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Theory of algorithms

Theory of algorithms refers to the  branch of mathematics which focuses on the ideas and methods to be applied on non costructuve objects only if these are encoded as constructive objects by dealing with  general properties of algorithms.The important part of theory of algorithm is general properties of encodings which is basically the theory of enumeration.It is of two types i.e descriptive (qualitative) and the metric (quantitative) theory.

Few Topics are:

  • Running time, Insertion sort
  •  binary search, merge two sorted arrays
  • Merge sort+
  • Binary trees, heaps,Heapsort
  • Priority queues, Quicksort
  • Analysis of quicksort, Randomized Quicksort
  • Radix sort, Bucket sort
  •  stack and queues
  • Linked lists
  • Pointers and objects, representing rooted trees
  • Hash tables, hashing with chaining
  • Hash functions
  • Binary search trees
  • Red-black trees
  • Graph representations, BFS
  • DFS
  •  Topological sort

Sample Assignment

15-7 Viterbi algorithm
 
We can use dynamic programming on a directed graph G D .V;E/ for speech recognition. Each edge .u; / 2 E is labeled with a sound  .u; / from a finite set † of sounds. The labeled graph is a formal model of a person speaking a restricted language. Each path in the graph starting from a distinguished vertex 0 2 V corresponds to a possible sequence of sounds produced by the model. We define the label of a directed path to be the concatenation of the labels of the edges on that path. a. Describe an efficient algorithm that, given an edge-labeled graph G with distinguished vertex 0 and a sequence s D h 1;  2; : : : ;  ki of sounds from †, returns a path in G that begins at 0 and has s as its label, if any such path exists. Otherwise, the algorithm should return NO-SUCH-PATH. Analyze the running time of your algorithm. (Hint: You may find concepts from Chapter 22 useful.) Now, suppose that every edge .u; / 2 E has an associated nonnegative probability p.u; / of traversing the edge .u; / from vertex u and thus producing the corresponding sound. The sum of the probabilities of the edges leaving any vertex equals 1. The probability of a path is defined to be the product of the probabilities of its edges. We can view the probability of a path beginning at 0 as theprobability that a “random walk” beginning at 0 will follow the specified path, where we randomly choose which edge to take leaving a vertex u according to the probabilities of the available edges leaving u. b. Extend your answer to part (a) so that if a path is returned, it is a most probable path starting at 0 and having label s. Analyze the running time of your algorithm.
 
Theory of algorithms
 
  • Reductions,Greedy algorithms,Minimum spanning trees,Dynamic programming,Shortest paths algorithms,Bellman-Ford,Floyd-Warshall, Depth First Search (DFS)
  • strongly connected components,topological sort,Maximum flow algorithms of Ford-Fulkerson Dinitz,Applications to matching and assignment problems
  • Randomized algorithms,String matching,Introduction to complexity theory,complexity classes,Cook-Levin theorem,techniques for proving NP-completenes
  • Induction,Recurrence relations,Big-Oh and little-Oh notation,Merge sort,Graph Algorithms,Depth-first search,strongly connected components
  • Breadth-first search,Dijkstra's algorithm,Greedy Algorithms,Minimum spanning tree,Union find,Set cover,Huffman coding,Dynamic Programming
  • Longest common subsequence,Traveling salesman,Divide and Conquer,Integer multiplication,Matrix multiplication,Hashing,Balls into bins problems
  • Bloom filters,Document similarity,Linear Programming,Problem definitions and solution techniques,Reductions,Maximum matching, Randomized Algorithms
  • Primality testing and factoring, RSA,Random walks and 2-SAT,NP-completeness review,Basic NP-complete problems,Novel approaches to NP-complete problems

Theory of algorithms includes:

  • Approximation algorithms,Heuristic algorithms,Stable matching, implementation, running times,Graphs and basic graph algorithms,Greedy algorithms for optimization problems
  • Divide-and-conquer,fast multiplication of integers,matrices and polynomials,Dynamic programming ,Max-flow/min-cut, polynomial time algorithms,introduction to NP
  • Lower bounds and approximation algorithms,Local search and heuristic approaches,Randomized algorithms, median-finding and order statistics,Introduction and document distance
  • More document distance, mergesort,Binary search trees,Airplane scheduling, binary search trees,Balanced binary search trees,Hashing: chaining, hash functions 
  • table doubling, Karp-Rabin,open addressing ,Sorting:heaps , lower bounds, linear-time sorting,stable sorting, radix sort ,Searching: graph search, representations, and applications
  • breadth-first search and depth-first search , topological sort and NP-completeness ,Shortest paths,Shortest paths: intro ,Bellman-Ford ,Dijkstra, Dijkstra speedups
  • Dynamic programming:memoization, Fibonacci, Crazy Eights, guessing, longest common subsequence, parent pointers ,text justification, parenthesization, knapsack
  • pseudopolynomial time, Tetris training,piano fingering, structural DP (trees), vertex cover, dominating set, and beyond,Numerics,Beyond 6.006: follow-on classes, geometric folding algorithms

 

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