݁ՌΛಘΒΕ͍ͯΔɻ

Slide 26

Slide 26 text

1BSUJUJPO-FGU3JHIU QBSUJUJPOؔ਺Λར༻ͯ͠࠶ؼతʹγʔέϯεΛύʔςΟγϣϯ͍͖ͯ͠·͢ɻ ྫ͑͹ԼهਤͷΑ͏ʹɺϐϘοτQͰЋΛύʔςΟγϣϯ͠ɺ࠶ؼతʹӈଆͷύʔςΟγϣϯΛϐ ϘοτRͱͯ͠ЌΛύʔςΟγϣϯ͍͖ͯ͠·͢ɻ Q͕Rʹ౳͘͠ͳ͍৔߹͸ɺ্هͷεςοϓΛ࠶ؼతʹద༻͠·͢ɻ࠷ऴతʹQRͱͳͬͨ࣌͸ɺಉ ͡஋ΛਅΜதʹूΊͨߏ੒ʹͰ͖͍ͯ·͢Լਤɻ ͪͳΈʹɺ͋Δ఺RͰЌ͕ۭʹͳͬͨ৔߹ɺQʹ౳͍͠ཁૉ͸ͳ͘ɺЋΛ࠷ॳʹ෼ׂͨ͠ͱ͖ʹ ෼ׂ͕ߦΘΕͨͱ݁࿦෇͚Δ͜ͱ͕Ͱ͖·͢ɻ

Slide 27

Slide 27 text

https://www.youtube.com/watch?v=jz-PBiWwNjc ͔ΒҾ༻

Slide 28

Slide 28 text

1JWPU4FMFDUJPO

Slide 29

Slide 29 text

+PIO5VLFZ`TOJOUIFS ϐϘοτͷબ୒ʹ͓͍ͯɺෆۉߧύʔςΟγϣϯॲཧΛආ͚ΔͨΊʹɺͳΔ΂͘தԝ஋Λબ୒͢Δ͜ ͱ͕ཧ૝Ͱ͢ɻҰํͰେ͖ͳೖྗͷ৔߹͸ద੾ͳϐϘοτΛબ୒͢Δͷ͸ύϑΥʔϚϯεͷ໰୊͕ൃ ੜ͠·͢ɻͦ͜Ͱ͜ͷ࿦จͰ͸༗໊ͳ+PIO5VLFZ`TOJOUIFSͱ͍͏ख๏ͰϐϘοτΛબ୒͠·͢ɻ ݫີͳதԝ஋Λܭࢉ͢Δʹ͸͢΂ͯͷσʔλϙΠϯτΛอଘ͠ɺͦΕΒΛฒ΂ସ͑ͯதؒ஋Λબ୒͢ Δඞཁ͕͋Γ·͕͢ɺ͜ͷํ๏Ͱ͋Ε͹ૉૣ͘େࡶ೺ͳதԝ஋ͷۙࣅ͕ಘΒΕ·͢ɻ

Slide 30

Slide 30 text

+PIO5VLFZ`TOJOUIFSJO(P ࣮ࡍͷ(Pͷ࣮૷Ͱ͸ϐϘοτΛબ୒͢Δαϒγʔέϯεͷ௕͞ʹΑͬͯϐϘοτબ୒ํ๏Λ੾Γସ ͑·͢ɻ௕͕͞ҎԼͳΒ୯७ʹதԝ஋ΛͱΓɺҎԼͳΒ୯७ʹ࠷ॳɺதԝɺ࠷ޙͷͭͷ஋ͷத ԝ஋ΛͱΓɺҎ্ͳΒ+PIO5VLFZ`TOJOUIFSΛར༻͠·͢ɻ

Slide 31

Slide 31 text

#BE1BUJUJPO

Slide 32

Slide 32 text

#BE1BUJUJPO 1BUUFSOEFGFBUJOH2VJDLTPSU͸ɺࢦఆͨ͠ϐϘοτύʔηϯλΠϧQ ೚ҙͷ஋ ΑΓ΋ෆۉߧͳ෼ׂ ૢ࡞ΛෆྑύʔςΟγϣϯͱݺͼ·͢ ͪͳΈʹϐϘοτύʔηϯλΠϧQ͕ͩͱཧ૝త͸ύʔ ςΟγϣϯʹͳΓ·͢ ɻྫ͑͹ԼهͷΑ͏ͳγʔέϯεͰϐϘοτύʔηϯλΠϧͷϙΠϯτ ࠓճ͸ Λ࠾༻ͯ͠͠·͏ͱ΄ͱΜͲύʔςΟγϣϯ͕ػೳ͠·ͤΜɻ

Slide 33

Slide 33 text

#BE1BUJUJPO :VWBM'JMNVT<>͸Qͷ࠷దͳέʔεͱൺֱͯ͠ΫΠοΫιʔτͷ଎౓௿ԼΛௐ΂·ͨ͠ɻ γϟϊϯͷ஋ΤϯτϩϐʔΛར༻ͯ͠ɺԼهͷΑ͏ͳάϥϑ͕ॻ͚·͢ɻ ͜ͷάϥϑ͔Βɺྫ͑͹Qͷ৔߹͚ͩͷ৔߹͸ɺཧ૝తͳ৔߹ͱൺ΂͚ͯͩ஗͘ͳΔ͜ͱ ͕Θ͔Γ·͢ɻQΛԼճΔͱύϑΥʔϚϯεͷ௿Լ͕ܹ͘͠ͳΔ͜ͱ΋Θ͔Γ·͢ɻ͜ͷͨΊɺ Qͷᮢ஋Λઃఆ͠ɺͦΕΑΓԼճΔෆྑύʔςΟγϣϯͷέʔε͕ଟ͚Ε͹ϑΥʔϧόοΫ͢Δઓུ ΛऔΓ·͢ɻ [3] https://cs.stackexchange.com/questions/31928/solving-recurrence-relation-with-two-recursive-calls/31930#31930

Slide 34

Slide 34 text

#BE1BUJUJPO ෆྑύʔςΟγϣϯΛෳ਺ճૺ۰͢ΔͷΛճආ͢ΔͨΊʹԼهεςοϓΛߦ͍·͢ɻ w ࠷ॳʹɺΧ΢ϯλʔͱͯ͠MPH αϒγʔέϯε௕ ʹઃఆ͠·͢ w ෆྑύʔςΟγϣϯʹૺ۰͢ΔͨͼʹɺΧ΢ϯλʔΛσΫϦϝϯτ͠·͢ w ࠶ؼݺͼग़͠ͷ։࢝࣌ʹΧ΢ϯλʔ͕ͷ৔߹ɺΫΠοΫιʔτͰ͸ͳ͘ɺώʔϓιʔτΛ࢖༻ ࿦จͰ͸͜ͷϐϘοτύʔηϯλΠϧQͷᮢ஋Λͱ͓ͯ͠Γ(P΋ʹͳ͍ͬͯ·͢ɻ ͜ͷεΩʔϜͷར఺͸ɺΞʔΩςΫνϟ͕มߋ͞Εͨ৔߹ ώʔϓιʔτͷ୅ΘΓʹผͷ࠷ѱͷέʔε ͷιʔτΞϧΰϦζϜΛ࢖༻͍ͯ͠Δ৔߹ͳͲ ʹɺͦΕʹ߹ΘͤͯQΛௐ੔Ͱ͖Δ͜ͱͰ͢ (Pͷ৔ ߹͸͕ϋʔυίʔυ͞Ε͍ͯͯมߋ͸Ͱ͖ͳ͍ ɻ

Slide 35

Slide 35 text

#BE1BUJUJPO (PͰ͸QERTPSUؔ਺ͷMJNJUҾ਺͕Χ΢ϯλʔ ʹ͋ͨΓɺ͜Ε͕ʹͳΔͱώʔϓιʔτʹ੾ ΓସΘΓ·͢ɻ

Slide 36

Slide 36 text

#BE1BUJUJPO ͪͳΈʹͪΐͬͱͨ͠޻෉ͱͯ͠Χ΢ϯλʔ͸γʔέϯεͷ௕͞ͷॴཁϏοτ਺ͱͯؒ͠ ઀తʹMPH@ O Λܭࢉ͍ͯ͠Δɻ

Slide 37

Slide 37 text

1BUUFSOEFGFBUJOH2VJDLTPSUJO(P

Slide 38

Slide 38 text

1BUUFSOEFGFBUJOH2VJDLTPSUJO(P (Pʹ࣮૷͞Εͨ1BUUFSOEFGFBUJOH2VJDLTPSU͸࿦จͱൺ΂ͯԼهͷΑ͏ͳมߋ͕Ճ͑ΒΕͯ ͍·͢ɻ w #MPDL2VJDLTPSUΛ࢖ͬͨ࠷దԽΛແޮ কདྷ௥Ճ͞ΕΔՄೳੑ͸͋Γͦ͏ w ϕϯνϚʔΫ݁Ռʹैͬͯɺ͍͔ͭ͘ͷύϥϝʔλʔΛมߋ

Slide 39

Slide 39 text

(Pʹ͓͚ΔϕϯνϚʔΫ͸ JTTVFͰఏग़͞Ε͍ͯ·͢ɻ΄ͱ ΜͲͷέʔεͰύϑΥʔϚϯε͕վળ͠ ͍ͯ·͢ɻ

Slide 40

Slide 40 text

$PODMVTJPO

Slide 41

Slide 41 text

$PODMVTJPO (Pʹ࣮૷͞Εͨ1BUUFSOEFGFBUJOH2VJDLTPSU͸ϕϯνϚʔΫ্༷ʑͳύλʔϯͰύϑΥʔϚ ϯε͕վળ͍ͯ͠Δɻ 1BUUFSOEFGFBUJOH2VJDLTPSU͸ओʹύʔςΟγϣϯॲཧ΍ϐϘοτબ୒ɺෆྑύʔςΟγϣϯ ݕ஌Ͱɺ2VJDLTPSUͷ࠷ѱέʔεΛճආͰ͖ΔΑ͏ʹ޻෉͞Ε͍ͯΔɻ (PʹΞοϓάϨʔυͯ͠TPSU͕ͲΕ͘Β͍ૣ͘ͳ͔ͬͨੋඇࢼͯ͠Έ͍ͯͩ͘͞ɻΑΓࡉ ͔͍޻෉΍ɺৄ͍͠ূ໌ͳͲ͸࿦จΛࢀর͍ͩ͘͞ɻ

Slide 42

Slide 42 text

ΤϜεϦʔͰ͸ࢸΔॴͰ(PΛ࠾༻ͯ͠։ൃΛ͍ͯ͠Δɻ΋ͪΖΜ(PΛ࣮຿Ͱ΍ͬͨ ͜ͱ͕ͳͯ͘΋׻ܴɻ ֤छ"1* ΞϓϦέʔγϣϯɺݕࡧɺϨίϝϯυɺϦϥϯΩϯά౳ Ϩίϝϯυ"1* σʔλ࿈ܞ#BUDI 1VTI௨஌ج൫ ֤छศརπʔϧͷ։ൃ (Pͷษڧձ΋։͔Ε͓ͯΓɺݱࡏ͸ʮ࣮༻(PݴޠʕγεςϜ։ൃͷݱ৔Ͱ஌͓ͬͯ ͖͍ͨΞυόΠεʯ<>ͷྠಡձ΋։͔Ε͍ͯΔɻιϑτ΢ΣΞΤϯδχΞ͚ͩͰͳ ͘ɺ.-ΤϯδχΞ΋(P͕ॻ͚Δͱ͍͏ҟ࣭ͳ؀ڥɻ ΤϜεϦʔ͸(PͰ։ൃ͍ͨ͠ϝϯόʔΛืूத [3] https://www.amazon.co.jp/dp/4873119693/ref=cm_sw_em_r_mt_dp_G4Z5RYS2FDVTVWXVJEYS?_encoding=UTF8&psc=1

Slide 43

Slide 43 text

(PͰ࠾༻͞Εͨ 1BUUFSOEFGFBUJOH2VJDLTPSUͷ঺հ )JSPNV/BLBNVSB !QPSJO 1BUUFSOEFGFBUJOH2VJDLTPSUJO(P