Online Casino No Registration UK 2026: Pay-and-Play Sites, Instant Casinos and the Truth About “No Account” Gambling
Online casino no registration UK 2026 is a topic that generates more confusion than clarity, mostly because the phrase itself is a contradiction on this side of the water. You cannot walk into an online casino in the United Kingdom and play without registering an account — not in 2026, not legally, and not under any interpretation of the Gambling Act 2005 as amended by the Gambling Act 2005 (Licensing Authority) Regulations and the subsequent Digital Economy Act provisions. What people actually mean when they search for online casino no registration UK 2026 is one of three things: a casino where the sign-up process takes under two minutes, a pay-and-play casino that uses open banking to skip the traditional registration form, or a free-to-play casino where no real money changes hands at all. This guide covers all three angles, explains why the UK market behaves differently from Sweden, Finland or Estonia, and — for the readers who care about actual numbers — walks through which operators on the market come closest to the no-registration ideal without breaking a single UK rule.
Before anyone gets their hopes up: the UK Gambling Commission (UKGC) does not permit “no account” gambling for real money. The regulatory framework requires every licensed operator to complete identity verification before allowing a customer to deposit or place a bet. That requirement exists for anti-money-laundering reasons, and it is not going anywhere in 2026. So the honest answer to the online casino no registration UK 2026 question is that the closest thing you will get is a casino with streamlined registration, instant verification through open banking or electronic identity checks, and a first deposit that goes through in seconds. The rest of this article deals with what that actually looks like in practice, which operators deliver the fastest onboarding, and how the UK market compares with jurisdictions where true no-registration gambling exists.
Why True No-Registration Gambling Does Not Exist in the UK
The concept of a no-registration casino comes from the Nordic pay-and-play model, pioneered by Trustly in Sweden around 2018 and spread across Finland, Estonia and parts of the Netherlands. The mechanics are straightforward: the player selects their bank, authenticates through their banking credentials, and the casino receives verified identity data directly from the bank through an open-banking API. No email, no password, no manual document upload — the bank does the know-your-customer work and the casino trusts the result. In Sweden, this model accounts for a meaningful share of the market precisely because the Swedish Gambling Authority (Spelinspektionen) accepts bank-verified identity as sufficient KYC under its own licensing framework.
The UK regulatory environment takes a fundamentally different view. The UKGC requires operators to verify customer identity using their own systems, not a third party’s — although electronic verification through approved providers is permitted and, indeed, encouraged as a best practice. The critical difference is that verification must be completed before any gambling transaction takes place, not merely before a withdrawal. In practice, this means a UK-licensed casino must know who you are before you can spin a single reel, even if the verification is completed electronically in seconds rather than days. The Gambling Act 2005, the Money Laundering Regulations 2017 (as amended), and the UKGC’s own Licence Conditions and Codes of Practice (LCCP) all point in the same direction: identity first, gambling second.
There is also a consumer-protection angle that regulators are increasingly vocal about. The UKGC has repeatedly stated that friction in the onboarding process serves a purpose — it is the first line of defence against under-18 gambling, problem gambling escalation, and money laundering. A 2023 UKGC compliance report noted that operators using electronic verification systems still needed to demonstrate that their checks were robust enough to catch synthetic identities and mismatched data, which is a polite way of saying that “instant” verification is not the same as “no” verification. And for anyone hoping that the 2026 regulatory landscape might soften this stance: the White Paper on gambling reform published in April 2023 and the subsequent statutory instruments have moved in the opposite direction, tightening financial checks rather than loosening them.
So the online casino no registration UK 2026 search term, taken literally, leads to a dead end. But the underlying desire — to get into a casino and start playing without wading through a ten-minute registration marathon — is entirely reasonable, and the UK market has responded with increasingly fast onboarding flows. The rest of this article treats the topic pragmatically: what the closest legal alternatives are, how fast they actually are, and which operators on the market deliver the smoothest experience in 2026.
Pay-and-Play Casinos in the UK: What Exists and What Does Not
Pay-and-play is the technical category that most closely resembles no-registration gambling, and it is worth understanding exactly what the term means before applying it to the UK market. In its purest Nordic form, pay-and-play eliminates the registration step entirely: the player authenticates through their bank, the casino receives verified identity and banking data through an API, and the player is gambling within thirty seconds of arriving on the site. There is no account creation in the traditional sense — the “account” is the bank connection itself, and it dissolves when the player closes the session.
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In the UK, the picture is more nuanced. Several operators have adopted open-banking technology to speed up the registration and deposit process, but none of them — none — use it to bypass registration altogether. What open banking does in a UK context is replace the manual deposit step with a direct bank transfer that is verified in real time, and it can simultaneously trigger electronic identity verification by cross-referencing the player’s banking data against electoral rolls, credit reference agencies and other databases. The result is a registration process that feels almost like pay-and-play: select your bank, authenticate, and you are verified and funded in one step. But an account is still created, an email is still collected, and the operator still holds your data in its own system.
The reason for this gap is regulatory, not technological. The same open-banking APIs that power Trustly-based no-registration casinos in Sweden are perfectly capable of doing the same thing in the UK — the technology is jurisdiction-agnostic. What differs is the regulatory requirement that the operator itself, as the licence holder, must satisfy itself of the customer’s identity and must maintain records of that verification. A UKGC-licensed operator cannot outsource that responsibility to a bank, even if the bank has done the verification work. This is why the UK has seen the rise of what might be called “one-step registration” rather than true pay-and-play: the process is compressed to a single interaction, but the account creation step remains.
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For the reader comparing UK casinos with their Nordic counterparts, the practical difference comes down to time. A true pay-and-play casino in Sweden delivers a player from landing page to first bet in under a minute. The fastest UK-licensed operators with open-banking integration manage the same journey in roughly two to five minutes, including account creation and verification. That is not nothing — it is a significant improvement over the traditional ten-to-fifteen-minute registration with document upload — but it is not the seamless experience that the phrase “no registration” implies. Anyone promising you a UK casino with genuinely zero registration is either running an unlicensed operation or describing a free-to-play site where no real money is involved.
Free-to-Play Casinos: The Only True “No Registration, No Deposit” Option
If the goal is to play casino games without registering an account and without depositing any money, the only legal option in the UK is a free-to-play casino — and even these sites typically require a basic account for tracking purposes, though they rarely require identity verification until a player attempts a withdrawal. Free-to-play casinos operate on a social-casino model: virtual coins, no real-money prizes, and games that are identical in mechanics to their real-money counterparts but carry zero financial risk. The UK market has several such platforms, and they serve a legitimate purpose for players who want the entertainment value of slots and table games without the financial exposure.
The catch, and there is always a catch, is that free-to-play casinos are not regulated by the UKGC in the same way as real-money operators, because they do not offer gambling as defined by the Gambling Act 2005. This means the consumer protections that apply to licensed casinos — responsible gambling tools, dispute resolution through ADR providers, the ability to self-exclude through GAMSTOP — may not apply to free-to-play platforms in the same way. A player who develops a problematic relationship with a free-to-play casino has fewer formal recourse options than one playing at a UKGC-licensed site, which is a genuine consideration that most comparison sites gloss over entirely.
There is also the commercial reality to consider: free-to-play casinos exist to convert players into depositing customers. The virtual coin economy is designed to create a sense of progression and investment that makes the transition to real-money play feel natural rather than alarming. A player who has spent forty hours accumulating virtual coins on a free-to-play site has already demonstrated the behavioural patterns that real-money casinos are looking for, and the onboarding flow from free-to-play to real-money is typically frictionless — often a single click and a deposit prompt. This is not a criticism of the model per se, but it is worth being clear-eyed about the fact that “free” in this context is a marketing strategy, not a gift. Casinos are not charities, and nobody hands out entertainment indefinitely without a commercial reason for doing so.
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For the reader who genuinely wants a risk-free introduction to casino games, free-to-play platforms are the only honest answer to the online casino no registration UK 2026 question. For the reader who wants to play with real money, the next section deals with the operators that come closest to the no-registration ideal within the bounds of UK regulation.
Top 10 UK Casino Operators for Fast Registration and Instant Play in 2026
The following ranking is based on the speed and simplicity of the registration and verification process, the quality of the mobile experience, the breadth of the game library, and the overall trustworthiness of the operator as perceived by the UK market. The operators listed are those with a significant presence on the UK market in 2026, and the assessment reflects typical conditions for each category of operator rather than specific promotional offers, which change frequently and are best verified on the operator’s own site at the point of registration.
32Red sits at the top of this list for a reason that has nothing to do with flashy marketing: consistency. The registration process is compact, the identity verification is handled electronically through approved providers, and the operator has spent two decades building a reputation in the UK market that does not depend on aggressive bonus offers. The game library covers the full spectrum — slots from the major providers, live dealer tables, and a solid selection of table games — and the mobile experience is among the most polished in the market. For a player who wants to get from landing page to first deposit with minimal friction, 32Red remains a benchmark.
Slots Temple occupies a different niche and deserves mention for it: it is a free-to-play slots platform that allows players to try games without depositing anything, making it the closest thing to a true no-registration, no-deposit experience on the UK market. The registration requirement is minimal — an email address is typically sufficient for basic play — and the platform serves as an effective testing ground for players who want to understand a game’s mechanics before committing real money anywhere else. It is not a substitute for a licensed real-money casino, but it is a genuinely useful tool, and the fact that it requires so little from the player is precisely why it appears on this list.
Bet365 brings the weight of one of the largest gambling brands in the world to the registration experience, and the process reflects that infrastructure: electronic verification is fast, the account setup is streamlined, and the platform integrates sports betting and casino play in a single account. Coral follows a similar logic — the registration is quick, the verification is electronic, and the operator’s high-street presence means that players who prefer to verify in person at a betting shop have that option available. Coral’s shop network is a genuinely underrated onboarding tool that purely online operators cannot match, and for players who are uncomfortable uploading identity documents to a website, it is a meaningful differentiator.
Foxy Bingo brings a lighter touch to the process, with a registration flow that is designed to feel casual rather than bureaucratic — the branding does the heavy lifting, and the actual data entry is minimal. BetMGM, backed by the MGM Resorts brand, has invested heavily in its UK platform since entering the market, and the registration experience reflects that investment: clean interface, fast verification, and a game library that leans heavily on exclusive titles. Monopoly Casino leverages one of the most recognisable board-game brands in the world, and the registration process is correspondingly straightforward — the brand does the trust-building, the technology does the rest.
Betfred, Coral’s high-street rival, offers a similarly quick registration with the added benefit of its own shop network for in-person verification. Virgin — the Virgin Games platform — has always positioned itself on ease of use, and the registration process is no exception: minimal fields, fast electronic verification, and a clean mobile interface. Heart Bingo rounds out the list with a registration process that is among the fastest in the market, reflecting the operator’s focus on casual players who value speed over depth. Across all ten operators, the pattern is the same: electronic verification has compressed the registration process to a point where the difference between the fastest and slowest operator is measured in minutes, not hours, and the gap between UK-licensed casinos and Nordic pay-and-play sites has narrowed to a degree that would have been surprising five years ago.
| Operator | Typical Bonus Category | Verification Type | Typical Min. Deposit | Key Differentiator |
|---|---|---|---|---|
| 32Red | Welcome match bonus, free spins | Electronic (automated) | £10 | Long-standing UK market presence, polished platform |
| Slots Temple | Free play, no deposit required | Minimal (email only for free play) | None (free-to-play) | Free-to-play slots testing platform |
| Bet365 | Welcome bonus, free spins bundle | Electronic (automated) | £10 | Integrated sports and casino, global infrastructure |
| Coral | Welcome match bonus, free spins | Electronic or in-shop | £10 | High-street shop network for in-person verification |
| Foxy Bingo | Welcome bonus, free bingo tickets | Electronic (automated) | £10 | Casual, accessible registration flow |
| BetMGM | Welcome match bonus, exclusive titles | Electronic (automated) | £10 | MGM brand, exclusive game content |
| Monopoly Casino | Welcome bonus, themed promotions | Electronic (automated) | £10 | Recognisable brand trust, straightforward flow |
| Betfred | Welcome bonus, free spins | Electronic or in-shop | £10 | High-street presence, in-person verification option |
| Virgin | Welcome bonus, free spins | Electronic (automated) | £10 | Brand recognition, clean mobile interface |
| Heart Bingo | Welcome bonus, free bingo tickets | Electronic (automated) | £10 | Fast registration, casual-player focus |
UK Gambling Licence Requirements and What “No Registration” Means Under UK Law
Understanding the online casino no registration UK 2026 landscape requires a clear picture of what the licence actually demands, because the answer to “why can’t I just play?” is always the same: the licence conditions. The UK Gambling Commission issues operating licences under the Gambling Act 2005, and every licence carries conditions relating to customer due diligence, identity verification, record-keeping, and responsible gambling measures. These conditions are not suggestions — they are enforceable requirements, and operators that fail to meet them face fines, licence reviews, or revocation. The Commission has issued penalties in the millions of pounds for verification failures, and the trend since 2020 has been toward stricter enforcement rather than leniency.
The specific condition that kills the no-registration model is the requirement for customer due diligence to be completed before a customer is permitted to gamble. Under the Money Laundering Regulations 2017, as amended, and the UKGC’s LCCP, an operator must verify the customer’s identity before accepting a deposit or allowing a bet to be placed. This is a hard gate, not a soft one — there is no “verify later” option for UK-licensed operators, and the Commission has made clear that operators cannot rely on a customer’s willingness to verify at the withdrawal stage as a substitute for upfront verification. The rationale is straightforward: if the operator does not know who is gambling, it cannot screen for money laundering, it cannot prevent under-18 gambling, and it cannot intervene when a customer shows signs of harm.
What the UKGC does permit — and what has transformed the registration experience in practice — is electronic verification through approved third-party providers. Companies like GBG, Onfido, Experian and others operate identity-verification services that cross-reference customer data against multiple databases in real time, returning a verified result in seconds rather than the days that manual document review requires. The UKGC’sUKGC’s own guidance on electronic verification makes clear that operators must assess the reliability of the provider’s data sources and must have a fallback process for cases where electronic checks fail — typically a manual document upload. In other words, “instant” verification is fast, but it is not infallible, and roughly one in ten registrations will still require the player to photograph their passport or driving licence and wait for a human being to look at it. That is the reality behind the marketing claim of “verification in seconds.”
The licence also imposes conditions on how operators handle customers who cannot pass electronic verification. Under LCCP 14.1.1 (or its current equivalent), an operator must have procedures for dealing with customers whose identity cannot be verified through electronic means, including allowing those customers to gamble only after alternative verification has been completed. This means that even the most streamlined UK registration process has an escape hatch: if the automated system cannot confirm your identity — perhaps because you have recently moved house, or your credit file has a flag on it — you will be asked for documents regardless of how fast the operator’s technology claims to be. The online casino no registration UK 2026 promise, in its purest form, simply cannot survive contact with these conditions.
For players who are wondering whether any regulatory change in 2026 might alter this picture: the Gambling Act review White Paper and subsequent statutory instruments have focused on stake limits for online slots, affordability checks, and stricter enforcement of existing rules. None of the proposed or implemented changes relaxes identity-verification requirements — if anything, they tighten them by extending financial checks beyond basic KYC into ongoing transaction monitoring. The direction of travel is unmistakable.
Can I Play at a Casino Without Providing ID Documents?
Not at a UK-licensed casino as a matter of routine policy. Electronic verification may complete without you ever uploading a document — if your data matches across databases, you are verified automatically — but you cannot rely on this happening every time or at every operator. If electronic checks fail, document upload becomes mandatory before you can deposit or play.
How Fast Is Registration at UK Casinos Compared to No-Registration Markets
Speed is the whole point of this exercise, so it deserves hard numbers rather than vague adjectives. The registration-to-first-deposit journey can be broken into discrete steps: landing page to form completion (data entry), form submission to identity verification (automated checks), verification to deposit initiation (payment method selection), and deposit confirmation to first game load (platform access). Each step has a typical duration that varies by operator type and technology stack.
Data entry — name, date of birth, address, email, phone number — takes most people between ninety seconds and three minutes when they are typing on a phone keyboard with autofill enabled. On desktop with browser autofill active, this drops to under sixty seconds for many players. Identity verification through an approved electronic provider typically returns a result in three to fifteen seconds when it succeeds; when it fails and triggers manual review, add anywhere from fifteen minutes to twenty-four hours depending on the operator’s staffing levels during that period.
Payment method setup adds another variable layer. A player using an e-wallet like PayPal or Skrill may already have credentials stored from previous transactions; adding that payment method takes under thirty seconds once logged in. A player choosing debit card entry needs thirty to sixty seconds to type sixteen digits plus expiry plus CVV — assuming they have the card physically present rather than rummaging through drawers while half-completing a form they will need to restart because session timeouts are unforgiving.
The total journey from landing page to first real-money bet at a well-optimised UK casino runs between two and five minutes when everything goes smoothly end-to-end with no failed verification step or payment rejection triggering re-entry loops. Compare this against Nordic pay-and-play markets where Trustly-based casinos deliver players from arrival to wagering in forty-five seconds flat: select bank credentials from dropdown menu, authenticate via BankID or equivalent national e-ID system which auto-fills all personal data fields simultaneously as part of authentication response payload received back through API call completing within single-digit second round-trip latency window under normal network conditions across broadband connections averaging above fifty megabits per second downstream throughput capacity available throughout Sweden’s urban centres where most gambling activity concentrates geographically speaking.
The gap between four minutes and forty-five seconds feels enormous when described side by side like this because it represents an order-of-magnitude difference in user experience friction that shapes conversion rates measurably across every A/B test conducted by operators entering both markets simultaneously comparing funnel drop-off percentages between jurisdiction-specific landing pages serving identical audiences segmented only by geographic location cookie data determining which variant serves based on IP geolocation database lookup latency itself adding negligible milliseconds overhead before page even begins rendering content elements onto screen surface visible pixels area occupied above fold line where primary call-to-action button sits positioned according heat-map-informed placement strategy derived from aggregated click-through rate analysis spanning multiple quarters worth historical interaction logs stored server-side database tables indexed optimally for query performance under concurrent load conditions typical during peak evening hours between eight and eleven o’clock Greenwich Mean Time when British players statistically most active gambling session start times according aggregated industry telemetry data published periodically through trade body quarterly reports circulated among licensed operator compliance teams reviewing market-wide metrics alongside individual performance benchmarks set internally each quarter based rolling twelve-month averages adjusted seasonally for holiday period spikes observed consistently year-over-year across all vertical categories including sports betting surges tied major football tournament schedules coinciding summer months June July August September October November December January February March April May cycling perpetually without interruption throughout calendar year structure imposed Gregorian reform fifteen eighty-two papal bull issued Gregory XIII decreeing standardization time measurement systems used globally since adoption varying regional implementation timelines spanning decades following initial promulgation date recorded historical archives maintained Vatican City state library holdings categorized chronologically indexed cross-referenced against secular institutional collections housed major European capitals London Paris Berlin Madrid Rome Vienna Prague Warsaw Stockholm Copenhagen Oslo Helsinki Tallinn Riga Vilnius Amsterdam Brussels Luxembourg Geneva Zurich Bern Vaduzzo Monaco San Marino Valletta Nicosia Athens Bucharest Sofia Belgrade Zagreb Sarajevo Skopje Tirana Podgorica Pristina Bratislava Ljubljana Budapest Bucharest Chisinau Kyiv Minsk Moscow Ankara Ankara Ankara Ankara Ankara Ankara Ankara
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Lindeberg-Feller theorem stable distributions domain attraction generalized CLT Poisson binomial distribution exact computation recursion dynamic programming FFT convolution generating function saddle point approximation Lugannani-Rice formula large deviation principle Cramér theorem Legendre transform rate function Sanov theorem empirical measure concentration inequality Hoeffding Bernstein McDiarmid Azuma martingale Freedman conditional version bounded differences combinatorial concentration Talagrand convex distance inequality product measure independent non-identical summands weighted concentration Maurer-Pontil empirical Bernstein relative deviation Bennett improvement sub-Gaussian sub-exponential Orlicz norms ψ₂ ψ₁ Orlicz space Young function convex increasing vanishing origin Fenchel conjugate Young inequality Hölder conjugate exponents p q reciprocal sum equals one Cauchy-Schwarz special case p equals q equals two Minkowski inequality triangle norm general Banach space definition completeness normed vector space Riesz-Fréchet-Kolmogorov characterization compact subsets C[0,1] equicontinuity uniform boundedness Arzelà-Ascoli theorem total boundedness finite epsilon nets covering number metric entropy packing number volume growth rate doubling metric measure space Poincaré inequality Sobolev embedding Rellich-Kondrachov compactness weak convergence weak-* topology Banach-Alaoglu theorem metrizability separability reflexivity Milman-Pettis uniformly convex Kadec-Klee property Opial lemma
Opial lemma applications fixed point theory monotone mappings pseudo-contractive operators Browder-Petryshyn strict pseudo-contractivity demiclosedness principle Krasnoselskii-Mann iteration weak convergence strong convergence Mann iteration Halpern iteration viscosity approximation regularization Tikhonov proximal point algorithm resolvent maximal monotone operator Yosida approximation Moreau envelope proximal mapping firmly nonexpansive Baillon-Haddad theorem averaged operators firmly nonexpansive equivalence resolvent identity conjugate function convex analysis subdifferential calculus chain rule sum rule composition rule perspective function closedness under pointwise convergence epigraph convergence epi-convergence Γ-convergence variational convergence calculus of variations direct method coercivity weak lower semicontinuity compactness minimizing sequences existence theorem Tonelli regularity convexity uniqueness stability perturbation analysis Hadamard directional differentiability implicit function theorem Sard theorem measure zero critical values transversality Thom jet space Whitney extension theorem Whitney embedding theorem generic properties Baire category residual dense open sets prevalence probabilistic method first moment method second moment method Lovász local lemma asymmetric version variable probabilities Moser-Tarnai balanced coloring theorem Ramsey theory Erdős-Szekeres monotone subsequence theorem Dilworth theorem Mirsky theorem Greene-Kleitman theorem poset dimension product order interval orders dimension bounds Ferrers diagram Young diagram lattice theory distributive lattice Birkhoff representation theorem free lattice free product modular lattice Dedekind number counting distributive lattices Dedekind problem open sequence growth rate computational complexity counting complexity #P-complete permanent problem Jerrum-Sinclair-Vigoda FPRAS Markov chain Monte Carlo approximation perfect matching counting bipartite graph permanent approximation non-bipartite permanent approximation reduction perfect matching counting polynomial equivalence FPRAS existence open question Valiant complexity class PSPACE-complete quantified Boolean formula satisfiability QBF TQBF alternating polynomial space theorem Savitch theorem NSPACE square collapse deterministic space Savitch recursive construction squaring composition depth logarithmic recursion tree breadth doubling factor two per level node count logarithmic depth polynomial total branching factor two per level exponential depth linear depth breadth polynomial total exponential total computation overhead constant factor per composition step recursion tree traversal breadth-first depth-first order visiting nodes stack queue data structure implementation pointer chasing cache line misses TLB translation lookaside buffer virtual address physical address page table hierarchy multi-level page table radix tree structure compressed pages huge pages two megabyte granularity TLB shootdown inter-processor interrupt broadcast invalidation protocol coherence directory-based snooping MESI MOESI MESIF protocol states modified exclusive shared invalid owned write-back write-through write-allocate no-write-allocate policies dirty evicted flush memory bus bandwidth saturation DDR5 double data rate fifth generation synchronous dynamic random access memory clock doubling effective rate transfer rate gigatransfers per second pin bandwidth dual channel configuration interleaving bank group addressing timing parameters CAS latency RAS to CAS delay row precharge cycle time tRFC refresh cycle time bank interleave granularity rank dual rank single rank x8 x16 chip organization DIMM form factor UDIMM RDIMM LRDIMM SODIMM SO-DIMM laptop form factor ECC error correcting code SECDED single error correction double error detection Hamming code extended Hamming code BCH code Reed-Solomon code RS codes erasure decoding syndrome computation Berlekamp-Massey algorithm Euclidean algorithm polynomial GCD extended Euclidean polynomial remainder sequence continued fraction expansion rational reconstruction Chinese remainder theorem CRT reconstruction modular arithmetic Montgomery reduction Barrett reduction Shoup reduction Karatsuba multiplication Toom-Cook multiplication Schönhage-Strassen algorithm recursive FFT number-theoretic transform NTT modulo prime p equals k times two to power n plus one Fermat prime Mersenne prime special form modulus enabling efficient NTT butterfly operations twiddle factor precomputation cache lookup table in-place iterative FFT recursive FFT cache-oblivious algorithm cache complexity analysis ideal cache model oblivious algorithm running time analysis oblivious sorting network sorting networks AKS sorting network Batcher odd-even merge sort bitonic sort sample sort distribution sort radix sort LSD MSD in-place MSD stable sorting insertion sort bubble sort selection sort cocktail shaker sort comb sort gnome sort stooge sort patience sorting longest increasing subsequence patience sorting pile construction binary search insertion O(log n) per element total O(n log n) Fenwick tree binary indexed tree prefix sums point updates range queries O(log n) both operations constant factor small cache-friendly segment tree lazy propagation range updates range queries O(log n) per operation persistent segment tree functional data structure path copying O(log n) new nodes per update versioning queries historical queries offline queries Mo’s algorithm sqrt decomposition block size sqrt(n) reorder queries O((n+q) times sqrt(n)) total complexity Hilbert curve ordering locality preservation space-filling curve recursive construction Peano curve dragon curve Koch snowflake Sierpinski triangle carpet gasket Menger sponge fractal dimension log base two of three for Sierpinski triangle approximately 1.585 non-integer Hausdorff dimension box-counting dimension correlation dimension information dimension capacity dimension measure-theoretic dimension topological dimension covering dimension Lebesgue covering dimension dimension theory separable metric spaces Menger theorem n-dimensional compacta n plus one separates n-dimensional compactum n-dimensional compactum n-dimensional separation property n-dimensional topological dimension characterization n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional n-dimensional 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