https://ocw.cs.pub.ro/ppcarte/ 2026-05-21T20:14:51+03:00 https://ocw.cs.pub.ro/ppcarte/ https://ocw.cs.pub.ro/ppcarte/lib/tpl/bootstrap3/images/favicon.ico text/html 2018-12-12T11:30:00+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:adt-formal&rev=1544607000&do=diff Abstract Datatypes (ADT) A tentative definition In mathematics, algebraic structures consist of a (carrier) set (say - the natural numbers) as well as operations on elements of the set, satisfying specific axioms (e.g. commutativity for addition). $ \lnot p \wedge q $$ p$$ q$$ p$$ q$$ p$$ q$$ p$$ q$$ Void : List$$ Cons : E \times List \rightarrow List$$ Void$$ Cons$$ Cons(1,Cons(2,Cons(3,Void)))$$ Cons(2,Cons(2,Void))$$ isEmpty : List \rightarrow \mathbb{B}$$ size : List \rightarrow \mathbb{N… text/html 2016-11-25T10:55:19+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:adt&rev=1480064119&do=diff Abstract Data Types - Intro A overview into correctness Consider the following list of position papers: * Goto statement considered harmful * Programming with abstract data types * Object Oriented Programming - A personal disaster * Lisp: Good News, Bad News, How to Win Big These papers/blog-posts share a strong view (not necessarily overlapping nor in opposition) regarding how programs should be developed in the right way$ !n$ text/html 2019-11-26T13:25:51+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:ammortized-analysis&rev=1574767551&do=diff Amortised analysis Motivation Recall our FIFO implementation relying on two lists, from the previous lecture. This implementation has several advantages: * it offers constant costs for insertion, removal (dequeue), retrieval; * the implementation allows $ \Theta(n)$$ n$$ O(n)$$ \Theta(n)$$ 1$$ S = op_1, \ldots, op_n $$ n$$ op_i$$ enqueue$$ dequeue$$ top$$ S$$ S$$ top$$ cost(S_1) \geq cost(S_2)$$ S_1$$ S_2$$ S_2$$ top$$ cost(S)= cost(ins_l) + cost(del_l) + cost(ins_r) + cost(del_r)$$ ins_… text/html 2016-10-31T14:31:10+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:beyondre&rev=1477917070&do=diff Beyond RE Recap In the last lecture: * we have introduced the classes $ R$ and $ RE$ , which classify decision problems into: * decidable (which can be solved by algorithms which always terminate). * semi-decidable (which can be solved by algorithms which terminate only if the answer is $ f$$ f^*$$ f^* \leq_T f$$ f$$ f^*$$ RE$$ RE$$ \overline{f}(n) = \left\{ \begin{array}{ll} 1 & \mbox{iff } f(n)=0\\0 & \mbox{iff } f(n)=1 \end{array} \right.$$ f_h$$ M$$ w$$ \overline{\overline{f}} = … text/html 2016-11-25T16:15:15+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:classes&rev=1480083315&do=diff Complexity Classes Preliminaries We define: $ DTIME(T(n))=\{ f : \mathbb{N} \rightarrow \{0,1\} \mid f \mbox{ is decidable in time } O(T(n))\}$ and $ NTIME(T(n))=\{ f:\mathbb{N} \rightarrow \{0,1\} \mid f \mbox{ is decidable by a } NTM \mbox{ in time } O(T(n))\}$ $ DTIME(T(n))$ is the class of problems which are solvable by a Turing Machine having execution time $ T(n)$ , while $ NTIME(T(n))$ is the class of problems which are solvable by a nondeterministic Turing Machine having executio… text/html 2019-10-15T06:53:16+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:decidability&rev=1571111596&do=diff Undecidable problems How many problems and how many ways of solving them? Consider the following sets: * $ \mathcal{M}$ - the set of all Turing Machines * $ \mathbb{H}\text{om}(\mathbb{N},\{0,1\})$ - the set of all decision problems. Note: in this lecture, we will interchangeably refer to problems as:$ f:\mathbb{N}\rightarrow\{0,1\}$$ f:\mathbb{\Sigma^*}\rightarrow\{0,1\}$$ \mathcal{M}$$ M\in\mathcal{M}$$ enc(M)\in\Sigma^*$$ \mathcal{M}$$ \Sigma$$ \Sigma_M$$ \Sigma$$ enc:\mathcal{M} \r… text/html 2021-01-13T09:54:40+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:exam&rev=1610524480&do=diff Examen AA 2020-2021 Examenul la Analiza Algoritmilor valorează 4 puncte. Pentru promovarea examenului este necesar 50% din punctaj (2 puncte). Cele 4 puncte vor fi obținute în cadrul a doua probe: * 1.5 puncte - proba scrisă. * 2.5 puncte - text/html 2021-01-13T10:06:42+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:examen-optional&rev=1610525202&do=diff Examen optional 2020-2021 Vineri 29 ianuarie, de la ora 10:00, in cadrul ultimului curs de AA, vom organiza un examen-interviu (oral) optional, la care studentii se pot inscrie pentru a degreva examenul oral din sesiune. Cu acordul studentilor, vom inregistra o parte din interviuri, astfel incat procedura sa poata fi transparenta pentru toti studentii. Cateva observatii vizavi de acest examen: text/html 2016-10-10T14:24:38+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:guidelines&rev=1476098678&do=diff Guidelines Resources & Content * The Wiki will contain: * all the material (syllabus) covered in the AA lecture, including definitions, proofs, examples. * all material covered in labs, including some solved exercises * The Lecture will feature: text/html 2017-01-11T16:10:50+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:induction&rev=1484143850&do=diff Structural induction Motivation In the last chapter, we have specified two ADTs: $ List$ and $ FIFO$ , as well as one $ FIFO$ implementation relying on lists. The following were open issues: * while selecting axioms for $ append$ and $ reverse$ , we claimed that they guarantee certain properties of the ADT hold$ T$$ O : T$$ O$$ T$$ E: X \rightarrow T$$ E$$ T$$ T$$ X$$ I : T \rightarrow T$$ I: X \times T \rightarrow T$$ I: T\times T\rightarrow T$$ I$$ T$$ T$$ \forall t \in T : P(t)$$ T$$ … text/html 2024-10-14T17:54:10+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture0&rev=1728917650&do=diff 3 Problems Challenge [3 Problems Challenge (slides)] [3 Problems Challenge - Summary (slides)] text/html 2024-11-11T16:07:17+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture1&rev=1731334037&do=diff Problems and Algorithms + Turing Machines [Problems and algorithms (lecture notes)] [Problems and algorithms (slides)] [Turing Machines (lecture notes)] [Turing Machines (slides)] text/html 2024-11-11T16:07:26+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture2&rev=1731334046&do=diff Computing [Computing (lecture notes)] [Computing (slides)] text/html 2024-11-11T16:07:47+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture3&rev=1731334067&do=diff Multitape TMs + Universal TMs [03a. Multitape Turing Machines (lecture notes)] [03a. Multitape Turing Machines (slides)] [03b. Universal Turing Machines (lecture notes)] [03b. Universal Turing Machines (slides)] text/html 2024-11-11T16:02:15+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture4&rev=1731333735&do=diff Undecidable Problems [04. Undecidable Problems (lecture notes)] [04. Undecidable Problems (slides)] text/html 2024-11-11T16:03:02+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture5&rev=1731333782&do=diff Reductions Lecture notes: [05. Reductions (lecture notes)] [05. Reductions (slides)] text/html 2024-11-11T16:05:32+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture6&rev=1731333932&do=diff Undecidability Results [06a. Rice's Theorem (lecture notes)] [06a. Rice's Theorem (slides)] [06b. Post's Correspondence Problem (lecture notes)] [06b. Post's Correspondence Problem (slides)] text/html 2024-11-22T11:52:02+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture7&rev=1732269122&do=diff Computational Resources [07. Computational resources (lecture notes)] [07. Computational resources (slides)] text/html 2024-11-22T11:52:13+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture8&rev=1732269133&do=diff Complexity Across Models [08. Complexity across models (lecture notes)] [08. Complexity across models (slides)] text/html 2024-11-22T11:52:26+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture9&rev=1732269146&do=diff "Hard?" problems [09. "Hard?" problems (lecture notes)] [09. "Hard?" problems (slides)] text/html 2024-11-22T11:52:38+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture10&rev=1732269158&do=diff Polynomial-time reductions [10. Polynomial-time reductions (lecture notes)] [10. Polynomial-time reductions (slides)] text/html 2024-11-22T11:51:14+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture11&rev=1732269074&do=diff P vs. NP [11. P vs. NP (lecture notes)] [11. P vs. NP (slides)] text/html 2024-12-03T13:15:39+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture12&rev=1733224539&do=diff Hardness and completeness [12. Hardness and completeness (lecture notes)] [12. Hardness and completeness (slides)] text/html 2024-12-06T12:21:40+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture13&rev=1733480500&do=diff Recurrence relations [13. Recurrence relations (lecture notes)] [13. Recurrence relations (slides)] text/html 2024-12-06T12:23:03+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture14&rev=1733480583&do=diff The Master Method [14. The Master Method (lecture notes)] [14. The Master Method (slides)] text/html 2024-12-13T11:41:43+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture15&rev=1734082903&do=diff Quicksort [15. Quicksort (lecture notes)] [15. Quicksort (slides)] text/html 2025-01-15T16:15:47+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture16&rev=1736950547&do=diff Amortized Analysis [16. Amortized Analysis (lecture notes)] [16. Amortized Analysis (slides)] text/html 2025-01-15T16:16:21+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture17&rev=1736950581&do=diff Algebraic Data Types [17. Algebraic Data Types (lecture notes)] text/html 2025-01-20T14:46:30+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lecture18&rev=1737377190&do=diff Structural Induction [18. Structural Induction (lecture notes)] [18. Structural Induction (slides)] text/html 2016-10-03T23:50:03+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:main&rev=1475527803&do=diff Algorithms and Complexity Theory * Introduction * Problems and Algorithms * Undecidable problems * Computability Theory * Complexity Theory * Bibliography * Rules * Macros text/html 2016-11-15T15:59:03+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:nondeterminism&rev=1479218343&do=diff Nondeterminism Intuition In the figure below, we show an ASCII map: ---- C3 --- dst --- C4 --- cliff / | | / | | src >>-- C1 ----------- C2 --- cliff Suppose a robot starts from position $ \delta$$ q_0$$ SAT$$ SAT$$ \psi$$ \psi = C_1 \wedge C_2 \wedge \ldots \wedge C_n$$ i:1 \leq i \leq n$$ C_i = L_{i1} \vee L_{i2} \vee \ldots \vee L_{im_i}$$ j:1 \leq j \leq m_i$$ L_{ij} = x$$ L_{ij}=\neg x$$ x$$ 1$$ I$$ I$$ \psi$$ \psi$$ \psi$$ n$$ … text/html 2019-11-04T13:52:07+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:notations&rev=1572868327&do=diff Asymptotic notations Motivation Let us consider the array sorting problem. Although it is not a decision problem, it is a sufficiently simple example. The sorting problem is sub-exponential: it can be solved in polynomial time (by many known algorithms and implicitly, by a Turing Machine).$ \Theta(g(n)) = \left \{ f : \mathbb{R} \rightarrow \mathbb{R} \left\lvert \begin{array}{ll} \exists c_1,c_2 \in \mathbb{R}^+ \\ \exists n_0 \in \mathbb{N} \end{array}, \forall n \geq n_0, c_1g(n) \leq f(n) … text/html 2021-01-17T22:48:25+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:pcp&rev=1610916505&do=diff PCP is undecidable In this section, we will prove that Post's Correspondence Problem is undecidable. Compared with the other proofs of this Chapter, PCP undecidability will be more involved. Basically, the proof is a two-step reduction from the halting problem$ \alpha_i, \beta_i$$ \beta_i$$ F=\{s_{yes},s_{no}\}$$ 0/1$$ M$$ w$$ w \neq \epsilon $$ \alpha_1, \ldots, \alpha_n$$\beta_1, \ldots, \beta_n$$ a_1a_2 \ldots a_k$$ a_1 = 1$$ a_i = 1, \ldots, n$$ \alpha_{a_1}\alpha_{a_2}\ldots \alpha_{a_k} =… text/html 2021-06-05T14:50:22+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:playground&rev=1622893822&do=diff Playground def f (v, x): n = len(v) if n == 0: return 1 return x*f(v[0:n/2], x-1) text/html 2016-11-10T18:50:35+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:recurrences&rev=1478796635&do=diff TODO text/html 2017-03-20T08:50:47+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:rules&rev=1489992647&do=diff REGULI Punctaj * Parcurs: 6p * Examen: 4p * pentru intrarea in examen sunt necesare minim 3p din punctajul de parcurs * pentru promovarea examenului sunt necesare minim 2p Punctajul pe parcurs consta din: * proiect: 3p * teste: 2p * text/html 2016-11-25T10:54:45+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:sat&rev=1480064085&do=diff SAT is NP-complete We have already shown in a previous lecture that $ SAT \in NP$ . It remains to show that $ SAT$ is NP-hard, i.e. $ \forall f \in NP$ we have $ f \leq_p SAT$ . Roadmap The difficulty of the proof relies in the fact that we cannot use the reduction technique shown in the previous lecture, since there is no choice of another NP-hard problem. Therefore, we need to prove that, for $ f \in NP$$ f$$ T(n) = n^k$$ k \in \mathbb{N}$$ f \leq_p SAT$$ w$$ \varphi_w$$ \forall w: f(w) =… text/html 2021-10-04T20:02:27+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:syllabus&rev=1633366947&do=diff Syllabus I Teoria computabilității 1. Introducere; probleme și algoritmi 2. Mașina Turing 3. Mașina Turing Universală 4. Acceptare și Decidabilitate 5. Probleme nedecidabile; problema terminării 6. Reduceri Turing 7. Teorema lui Rice text/html 2025-10-21T14:03:27+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:team&rev=1761044607&do=diff Analiza Algoritmilor 2024-2025 Echipa * Matei POPOVICI * Mihai-Valentin DUMITRU * Vlad-Andrei Badoiu * Vlad JUJA * Ștefan-Octavian STEREA * Alexandru-Petru TOADER * Cezar-Stelian ZLATEA * Andrei-Adrian RĂGMAN * Balcan Rares Stefan * Mihnea-Alex GHEORGHE * Andrei DÂRLĂU * Aureliu Valentin ANTONIE * Octavian-Alexandru Osnaga Orar Curs Zi Interval orar Vineri 10:00-12:00 Laboratoare Grupă Zi Interval orar Sală Asistent 321CBa Joi 16:00-18:00 EG209 Mihne… text/html 2025-12-29T13:14:20+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:tema&rev=1767006860&do=diff Deadline: 16 ianuarie 2026 Schelet de cod: Tema: Editor de Text Scopul acestei teme este optimizarea backend-ului unui editor de text. În varianta inițială, stocarea textului într-un array continuu impunea mutarea tuturor caracterelor la fiecare inserare sau ștergere, rezultând o complexitate de O(N). text/html 2018-11-06T23:44:12+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:tema_1&rev=1541540652&do=diff TEMA 1 Responsabili tema: Mihaela Catrina, Teodor Popescu * I. (0.5 x 6 = 3p) Pentru fiecare din urmatoarele recurente aplicati teorema Master (sau argumentati de ce nu poate fi aplicata, daca este cazul): * $ T(n) = T(\frac{n}{2}) + 2^n$ * $ T(n) = 2^nT(\frac{n}{2}) + n^n$ * $ T(n) = 2T(\frac{n}{4}) + n^{0.51}$ * $ T(n) = 64T(\frac{n}{8}) - n^2log n$ * $ T(n) = T(\frac{n}{2}) + n(2 - cos n)$ $ T(n) = 16T(\frac{n}{4}) + n!$$ f(n)$$ T(n) = \theta(f(n))$$ T(n) = \frac{… text/html 2018-11-06T19:25:48+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:tema_1_2018&rev=1541525148&do=diff TEMA 1 Responsabili tema: Mihaela Catrina, Teodor Popescu * I. (0.5 x 6 = 3p) Pentru fiecare din urmatoarele recurente aplicati teorema Master (sau argumentati de ce nu poate fi aplicata, daca este cazul): * $ T(n) = T(\frac{n}{2}) + 2^n$ * $ T(n) = 2^nT(\frac{n}{2}) + n^n$ * $ T(n) = 2T(\frac{n}{4}) + n^{0.51}$ * $ T(n) = 64T(\frac{n}{8}) - n^2log n$ * $ T(n) = T(\frac{n}{2}) + n(2 - cos n)$ $ T(n) = 16T(\frac{n}{4}) + n!$$ f(n)$$ T(n) = \theta(f(n))$$ T(n) = \frac{… text/html 2019-11-23T11:18:58+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:tema_1_2019&rev=1574500738&do=diff TEMA 1 Responsabili tema: Peticila Alexandru, Peticila Constantin * I. (0.4 x 5 = 2p) Pentru fiecare din urmatoarele recurente aplicati teorema Master (sau argumentati de ce nu poate fi aplicata, daca este cazul): * $ T(n) = 4T(\frac{n}{2}) + n^{2.5}$ * $ T(n) = 4T(\frac{n}{2}) + n^2 log(n)$ * $ T(n) = T(\sqrt{n}) + \Theta(log( log( n)))$ * $ T(n) = T(\frac{n}{2}) + n(2 - cos n)$ $ T(n) = 3T(\frac{n}{3}) + \frac{n}{log (n)}$$ f(n)$$ T(n) = \Theta(f(n))$$ T(n) = T(n - a) … text/html 2017-10-29T00:14:46+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:tema_2&rev=1509225286&do=diff Tema 2 Enunț: Last update: 29.10.2017 00:14 Precizari Tema va fi redactată in echipe de câte 2 persoane. Echipele vor fi stabilite in cadrul laboratorului. Tema va fi predată la cursul din data de 10 noiembrie 2017 pentru primirea punctajului întreg sau la text/html 2017-10-20T10:27:25+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:tema_3&rev=1508484445&do=diff Tema 3 - Reduceri Polinomiale [Enunt] text/html 2017-12-10T00:41:42+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:tema_4&rev=1512859302&do=diff Tema 4 - Reduceri polinomiale text/html 2016-11-02T09:02:18+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:timespace&rev=1478070138&do=diff Time and space In Computability Theory, we have used classes such as $ R$ and $ RE$ in order to establish a hierarchy of hardness for problem solvability (in terms of the Turing Machine). We now take into account the resources spent by the Turing Machine, during the computation process.$ M$$ \mathcal{T}_M, \mathcal{S}_M : \Sigma^* \rightarrow \mathbb{N}$$ \mathcal{T}_M(w)$$ \mathcal{S}_M(w)$$ M$$ w$$ \mathbf{while} \mbox{ } n \lt 100$$ \quad n=n+1$$ \mathbf{return} \mbox{ } 1$$ Alg$$ n=0$$ n…