The document discusses divide and conquer algorithms and solving recurrences. It covers asymptotic notations, examples of divide and conquer including finding the largest number in a list, recurrence relations, and methods for solving recurrences including iteration, substitution, and recursion trees. The iteration method involves unfolding the recurrence into a summation. The recursion tree method visually depicts recursive calls in a tree to help solve the recurrence. Divide and conquer algorithms break problems into smaller subproblems, solve the subproblems recursively, and combine the solutions.
The document describes the quicksort algorithm. Quicksort works by:
1) Partitioning the array around a pivot element into two sub-arrays of less than or equal and greater than elements.
2) Recursively sorting the two sub-arrays.
3) Combining the now sorted sub-arrays.
In the average case, quicksort runs in O(n log n) time due to balanced partitions at each recursion level. However, in the worst case of an already sorted input, it runs in O(n^2) time due to highly unbalanced partitions. A randomized version of quicksort chooses pivots randomly to avoid worst case behavior.
This document discusses the merge sort algorithm for sorting a sequence of numbers. It begins by introducing the divide and conquer approach, which merge sort uses. It then provides an example of how merge sort works, dividing the sequence into halves, sorting the halves recursively, and then merging the sorted halves together. The document proceeds to provide pseudocode for the merge sort and merge algorithms. It analyzes the running time of merge sort using recursion trees, determining that it runs in O(n log n) time. Finally, it covers techniques for solving recurrence relations that arise in algorithms like divide and conquer approaches.
The document discusses minimum spanning trees (MST) and two algorithms for finding them: Prim's algorithm and Kruskal's algorithm. Prim's algorithm operates by building the MST one vertex at a time, starting from an arbitrary root vertex and at each step adding the cheapest connection to another vertex not yet included. Kruskal's algorithm finds the MST by sorting the edges by weight and sequentially adding edges that connect different components without creating cycles.
The document discusses different sorting algorithms including merge sort and quicksort. Merge sort has a divide and conquer approach where an array is divided into halves and the halves are merged back together in sorted order. This results in a runtime of O(n log n). Quicksort uses a partitioning approach, choosing a pivot element and partitioning the array into subarrays of elements less than or greater than the pivot. In the best case, this partitions the array in half at each step, resulting in a runtime of O(n log n). In the average case, the runtime is also O(n log n). In the worst case, the array is already sorted, resulting in unbalanced partitions and a quadratic runtime of O(n^2
Describes basic understanding of priority queues, their applications, methods, implementation with sorted/unsorted list, sorting applications with insertion sort and selection sort with their running times.
A presentation on prim's and kruskal's algorithmGaurav Kolekar
This slides are for a presentation on Prim's and Kruskal's algorithm. Where I have tried to explain how both the algorithms work, their similarities and their differences.
The document discusses the divide and conquer algorithm design technique. It begins by explaining the basic approach of divide and conquer which is to (1) divide the problem into subproblems, (2) conquer the subproblems by solving them recursively, and (3) combine the solutions to the subproblems into a solution for the original problem. It then provides merge sort as a specific example of a divide and conquer algorithm for sorting a sequence. It explains that merge sort divides the sequence in half recursively until individual elements remain, then combines the sorted halves back together to produce the fully sorted sequence.
The document describes the quicksort algorithm. Quicksort works by:
1) Partitioning the array around a pivot element into two sub-arrays of less than or equal and greater than elements.
2) Recursively sorting the two sub-arrays.
3) Combining the now sorted sub-arrays.
In the average case, quicksort runs in O(n log n) time due to balanced partitions at each recursion level. However, in the worst case of an already sorted input, it runs in O(n^2) time due to highly unbalanced partitions. A randomized version of quicksort chooses pivots randomly to avoid worst case behavior.
This document discusses the merge sort algorithm for sorting a sequence of numbers. It begins by introducing the divide and conquer approach, which merge sort uses. It then provides an example of how merge sort works, dividing the sequence into halves, sorting the halves recursively, and then merging the sorted halves together. The document proceeds to provide pseudocode for the merge sort and merge algorithms. It analyzes the running time of merge sort using recursion trees, determining that it runs in O(n log n) time. Finally, it covers techniques for solving recurrence relations that arise in algorithms like divide and conquer approaches.
The document discusses minimum spanning trees (MST) and two algorithms for finding them: Prim's algorithm and Kruskal's algorithm. Prim's algorithm operates by building the MST one vertex at a time, starting from an arbitrary root vertex and at each step adding the cheapest connection to another vertex not yet included. Kruskal's algorithm finds the MST by sorting the edges by weight and sequentially adding edges that connect different components without creating cycles.
The document discusses different sorting algorithms including merge sort and quicksort. Merge sort has a divide and conquer approach where an array is divided into halves and the halves are merged back together in sorted order. This results in a runtime of O(n log n). Quicksort uses a partitioning approach, choosing a pivot element and partitioning the array into subarrays of elements less than or greater than the pivot. In the best case, this partitions the array in half at each step, resulting in a runtime of O(n log n). In the average case, the runtime is also O(n log n). In the worst case, the array is already sorted, resulting in unbalanced partitions and a quadratic runtime of O(n^2
Describes basic understanding of priority queues, their applications, methods, implementation with sorted/unsorted list, sorting applications with insertion sort and selection sort with their running times.
A presentation on prim's and kruskal's algorithmGaurav Kolekar
This slides are for a presentation on Prim's and Kruskal's algorithm. Where I have tried to explain how both the algorithms work, their similarities and their differences.
The document discusses the divide and conquer algorithm design technique. It begins by explaining the basic approach of divide and conquer which is to (1) divide the problem into subproblems, (2) conquer the subproblems by solving them recursively, and (3) combine the solutions to the subproblems into a solution for the original problem. It then provides merge sort as a specific example of a divide and conquer algorithm for sorting a sequence. It explains that merge sort divides the sequence in half recursively until individual elements remain, then combines the sorted halves back together to produce the fully sorted sequence.
The document discusses graph traversal algorithms breadth-first search (BFS) and depth-first search (DFS). It provides examples of how BFS and DFS work, including pseudocode for algorithms. It also discusses applications of BFS such as finding shortest paths and detecting bipartitions. Applications of DFS include finding connected components and topological sorting.
This document contains a presentation on solving the coin change problem using greedy and dynamic programming algorithms. It introduces the coin change problem and provides an example. It then describes the greedy algorithm approach and how it works for some cases but fails to find an optimal solution in other cases when coin values are not uniform. The document next explains dynamic programming, its four step process, and how it can be applied to the coin change problem to always find an optimal solution using a bottom-up approach and storing results of subproblems to build the final solution.
The document discusses asymptotic notations that are used to describe the time complexity of algorithms. It introduces big O notation, which describes asymptotic upper bounds, big Omega notation for lower bounds, and big Theta notation for tight bounds. Common time complexities are described such as O(1) for constant time, O(log N) for logarithmic time, and O(N^2) for quadratic time. The notations allow analyzing how efficiently algorithms use resources like time and space as the input size increases.
Merge sort is a divide and conquer algorithm that divides an array into halves, recursively sorts the halves, and then merges the sorted halves back together. The key steps are:
1. Divide the array into equal halves until reaching base cases of arrays with one element.
2. Recursively sort the left and right halves by repeating the divide step.
3. Merge the sorted halves back into a single sorted array by comparing elements pairwise and copying the smaller element into the output array.
Merge sort has several advantages including running in O(n log n) time in all cases, accessing data sequentially with low random access needs, and being suitable for external sorting of large data sets that do not fit in memory
John von Neumann invented the merge sort algorithm in 1945. Merge sort follows the divide and conquer paradigm by dividing the unsorted list into halves, recursively sorting each half through merging, and then merging the sorted halves back into a single sorted list. The time complexity of merge sort is O(n log n) in all cases (best, average, worst) due to its divide and conquer approach, while its space complexity is O(n) to store the temporary merged list.
The document discusses divide and conquer algorithms. It describes divide and conquer as a design strategy that involves dividing a problem into smaller subproblems, solving the subproblems recursively, and combining the solutions. It provides examples of divide and conquer algorithms like merge sort, quicksort, and binary search. Merge sort works by recursively sorting halves of an array until it is fully sorted. Quicksort selects a pivot element and partitions the array into subarrays of smaller and larger elements, recursively sorting the subarrays. Binary search recursively searches half-intervals of a sorted array to find a target value.
The document discusses the divide-and-conquer algorithm design paradigm. It explains that a problem is divided into smaller subproblems, the subproblems are solved independently, and then the solutions are combined. Recurrence equations can be used to analyze the running time of divide-and-conquer algorithms. The document provides examples of solving recurrences using methods like the recursion tree method and the master theorem.
The document discusses the divide and conquer algorithm design paradigm. It begins by defining divide and conquer as recursively breaking down a problem into smaller sub-problems, solving the sub-problems, and then combining the solutions to solve the original problem. Some examples of problems that can be solved using divide and conquer include binary search, quicksort, merge sort, and the fast Fourier transform algorithm. The document then discusses control abstraction, efficiency analysis, and uses divide and conquer to provide algorithms for large integer multiplication and merge sort. It concludes by defining the convex hull problem and providing an example input and output.
The document discusses the knapsack problem and greedy algorithms. It defines the knapsack problem as an optimization problem where given constraints and an objective function, the goal is to find the feasible solution that maximizes or minimizes the objective. It describes the knapsack problem has having two versions: 0-1 where items are indivisible, and fractional where items can be divided. The fractional knapsack problem can be solved using a greedy approach by sorting items by value to weight ratio and filling the knapsack accordingly until full.
This document discusses stacks and queues as linear data structures. It defines stacks as last-in, first-out (LIFO) collections where the last item added is the first removed. Queues are first-in, first-out (FIFO) collections where the first item added is the first removed. Common stack and queue operations like push, pop, insert, and remove are presented along with algorithms and examples. Applications of stacks and queues in areas like expression evaluation, string reversal, and scheduling are also covered.
Divide and Conquer Algorithms - D&C forms a distinct algorithm design technique in computer science, wherein a problem is solved by repeatedly invoking the algorithm on smaller occurrences of the same problem. Binary search, merge sort, Euclid's algorithm can all be formulated as examples of divide and conquer algorithms. Strassen's algorithm and Nearest Neighbor algorithm are two other examples.
The document discusses optimal binary search trees (OBST) and describes the process of creating one. It begins by introducing OBST and noting that the method can minimize average number of comparisons in a successful search. It then shows the step-by-step process of calculating the costs for different partitions to arrive at the optimal binary search tree for a given sample dataset with keys and frequencies. The process involves calculating Catalan numbers for each partition and choosing the minimum cost at each step as the optimal is determined.
Mergesort is a divide and conquer algorithm that works as follows:
1) Recursively sort the left and right halves of the array.
2) Merge the two sorted halves into a new sorted array.
3) Repeat until the entire array is sorted.
It has superior time complexity of O(n log n) in all cases but requires O(n) additional space for the auxiliary array used during merging.
1. Asymptotic notation such as Big-O, Omega, and Theta are used to describe the running time of algorithms as the input size n approaches infinity, rather than giving the exact running time.
2. Big-O notation gives an upper bound and describes worst-case running time, Omega notation gives a lower bound and describes best-case running time, and Theta notation gives a tight bound where the worst and best cases are equal up to a constant.
3. Common examples of asymptotic running times include O(1) for constant time, O(log n) for logarithmic time, O(n) for linear time, and O(n^2) for quadratic time.
- Recurrences describe functions in terms of their values on smaller inputs and arise when algorithms contain recursive calls to themselves.
- To analyze the running time of recursive algorithms, the recurrence must be solved to find an explicit formula or bound the expression in terms of n.
- Examples of recurrences and their solutions are given, including binary search (O(log n)), dividing the input in half at each step (O(n)), and dividing the input in half but examining all items (O(n)).
- Methods for solving recurrences include iteration, substitution, and using recursion trees to "guess" the solution.
Given two integer arrays val[0...n-1] and wt[0...n-1] that represents values and weights associated with n items respectively. Find out the maximum value subset of val[] such that sum of the weights of this subset is smaller than or equal to knapsack capacity W. Here the BRANCH AND BOUND ALGORITHM is discussed .
The document discusses time and space complexity analysis of algorithms. Time complexity measures the number of steps to solve a problem based on input size, with common orders being O(log n), O(n), O(n log n), O(n^2). Space complexity measures memory usage, which can be reused unlike time. Big O notation describes asymptotic growth rates to compare algorithm efficiencies, with constant O(1) being best and exponential O(c^n) being worst.
An array is a data structure that stores fixed number of items of the same type. It allows fast access of elements using indices. Basic array operations include traversing elements, inserting/deleting elements, searching for elements, and updating elements. Arrays are zero-indexed and elements are accessed via their index.
Binary search is an algorithm for finding an element in a sorted array. It works by recursively checking the middle element, dividing the array in half, and searching only one subarray. The time complexity is O(log n) as the array is divided in half in each step.
The document discusses graph traversal algorithms breadth-first search (BFS) and depth-first search (DFS). It provides examples of how BFS and DFS work, including pseudocode for algorithms. It also discusses applications of BFS such as finding shortest paths and detecting bipartitions. Applications of DFS include finding connected components and topological sorting.
This document contains a presentation on solving the coin change problem using greedy and dynamic programming algorithms. It introduces the coin change problem and provides an example. It then describes the greedy algorithm approach and how it works for some cases but fails to find an optimal solution in other cases when coin values are not uniform. The document next explains dynamic programming, its four step process, and how it can be applied to the coin change problem to always find an optimal solution using a bottom-up approach and storing results of subproblems to build the final solution.
The document discusses asymptotic notations that are used to describe the time complexity of algorithms. It introduces big O notation, which describes asymptotic upper bounds, big Omega notation for lower bounds, and big Theta notation for tight bounds. Common time complexities are described such as O(1) for constant time, O(log N) for logarithmic time, and O(N^2) for quadratic time. The notations allow analyzing how efficiently algorithms use resources like time and space as the input size increases.
Merge sort is a divide and conquer algorithm that divides an array into halves, recursively sorts the halves, and then merges the sorted halves back together. The key steps are:
1. Divide the array into equal halves until reaching base cases of arrays with one element.
2. Recursively sort the left and right halves by repeating the divide step.
3. Merge the sorted halves back into a single sorted array by comparing elements pairwise and copying the smaller element into the output array.
Merge sort has several advantages including running in O(n log n) time in all cases, accessing data sequentially with low random access needs, and being suitable for external sorting of large data sets that do not fit in memory
John von Neumann invented the merge sort algorithm in 1945. Merge sort follows the divide and conquer paradigm by dividing the unsorted list into halves, recursively sorting each half through merging, and then merging the sorted halves back into a single sorted list. The time complexity of merge sort is O(n log n) in all cases (best, average, worst) due to its divide and conquer approach, while its space complexity is O(n) to store the temporary merged list.
The document discusses divide and conquer algorithms. It describes divide and conquer as a design strategy that involves dividing a problem into smaller subproblems, solving the subproblems recursively, and combining the solutions. It provides examples of divide and conquer algorithms like merge sort, quicksort, and binary search. Merge sort works by recursively sorting halves of an array until it is fully sorted. Quicksort selects a pivot element and partitions the array into subarrays of smaller and larger elements, recursively sorting the subarrays. Binary search recursively searches half-intervals of a sorted array to find a target value.
The document discusses the divide-and-conquer algorithm design paradigm. It explains that a problem is divided into smaller subproblems, the subproblems are solved independently, and then the solutions are combined. Recurrence equations can be used to analyze the running time of divide-and-conquer algorithms. The document provides examples of solving recurrences using methods like the recursion tree method and the master theorem.
The document discusses the divide and conquer algorithm design paradigm. It begins by defining divide and conquer as recursively breaking down a problem into smaller sub-problems, solving the sub-problems, and then combining the solutions to solve the original problem. Some examples of problems that can be solved using divide and conquer include binary search, quicksort, merge sort, and the fast Fourier transform algorithm. The document then discusses control abstraction, efficiency analysis, and uses divide and conquer to provide algorithms for large integer multiplication and merge sort. It concludes by defining the convex hull problem and providing an example input and output.
The document discusses the knapsack problem and greedy algorithms. It defines the knapsack problem as an optimization problem where given constraints and an objective function, the goal is to find the feasible solution that maximizes or minimizes the objective. It describes the knapsack problem has having two versions: 0-1 where items are indivisible, and fractional where items can be divided. The fractional knapsack problem can be solved using a greedy approach by sorting items by value to weight ratio and filling the knapsack accordingly until full.
This document discusses stacks and queues as linear data structures. It defines stacks as last-in, first-out (LIFO) collections where the last item added is the first removed. Queues are first-in, first-out (FIFO) collections where the first item added is the first removed. Common stack and queue operations like push, pop, insert, and remove are presented along with algorithms and examples. Applications of stacks and queues in areas like expression evaluation, string reversal, and scheduling are also covered.
Divide and Conquer Algorithms - D&C forms a distinct algorithm design technique in computer science, wherein a problem is solved by repeatedly invoking the algorithm on smaller occurrences of the same problem. Binary search, merge sort, Euclid's algorithm can all be formulated as examples of divide and conquer algorithms. Strassen's algorithm and Nearest Neighbor algorithm are two other examples.
The document discusses optimal binary search trees (OBST) and describes the process of creating one. It begins by introducing OBST and noting that the method can minimize average number of comparisons in a successful search. It then shows the step-by-step process of calculating the costs for different partitions to arrive at the optimal binary search tree for a given sample dataset with keys and frequencies. The process involves calculating Catalan numbers for each partition and choosing the minimum cost at each step as the optimal is determined.
Mergesort is a divide and conquer algorithm that works as follows:
1) Recursively sort the left and right halves of the array.
2) Merge the two sorted halves into a new sorted array.
3) Repeat until the entire array is sorted.
It has superior time complexity of O(n log n) in all cases but requires O(n) additional space for the auxiliary array used during merging.
1. Asymptotic notation such as Big-O, Omega, and Theta are used to describe the running time of algorithms as the input size n approaches infinity, rather than giving the exact running time.
2. Big-O notation gives an upper bound and describes worst-case running time, Omega notation gives a lower bound and describes best-case running time, and Theta notation gives a tight bound where the worst and best cases are equal up to a constant.
3. Common examples of asymptotic running times include O(1) for constant time, O(log n) for logarithmic time, O(n) for linear time, and O(n^2) for quadratic time.
- Recurrences describe functions in terms of their values on smaller inputs and arise when algorithms contain recursive calls to themselves.
- To analyze the running time of recursive algorithms, the recurrence must be solved to find an explicit formula or bound the expression in terms of n.
- Examples of recurrences and their solutions are given, including binary search (O(log n)), dividing the input in half at each step (O(n)), and dividing the input in half but examining all items (O(n)).
- Methods for solving recurrences include iteration, substitution, and using recursion trees to "guess" the solution.
Given two integer arrays val[0...n-1] and wt[0...n-1] that represents values and weights associated with n items respectively. Find out the maximum value subset of val[] such that sum of the weights of this subset is smaller than or equal to knapsack capacity W. Here the BRANCH AND BOUND ALGORITHM is discussed .
The document discusses time and space complexity analysis of algorithms. Time complexity measures the number of steps to solve a problem based on input size, with common orders being O(log n), O(n), O(n log n), O(n^2). Space complexity measures memory usage, which can be reused unlike time. Big O notation describes asymptotic growth rates to compare algorithm efficiencies, with constant O(1) being best and exponential O(c^n) being worst.
An array is a data structure that stores fixed number of items of the same type. It allows fast access of elements using indices. Basic array operations include traversing elements, inserting/deleting elements, searching for elements, and updating elements. Arrays are zero-indexed and elements are accessed via their index.
Binary search is an algorithm for finding an element in a sorted array. It works by recursively checking the middle element, dividing the array in half, and searching only one subarray. The time complexity is O(log n) as the array is divided in half in each step.
In divide and conquer, we will see
1.- Why Divide and Conquer?
2.- The Gauss Trick
3.- Recursion is the base of Divide and Conquer
4.- Induction to prove the correctness of algorithms
5.- The use of the Asymptotic notation
6.- Why the worst case?
7.- Some tricks to calculate upper and lower bounds for recursions:
- The substitution method
- The tree method
- The Master Theorem
This document discusses algorithms and their analysis. It defines an algorithm as a step-by-step procedure to solve a problem or calculate a quantity. Algorithm analysis involves evaluating memory usage and time complexity. Asymptotics, such as Big-O notation, are used to formalize the growth rates of algorithms. Common sorting algorithms like insertion sort and quicksort are analyzed using recurrence relations to determine their time complexities as O(n^2) and O(nlogn), respectively.
This document discusses using recurrence relations to model problems involving counting techniques. It provides examples of modeling problems related to bacteria population growth, rabbit population growth, the Tower of Hanoi puzzle, and valid codeword enumeration. For each problem, it defines the recurrence relation and initial conditions, derives a closed-form solution, and proves its correctness using mathematical induction. Recurrence relations provide a way to define sequences and solve problems recursively by relating terms to previous terms in the sequence.
The document discusses recurrence relations and methods for solving them. It defines a recurrence relation as an equation that expresses the terms of a sequence in terms of previous terms. It provides examples of homogeneous and non-homogeneous recurrence relations. For homogeneous relations, it describes guessing a solution of the form T(n)=x^n and finding the characteristic equation. For non-homogeneous relations, it explains adding the non-recursive term to the characteristic equation. It then works through examples of solving both homogeneous and non-homogeneous recurrence relations.
This document discusses various divides that exist in society such as between the rich and poor, different races, and gangs. It notes that divisions are created among people and provides an example where the ratio of poor to rich changed from 99 to 1 initially to 4 to 1, showing that divisions disproportionately benefit the rich.
How do you increase the effectiveness of committees? Use good governance practices, your vision, board evaluations, bylaws, and your strategic plan to identify the need for committees, then set your purpose and goals to attract the right people and become more accountable.
Chapter 2 - Protocol Architecture, TCP/IP, and Internet-Based Applications 9eadpeer
1. Protocol architectures break communication tasks into modular layers to allow for independent development and changes without affecting other layers. TCP/IP and OSI are examples of protocol architectures.
2. The TCP/IP protocol architecture has four layers - physical, network access, internet, and transport. Example protocols are Ethernet, IP, and TCP.
3. The OSI reference model standardized a seven-layer architecture to provide a framework for protocol standardization. Each layer provides services to the layer above and relies on the layer below.
A recurrence relation defines a sequence based on a rule that gives the next term as a function of previous terms. There are three main methods to solve recurrence relations: 1) repeated substitution, 2) recursion trees, and 3) the master method. Repeated substitution repeatedly substitutes the recursive function into itself until it is reduced to a non-recursive form. Recursion trees show the successive expansions of a recurrence using a tree structure. The master method provides rules to determine the time complexity of divide and conquer recurrences.
Algorithm Design and Complexity - Course 3Traian Rebedea
The document provides an overview of recursive algorithms and complexity analysis. It discusses recursive algorithms, divide and conquer design technique, and several examples of recursive algorithms including Towers of Hanoi, Merge Sort, and Quick Sort. For recursive algorithms, it explains how to analyze their running time using recurrence relations. It then covers four methods for solving recurrence relations: iteration, recursion trees, substitution method, and master theorem. The substitution method and master theorem are described as the most rigorous mathematical approaches.
This document discusses recurrence relations and their use in defining sequences. It introduces key concepts like recurrence relations, initial conditions, explicit formulas, and solving recurrence relations using techniques like backtracking or finding the characteristic equation. As examples, it examines the Fibonacci sequence and linear homogeneous recurrence relations of varying degrees.
This document summarizes key aspects of protocol architecture, TCP/IP, and internet-based applications. It discusses the need for a protocol architecture to break communication tasks into layers. It then describes the layered TCP/IP protocol architecture and its components, including the physical, network access, internet, transport, and application layers. It also summarizes TCP and IP addressing requirements and operation, as well as standard TCP/IP applications like SMTP, FTP, and Telnet. Finally, it contrasts traditional data-based applications with newer multimedia applications involving large amounts of real-time audio and video data.
The document discusses recurrence relations and their applications. It begins by defining a recurrence relation as an equation that expresses the terms of a sequence in terms of previous terms. It provides examples of recurrence relations and their solutions. It then discusses solving linear homogeneous recurrence relations with constant coefficients by finding the characteristic roots and obtaining an explicit formula. Applications discussed include financial recurrence relations, the partition function, binary search, and the Fibonacci numbers. It concludes by discussing the case when the characteristic equation has a single root.
Dynamic programming is an algorithm design technique that solves problems by breaking them down into smaller overlapping subproblems and storing the results of already solved subproblems, rather than recomputing them. It is applicable to problems exhibiting optimal substructure and overlapping subproblems. The key steps are to define the optimal substructure, recursively define the optimal solution value, compute values bottom-up, and optionally reconstruct the optimal solution. Common examples that can be solved with dynamic programming include knapsack, shortest paths, matrix chain multiplication, and longest common subsequence.
Divide and Conquer - Part II - Quickselect and Closest Pair of PointsAmrinder Arora
This document discusses divide and conquer algorithms. It covers the closest pair of points problem, which can be solved in O(n log n) time using a divide and conquer approach. It also discusses selection algorithms like quickselect that can find the median or kth element of an unsorted array in linear time O(n) on average. The document provides pseudocode for these algorithms and analyzes their time complexity using recurrence relations. It also provides an overview of topics like mergesort, quicksort, and solving recurrence relations that were covered in previous lectures.
The document discusses the analysis of algorithms. It begins by defining an algorithm and describing different types. It then covers analyzing algorithms in terms of correctness, time efficiency, space efficiency, and optimality through theoretical and empirical analysis. The document discusses analyzing time efficiency by determining the number of repetitions of basic operations as a function of input size. It provides examples of input size, basic operations, and formulas for counting operations. It also covers analyzing best, worst, and average cases and establishes asymptotic efficiency classes. The document then analyzes several examples of non-recursive and recursive algorithms.
Query Processing and Optimisation - Lecture 10 - Introduction to Databases (1...Beat Signer
This document discusses query processing and optimization in databases. It covers the basic steps of query processing including parsing, optimization, and evaluation. It also describes different algorithms for query operations like selection, join, and sorting that are used to process queries efficiently. The goals of query optimization are to select the most efficient query execution plan based on the given data and minimize the number of disk accesses.
The document summarizes the OSI model and TCP/IP protocol suite. It describes the seven layers of the OSI model and their functions. It then explains that the TCP/IP protocol suite has five layers that correspond to the bottom four layers of the OSI model, with the top three OSI layers combined into a single application layer in TCP/IP. It also discusses the different types of addresses used in each layer, including physical, logical, and port addresses.
The document discusses techniques used by a database management system (DBMS) to process, optimize, and execute high-level queries. It describes the phases of query processing which include syntax checking, translating the SQL query into an algebraic expression, optimization to choose an efficient execution plan, and running the optimized plan. Query optimization aims to minimize resources like disk I/O and CPU time by selecting the best execution strategy. Techniques for optimization include heuristic rules, cost-based methods, and semantic query optimization using constraints.
This document discusses analyzing recursive algorithms and forming recurrence relations. It provides examples of writing recurrence relations for recursive functions. The key steps are:
1) Identify the base case(s) where recursive calls stop.
2) Express the work done and size of subproblems at each recursive call.
3) Derive the recurrence relation relating the function at different inputs sizes.
The recurrence relation captures the work at each level of recursion and sums the costs to determine overall runtime. Analyzing recurrences helps understand the asymptotic complexity of recursive algorithms.
This document discusses recurrence relations and methods for solving recurrences. It introduces recurrence relations and examples. It covers the substitution method, iteration method, and Master Theorem for solving recurrences. The Master Theorem is a technique for solving divide-and-conquer recurrences to determine asymptotic tight bounds. Examples are provided to demonstrate applying these techniques.
In computer science, divide and conquer (D&C) is an algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
In computer science, merge sort (also commonly spelled mergesort) is an O(n log n) comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the implementation preserves the input order of equal elements in the sorted output. Mergesort is a divide and conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up mergesort appeared in a report by Goldstine and Neumann as early as 1948.
The document discusses the divide and conquer algorithm design technique. It begins by defining divide and conquer as breaking a problem down into smaller subproblems, solving the subproblems, and then combining the solutions to solve the original problem. It then provides examples of applying divide and conquer to problems like matrix multiplication and finding the maximum subarray. The document also discusses analyzing divide and conquer recurrences using methods like recursion trees and the master theorem.
Here are the key steps:
1. Guess the solution: T(n) = O(n log n)
2. Set the induction goal: T(n) ≤ c n log n for some c > 0 and n ≥ n0
3. Apply the induction hypothesis: T(n/2) ≤ c (n/2) log(n/2)
4. Substitute into the recurrence: T(n) = 2T(n/2) + n ≤ 2c(n/2)log(n/2) + n = cn log n
5. Simplify and show it meets the induction goal.
Therefore, by mathematical induction, the solution T(n) =
The document discusses recursion, which is a method for solving problems by breaking them down into smaller subproblems. It provides examples of recursive algorithms like summing a list of numbers, calculating factorials, and the Fibonacci sequence. It also covers recursive algorithm components like the base case and recursive call. Methods for analyzing recursive algorithms' running times are presented, including iteration, recursion trees, and the master theorem.
The document discusses recursion, including:
1) Recursion involves breaking a problem down into smaller subproblems until a base case is reached, then building up the solution to the overall problem from the solutions to the subproblems.
2) A recursive function is one that calls itself, with each call typically moving closer to a base case where the problem can be solved without recursion.
3) Recursion can be linear, involving one recursive call, or binary, involving two recursive calls to solve similar subproblems.
The document discusses algorithms and their analysis. It defines an algorithm as a step-by-step procedure for solving a problem and analyzes the time complexity of various algorithms. Key points made include:
1) Algorithms are analyzed based on how many times their basic operation is performed as a function of input size n.
2) Common time complexities include O(n) for sequential search, O(n^3) for matrix multiplication, and O(log n) for binary search.
3) Recursive algorithms like fibonacci are inefficient and iterative versions improve performance by storing previously computed values.
This document discusses the divide and conquer algorithm called merge sort. It begins by explaining the general divide and conquer approach of dividing a problem into subproblems, solving the subproblems recursively, and then combining the solutions. It then provides an example of how merge sort uses this approach to sort a sequence. It walks through the recursive merge sort algorithm on a sample input. The document explains the merge procedure used to combine the sorted subproblems and proves its correctness. It analyzes the running time of merge sort using recursion trees and determines it is O(n log n). Finally, it introduces recurrence relations and methods like substitution, recursion trees, and the master theorem for solving recurrences.
The document discusses using recursion trees to analyze divide and conquer algorithms. It provides an example of using a recursion tree to solve the recurrence relation for merge sort. The recursion tree shows the successive divisions of the problem into smaller subproblems until the base case is reached. Each node represents the cost of a subproblem, and the total cost is calculated by summing the costs at each level of the tree. For merge sort, the tree has log n levels, with a total cost of O(n log n).
The document discusses different methods for analyzing recursive functions, including:
1. The recursion-tree method, which represents each subproblem as a node with costs summed at each level and total. This is suitable for divide-and-conquer recurrences.
2. An example of using the recursion-tree method to solve T(n)=3T(n/4)+Θ(n^2), showing the tree has height log4n, Θ(nlog43) leaf nodes each costing T(1), and total cost O(n^2).
3. Another example of T(n)=T(n/3)+T(2n/3)+O(n
The document discusses the dynamic programming approach to solving the Fibonacci numbers problem and the rod cutting problem. It explains that dynamic programming formulations first express the problem recursively but then optimize it by storing results of subproblems to avoid recomputing them. This is done either through a top-down recursive approach with memoization or a bottom-up approach by filling a table with solutions to subproblems of increasing size. The document also introduces the matrix chain multiplication problem and how it can be optimized through dynamic programming by considering overlapping subproblems.
1) The document describes the divide-and-conquer algorithm design paradigm. It can be applied to problems where the input can be divided into smaller subproblems, the subproblems can be solved independently, and the solutions combined to solve the original problem.
2) Binary search is provided as an example divide-and-conquer algorithm. It works by recursively dividing the search space in half and only searching the subspace containing the target value.
3) Finding the maximum and minimum elements in an array is also solved using divide-and-conquer. The array is divided into two halves, the max/min found for each subarray, and the overall max/min determined by comparing the subsolutions.
1) The document describes the divide-and-conquer algorithm design paradigm. It splits problems into smaller subproblems, solves the subproblems recursively, and then combines the solutions to solve the original problem.
2) Binary search is provided as an example algorithm that uses divide-and-conquer. It divides the search space in half at each step to quickly determine if an element is present.
3) Finding the maximum and minimum elements in an array is another problem solved using divide-and-conquer. It recursively finds the max and min of halves of the array and combines the results.
This document discusses the divide and conquer algorithm design strategy and provides an analysis of the merge sort algorithm as an example. It begins by explaining the divide and conquer strategy of dividing a problem into smaller subproblems, solving those subproblems recursively, and combining the solutions. It then provides pseudocode and explanations for the merge sort algorithm, which divides an array in half, recursively sorts the halves, and then merges the sorted halves back together. It analyzes the time complexity of merge sort as Θ(n log n), proving it is more efficient than insertion sort.
This document discusses recursion, which is a fundamental concept in computer science and mathematics where a function calls itself. It provides examples of common recursive definitions like factorials and Fibonacci numbers. It explains how recursive programs work by dividing the problem into smaller subproblems until a base case is reached. The document also introduces the "divide and conquer" paradigm where most recursive programs make two recursive calls on halves of the input to solve problems more efficiently than iterative approaches.
This document discusses the merge sort algorithm for sorting a sequence of numbers. It begins by introducing the divide and conquer approach and defining the sorting problem. It then describes the three steps of merge sort as divide, conquer, and combine. It provides pseudocode for the merge sort and merge algorithms. Finally, it analyzes the running time of merge sort, showing that it runs in O(n log n) time using the recursion tree method.
Online train ticket booking system project.pdfKamal Acharya
Rail transport is one of the important modes of transport in India. Now a days we
see that there are railways that are present for the long as well as short distance
travelling which makes the life of the people easier. When compared to other
means of transport, a railway is the cheapest means of transport. The maintenance
of the railway database also plays a major role in the smooth running of this
system. The Online Train Ticket Management System will help in reserving the
tickets of the railways to travel from a particular source to the destination.
Better Builder Magazine brings together premium product manufactures and leading builders to create better differentiated homes and buildings that use less energy, save water and reduce our impact on the environment. The magazine is published four times a year.
Sri Guru Hargobind Ji - Bandi Chor Guru.pdfBalvir Singh
Sri Guru Hargobind Ji (19 June 1595 - 3 March 1644) is revered as the Sixth Nanak.
• On 25 May 1606 Guru Arjan nominated his son Sri Hargobind Ji as his successor. Shortly
afterwards, Guru Arjan was arrested, tortured and killed by order of the Mogul Emperor
Jahangir.
• Guru Hargobind's succession ceremony took place on 24 June 1606. He was barely
eleven years old when he became 6th Guru.
• As ordered by Guru Arjan Dev Ji, he put on two swords, one indicated his spiritual
authority (PIRI) and the other, his temporal authority (MIRI). He thus for the first time
initiated military tradition in the Sikh faith to resist religious persecution, protect
people’s freedom and independence to practice religion by choice. He transformed
Sikhs to be Saints and Soldier.
• He had a long tenure as Guru, lasting 37 years, 9 months and 3 days
Particle Swarm Optimization–Long Short-Term Memory based Channel Estimation w...IJCNCJournal
Paper Title
Particle Swarm Optimization–Long Short-Term Memory based Channel Estimation with Hybrid Beam Forming Power Transfer in WSN-IoT Applications
Authors
Reginald Jude Sixtus J and Tamilarasi Muthu, Puducherry Technological University, India
Abstract
Non-Orthogonal Multiple Access (NOMA) helps to overcome various difficulties in future technology wireless communications. NOMA, when utilized with millimeter wave multiple-input multiple-output (MIMO) systems, channel estimation becomes extremely difficult. For reaping the benefits of the NOMA and mm-Wave combination, effective channel estimation is required. In this paper, we propose an enhanced particle swarm optimization based long short-term memory estimator network (PSOLSTMEstNet), which is a neural network model that can be employed to forecast the bandwidth required in the mm-Wave MIMO network. The prime advantage of the LSTM is that it has the capability of dynamically adapting to the functioning pattern of fluctuating channel state. The LSTM stage with adaptive coding and modulation enhances the BER.PSO algorithm is employed to optimize input weights of LSTM network. The modified algorithm splits the power by channel condition of every single user. Participants will be first sorted into distinct groups depending upon respective channel conditions, using a hybrid beamforming approach. The network characteristics are fine-estimated using PSO-LSTMEstNet after a rough approximation of channels parameters derived from the received data.
Keywords
Signal to Noise Ratio (SNR), Bit Error Rate (BER), mm-Wave, MIMO, NOMA, deep learning, optimization.
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Data Communication and Computer Networks Management System Project Report.pdfKamal Acharya
Networking is a telecommunications network that allows computers to exchange data. In
computer networks, networked computing devices pass data to each other along data
connections. Data is transferred in the form of packets. The connections between nodes are
established using either cable media or wireless media.
5. Problems – Big Oh, Big Omega, Big Thetha
• First two functions are linear and hence have a lower order
of growth than g(n) = n2, while the last one is quadratic and
hence has the same order of growth as n2
• Functions n3 and 0.00001n3 are both cubic and hence have a
higher order of growth than n2, and so has the fourth-
degree polynomial n4 + n + 1
6. Problems – Big Oh, Big Omega, Big Thetha
• Ω(g(n)), stands for the set of all functions with a higher or
same order of growth as g(n) (to within a constant multiple,
as n goes to infinity).
• Θ(g(n)) is the set of all functions that have the same order of
growth as g(n) (to within a constant multiple, as n goes to
infinity). Every quadratic function an2 + bn + c with a > 0 is in
Θ(n2).
8. Divide-and-Conquer
• The most-well known algorithm design strategy:
1. Divide instance of problem into two or more smaller
instances
2. Solve smaller instances recursively
(Base case: If the sub-problem sizes are small enough, solve
the sub-problem in a straight forward or direct manner).
3. Obtain solution to original (larger) instance by combining
these solutions
• Type of recurrence relation
11. Example
Algorithm : Largest Number
Input : A non-empty list of numbers L
Output : The largest number in the list L
Comment : Divide and Conquer
If L.size == 1 then
return L.front
Largest 1 <- LargestNumber (L.front .. L.mid)
Largest 2 <- LargestNumber (L.mid … L.back)
If (Largest 1 > Largest 2) then
Largest <- Largest 1
Else
Largest <- Largest 2
Return Largest
12. Recurrence Relation
• Any problem can be solved either by writing recursive
algorithm or by writing non-recursive algorithm.
• A recursive algorithm is one which makes a recursive call to
itself with smaller inputs.
• We often use a recurrence relation to describe the running
time of a recursive algorithm.
• Recurrence relations often arise in calculating the time and
space complexity of algorithms.
13. Recurrence Relations contd..
• A recurrence relation is an equation or inequality that
describes a function in terms of its value on smaller inputs or
as a function of preceding (or lower) terms.
1. Base step:
– 1 or more constant values to terminate recurrence.
– Initial conditions or base conditions.
2. Recursive steps:
– To find new terms from the existing (preceding) terms.
– The recurrence compute next sequence from the k
preceding values .
– Recurrence relation (or recursive formula).
– This formula refers to itself, and the argument of the
formula must be on smaller values (close to the base
value).
14. Recurrence Formula :
Ex 1 Fibonacci Sequence
• Recurrence has one or more initial conditions and a
recursive formula, known as recurrence relation.
• Fibonacci sequence f0,f1,f2…. can be defined by the
recurrence relation
• (Base Step)
– The given recurrence says that if n=0 then f0=1 and if n=1
then f1=1.
– These two conditions (or values) where recursion does
not call itself is called a initial conditions (or Base
conditions).
15. Ex : Fibonacci Sequence contd..
• (Recursive step): This step is used to find new
terms f2,f3….from the existing (preceding)
terms, by using the formula
• This formula says that “by adding two
previous sequence (or term) we can get the
next term”.
17. Recurrence Relation for
Factorial Computation
• M(n)
– denoted the number of multiplication required to
execute the n!
• Initial condition
– M(1) = 0 ; BASE Step
• n > 1
– Performs 1 Multiplication + FACT recursively called
with input n-1
20. Example 3
• Let T(n) denotes recurrence relation - number
of times the statement x=x+1 is executed in
the algorithm
x+1 to be executed T(n/2) additional times
21. Example 3
Performs TWO recursive
calls each with the
parameter at line 4, and
some constant number of
basic operations
22. 22
Recurrences and Running Time
• An equation or inequality that describes a function in terms of
its value on smaller inputs.
T(n) = T(n-1) + n
• Recurrences arise when an algorithm contains recursive calls to
itself
• What is the actual running time of the algorithm?
• Need to solve the recurrence
– Find an explicit formula of the expression
– Bound the recurrence by an expression that involves n
23. 23
Recurrent Algorithms - BINARY-SEARCH
• for an ordered array A, finds if x is in the array A[lo…hi]
Alg.: BINARY-SEARCH (A, lo, hi, x)
if (lo > hi)
return FALSE
mid (lo+hi)/2
if x == A[mid]
return TRUE
if ( x < A[mid] )
BINARY-SEARCH (A, lo, mid-1, x)
if ( x > A[mid] )
BINARY-SEARCH (A, mid+1, hi, x)
12111097532
1 2 3 4 5 6 7 8
mid
lo hi
24. 24
Example
• A[8] = {1, 2, 3, 4, 5, 7, 9, 11}
– lo = 1 hi = 8 x = 7
mid = 4, lo = 5, hi = 8
mid = 6, A[mid] = x Found!119754321
119754321
1 2 3 4 5 6 7 8
8765
25. 25
Another Example
• A[8] = {1, 2, 3, 4, 5, 7, 9, 11}
– lo = 1 hi = 8 x = 6
mid = 4, lo = 5, hi = 8
mid = 6, A[6] = 7, lo = 5, hi = 5119754321
119754321
1 2 3 4 5 6 7 8
119754321 mid = 5, A[5] = 5, lo = 6, hi = 5
NOT FOUND!
119754321
low high
low
lowhigh
high
26. 26
Analysis of BINARY-SEARCH
Alg.: BINARY-SEARCH (A, lo, hi, x)
if (lo > hi)
return FALSE
mid (lo+hi)/2
if x = A[mid]
return TRUE
if ( x < A[mid] )
BINARY-SEARCH (A, lo, mid-1, x)
if ( x > A[mid] )
BINARY-SEARCH (A, mid+1, hi, x)
• T(n) = c +
– T(n) – running time for an array of size n
constant time: c2
same problem of size n/2
same problem of size n/2
constant time: c1
constant time: c3
T(n/2)
27. 27
Methods for Solving Recurrences
• Iteration method
–(unrolling and summing)
• Substitution method
• Recursion tree method
• Master method
28. • Iteration Method :
– Converts the recurrence into a summation and then relies
on techniques for bounding summations to solve the
recurrence.
• Substitution Method :
– Guess a asymptotic bound and then use mathematical
induction to prove our guess correct.
• Recursive Tree Method :
– Graphical depiction of the entire set of recursive
invocations to obtain guess and verify by substitution
method.
• Master Method :
– Cookbook method for determining asymptotic solutions to
recurrences of a specific form.
Method of Solving Recurrences
29. 29
The Iteration Method
• Convert the recurrence into a summation and
try to bound it using known series
– Iterate the recurrence until the initial condition is
reached.
– Use back-substitution to express the recurrence in
terms of n and the initial (boundary) condition.
34. Binary Search – Running Time
T(n) = c + T(n/2)
T(n) = c + T(n/2)
= c + c + T(n/4)
= c + c + c + T(n/8)
Assume n = 2k
T(n) = c + c + … + c + T(1) = c log2 n + T(1)
= O(log n)k times
41. Example Recurrences
• T(n) = T(n-1) + n
– Recursive algorithm that loops through the input to eliminate one item
• T(n) = T(n/2) + c
– Recursive algorithm that halves the input in one step
• T(n) = T(n/2) + n
– Recursive algorithm that halves the input but must examine every item in
the input
• T(n) = 2T(n/2) + 1
– Recursive algorithm that splits the input into 2 halves and does a constant
amount of other work
• T(n) = T(n/3) + T(2n/3) + n
44. Divide and Conquer – Recurrence form
T(n) – running time of problem of size n.
If the problem size is small enough (say, n ≤ c for some constant c), we have a base case.
The brute-force (or direct) solution takes constant time: Θ(1)
Divide into “a” sub-problems, each 1/b of the size of the original problem of size n.
Each sub-problem of size n/b takes time T(n/b) to solve
Total time to solve “a” sub-problems = spend aT(n/b)
D(n) is the cost(or time) of dividing the problem of size n.
C(n) is the cost (or time) to combine the sub-solutions.
45. Iteration Method
Unroll (or substitute) the given recurrence back to itself until a
regular pattern is obtained (or series).
Steps to solve any recurrence:
1. Expand the recurrence
2. Express the expansion as a summation by plugging the recurrence
back into itself until you see a pattern.
3. Evaluate the summation by using the arithmetic or geometric
summation formulae
46. Recursion Tree Method
A convenient way to visualize what
happens when a recurrence is iterated.
Pictorial representation of how recurrences
is divided till boundary condition
Used to solve a recurrence of the form
47. Steps for solving a recurrence using recursion Tree:
Step1: Make a recursion tree for a
given recurrence as follow:
a) Put the value of f(n) at root node
of a tree and make “a” no of child
node of this root value f(n).
49. Steps for solving a recurrence using recursion Tree:
b) Find the value of T(n/b)
50. c) Expand a tree one more level (i.e. up to (at
least) 2 levels)
Steps for solving a recurrence using recursion Tree:
51. Recursive Tree method
Step2: (a) Find per level cost of a tree
Per level cost = Sum of the cost of each node at that level
Ex : Per level cost at level 1 = Row Sum
Total (final) cost of the tree = Sum of costs of all
these levels. – Column Sum
52. Example 1
• Solve recurrence T(n) = 2 T(n/2) + n using
recursive tree method
53. Example 1
• Solve recurrence T(n) = 2 T(n/2) + n using
recursive tree method
54.
55.
56. Per level cost = Sum of cost at each level = Row sum
Ex: Depth 2
Total cost is the sum of the costs of all levels (called Column
sum), which gives the solution of a given Recurrence
57. To find Total number of terms -> Height of the tree
k represents the height of tree = log2n
58. Example 2
• Solve recurrence
using recursive tree method
We always omit floor & ceiling function while
solving recurrence. Thus given recurrence can
be written
59.
60.
61. In this way, you can extend a tree up to Boundary condition
(when problem size becomes 1)
66. Example 3
• To move n disks (n > 1) from peg A to C
– Move (n-1) disks recursively from peg A to peg B
using peg C as auxillary = M(n-1) moves
– Move the nth disk directly (last) from peg A to peg
C = 1 Move
– Move (n-1) disk recursively from peg B to peg C
using peg A as auxillary = M(n-1) moves
67. Recurrence Relation for the Towers of
Hanoi
Given: T(1) = 1
T(n) = 2 T( n-1 ) +1
N No.Moves
1 1
2 3
3 7
4 15
5 31
79. Quick Sort
Divide:
• A [p. . r] is partitioned (rearranged) into A [p..q-1] and A [q+1,..r],
• Each element in the left subarray A[p…q-1] is ≤ A [q] and
• A[q] is ≤ each element in the right subarray A[q+1…r]
• PARTITION procedure (Divide Step); returns the index q, where
the array gets partitioned.
Conquer:
• These two subarray A [p…q-1] and A [q+1..r] are sorted by
recursive calls to QUICKSORT.
Combine:
• Since the subarrays are sorted in place, so there is no need to
combine the subarrays.
82. Pseudo Code of Quick Sort
PARTITION (A, p, r) {
x ← A[r] /* select last element as pivot */
i ← p – 1 /* i is pointing one position before than p initially */
for j ← p to r − 1 do
{
if (A[j] ≤ x)
{
i ← i + 1
swap(A[i],A[j])
}
}/* end for */
swap(A [i + 1] ,A[r])
return ( i+1)
} /* end module */
90. QUICK-SORT
Fastest known Sorting algorithm in practice
Running time of Quick-Sort depends on the nature of its input data
Worst case (when input array is already sorted) O(n2)
Best Case (when input data is not sorted) Ω(nlogn)
Average Case (when input data is not sorted & Partition of array is not
unbalanced as worst case) : Θ(nlogn)
91. 91
Master’s method
• “Cookbook” for solving recurrences of the form:
where, a ≥ 1, b > 1, and f(n) > 0
Idea: compare f(n) with nlog
b
a
• f(n) is asymptotically smaller or larger than nlog
b
a by a
polynomial factor n
• f(n) is asymptotically equal with nlog
b
a
)()( nf
b
n
aTnT
92. 93
Master’s method
• “Cookbook” for solving recurrences of the form:
where, a ≥ 1, b > 1, and f(n) > 0
Case 1: if f(n) = O(nlog
b
a -) for some > 0, then: T(n) = (nlog
b
a)
Case 2: if f(n) = (nlog
b
a), then: T(n) = (nlog
b
a lgn)
Case 3: if f(n) = (nlog
b
a +) for some > 0, and if
af(n/b) ≤ cf(n) for some c < 1 and all sufficiently large n, then:
T(n) = (f(n))
)()( nf
b
n
aTnT
regularity condition
93. 94
Examples
T(n) = 2T(n/2) + n
a = 2, b = 2, log22 = 1
Compare nlog
2
2 with f(n) = n
f(n) = (n) Case 2
T(n) = (nlgn)
94. 95
Examples
T(n) = 2T(n/2) + n2
a = 2, b = 2, log22 = 1
Compare n with f(n) = n2
f(n) = (n1+) Case 3 verify regularity cond.
a f(n/b) ≤ c f(n)
2 n2/4 ≤ c n2 c = ½ is a solution (c<1)
T(n) = (n2)
95. 96
Examples (cont.)
T(n) = 2T(n/2) +
a = 2, b = 2, log22 = 1
Compare n with f(n) = n1/2
f(n) = O(n1-) Case 1
T(n) = (n)
n
96. 97
Examples
T(n) = 2T(n/2) + nlgn
a = 2, b = 2, log22 = 1
• Compare n with f(n) = nlgn
– seems like case 3 should apply
• f(n) must be polynomially larger by a factor of n
• In this case it is only larger by a factor of lgn
97. 98
Examples
T(n) = 3T(n/4) + nlgn
a = 3, b = 4, log43 = 0.793
Compare n0.793 with f(n) = nlgn
f(n) = (nlog
4
3+) Case 3
Check regularity condition:
3(n/4)lg(n/4) ≤ (3/4)nlgn = c f(n),
c=3/4
T(n) = (nlgn)
98. Substitution Method
• Step1
– Guess the form of the Solution.
• Step2
– Prove your guess is correct by using Mathematical
Induction
99. Mathematical Induction
• Proof by using Mathematical Induction of a
given statement (or formula), defined on the
positive integer N, consists of two steps:
1. (Base Step): Prove that S(1) is true
2. (Inductive Step): Assume that S(n) is true,
and prove that S(n+1) is true for all n>=1
102. Substitution method
• Guess a solution
– T(n) = O(g(n))
– Induction goal: apply the definition of the
asymptotic notation
• T(n) ≤ c g(n), for some c > 0 and n ≥ n0
– Induction hypothesis: T(k) ≤ c g(k) for all k < n
(strong induction)
• Prove the induction goal
– Use the induction hypothesis to find some
values of the constants c and n0 for which the
induction goal holds
103. Substitution Method
• T(n) = 2 if 1<=n<3
3T(n/3)+n if n>=3
• Guess the solution is T(n) = O(nlogn)
• Prove by mathematical induction
To prove : T(n) = O(nlogn)
T(n) <=cnlogn , n>=n0
Induction hypothesis : Let n > n0 and assume k < n
T(k) <=cklog(k)
104. Substitution method
• Let’s take k = (n/3), T(n/3) <= c(n/3) log (n/3)
• To show T(n) <=cnlogn
T(n) = 3T(n/3) + n ( By recurrence for T)
T(n) = 3c(n/3)log(n/3) + n (By Induction hypothesis)
T(n) = cn(log n -1) + n
T(n) = cnlogn – cn + n
• To obtain T(n) <=cnlogn we need to have –cn+n <=0
, so c >=1 Induction step is cleared
• To determine n0, base step T(n0) <=cn0(logn0)
105. Advantages of Divide and Conquer
• Solving difficult problems
• Algorithm Efficiency
– Size n/b at each stage
• Parallelism
– Sub problems – multiprocessor
• Memory Access
– Make efficient use of memory cache
106. Disadvantages of D & C
• Recursion is slow
• Overhead of repeated subroutine calls
• With large recursive base cases, overhead can
be become negligible
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