Dynamic Programming - Part II

Oct 6, 20146 likes5,576 views

The document is a lecture on dynamic programming, focusing on various algorithms such as maximum value contiguous subarray, longest increasing subsequence, and coin change. It details algorithmic approaches, including brute force, greedy variations, and dynamic programming solutions with time complexities. Additionally, it includes sample runs and insights on optimal substructure in dynamic programming.

Design and
Analysis of
Algorithms
DYNAMIC PROGRAMMING
PART II
Maximum Value Contiguous Subarray
Maximum Increasing Subsequence
Coin Change
Algorithms Dynamic Programming - Part II 1

 Instructor
Prof. Amrinder Arora
amrinder@gwu.edu
Please copy TA on emails
Please feel free to call as well

 Available for study sessions
Science and Engineering Hall
GWU
Algorithms Dynamic Programming - Part II 2
LOGISTICS

Algorithms
Analysis
Asymptotic
NP-
Completeness
Design
D&C
Greedy
DP
Graph
B&B
Applications
Algorithms Dynamic Programming - Part II 3
WHERE WE ARE
 Done
 Done
 Finishing today..
 Done

 http://video.google.com/videoplay?docid=16073867
32227802988#
 http://people.csail.mit.edu/bdean/6.046/dp/
Algorithms Dynamic Programming - Part II 4
DP (CONT.)

 Given an Array A(1..n)
 To find A(i..j) such that the sum of the terms in the
subarray is maximized.
 Insights
 If no negative terms in the array, then of course, we just select
the entire array.
 So, this problem is meaningful only when we have negative
numbers
 We can find sum of all contiguous subarrays in O(n3) time.
 Can we do better?
Algorithms Dynamic Programming - Part II 5
MAXIMUM VALUE CONTIGUOUS
SUBSEQUENCE

MaxValue = - infinity
for i = 1 to n {
for j = i to n {
currSum = findSubArraySum(i,j)
if CurrSum > MaxValue, then MaxValue = CurrSum
}
}
Procedure FindSubArraySum(i,j)
Double sum = 0
For int k = i to j {
sum = sum + A[k]
}
return sum
}
Algorithms Dynamic Programming - Part II 6
ALGORITHM MVCS1
Brute Force
Search!
O(n3) time

Procedure InitSubArraySums() {
for int i = 1 to n {
double sum = 0
for int k = i to n {
sum = sum + A(k)
SubArraySum[i][k] = sum
}
}
}
InitSubArraySums();
MaxValue = - infinity
for i = 1 to n {
for j = i to n {
currSum = SubArraySum[i][j]
If currSum > MaxValue, then MaxValue = currSum
}
}
Algorithms Dynamic Programming - Part II 7
ALGORITHM MVCS2
Still a full search,
but don’t
recompute sums
O(n2) time

 What can be a greedy variation of MVCS algorithm?
 What can be a D&C version of MVCS algorithm?
 How about Dynamic Programming?
Algorithms Dynamic Programming - Part II 8
USING OTHER TECHNIQUES FOR MVCS

Using Dynamic Programming Template
1. N: MVCS(i): value of MVCS ending at position i.
2. O: Prove Optimal Substructure Holds (What is our
claim?)
3. R: MVCS(i) = max {MVCS(i-1) + A[i],A[i]}
4. A: Start from left, and keep track of maximum
MVCS(i) encountered.
1. For I = 1 to n
Algorithms Dynamic Programming - Part II 9
MVCS3
O(n) time

 Using the Array used in class
 A[]: 22, -17, -71, 12, -22, 81, -10, 63
 MVCS[]: 22, 5, -66, 12, -10, 81, 71, 134
Algorithms Dynamic Programming - Part II 10
SAMPLE RUN OF MVCS3

 Given array A(1..n)
 Find a subsequence (not necessarily contiguous) that
is strictly increasing.
 For example
 {1, 7, 2, 8, 4, 1, 6, 11, 3, 15, 5, 12, 14}
Algorithms Dynamic Programming - Part II 11
LONGEST INCREASING SUBSEQUENCE
Longest subsequence is of length 7:
{1, .., 2, .., 4, .., 6, 11, .., .., .., 12, 14}

 Given array A(1..n)
 LIS(i) = longest strictly increasing subsequence that
ends at i.
 LIS(i) = max {LIS(j)} + 1, such that j < i, and A[j] <
A[i]
[Or A[j] ≤ A[i] if we are looking for that condition]
Algorithms Dynamic Programming - Part II 12
LONGEST INCREASING SUBSEQUENCE

 What is the time complexity of this algorithm?
Algorithms Dynamic Programming - Part II 13
LIS

 Using the Array used in Class
 {1, 7, 2, 8, 4, 1, 6, 11, 3, 15, 5, 12, 14}
 {1, 2, 2, 3, 3, 1, 4, 5, 3, 6, 4, 6, 7}
 What are the LIS values?
Algorithms Dynamic Programming - Part II 14
SAMPLE RUN OF LIS

 Given a set of coins with values v1, v2, … vn such that
v1 = 1, v1 < v2 < v3 < v4 < … < vn
 We want to make change for a given value S using as
few coins as possible
Algorithms Dynamic Programming - Part II 15
COIN CHANGE

 http://people.csail.mit.edu/bdean/6.046/dp/
Algorithms Dynamic Programming - Part II 16
AN INTERESTING SET OF PROBLEMS

Algorithms Dynamic Programming - Part II 17
80% OF
SUCCESS IS
SHOWING UP.

Algorithms Dynamic Programming - Part II 18
REST 80% OF
SUCCESS IS
BEING
PRESENT!

CS 6212
Analysis
Asymptotic
NP-
Completeness
Design
D&C
Greedy
DP
Graph
B&B
Applications
Algorithms Dynamic Programming - Part II 19
WHERE WE ARE
 Done
 Done
 Done
 Done

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Convex Hull - Chan's Algorithm O(n log h) - Presentation by Yitian Huang and ...Amrinder Arora

The document discusses the concept of the convex hull, defined as the smallest convex polygon containing a set of points in a plane. It introduces three algorithms for computing convex hulls: Jarvis's algorithm (O(nh)), Graham's algorithm (O(n log n)), and Chan's algorithm (O(n log h)), with Chan's being a combination of the former two and output-sensitive. The summary includes the working principles of each algorithm and emphasizes the importance of determining the optimal value of 'm' in Chan's algorithm.

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The document discusses the ARIMA forecasting algorithm, highlighting its importance for businesses to accurately predict future market trends and avoid losses. It outlines the algorithm's procedure, including data analysis, model parameter estimation, and residual verification for effective forecasting. The ARIMA technique is shown to be more accurate than traditional forecasting methods, with applications in various industries, though it requires historical data for optimal results.

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The document discusses the secretary problem, which involves selecting the best candidate from n applicants interviewed in random order. A stopping rule is introduced where the interviewer rejects the first r-1 applicants and selects the next candidate superior to the best among them, with an optimal stopping number of r determined to maximize the probability of hiring the best candidate. The conclusion is that rejecting the first n/e applicants leads to a probability of about 0.368 for selecting the best candidate.

Proof of O(log *n) time complexity of Union find (Presentation by Wei Li, Zeh...Amrinder Arora

The document discusses the union find algorithm and its time complexity. It defines the union find problem and three operations: MAKE-SET, FIND, and UNION. It describes optimizations like union by rank and path compression that achieve near-linear time complexity of O(m log* n) for m operations on n elements. It proves several lemmas about ranks and buckets to establish this time complexity through an analysis of the costs of find operations.

Proof of Cook Levin Theorem (Presentation by Xiechuan, Song and Shuo)Amrinder Arora

The document provides a proof of the Cook-Levin theorem, establishing that the satisfiability problem (SAT) is NP-complete. It describes the process of constructing a CNF formula that represents the acceptance of a non-deterministic Turing machine for a given language and highlights the significance of the theorem in demonstrating that all NP-complete problems can be reduced to SAT. The proof includes a time analysis showing that the constructed CNF formula has polynomial size.

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The document provides an overview of online algorithms in machine learning, particularly focusing on the classification problem, where a model is created from a training set to predict unseen data. It discusses various classification techniques, including decision trees and support vector machines, and presents the 'weighted majority' algorithms, which help improve decision-making by leveraging multiple expert predictions. The text highlights the practical applications of these algorithms, such as in recommendation systems and targeting strategies.

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This document discusses the design and analysis of algorithms with a focus on NP-completeness, explaining key concepts such as the classes P and NP, the significance of Turing machines, and methods to show a problem is NP-complete. It details the properties and challenges of common NP-complete problems like the satisfiability problem and provides examples of reductions between different problems. The material includes instructions on proving NP-completeness, notably through polynomial time reductions.

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This document presents algorithmic puzzles and their solutions. It discusses puzzles involving counterfeit coins, uneven water pitchers, strong eggs on tiny floors, and people arranged in a circle. For each puzzle, it provides the problem description, an analysis or solution approach, and sometimes additional discussion. The document is a presentation on algorithmic puzzles given by Amrinder Arora, including their contact information.

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The document presents an overview of the Divide and Conquer (D&C) algorithm design paradigm, detailing its recursive approach to solving complex problems by breaking them into smaller subproblems. It covers essential algorithms like binary search, merge sort, and quicksort, and discusses recurrence relations used to analyze their time complexities. Various methods for solving these recurrence relations, such as substitution and the master theorem, are also introduced.

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The document outlines a course on advanced data structures, specifically focusing on disjoint set data structures, their applications, and algorithms like Kruskal's for minimum spanning trees. It discusses the union-find operations, performance optimizations through path compression, and the implementation of Bloom filters for approximate set membership. The course is taught by Professor Amrinder Arora at George Washington University.

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The document discusses various priority queue data structures like binary heaps, binomial heaps, and Fibonacci heaps. It begins with an overview of binary heaps and their implementation using arrays. It then covers operations like insertion and removal on heaps. Next, it describes binomial heaps and their properties and operations like union and deletion. Finally, it discusses Fibonacci heaps and how they allow decreasing a key in amortized constant time, improving algorithms like Dijkstra's.

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The document provides an overview of R-trees, a type of spatial data structure used for efficient indexing and retrieval of spatial objects, particularly in multidimensional spaces. It covers R-tree properties, insertion routines, splitting and condensing nodes, and querying strategies, emphasizing the importance of dynamic adjustments for maintaining optimal search performance. The course is part of an advanced data structures lecture by Professor Amrinder Arora at George Washington University.

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This document discusses splay trees, a self-organizing binary search tree data structure. Splay trees perform rotations to move recently accessed elements closer to the root of the tree after operations like insertion, deletion and search. While splay trees do not guarantee an upper bound on tree height like AVL or red-black trees, the amortized time of all operations is O(log n). Splaying restructures the tree in a way that frequently accessed elements are faster to access in the future.

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This lecture discusses B-trees, a data structure optimized for disk access that addresses limitations of traditional binary trees by keeping keys balanced across nodes. Key properties include limitations on the number of child nodes and keys per node, ensuring all leaves are at the same level, and emphasizing efficient insertions and deletions. B-trees allow for high utilization of disk space and effective access patterns, making them ideal for large datasets exceeding RAM capacities.

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The document describes a course on advanced data structures. It provides information on the instructor, teaching assistant, topics to be covered including AVL and Red Black Trees, objectives of learning deletions, insertions and searches in logarithmic time. It also lists credits to other professors and researchers. The document then goes into details about balanced binary search trees, describing properties of AVL Trees and Red Black Trees to ensure the tree remains balanced during operations.

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This document introduces the course CS 6213 - Advanced Data Structures. It discusses what data structures are, how they are designed to efficiently support specific operations, and provides examples. Common data structures like stacks, queues, linked lists, trees, and graphs are introduced along with their basic operations and implementations. Real-world applications of these data structures are also mentioned.

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