Nkitahodi nnianim asɛm a ɛfa quadtrees ho

Nea Enti a Quadtrees Ho Hia Sen sɛnea Wosusuw

Bere biara a wobɛ pinch-to-zoom wɔ dijitaal map so, abisa adidibea ahorow a ɛbɛn hɔ, anaa wobɛhwɛ bere ankasa mu fleet tracker update kar icons du du pii a wo browser no ntwitwiw ma ennyae no, hokwan pa wɔ hɔ sɛ quadtree reyɛ heavy lifting no wɔ akyi. Quadtrees yɛ saa data nhyehyɛe ahorow a ɛyɛ fɛ a nnipa dodow no ara nte ho asɛm da no mu biako, nanso wɔde komm tumi ma nhyehyɛe ahorow a ɛho hia sen biara wɔ adwumayɛ mu wɔ nnɛyi softwea mu no bi — efi video so agodie a wohu so kosi asase ho nsɛm nhyehyɛe ahorow a edi ahunmu nsɛmmisa ɔpepem pii ho dwuma wɔ sekan biara mu. Sɛnea wɔyɛ adwuma no ntease mma wobɛyɛ developer pa kɛkɛ; ɛsakra sɛnea wususuw sɛnea wobɛhyehyɛ na woahwehwɛ ahunmu data mu no titiriw. Sɛ́ ebia worekyekye delivery logistics platform, beae a egyina analytics dashboard, anaasɛ worebɔ mmɔden ara kwa sɛ wobɛkyerɛ data nsɛntitiriw 50,000 wɔ canvas so a worensɛe browser no, quadtrees de ano aduru a ɛyɛ mmerɛw na ɛyɛ adwuma yiye ma ɛyɛ nwonwa ma.

Dɛn Pɛpɛɛpɛɛ ne Quadtree?

Quadtree yɛ dua data nhyehyeɛ a emu node biara wɔ mma baanan pɛpɛɛpɛ, a emu biara gyina hɔ ma quadrant baako a ɛwɔ afã mmienu ahunmu. Fa no sɛ wobɛfa ɔmantam a ɛyɛ ahinanan na woakyekyɛ mu ahinanan anan a ɛyɛ pɛ — atifi fam atɔe, atifi fam apuei, anafo fam atɔe, ne anafo fam apuei. Wobetumi akyekyɛ saa ahinanan no mu biara mu bio ayɛ no ahinanan anan foforo, ne nea ɛkeka ho, mpɛn pii, kosi sɛ wubedu tebea bi a wubegyina mu. Saa gyinabea tebea no taa yɛ bun a ɛsen biara anaasɛ aboboano a ɛkyerɛ data nsɛntitiriw dodow a node biako betumi akura ansa na ɛho hia sɛ ɛpaapae.

Afɛfɛde a ɛwɔ saa kwan yi so no gyina ne su a ɛma ɛyɛ nsakrae no so. Mmeae a data nsɛntitiriw ayɛ hɔ ma no mu kyekyɛ yɛ no nkwammoaa a ɛyɛ fɛ na ɛyɛ fɛ, bere a mmeae a ɛnyɛ kɛse no da so ara yɛ mmeae akɛse a wɔankyekyɛ mu. Dua anan a ɛkora mmeae a kɔfe sotɔɔ 10,000 wɔ wɔ ɔman bi mu nyinaa bɛma wɔayɛ mpaapaemu a emu dɔ na ɛkɔ akyiri wɔ Manhattan so — baabi a ebia sotɔɔ 300 na ɛwɔ kilomita ahinanan kakraa bi mu — bere a ɛbɛma Wyoming nkuraase mmeae akɛse atra hɔ sɛ node biako a wɔanpaapae mu a zero anaa point biako wom no. Saa adaptive resolution yi ne nea ɛma quadtrees nya tumi kɛse sɛ wɔde toto grid a ɛyɛ tratraa ho a, ɛbɛsɛe memory pii wɔ cells a hwee nni mu so.

Raphael Finkel ne J.L. Bentley na wodii kan kaa adwene no ho asɛm wɔ afe 1974 mu, na efi saa bere no, ayɛ nkorabata ayɛ no ahorow pii: point quadtrees sie ankorankoro coordinate pairs, region quadtrees gyina hɔ ma mmeae a ɛwɔ ahunmu (ɛho wɔ mfaso ma mfonini a wɔde hyɛ mu), na edge quadtrees di nkyerɛwde ne curves ho dwuma. Nsonsonoeɛ biara yɛ yie ma dwumadie ahodoɔ, nanso core recursive subdivision nnyinasosɛm no da so ara yɛ pɛ wɔ wɔn nyinaa mu.

Sɛnea Insertion ne Querying Yɛ Adwuma

Sɛ wode nsɛntitiriw bi bɛhyɛ quadtree mu a, wufi ase wɔ ntini node no so na wokyerɛ quadrants anan no mu nea nsɛntitiriw no hwe ase. Afei wosan kɔ saa quadrant no ba node no mu na wosan yɛ adeyɛ no bio. Sɛ wudu ahaban node bi a ɛntra ne tumi so (wɔtaa de si nsɛntitiriw 1 anaa 4) a, wode nsɛntitiriw no sie hɔ ara kwa. Sɛ ahaban no tumi yɛ adwuma dedaw a, ɛpaapae yɛ mmofra baanan, san kyekyɛ ne nsɛntitiriw a ɛwɔ hɔ dedaw no mu ma wɔn, na afei ɛde nsɛntitiriw foforo no hyɛ abofra a ɔfata no mu. Saa dwumadie yi taa wie wɔ O(log n) berɛ mu ma nkyekyɛmu a ɛkari pɛ, ɛwom sɛ tebea a enye koraa a ɛwɔ data a wɔaboaboa ano kɛseɛ betumi asɛe adwumayɛ.

Range querying — hwehwɛ nsɛntitiriw nyinaa wɔ beae a ɛyɛ ahinanan a wɔde ama mu — ne baabi a nnua anan hyerɛn ampa. Sɛ anka wobɛhwɛ beaeɛ baako biara wɔ wo dataset no mu (O(n) dwumadie bi), wofiri aseɛ wɔ nhini no so na wobisa asɛm tiawa bi wɔ node biara so: so saa node yi hyeɛ ne me hwehwɛ ahinanan no twam? Sɛ ɛnte saa a, wotwitwa dua ketewa no nyinaa — ebetumi ayi nsɛntitiriw mpempem pii afi nsusuwii mu wɔ ntotoho biako mu. Sɛ nhyiam bi wɔ hɔ a, wosan kɔ mmofra a ɛfa ho no mu. Nsɛntitiriw a wohu wɔ ahaban node a ɛtɔ hwehwɛ ahinanan no mu no nya bi ka nea efi mu ba no ho.

Susuw nhwɛso a mfaso wɔ so ho: wowɔ dataset a ɛwɔ mmeae 100,000 a adetɔfo wɔ na ɛsɛ sɛ wohwehwɛ obiara a ɔwɔ kilomita 5 kwan a ɛda sotɔɔ foforo a wobebue no mu. Brute-force kwan a wɔfa so yɛ no hwehwɛ sɛ wobu akontaa 100,000 wɔ akyirikyiri. Quadtree a wɔayɛ no yiye betumi atew ɛno so akɔ 200-500 pɛ checks denam asasesin nyinaa a ɛda adi pefee sɛ ɛne wo hwehwɛ beae no nhyia a wobeyi afi hɔ ntɛmntɛm no so. Ɛno yɛ adwumayɛ mu nkɔsoɔ a ɛyɛ 200x anaa nea ɛboro saa — nsonsonoeɛ a ɛda asɛmmisa a ɛgye milisekɔn 800 ne milisekɔn 4 ntam.

Wiase Ankasa Dwumadie a Ɛyɛ adwuma wɔ Quadtrees

Nnua anan a wɔde di dwuma no trɛw kɔ akyiri sen kɔmputa ho adesua a wɔde di dwuma wɔ adesua mu. Wɔyɛ fapem ma nhyehyɛe ahorow a nnipa ɔpepepem pii de di dwuma da biara da, mpɛn pii no wonnim.

• Mapping ne navigation: Nnwuma te sɛ Google Maps ne Mapbox de tile nhyehyɛe a ɛte sɛ quadtree di dwuma de som map mfonini. Zoom level biara kyekyɛ tiles mu ma ɛyɛ mmofra baanan, ɛno nti na map tile coordinates di z/x/y nhyehyɛe a ɛkyerɛ quadtree addressing akyi. Sɛ wo zoom kɔ kuropɔn block bi mu a, tiles a ɛfa ho a ɛwɔ high-resolution nkutoo na ɛde adesoa — wiase nkaeɛ no tra coarse resolution.
• Nneɛma a ɛbɔ wɔ agodie mu a wɔhunu: Agodie engine ahodoɔ de quadtrees (ne wɔn 3D yɔnko, octrees) di dwuma de hunu yie berɛ a nneɛma abɔ. Sɛ anka ɛbɛsɔ nneɛma mmienu biara ahwɛ — O(n2) daeɛ bɔne a ɛwɔ entities 1,000 wɔ screen so — engine no hwɛ nneɛma a wɔkyɛ quadtree cell koro no ara nko ara, na ɛtew checks so kɔ dodoɔ a wɔtumi di ho dwuma.
• Mfonini nhyɛsoɔ: Region quadtrees tumi mia mfonini denam piksel a ɛbɛn a ɛkyɛ kɔla a ɛte saa ara a wɔde bɛka abom ayɛ no block akɛseɛ so. Eyi ne nnyinaso a ɛwɔ nhyɛso nhyehyɛe ahorow bi a ɛma nhyɛso nsusuwii 10:1 bere a ɛkura aniwa mu nokwaredi mu wɔ mmeae a nsɛm a ɛkɔ akyiri kakraa bi.
• Ahyɛn sohwɛ ne nneɛma a wɔde kɔ: Nnwumakuw a wɔde nneɛma kɔma no de spatial indexing di dwuma de ma karkafo ne nneɛma a wɔkra a ɛbɛn wɔn no hyia wɔ bere ankasa mu. Quadtree ma dispatch system bi bua asɛmmisa a ɛne "draiver 5 bɛn na ɛbɛn beae a wɔde pickup yi kɛse?" wɔ kar mpempem pii a wɔreyɛ wɔn GPS gyinabea foforo wɔ sikani kakraa biara mu.
• Geospatial analytics: Platforms a ɛboaboa adwumayɛ ho nsɛm a egyina beaeɛ so ano — adetɔfoɔ dodoɔ ho mfonini, adetɔn asasesin a wɔyɛ no yie, store placement analysis — de wɔn ho to spatial data nhyehyeɛ so na ama saa nsɛmmisa yi ayɛ nkitahodiɛ mmom sen sɛ wɔbɛyɛ no batch-processed.

a wɔde ahyɛ mu

Nhumu titiriw a ɛwɔ quadtrees akyi ne sɛ, ɛho nhia sɛ ahunmu nsɛmmisa dodow no ara hwehwɛ data dodow no ara mu. Ɛdenam nhyehyɛe a wobɛhyehyɛ ahunmu wɔ nhyehyɛe kwan so no, wodan brute-force hwehwɛ ahorow kɔ traversals a wɔde wɔn ani asi so — wodannan sikani ma ɛyɛ milisekɔn na woma bere ankasa mu nkitahodi tumi yɛ yiye wɔ dataset akɛse mpo mu.

Quadtree a Wɔbɛsi Fi Mfitiaseɛ

Mfitiaseɛ quadtree a wɔde bedi dwuma no yɛ nwonwa sɛ wotumi bɛn, mpo ma ntamgyinafoɔ developers. Nkyekyɛm titiriw no hia nneɛma kakraa bi pɛ: hye (beae a ɛyɛ ahinanan a node no kata so), tumi (nsɛntitiriw a ɛsen biara ansa na wɔapaapae mu), nsɛntitiriw nhyehyɛe, ne nsɛm a ɛfa mmofra node anan ho (mfiase no na ɛnyɛ hwee). Wobetumi akyerɛw insert dwumadie no nyinaa wɔ koodu nkyerɛwdeɛ 30 ase wɔ kasa dodoɔ no ara mu.

Adwuma a wɔpaapae mu no yɛ mmofra node foforo anan, a emu biara kata ɔwofo no hye no fã biako so. Wɔ ɔwofo a ɔwɔ ɔhye (x, y, trɛw, ɔsorokɔ) fam no, atifi fam apuei abofra no nya (x + trɛw/2, y, trɛw/2, sorokɔ/2), atifi fam atɔe nya (x, y, trɛw/2, sorokɔ/2), ne nea ɛkeka ho. Sɛ wɔkyekyɛ mu wie a, wɔsan kyekyɛ nsɛntitiriw a ɛwɔ hɔ dedaw no mu ma mmofra a wɔfata. Mfomso a ɛtaa ba ne sɛ wo werɛ fi sɛ wobɛpopa ɔwofo no nsɛntitiriw nhyehyɛe no bere a wɔasan akyekyɛ no akyi, a ɛma nea efi mu ba no yɛ abien wɔ nsɛmmisa bere mu.

Wɔ production dwumadie mu no, optimizations dodoɔ bi ho hia. Sɛ wode node tumi no si nsɛntitiriw 4-8 a, ɛtaa yɛ adwuma sen tumi a ɛyɛ 1, efisɛ ɛtew dua no mu dɔ ne node nneɛma a ɛwɔ soro no so. Sɛ wode maximum depth limit (mpɛn pii no 8-12 levels) ka ho a, esiw pathological cases a nsɛntitiriw pii kyɛ coordinates a ɛyɛ pɛ no kwan sɛ wɔbɛbɔ nnua a emu dɔ a enni ano. Na wɔ datasets a ɛyɛ nnam a nsɛntitiriw tu — te sɛ kar akyidi — wobɛpɛ sɛ wonya afiri a wɔde yi fi hɔ anaasɛ ɔkwan a wobɛfa so asan asi dua no bere ne bere mu, efisɛ nnua anan ntumi nkari pɛ te sɛ nnua kɔkɔɔ-tuntum yɛ.

Quadtrees wɔ Adwumayɛbea ne Nhwehwɛmu mu

Nnɛyi adwumayɛ nhyehyɛe ahorow no di ahunmu data ho dwuma kɛse, sɛ́ ɛyɛ mmeae a adetɔfo wɔ, mmeae a wɔde nneɛma kɔ, nsasesin a wɔtɔn, anaa agyapade akyi di. Asɛnnennen no nyɛ sɛ wobɛkora saa data yi so kɛkɛ — ɛrema ayɛ nea wotumi bisa wɔ bere ankasa mu wɔ nsenia mu. Sɛ ɛhia sɛ adwuma bi a ɛreyɛ adwuma wɔ nkuropɔn 50 mu no yɛ adetɔfoɔ dodoɔ ho mfonini wɔ n’adwene mu, wɔyɛ akwan a wɔfa so de nneɛma kɔma, anaa wɔhwehwɛ mpɔtam hɔ adetɔn adwumayɛ mu a, ahunmu indexing nhyehyɛeɛ a ɛwɔ aseɛ no na ɛkyerɛ sɛ ebia dashboard no bɛfa milisekɔn 200 anaa sikɔne 20.

Eyi yɛ adeɛ baako nti a platforms te sɛ Mewayz — a ɛka module 207 a ɛfa CRM, invoicing, fleet management, booking, ne analytics bom yɛ adwuma OS baako — nya mfasoɔ firi spatial data a ɛdi mu yie wɔ hood ase. Sɛ ɛhia sɛ fleet management module bi da kar 500 a ɛyɛ nnam adi wɔ asase mfonini so, anaasɛ sɛ CRM module bi yɛ mmeae a wɔde di dwuma 138,000+ ho mfonini wɔ n’adwenem ma asasesin nhyehyɛe a, naive akwan no ntumi nkɔ soro kɛkɛ. Spatial indexing nhyehyeɛ te sɛ quadtrees (anaasɛ wɔn database a ɛne no sɛ, te sɛ PostGIS R-trees ne MySQL spatial indexes) ma ɛyɛ yie sɛ wɔde saa nneɛma yi bɛma a enhia sɛ adwumayɛkuo-grade hardware.

Wɔ nnwuma a wɔresusuw platform ahorow ho no, takeaway no yɛ nea mfaso wɔ so: nnwinnade a edi beae ne beae data ho dwuma yiye no mfa fancy algorithms redi dwuma kɛkɛ esiane ɛno nti. Wɔreyɛ nsonsonoe a ɛda nhyehyɛe a wɔde kyerɛw nneɛma a ebetumi akyerɛ wɔn a wɔde nnwuma ma a ɛwɔ hɔ ntɛm ara wɔ kilomita 10 mu ne nea egye sikani 8 na wɔde nea efi mu ba koro no ara ahyɛ mu no ntam. Adwumayɛ wɔ saa gyinabea yi kyerɛ ase tẽẽ kɔ osuahu a ɔde di dwuma ne, awiei koraa no, sika a wonya mu.

Quadtrees vs. Ahunmu Data Nhyehyɛe Afoforo

Ɛnyɛ Quadtrees nko ara a wobɛfa so ayɛ spatial indexing, na sɛ wote akwan foforo no ase a, ɛboa wo ma wopaw adwinnade a ɛfata. R-nnua, a wɔde di dwuma kɛse wɔ database te sɛ PostGIS ne SQLite R*Tree module mu no, hyehyɛ data ma ɛyɛ minimum bounding rectangles na di range queries ne nearest-neighbor searches ho dwuma yiye. Wɔtaa yɛ adwuma sene quadtrees ma disk-based storage ɛfiri sɛ wɔtew I/O dwumadie so, ɛno nti na spatial database dodoɔ no ara de R-tree variants di dwuma wɔ mu sene quadtrees.

K-d nnua kyekyɛ ahunmu denam alternating axis-aligned splits (nea edi kan denam x, afei y, afei x bio) na ɛyɛ papa ma nea ɛbɛn-afipamfo hwehwɛ wɔ nsusuwii a ɛkɔ fam mu. Wɔtaa yɛ adwuma sen quadtrees bere a dimensionality no sua na dataset no yɛ static, nanso ɛyɛ den sɛ wobɛma wɔayɛ no foforo dynamically. Geohashes fa ɔkwan soronko so koraa, ɛde latitude ne longitude kɔ ahama baako mu a prefixes a wɔkyɛ kyerɛ beaeɛ a ɛbɛn — ɛma ɛyɛ papa ma database indexing ne caching nanso ɛnyɛ mmerɛw mma range queries a wɔpɛ.

Quadtrees kura wɔn ankasa de wɔ tebea horow a edi agoru ma wɔn ahoɔden mu: in-memory spatial indexing, dynamic datasets a wɔtaa de hyɛ mu na wɔpopa, visualization applications a hierarchical grid structure no yɛ maps wɔ abɔde mu kɔ zoom levels, ne tebea horow a ɛyɛ mmerɛw sɛ wɔde bedi dwuma no ho hia. Wɔ anim-awieɛ aplikeshɔn a ɛkyerɛ data nsɛntitiriw 10,000 wɔ canvas a ɛwɔ pan-and-zoom so no, quadtree a wɔde adi dwuma wɔ JavaScript nkyerɛwdeɛ 100 mu no bɛyɛ adwuma asen ano aduru biara a database gyina akyi denam network latency a ɛbɛyi afiri hɔ ara kwa so.

Mfiase: Anamɔn a Edi Hɔ a Ɛyɛ Mfaso

Sɛ wopɛ sɛ wo nteaseɛ a wowɔ wɔ nnua anan ho no mu dɔ sen sɛ wobɛkenkan ho asɛm a, ɔkwan a ɛtu mpɔn paa ne sɛ wobɛkyekyere bi wɔ aniwa so. Yɛ canvas application a ɛnyɛ den a sɛ wo klik a, ɛde nsɛntitiriw ka ho, na hwɛ dua no mu nkyekyɛmu wɔ bere ankasa mu. Fa range-query rectangle a wubetumi atwe akɔ akyirikyiri na woatwe adwene asi nsɛntitiriw a ebehu no so dua ka ho. Saa nsaano nkitahodi yi kyekye nkate a akenkan dodow biara ntumi nhyia — ntɛm ara wubehu nea enti a data a wɔaboaboa ano no ma nnua a emu dɔ ne sɛnea twitwa suban bere a wɔrebisabisa nsɛm no yi ahunmu akɛse fi hɔ.

Wɔ production applications ho no, susuw akwankyerɛ yi ho: sɛ wo data te database mu a, fa spatial indexing a wo database de ma (PostGIS, MySQL Spatial, MongoDB 2dsphere indexes) di dwuma sen sɛ wode quadtrees bedi dwuma wɔ application code mu. Sɛ woreyɛ client-side visualization anaa in-memory processing a, nwomakorabea te sɛ d3-quadtree ma JavaScript anaa pyquadtree ma Python ma wo implementations a wɔasɔ ahwɛ wɔ ɔko mu. Na sɛ worekyekye platform a ɛdi beaeɛ data biara ho dwuma — ɛfiri adetɔfoɔ address so kɔsi delivery routing so kɔsi asasesin sohwɛ so — fa bere no to hɔ na woate spatial indexing ase, ɛfiri sɛ ɛbɛkyerɛ nea wo application no bɛtumi ayɛ wɔ scale mu titire.

Nnua anan gyina hɔ ma nnyinasosɛm a ɛtrɛw wɔ kɔmputa ho nyansahu mu: sɛ nhyehyɛe a wopaw ma wo data no na ɛkyerɛ nsɛmmisa a wubetumi abua yiye. Nsɛntitiriw a wɔahyehyɛ no tratraa betumi abua "ma me nsɛntitiriw no nyinaa," nanso quadtree betumi abua "ma me nsɛntitiriw a ɛbɛn ha nyinaa" — na ebetumi ayɛ no ntɛmntɛm sɛnea ɛbɛyɛ a ɛbɛte nka sɛ ɛyɛ ntɛm ara. Wɔ wiase a adwumayɛ ho nsɛm 73% wɔ beaeɛ bi sɛdeɛ nnwuma akontabuo kyerɛ no, saa tumi no nyɛ adesua nko ara. Ɛyɛ akansi mu mfaso.

Nsɛmmisa a Wɔtaa Bisa

Dɛn ne quadtree na ɛyɛ adwuma dɛn?

Quadtree yɛ data nhyehyeɛ a egyina dua so a ɛkyekyɛ ahunmu a ɛwɔ afã mmienu mu mpɛn pii ma ɛyɛ quadrants anan a ɛyɛ pɛ. Node biara tumi kura data nsɛntitiriw dodow bi ansa na wɔakyekyɛ mu ayɛ no mmofra node anan. Saa nkyekyɛmu a ɛkɔ soro yi ma ahunmu nsɛmmisa — te sɛ nsɛntitiriw nyinaa a wobɛhwehwɛ wɔ beaeɛ bi a wɔde ama no — yɛ ntɛmntɛm yie, ɛtew hwehwɛ berɛ so firi linear kɔ logarithmic wɔ tebea dodoɔ no ara a mfasoɔ wɔ mu.

Ɛhe na wɔtaa de nnua anan di dwuma wɔ wiase ankasa dwumadie mu?

Quadtrees tumi ma nhyehyɛe ahorow pii a dijitaal asase mfonini a ɛwɔ pinch-to-zoom dwumadie, bere ankasa mu po so ahyɛn akyi dashboards, video agodie engine a ɛkyerɛ sɛ obi abɔ abira, ne asase ho nsɛm nhyehyɛe a ɛdi ahunmu nsɛmmisa ɔpepem pii ho dwuma wɔ sekan biara mu. Application biara a ɛhia sɛ ɛhwehwɛ, de hyɛ mu, anaa ɛhwɛ nneɛma a wɔakyekyɛ wɔ afã mmienu atenaeɛ so yie no bɛtumi anya mfasoɔ afiri quadtree indexing mu.

Sɛ wɔde nnua anan toto ahunmu data nhyehyɛe afoforo ho dɛn?

Nea ɛnte sɛ grids a ɛyɛ tratraa no, quadtrees sesa wɔn resolution ma ɛne data density hyia — mmeae a ɛnyɛ den no tra hɔ a ɛyɛ mmerɛw bere a mmeae a nnipa ahyɛ mu ma no kyekyɛ kɔ akyiri. Sɛ wɔde toto k-d nnua ho a, quadtrees yɛ mmerɛw sɛ wɔde bedi dwuma na ɛfata yiye ma 2D data a wɔakyekyɛ no pɛpɛɛpɛ. R-nnua di mpɔtam a ɛka bom ho dwuma fɛfɛɛfɛ, nanso quadtrees di nkonim wɔ ahoɔhare a wɔde hyɛ mu na ɛnyɛ den sɛ wɔde bɛyɛ pɛpɛɛpɛ ama bere ankasa mu adwumayɛ.

So quadtrees betumi aboa ma adwumayɛ ayɛ yiye wɔ adwumayɛ softwea mu?

Ɛyɛ saa koraa. Adwumayɛ adwinnade biara a ɛdi beae ho nsɛm, ahunmu nhwehwɛmu, anaa nkitahodi dashboard ahorow ho dwuma no nya mfaso fi quadtree optimization mu. Platforms te sɛ Mewayz, 207-module adwumayɛ OS a efi ase fi $19/mo, de data nhyehyɛe a etu mpɔn di dwuma wɔ akyi de osuahu ahorow a ɛyɛ ntɛm, a ɛyɛ mmuae ma — efi store locator maps so kosi bere ankasa mu nhwehwɛmu so wɔ data nsɛntitiriw mpempem pii so.