Prove that you cannot schedule this set of intervals with two processors Depth: maximum number of intervals active Algorithm • Sort by start times • Suppose maximum depth is d, create d slots • Schedule items in increasing order, assign each item to an open slot • Correctness proof: When we reach an item, we always have an open slot Greedy Graph Colorin Interval Scheduling: Greedy Algorithm Greedy algorithm. Consider jobs in increasing order of finish time.Take each job provided it's compatible with the ones already taken. Sort jobs by finish times so that f1 ≤ f2 ≤... ≤ fn. set of jobs selecte CSC 373 - Algorithm Design, Analysis, and Complexity Summer 2016 Lalla Mouatadid Greedy Algorithms: Interval Scheduling De nitions and Notation: A graph G is an ordered pair (V;E) where V denotes a set of vertices, sometimes called nodes, and E the corresponding set of edges (lines connecting the vertices). Formally V = fv 1;v 2;:::; Fig. 2: An example of the greedy algorithm for interval scheduling. The nal schedule is f1;4;7g. Second, we consider optimality. The proof's structure is worth noting, because it is common to many correctness proofs for greedy algorithms. It begins by considering an arbitrary solution, which may assume to be an optimal solution. If it is equal to the greedy solution

Interval Scheduling: Correctness Theorem. Greedy algorithm is optimal. Proof. Assume greedy is different from OPT. Let's see what's different. Let 1, 2, denote the set of jobs selected by greedy. Let 1, 2, denote set of jobs in the optimal solution with 1= 1, 2= 2 ** Interval Scheduling: Analysis Theorem 4**.3. Greedy algorithm is optimal. Pf. (by contradiction: exchange argument) Suppose Greedy is not optimal. Let i1, i2, ik denote set of jobs selected by Greedy. Let j1, j2, jm denote set of jobs in the optimal solution. Consider OPT solution that follows Greedy as long as possible (up to r), s

Interval SchedulingInterval PartitioningMinimising Lateness Algorithm Design I Start discussion of di erent ways of designing algorithms. I Greedy algorithms, divide and conquer, dynamic programming. I Discuss principles that can solve a variety of problem types. I Design an algorithm, prove its correctness, analyse its complexity. I Greedy algorithms: make the current best choice The greedy algorithm stays ahead Exchange argument Data structures: Priority queue Union-ﬁnd data structure. Interval Scheduling Given: A set of activities competing for time intervals on a certain resource (E.g., classes to be scheduled competing for a classroom) Goal: Schedule as many non-conﬂicting activities as possible. Interval Scheduling Given: A set of activities competing. 3 An overview of greedy algorithms Informally, a greedy algorithm is an algorithm that makes locally optimal deci-sions, without regard for the global optimum. An important part of designing greedy algorithms is proving that these greedy choices actually lead to a glob-ally optimal solution. One common way of formally describing greedy algorithms is in terms op Greedy algorithms, divide and conquer, dynamic programming. Discuss principles that can solve a variety of problem types. Design an algorithm, prove its correctness, analyse its complexity. Greedy algorithms: make the current best choice. T. M. Murali February 10, 15, 2021 CS 4104: Greed is Good. Interval SchedulingInterval PartitioningMinimising Lateness Algorithm Design Start discussion of.

I have found many proofs online about proving that a greedy algorithm is optimal, specifically within the context of the interval scheduling problem. On the second page of Cornell's Greedy Stays Ahead handout, I don't understand a few things: All of the proofs make the base case seem so trivial (when r=1). I don't understand the context behind the base case - perhaps in more easy to understand terms It's Not for the same reason as in the greedy algorithm for unweighted interval scheduling. Instead, it's because it allows us to consider only a small number of subproblems (O(n)), vs the exponential number that seem to be needed if the jobs aren't ordered (seemingly, any of the 2n possible subsets might be relevant) Don't believe me? Think about the analogous problem for weighte an interval can be scheduled with a start time equal to the previous tasks nish time. Output: A schedule A of maximum size such that A I and no two intervals in A con ict. 1.2 Algorithm 1.Sort the intervals in I by increasing nish time 2.Initialize A = ; 3.Iterate through the intervals in I (a)If the current interval does not con ict with any interval in A, add it to Greedy algorithms usually involve a sequence of choices. The basic proof strategy is that we're going to try to prove that the algorithm never makes a bad choice. Greedy algorithms can't backtrack -- once they make a choice, they're committed and will never undo that choice -- so it's critical that they never make a bad choice

- Lecture 6: Greedy algorithms 4 Interval scheduling Input: set of intervals on the line, represented by pairs of points (ends of intervals) Output: finding the largest set of intervals such that none two of them overlap Greedy algorithm: •Select intervals one after another using some rul
- Although easy to devise, greedy algorithms can be hard to analyze. The correctness of a greedy algorithm is often established via proof by contradiction, and that is always the most di cult part for designing a greedy algorithm. In this lecture, we will demonstrate greedy algorithms for solving interval scheduling problem and prove its correctness
- The interesting thing to realize about the interval scheduling problem is that there exists a greedy algorithm to solve the interval scheduling problem that can be proven to always find the optimal solution. The optimal algorithm is 2 steps: Sort the list of intervals \((s_i,f_i)\) by finishing time \(f_i\). While elements remain in the list, add the soonest-ending interval compatible with.
- Greedy algorithm stays ahead (e.g. Interval Scheduling). Show that after each step of the greedy algorithm, its solution is at least as good as any other algorithm's. Structural (e.g. Interval Partition). Discover a simple structural bound asserting that every possible solution must have a certain value. Then show that your algorithm alway

Interval Scheduling: Proving the simple wrong Greedy algorithms are easy to design, but hard to prove correct Usually, a counterexample is the best way to do this Interval scheduling provided an example where it was easy to come up with a simple greedy algorithm. { However, we were able to show the algorithm non-optimal by using a counterexample knapsack and interval scheduling problem and analyze their correctness. 2 Introduction to Greedy Algorithms Today we discuss greedy algorithms. This is the third algorithm design technique we have covered in the class and is the last one. Previously, we saw dynamic programming and divide-and-conquer. All these paradigms attempt to break a large, complicated problems into many smaller ones. In. Greedy Algorithm Greedy algorithm works: proof of correctness Interval scheduling: induction on step Optimal loading: induction on input size Scheduling to minimum lateness: exchange argument Greedy algorithm does not work: find a counter-example Coin changing problem 4/5

So what would a template for a **greedy** **algorithm** look like for our **interval** **scheduling** problem? Here's a template that probably puts it all together and gives you a good sense of what I mean by **greedy**, at least in this context. So before we even get into particulars of selection strategies, let me give you a template for **greedy** **interval**. This proof of optimality for Prim's algorithm uses an argument called an exchange argument. General structure is as follows * Assume the greedy algorithm does not produce the optimal solution, so the greedy and optimal solutions are different. Show how to exchange some part of the optimal solution with some part of the greedy solution in a way that improves the optimal solution. Reach a. This article will solve a classical greedy algorithm problem: Interval Scheduling. Given a series of closed intervals [start, end], you should design an algorithm to compute the number of maximum subsets without any overlapping. int intervalSchedule(int[][] intvs) {

Interval Scheduling What is the largest solution? Greedy Algorithm for Scheduling Let T be the set of tasks, construct a set of independent tasks I, A is the rule determining the greedy algorithm I = { } While (T is not empty) Select a task t from T by a rule A Add t to I Remove t and all tasks incompatible with t from T. Simulate the greedy algorithm for each of these heuristics Schedule. Interval Scheduling Maximization (Proof w/ Exchange Argument) - YouTube * Many scheduling problems can be solved using greedy algorithms*. Problem statement: Given N events with their starting and ending times, find a schedule that includes as many events as possible. It is not possible to select an event partially. Consider the below events: In this case, the maximum number of events is two

There are four main steps for a greedy stays ahead proof. Step 1: Deﬁne your solutions.Describe the form your greedy solution takes, and what form someother solution takes (possibly the optimal solution). For example, letAbe the solution con-structed by the greedy algorithm, and letObe a (possibly optimal) solution Interval Partitioning ( Greedy Algorithm ) - Algorithms - YouTube. Thanks for subscribing!---This video is about a greedy algorithm for interval partitioning. With this algorithm you can minimize. Proof Notes - Interval Scheduling How can we prove that the greedy algorithm with earliest nish time is correct? We will prove this with an approach called the stay-ahead proof approach, which is used with greedy algorithms. Note that it's an approach (idea), not a technique like proof by contradiction or proof by induction. The main idea behind the stay-ahead approach is to nd the super.

Interval Scheduling: Correctness Theorem. Greedy algorithm is optimal. Proof. Assume greedy is different from OPT. Let's see what's different. Let 1, 2, denote the set of jobs selected by greedy. Let 1, 2, denote set of jobs in the optimal solution wit greedy algorithms for solving interval scheduling and optimal encoding and analyze their correct-ness. Although easy to devise, greedy algorithms can be hard to analyze. The correctness is often established via proof by contradiction. We demonstrate greedy algorithms for solving fractional knapsack and interval scheduling problem and analyze their correctness. 2 Interval Scheduling *repeated.

Greedy Algorithm for Interval Scheduling Idea: greedy by minimum ﬁnish time Algorithm: Minimum Finish Time S = ∅ Sort jobs by increasing ﬁnish time. For each job j (in sorted order): •if j if compatible with S schedule j (S ←S ∪{j}) •else discard j Analysis Runtime First attempt:: T(n) ≤nlogn+ X i i ≈nlogn+n2 = O(n2) A strategy for designing greedy algorithms and proving optimality In the INTERVAL SCHEDULING problem, we are given a set R of n intervals [s i,f i), 1 ≤ i ≤ n. We are supposed to output a maximum-size subset of R that are mutually non-overlapping. Our algorithm A takes R as input and outputs A(R) - a subset of non-overlapping intervals. The algorithm is a recursive one: (1) let I ∈. Greedy stays ahead - the interval scheduling example Exchange argument - job scheduling Greedy graph algorithms: shortest path, spanning tree and arborescence When greedy works - matroids Greedy algorithms: there is no exact de nition. Algorithms where the solution is found through a sequence of locally optimal steps. 1.2.1 Greedy Stays Ahead - Interval Scheduling Problem.Interval scheduling. Exakte Algorithmen Interval Scheduling 5 / 80. Die Berechnung des Huffman Code Exakte Algorithmen Huffman Code 6 / 80. HUFFMAN CODE -Eine Datei D ist aus Buchstaben eines Alphabets aufgebaut.-Für jeden Buchstaben a 2 ist H(a) die Häuﬁgkeit von a.-Bestimme einen Präﬁxcode, der D optimal komprimiert. Huffman's Algorithmus baut einen binären (Kodier-)Baum B Bestimme die beiden.

Greedy Algorithms • No clear definition, but essentially: Proof (by induction on E): Corollary: Earliest finishing time algorithm is optimal. Algorithm Theory, WS 2012/13 Fabian Kuhn 9 Weighted Interval Scheduling Weighted version of the problem: • Each interval has a weight • Goal: Non‐overlapping set with maximum total weight Earliest finishing time greedy algorithm fails. ** We have seen that a particular greedy algorithm produces an optimal solution to the Interval Scheduling Problem, where the goal is to accept as large a set of nonoverlapping intervals as possible**. The Weighted Interval Scheduling Problem is a strictly more general version, in which each interval has a certain value (or weight), and we want to accept a set of maximum value. Designing a. Interval Scheduling: Greedy Algorithms Greedy template. Consider jobs in some order. Take each job provided it's compatible with the ones already taken. breaks earliest start time breaks shortest interval breaks fewest conflicts 6 Greedy algorithm. Consider jobs in increasing order of finish time Lemma: The Greedy algorithm assigns to every interval a label. No two overlapping interals receive the same label. Proof: We first argue that each interval is assigned a label. Consider interval I i and suppose that there exist exactly t intervals among I 1,I i-1 that overlap with I i. These t intervals, together with I i, pass over a common. Interval Scheduling: Greedy Algorithm Greedy template Consider jobs in some natural order, then take each job provided it's compatible with the ones already taken. Selection strategy is short-sighted ; the order might not be optimal Candidate selection strategies [Earliest start time] Consider jobs in ascending order of si [Earliest finish time]Consider jobs in ascending order of fi.

** So what would a template for a greedy algorithm look like for our interval scheduling problem? Here's a template that probably puts it all together and gives you a good sense of what I mean by greedy, at least in this context**. So before we even get into particulars of selection strategies, let me give you a template for greedy interval. Correctness of Algorithm • Set output consists of compatible requests • By construction! • We want to prove our solution is optimal (schedules the maximum number of jobs) • Let be an optimal set of jobs.Goal: show ,i.e., greedy also selects the same number of jobs and thus is optimal • Proof technique to prove optimality: • Greedy always stays ahead (or rather never falls behind

- Interval Scheduling: Greedy Algorithms Greedy template. Consider jobs in some natural order. Take each job provided it's compatible with the ones already taken. [Earliest start time]Consider jobs in ascending order of sj. [Earliest finish time]Consider jobs in ascending order of fj. [Shortest interval]Consider jobs in ascending order of fj-sj. [Fewest conflicts]For each job j, count the number.
- g is slower, but ﬁnds the optimum solution. 5.2 Greedy Algorithms Basics. It is hard to formally deﬁne what is meant by a Greedy Algorithm, but one generally has these important features: it builds up a solution in small steps
- We now prove by contradiction that the above greedy algorithm uses exactly d resources to schedule a set of requests R, where d is the depth of the set R. Suppose the greedy algorithm uses more than d resources. Let j be the first request that is assigned to the resource d+1. Since we process requests according to non-decreasing start times, there are d requests that start (at or) before j and.
- A greedy algorithm starting at a or b always gives the optimal solution. Proof: consider the set S a of all the subintervals covering a. Clearly, one of them has to belong to the optimal solution. If we replace it with a subinterval (a max,b max) from S a whose right endpoint b max is maximal in S a (reaches furthest to the right), the remaining uncovered interval (b max,b) will be a subset of.
- ates was a kind of 'stays ahead' argument. 4.1 Interval Scheduling . When trying to figure out a greedy algorithm for the interval scheduling problem of section 4.1 of our text, it's a challenge to figure out what might be a good rule-of-thumb (heuristic) for choosing each interval to be added to the set.

* Interval Scheduling*. We have a set of jobs J= {a,b,c,d,e,f,g}. Let j in J be a job than its start at sj and ends at fj. Two jobs are compatible if they don't overlap. A picture as example: The goal is to find the maximum subset of mutually compatible jobs. There are several greedy approaches for this problem: Earliest start time: Consider jobs in ascending order of sj; Earliest finish time. Interval Scheduling: Greedy Algorithm Greedy algorithm. allocate d labels(d = depth) sort the intervals by starting time: I 1,I 2,..,I n forj = 1 to n for each interval I i that precedes and overlaps with I j exclude its label for I j pick a remaining label for I

by the greedy and optimal algorithm respectively, ordered by ﬁnish times. Lemma 1. For all , we have: . Lemma 2. The greedy algorithm returns an optimal set of jobs , that is, . Proof. (By contradiction) Suppose is not optimal, then the optimal set must select more jobs, that is, . That is, there is a job that starts after end CS 312: Algorithms Dan Sheldon Mount Holyoke College Last Compiled: October 1, 2018 Greedy Algorithms We are moving on to our study of algorithm design techniques: I Greedy I Divide-and-conquer I Dynamic programming I Network ﬂow Let's jump right in, then characterize later what is means to be greedy. Interval Scheduling Interval scheduling: Algorithm Analysis (8) Proof. But this cannot happen: rather than choosing a later-ﬁnishing interval, the greedy algorithm always has the option (at worst) of choosing jr and thus fulﬁlling the induction step. We know that: f(jr 1) s(jr) Continuing with the induction f(ir 1) f(jr 1) we get f(ir 1) s(jr

- g the greedy solution isn't optimal and using the fact that greedy stays ahead to de-rive a contradiction. When writing up a proof of this form, you don't need to explicitly enumerate these steps (we didn't do this in lecture, if you.
- Interval scheduling: greedy algorithms Greedy template. Consider jobs in some natural order. Take each job provided it's compatible with the ones already taken. counterexample for earliest start time counterexample for shortest interval counterexample for fewest conßicts 12 Proposition . Can implement earliest-finish-time first in O(n log n) time. ~ Keep track of job j* that was added last to.
- Interval Scheduling: Greedy Algorithms Greedy template. Consider jobs in some natural order. Take each job provided it's compatible with the ones already taken. Possible orders: [Earliest start time]Consider jobs in ascending order of s j. [Earliest finish time]Consider jobs in ascending order of f j. [Shortest interval]Consider jobs in ascending order of f j-s j. [Fewest conflicts]For each.
- Exercises: interval scheduling [Last updated March 3] Questions. 1. Consider the greedy algorithm (#3, see lecture) for interval scheduling which picks the compatible interval that finishes as soon as possible. We saw in class that this gives an optimal solution for the problem of finding the maximum number of intervals. Does it also find an.

Minimizing Maximum Lateness: Greedy Algorithm Greedy algorithm. Earliest deadline first. Observation. The greedy schedule has no idle time. d j 6 t j 3 1 8 2 2 9 1 3 9 4 4 14 3 5 15 2 6 time required deadline job numbe Interval Scheduling Interval Scheduling Problem Input: A set of n activities, each one starting on time s i and nishing on time f i. Output: Largest possible quantity of activities without overlapping Two intervals i and j overlap if there is a time k where both are active. Pedro Ribeiro (DCC/FCUP) Greedy Algorithms 2018/2019 18 / 4 Introduction to Algorithms Outline for Greedy Algorithms CS 482 Spring 2006 Greedy Stays Ahead Main Steps There are four main steps for a greedy stays ahead proof. Step 1: Deﬁne your solutions. Describe the form your greedy solution takes, and what form some other solution takes (possibly the optimal solution). For example, let A be the solution con-structed by the greedy algorithm, and let.

** Interval Scheduling Problem**. The interval scheduling problem is described as follows: Given a set {1, 2, , n} of n requests, where ith request starts at time s (i) and finishes at time f (i), find a maximum-size subset of compatible requests. Two requests i and j are compatible if they do not overlap i.e., either f (i) <= s (j) or f (j) <= s. 2 Greedy algorithms 1. Introduction 2. Basic examples • The coin-changing problem • The Traveling Salesman Problem 3. Scheduling problems • Interval scheduling

Minimizing Lateness: Greedy Algorithms 11 01 23 45 67 8910 1112 13 14 15 d1 = 6 d2 = 8 d3 = 9 d4 = 9 d5 = 14 d6 = 15 max lateness = 1 Sort n jobs by deadline so that d1 d2 dn t 0 for j = 1 to n Assign job j to interval [t, t + tj] sj t, fj t + tj t t + tj output intervals [sj, fj] Minimizing Lateness: Greedy Algorithm Greedy algorithm. Greedy Algorithms. Greedy is an algorithmic paradigm that builds up a solution piece by piece, always choosing the next piece that offers the most obvious and immediate benefit. So the problems where choosing locally optimal also leads to global solution are best fit for Greedy. For example consider the Fractional Knapsack Problem · Interval Scheduling. Greedy Algorithm to find the maximum number of mutually compatible jobs. Problem Statement. Job j starts at s(j) and finishes at f(j) 2 jobs are compatible if they do not overlap (2nd job starts after or at the same time as the 1st one finishes); Goal: find the maximum number of mutually compatible job

Greedy Algorithm proofs 4 Relationship between Dynamic Programming and Greedy Interval scheduling! Set S of jobs. Interval Scheduling Greedy algorithm! Go through lectures (in order of start time)! If an empty lecture hall is available, schedule the lecture! If not, create a new lecture hall In detail! Sort lectures by start time numClassrooms = 0 for j = 1 to n if lecture j is compatible. The activity selection problem is also known as the Interval scheduling maximization problem A pseudocode sketch of the iterative version of the algorithm and a proof of the optimality of its result are included below . Algorithm. Greedy-Iterative-Activity-Selector (A, s, f): Sort A by finish times stored in S = {A [1]} k = 1 n = A. length for i = 2 to n: if s [i] ≥ f [k]: S = S U {A [i.

Prove that for a given set of boxes with speci ed weights w i that the greedy algorithm currently in use actually minimizes the number of trucks that are needed. (Hint: Your proof should follow the same type of analysis as we used for the interval scheduling problem in class Consider the Shortest-First greedy algorithm for interval scheduling: choose the shortest interval I, delete all the intervals that intersect with I, and repeat until no intervals remain. We saw in class that this algorithm does not always find the optimum (maximum) solution. But what about its approximation ratio ρ(n)? Either show an example where ρ(n) is unboundedly large, or prove that ρ. We will understand the basic design principles for **greedy** **algorithms** and learn about a few **algorithms** for **greedy** **scheduling** and Huffman codes. We will also learn some interesting cases when being **greedy** provides a guaranteed approximation to the actual solution. 7 hours to complete. 5 videos (Total 114 min), 4 readings, 4 quizzes. See All. 5 videos. Introduction to **Greedy** **Algorithms** 20m. class Solution: def minMeetingRooms(self, intervals: List[List[int]]) -> int: # If there is no meeting to schedule then no room needs to be allocated. if not intervals: return 0 # The heap initialization free_rooms = [] # Sort the meetings in increasing order of their start time. intervals.sort(key= lambda x: x[0]) # Add the first meeting. We have to give a new room to the first meeting. heapq. Unit -3 Greedy algorithms (Interval Scheduling- Optimal Caching).pdf . 47 0 0 0

Greedy algorithms have some advantages and disadvantages: It is quite easy to come up with a greedy algorithm (or even multiple greedy algorithms) for a problem. Analyzing the run time for greedy algorithms will generally be much easier than for other techniques (like Divide and conquer). For the Divide and conquer technique, it is not clear. Greedy Interval Scheduling •Step 1: Identify a greedy choice property -Options: •Shortest interval •Fewest conflicts •Earliest start •Earliest end 3 Prove using Exchange Argument. Interval Scheduling Algorithm 4 Find event ending earliest, add to solution, Remove itandall conflicting events, Repeat until all events removed, return solution. Interval Scheduling Algorithm Find event. Interval Scheduling: Greedy Algorithms Greedy template. Consider jobs in some natural order. Take each job provided it's compatible with the ones already taken. counterexample for earliest start time counterexample for shortest interval counterexample for fewest conflicts 5 Greedy algorithm. Consider jobs in increasing order of finish time

Interval Scheduling Problem Proof the Optimality of Greedy Strategy Let A be the solution from the greedy strategy Let O be the optimal solution In order to prove A is optimal, we should simply show that |A|=|O| Computer Algorithms Design and Analysis j r-1 j r i r-1 i r Let A={i 1,i 2i k}, O={j 1,j 2j m}, our goal is to prove k=m Assume requests in A and O are ordered in the. 3 Greedy Algorithms interval scheduling a greedy algorithm the interval partitioning problem CS 401/MCS 401 Lecture 5 Computer Algorithms I Jan Verschelde, 27 June 2018 Computer Algorithms I (CS 401/MCS 401) Directed Graphs; Interval Scheduling L-5 27 June 2018 1 / 57 . Directed Graphs; Interval Scheduling 1 Directed Graphs representing directed graphs search algorithms and strong connectivity. Interval Scheduling: Greedy Algorithms Greedy template. Consider jobs in some order. Take each job provided it's compatible with the ones already taken. breaks earliest start time breaks shortest interval breaks fewest conflicts 6 Greedy algorithm. Consider jobs in increasing order of finish time. Take each job provided it's compatible with the ones already taken. Implementation. O(n log n.

Interval scheduling: greedy algorithms Greedy template. Consider jobs in some natural order. Take each job provided it′s compatible with the ones already taken. 20 counterexample for earliest start time counterexample for shortest interval counterexample for fewest conﬂict CSE 441/541 Lecture Notes 2 - Greedy Algorithms Weixiong Zhang 1 A Scheduling Problem You manage a ginormous space telescope. Lots of astronomers want to use it to make observations. Each astronomer's project p i requires use of the telescope starting at a xed time s i (when their grant starts) and running for ' i days. Only one project can. Question 3 (6pt) In class we discussed an greedy algorithm for Interval Scheduling that works by always selecting requests that ﬁnish as early as possible. Another greedy algorithm that also produces an optimal solution is to always select requests that start as late as possible. (a) (2pt) Give pseudo-code for the algorithm. (b) (4pt) Proof that the algorithm always output an optimal. Greedy algorithms ; Interval scheduling; 3 Greedy algorithms paradigm. Algorithm is greedy if ; it builds up a solution in small steps ; it chooses a decision at each step myopically to optimize some underlying criterion ; Analyzing optimal greedy algorithms by showing that ; in every step it is not worse than any other algorithm, or ; every algorithm can be gradually transformed to the greedy.

Algorithmic approaches: scheduling case study1 A Greedy Algorithm for Scheduling Jobs with Deadlines and Proﬁts The setting is that we have n jobs, each of which takes unit time, and a processor on which we would like to schedule them in as proﬁtable a manner as possible. Each job has a proﬁt associated with it, as well as a deadline; if the job is not scheduled by the deadline, then we. Lecture 3 { Greedy algorithms and Matroids Uriel Feige Department of Computer Science and Applied Mathematics The Weizman Institute Rehovot 76100, Israel uriel.feige@weizmann.ac.il March 31, 2014 1 Greedy algorithms When searching for the optimal solution to a problem that has many feasible solutions, search may progress in one of several ways, depending on the structure of the problem. As a. Notes on Greedy Algorithms. SCREENCAST Introducing Greedy Algorithms and An Algorithm for The Interval Scheduling Problem (MP4) SCREENCAST Proving That The Earliest-Finish-Time-First Algorithm Works (MP4) Worked Example of The Interval Scheduling Algorithm of Section 4.1 (PDF) Worked Example of The Interval Scheduling Algorithm of Section 4.1 (PowerPoint Download) Discussion about Greedy.

Interval scheduling Weighted activity selection Minimizing lateness Optimal pre x codes data compression pre x codes Hu man code Approximation algorithms Greedy Algorithms Greedy algorithms are mainly designed to solve combinatorial optimization problems: Given an input, we want to compute an optimal solution according to someobjective function algorithm, the proof is the same as the proof for binary Huffman algorithm is not enough to get full credit. Extra Credit (16.1-3). Suppose that we have a set of activities to schedule among a large number of lecture halls. We wish to schedule all the activities using as few lecture halls as possible. Give an efficient greedy greedy.

Proof methods and greedy algorithms Magnus Lie Hetland Lecture notes, May 5th 2008 ⇤ 1 Introduction This lecture in some ways covers two separate topics: (1) how to prove al-gorithms correct, in general, using induction; and (2) how to prove greedy algorithms correct. Of course, a thorough understanding of induction is a foundation for the more advanced proof techniques, so the two are related Proof. Simple proof by contradiction - if f(i. j) >s(i. j+1), interval j and j +1 intersect, which is a contradiction of Step 2 of the algorithm! Claim 2. Given list of intervals L, greedy algorithm with earliest ﬁnish time produces k. ∗ intervals, where k ∗ is optimal. Proof. ∗Induction on k. Base case: k. Every online algorithm for interval scheduling on related machines can be turned into a greedy online algorithm that accepts at least the same number of intervals on every instance. Proof . To show that an arbitrary online algorithm A can be turned into a greedy online algorithm that performs as least as good on every instance, we show how we can modify the algorithm such that it uses the. 1 Interval scheduling (Activity selection) Problem: Given a set A = fA 1;A 2; ;A ngof n activities with start and nish times (s i;f i), 1 i n, nd a maximal set S of non-overlapping activities. This is a special case of the weighted-interval scheduling problem, where all intervals have the same weight. A possible practical interpretation: corresponds to scheduling the maximal number of classes.