lecture 12
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The document discusses binary search trees (BSTs) and their use as a data structure for dynamic sets. It covers BST properties, operations like search, insert, delete and their running times. It also discusses using BSTs to sort an array in O(n lg n) time by inserting elements, similar to quicksort. Maintaining a height of O(lg n) is important for efficient operations.
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The document discusses binary trees and binary search trees. It begins with an overview of binary tree operations and traversals, including in-order, pre-order and post-order traversal. It then covers binary search trees, including their properties and operations like searching, finding the minimum/maximum values, and successors/predecessors. The document provides examples and pseudocode for traversing and performing operations on binary trees and binary search trees, including inserting and deleting nodes.
1st lecture.ppt
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DS Complete notes for Computer science and Engineering
The document provides information about data structures using C programming language. It discusses various topics like arrays, linked lists, stacks, queues, trees and graphs. It provides the syllabus, contents and references for the course on data structures. The document contains lecture notes on different data structure topics with examples and algorithms for common operations like search, insertion, deletion on arrays and linked lists.
lecture 10
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2. Counting sort runs in O(n+k) time where k is the range of input values, and is not a comparison sort.
3. Radix sort treats input as d-digit numbers in some base k and uses counting sort to sort on each digit, achieving O(dn+dk) time which is O(n) when d and k are constants.
4. A randomized selection algorithm finds the ith order statistic in expected O(n) time using randomized partition.
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CS-102 BST_27_3_14v2.pdf
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- Double hashing, which uses a second hash function to determine probe distances when collisions occur, reducing clustering compared to linear probing.
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DS Complete notes for Computer science and Engineering
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lecture 30
The document reviews concepts related to NP-completeness, including reductions between problems. It provides examples of reducing the directed Hamiltonian cycle problem to the undirected version. It also reduces 3-SAT to the clique problem by transforming a Boolean formula to a graph, then further reduces clique to vertex cover. Hundreds of problems have been shown to be NP-complete through relatively simple reductions like these that leverage previous results.
lecture 29
The document discusses reductions between problems to prove NP-completeness. It first reviews P, NP, reductions, NP-hard and NP-complete problems. It then walks through reducing directed Hamiltonian cycle to undirected Hamiltonian cycle to traveling salesman problem (TSP) to prove that TSP is NP-complete in 3 steps or less.
lecture 28
This document discusses NP-completeness and algorithms. It begins by reviewing tractable versus intractable problems, and the complexity classes P and NP. P contains problems solvable in polynomial time, while NP includes problems verifiable in polynomial time. The document then discusses NP-complete problems, which are the hardest problems in NP. It notes that reducing one problem to another shows the first problem is no harder than the second. The document concludes by outlining the steps to prove a problem is NP-complete: choosing a known NP-complete problem, reducing it to the target problem in polynomial time, and showing the target is in NP.
lecture 27
The document discusses NP-completeness and algorithms. It introduces dynamic programming and greedy algorithms. It then discusses the activity selection problem and shows it can be solved greedily by choosing the activity with the earliest finish time at each step. Finally, it defines the classes P and NP, introduces the concept of reductions to show problems are NP-complete, and states that if any NP-complete problem can be solved in polynomial time, then P=NP.
lecture 26
The document discusses various algorithms techniques including greedy algorithms, dynamic programming, and their application to problems like the activity selection problem and knapsack problem. It provides examples of optimal subproblems, overlapping subproblems, and how dynamic programming and greedy algorithms can be used to solve problems exhibiting these properties.
lecture 25
The document describes the 0-1 knapsack problem and how to solve it using dynamic programming. The 0-1 knapsack problem involves packing items of different weights and values into a knapsack of maximum capacity to maximize the total value without exceeding the weight limit. A dynamic programming algorithm is presented that breaks the problem down into subproblems and uses optimal substructure and overlapping subproblems to arrive at the optimal solution in O(nW) time, improving on the brute force O(2^n) time. An example is shown step-by-step to illustrate the algorithm.
lecture 24
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lecture 23
The document discusses dynamic programming and amortized analysis. It reviews how dynamic tables use amortized analysis to achieve an overall cost of O(1) per insertion by occasionally doubling the table size and reinserting all elements. This results in a worst case cost of O(n) for a single insertion but averages to O(1) over many insertions. It also discusses using an accounting method with a $3 charge per insertion to pay for future table resizes, achieving an amortized cost of O(1) per operation. Finally, it introduces dynamic programming and uses the longest common subsequence problem to illustrate how it breaks problems into optimal subrecurring subproblems.
lecture 22
The document discusses algorithms and data structures covered in a CS algorithms course, including Dijkstra's algorithm, Bellman-Ford algorithm, shortest paths in DAGs, Kruskal's algorithm for minimum spanning trees, and disjoint-set union data structures. It provides examples and explanations of how each algorithm works.
lecture 21
The document discusses algorithms for solving the single-source shortest path problem, including Dijkstra's algorithm, Bellman-Ford algorithm, and shortest paths in directed acyclic graphs (DAGs). It reviews the key ideas behind relaxation and how algorithms like Bellman-Ford and Dijkstra's use relaxation to iteratively improve the upper bound on shortest path distances. Running times of the algorithms are analyzed, with Dijkstra's algorithm taking O(ElogV) time and Bellman-Ford taking O(VE) time.
lecture 20
The document describes Prim's algorithm for finding a minimum spanning tree (MST) of a weighted undirected graph. It shows the pseudocode for Prim's algorithm and step-by-step workings on an example graph to build up the MST. It notes that the key operation is decreasing the key value, which is called once for each edge in the MST, and the running time depends on the priority queue implementation, being O(ElgV) for a binary heap and O(VlgV+E) for a Fibonacci heap.
lecture 19
The document discusses depth-first search (DFS) graph algorithms. DFS explores a graph by going as deep as possible at each step and backtracking when no further progress can be made. It distinguishes between different types of edges discovered: tree edges connect newly found nodes, back edges connect to ancestors, and cross/forward edges connect between subtrees or ancestors and descendants. DFS runs in O(V+E) time and can detect cycles in O(V) time by checking for back edges.
lecture 18
The document discusses the algorithms of breadth-first search (BFS) and depth-first search (DFS) on graphs. It provides pseudocode for BFS and DFS, and examples of running the algorithms on sample graphs. Key points include:
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- DFS uses recursion to explore as deep as possible before backtracking. It identifies tree edges, back edges, and forward edges.
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lecture 17
The document discusses interval trees and breadth-first search (BFS) algorithms. Interval trees are used to maintain a set of intervals and efficiently find overlapping intervals given a query. BFS is a graph search algorithm that explores all neighboring vertices of a starting node before moving to neighbors of neighbors. BFS builds a breadth-first tree and calculates the shortest path distances from the source node in O(V+E) time and space.
lecture 16
The document discusses algorithms for finding minimum spanning trees in graphs. It describes two common algorithms: Kruskal's algorithm and Prim's algorithm. Kruskal's algorithm sorts the edges by weight and builds up the spanning tree by adding edges that do not create cycles. Prim's algorithm maintains a set of vertices in the spanning tree and iteratively adds the lowest weight edge connecting a vertex outside the set to one inside. Both run in O((V+E)logV) time using efficient data structures.
lecture 15
The document discusses two algorithms for matrix multiplication and finding the median of an unsorted list:
1) Strassen's algorithm improves on the traditional O(n^3) matrix multiplication algorithm by using divide and conquer to achieve O(n^lg7) time complexity.
2) Finding the median can be done in expected O(n) time using quickselect, or deterministically in O(n) time by choosing the median of medians as the pivot.
lecture 14
The document discusses red-black trees and their operations. It reviews the properties of red-black trees that guarantee logarithmic height. It then describes that inserting nodes can violate properties and requires recoloring nodes or rotating the tree. The key steps of insertion involve adding the new node as red, then fixing any violations by moving them up the tree through recoloring and rotations.
lecture 9
The document discusses various sorting algorithms and their time complexities, including:
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- Merge sort and heap sort run in O(nlogn) time in the worst case.
- Any comparison-based sorting algorithm requires Ω(nlogn) time.
- Counting sort and radix sort can run in O(n) time by avoiding comparisons, but have additional requirements on the key range.
lecture 8
The document discusses analyzing the average case runtime of quicksort. It shows that:
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2) This is proved rigorously by modeling quicksort as alternating best and worst case partitions, showing the cost is absorbed by subsequent partitions.
3) A recurrence is written and solved to formally prove the average case runtime is O(n log n).
lecture 7
The document discusses the quicksort algorithm. It begins with an overview of quicksort and how it works. It then goes into more detail about the partition function, providing pseudocode and analyzing its running time. The document analyzes quicksort's average-case running time of O(n log n) by modeling it as a recurrence relation and solving the relation. It also tightly bounds a key summation term in the analysis.
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