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--- teaching:ads12526_lecture [2026/03/16 21:51] – Martin Koutecky
+++ teaching:ads12526_lecture [2026/05/11 14:04] (aktuální) – Martin Koutecky
@@ Řádek 15: / Řádek 15: @@
 | 9. 3 | Finish SCC algorithm **[ALG 5.9]**, **[A, Chap 3]**; begin shortest paths: BFS **[ALG 5.2]**, Dijkstra **[ALG 6.0-6.2]**, **[A, Chap 4]**|
 | 16. 3. | binary, $d$-ary, Fibonacci heaps for Dijkstra; general relaxation algorithm, start Bellman-Ford **[ALG 6.1-6.3]**, **[A, Chap 4]**.|
-| 23. 3. |//Plan: Bellman-Ford **[A 4]** (Also see **[JE 8, 9]** but that's a lot more information than necessary.) Beginning of Min Spanning Trees: cut lemma, if edge weights distinct, MST is unique; Jarník's algorithm; Borůvka's algorithm **[A 5]**, **[JE 7]**.)//|
+| 23. 3. | Bellman-Ford **[ALG 6.3]**, **[A 4]** //(Also see **[JE 8, 9]** but that's a lot more information than necessary.)// Start Min Spanning Trees: cut lemma, if edge weights distinct, MST is unique; Jarník's algorithm; Borůvka's algorithm **[ALG 7-7.3]**, **[A 5]**, **[JE 7]**.)|
+| 30. 3. | Kruskal's algorithm; Union-Find problem and data structure via forests ("bushes"), $\mathcal{O}(\log n)$ Union and Find **[ALG 7.4]**, **[A 5.1.4]**. Start data structures: Binary Search Trees; perfectly balanced BSTs have depth $\mathcal{O}(\log n)$ **[ALG 8.1]** ([[https://ksvi.mff.cuni.cz/~dingle/2021-2/algs/notes_10.html|intro to Algs notes]])|
+| 6. 4. | //Easter Monday — no lecture.//|
+| 13. 4. | AVL trees: depth lower bound $\Omega(\log n)$ and upper bound $\mathcal{O}(\log n)$; FIND runs in $\mathcal{O}(\log n)$; INSERT and DELETE via rotations also run in $\mathcal{O}(\log n)$. **[ALG 8.2]** |
+| 20. 4. | $(a,b)$-trees and brief mention of Left-Leaning Red-Black (LLRB) trees **[ALG 8.3]** ([[https://sedgewick.io/wp-content/themes/sedgewick/papers/2008LLRB.pdf|original LLRB paper]], [[https://sedgewick.io/wp-content/uploads/2022/03/2008-09LLRB.pdf|slides]], [[https://read.seas.harvard.edu/~kohler/notes/llrb.html|controversy]]). Tries **[ALG 4.3]**.|
+| 27. 4. | Hashing with chaining **[ALG 11.3]**, open addressing (linear probing, double hashing) **[ALG 11.4]**, universal hashing **[ALG 11.5]**. Brief mention of amortized complexity and expanding arrays **[ALG 9.1]**.|
+| 4. 5. | Divide and conquer: analysis of Mergesort via the substitution method and the recursion tree technique **[ALG 10.2]**, Karatsuba's algorithm for multiplication of $n$-digit numbers in time $\mathcal{O}(n^{\log_2 3})$ **[ALG 10.3]**, started the Master theorem **[ALG 10.4]** //(to be finished next time)//.|
+| 11. 5. | Finish Master theorem **[ALG 10.4]**. QuickSelect, QuickSort **[ALG 10.6-10.7]**, finding the $k$-th smallest element in linear time **[ALG 10.8]**|
+| 18. 5. | //Plan: dynamic programming: Longest Increasing Subsequence in $O(n \log n)$ time. Edit Distance in $O(nm)$. Floyd-Warshall algorithm for All-Pairs Shortest Paths (APSP). **[ALG 12]**, [[https://jeffe.cs.illinois.edu/teaching/algorithms/book/03-dynprog.pdf|JE 3]].//|
 /*
-| 2. 4. | Finish Borůvka's algorithm; Kruskal's algorithm for MSTs, Union-Find problem and data structure. **[A 5.1.4]**. Intro to Data structures. Binary Search Trees ([[https://ksvi.mff.cuni.cz/~dingle/2021-2/algs/notes_10.html|intro to Algs notes]]) |
+| 2. 4. |
 | 9. 4. | Perfectly balanced trees (//balancing turns out to be too expensive, this is a dead-end//); depth-balanced trees = AVL trees. [[https://courses.cs.washington.edu/courses/cse373/19su/files/lectures/slides/lecture09.pdf|slides from U of Washington]]). **{{ :teaching:ads12122:ads_bsts.pdf |OneNote notes}}**. AVL tree rebalancing. Possible resources for AVL trees: [[https://www.programiz.com/dsa/avl-tree|1]], [[http://gtu-paper-solution.com/Paper-Solution/DataStructure-2130702/AVL%20Trees/Summer-2019/Question-4-c-OR|2]], [[https://courses.cs.washington.edu/courses/cse373/19su/files/lectures/slides/lecture09.pdf|3]]|
 | 16. 4. | $(a,b)$-trees [[https://cs.lmu.edu/~ray/notes/abtrees/|1]], [[http://www14.in.tum.de/lehre/2016WS/ea/split/sub-ab-trees-single.pdf|2]], hashing [[https://jeffe.cs.illinois.edu/teaching/algorithms/notes/05-hashing.pdf|hashing notes from Jeff E]]. **In tutorials: we showed a construction of a $1$-universal family of hashing functions, see [[https://research.koutecky.name/db/_media/teaching:ads12324:09-cv.pdf|here]]**. This theorem may appear at the exam.|
@@ Řádek 32: / Řádek 40: @@
 {{page>teaching:bits:resources_ads1}}
-/*
-{{ :teaching:ads12324:exam2324.pdf | Exam topics and format (PDF).}}
-====== ADS1 Exam / 2024 ======
+/* {{ :teaching:ads12324:exam2324.pdf | Exam topics and format (PDF).}} */
+====== ADS1 Exam / 2026 ======
 The exam will consist of:
@@ Řádek 65: / Řádek 73: @@
   * Dijkstra's algorithm, $d$-ary heaps
   * Bellman-Ford's algorithm
-  * Floyd-Warshall's algorithm (small topic)
+  * Floyd-Warshall's algorithm (small topic -- covered with dynamic programming)
   * Jarník's algorithm; cut lemma
   * Borůvka's algorithm
@@ Řádek 72: / Řádek 80: @@
   * AVL trees
   * $(a,b)$-trees
+  * Tries (FIND, INSERT, DELETE in $O(L)$ for a string of length $L$)
   * Hashing with chaining (each bucket is a linked list)
-  * Universal hashing
+    * Hashing with open addressing: just know that it exists, with various schemes (collisions are handled by putting elements elsewhere in the same array, not by creating a chain)
+  * Universal hashing -- construction of a $c$-universal family
+  * Expanding arrays -- brief mention, be able to prove that inserting $n$ elements into an expanding array (start from size 1, if it would overflow $\to$ double size and re-insert) takes time $O(n)$.
   * Master theorem -- analyzing the complexity of Divide & Conquer algorithms using the recursion tree
   * Integer multiplication using Karatsuba's algorithm
-  * MomSelect (finding the $k$-th smallest element in an array in linear time)
+  * LinearSelect (finding the $k$-th smallest element in an array in linear time)
   * QuickSelect with random pivots works in expected $\mathcal{O}(n)$ time, QuickSort with random pivots in expected $\mathcal{O}(n \log n)$ time.
   * Edit Distance dynamic programming algorithm
   * Longest Increasing Subsequence dynamic programming algorithm
-  * Lower bounds: searching in a sorted array is $\Omega(\log n)$, sorting is $\Omega(n \log n)$ (only applies to deterministic and comparison-based algorithms)
-*/
+/*  * Lower bounds: searching in a sorted array is $\Omega(\log n)$, sorting is $\Omega(n \log n)$ (only applies to deterministic and comparison-based algorithms) */