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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. Mathematic **Induction** for **Greedy Algorithm Proof** template for **greedy algorithm** 1 Describe the correctness as a proposition about natural number n, which claims **greedy algorithm** yields correct solution. Here, n could be the **algorithm** steps or input size. 2 Prove the proposition is true for all natural number. **Induction** basis: from the smallest. **GREEDY** **ALGORITHMS** The **proof** of the correctness of a **greedy** **algorithm** is based on three main steps: 1: The **algorithm** terminates, i.e. the while loop performs a ﬁnite number of iterations. 2: The partial solution produced at every iteration of the **algorithm** is a subset of an optimal solution, i.e. for each.**Greedy** is an algorithmic paradigm that builds up a solution piece by piece, always. Observation. **Greedy algorithm** never schedules two incompatible lectures in the same classroom. Theorem. **Greedy algorithm** is optimal. Pf. Let d = number of classrooms that the **greedy algorithm** allocates. Classroom d is opened because we needed to schedule a job, say j, that is incompatible with all d-1 other classrooms. 4. TWO BASIC **GREEDY** CORRECTNESS **PROOF** METHODS 4 4 Two basic **greedy** correctness **proof** methods The material in this section is mainly based on the chapter from **Algorithm** Design [4]. 4.1 Staying ahead Summary of method If one measures the **greedy** **algorithm's** progress in a step- by -step fashioin, one sees that it does better than any other **algorithm** at. Mar 29, 2022 · See, here each coin of a given denomination can come an infinite number of times. (Repetition allowed), this is what we call UNBOUNDED KNAPSACK. Theorem A Greedy-Activity-Selector solves the activity-selection problem. Proof The proof is by induction on n. For the base case, let** n **=1. The statement trivially holds. For the induction step, let n 2, and assume that the claim holds for all values of n less than the current one. We may assume that the activities are already sorted according to. **Greedy algorithms** is a paradigm that describes the approach on how you can go about solving a problem. You might have heard about a lot of algorithmic techniques while learning how to solve problems in computer science. The others being: Brute Force Divide and Conquer Dynamic Programming In this post we shall understand what a >**greedy**</b> <b>**algorithm**</b>. – Common techniques are by mathematical **induction** & contradiction 18 **Proof by Induction** • The **induction** base: – is the **proof** that the statement is true for initial value ... • **Greedy algorithm** for coin changing –Order coins in decreasing order –Select coins one at a time (divide x by denomination) –Solution: contains a = 3, b = 1,. So, the **algorithm** would be like. 1) Let, count=0 to count minimum number of coin used 2) Pick up coin with maximum denomination say, value x 3) while amount≥x amount=amount-x count=count+1 4) if amount=0 Go to Step 7 5) Pick up the next best denomination of coin and assign it to x 6) Go to Step 2 7) End. count is the minimum number. 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. 4. TWO BASIC **GREEDY** CORRECTNESS **PROOF** METHODS 4 4 Two basic **greedy** correctness **proof** methods The material in this section is mainly based on the chapter from **Algorithm** Design [4]. 4.1 Staying ahead Summary of method If one measures the **greedy algorithm's** progress in a step-by-step fashioin, one sees that it does better than any other **algorithm** at.**GREEDY**. **Induction** • There is an optimal solution that always picks the **greedy** choice - **Proof** by strong **induction** on J, the number of events - Base case: J L0or J L1. The **greedy** (actually, any) choice works. - Inductive hypothesis (strong) - Assume that the **greedy algorithm** is optimal for any Gevents for 0 Q J. We will prove A is optimal by a " **greedy** stays ahead" argument **Proof** on board. ... I **Proof** **by** **induction** on r I Base case (r =1): ir is the ﬁrst choice of the **greedy** **algorithm** ,. Theorem. Cashier's **algorithm** is optimal for U.S. coins: 1, 5, 10, 25, 100. Pf. [**by** **induction** on x] Consider optimal way to change ck ≤ x < ck+1 : **greedy** takes coin k.

Observation. **Greedy algorithm** never schedules two incompatible lectures in the same classroom. Theorem. **Greedy algorithm** is optimal. Pf. Let d = number of classrooms that the **greedy algorithm** allocates. Classroom d is opened because we needed to schedule a job, say j, that is incompatible with all d-1 other classrooms. **GREEDY ALGORITHMS** The **proof** of the correctness of a **greedy algorithm** is based on three main steps: 1: The **algorithm** terminates, i.e. the while loop performs a ﬁnite number of iterations. 2: The partial solution produced at every iteration of the **algorithm** is a subset of an optimal solution, i.e. for each. The **Greedy Algorithm** Stays Ahead **Proof by induction**: Base case(s):Verify that. **Proof by induction** BASE: There is an optimal solution that contains **greedy** activity 1 as first activity. Let A be an optimal solution with ... Huffman invented a **greedy algorithm** to construct an optimal prefix code called the Huffman code. An encoding is represented by a binary prefix tree:. **Induction** • There is an optimal solution that always picks the **greedy** choice - **Proof** by strong **induction** on J, the number of events - Base case: J L0or J L1. The **greedy** (actually, any) choice works.. **Proof** **by** **induction** on the **greedy** decision 2.Proof by **induction** on an exchange argument 1. Either by contraction 2. Or by exchanging. The **greedy** **algorithm** selects the available interval with smallest nish time; since interval j r is one of these available intervals, we have f(i r) f(j r). This completes the **induction** step. Therefore, for each r. **Proof by induction** BASE: There is an optimal solution that contains **greedy** activity 1 as first activity. Let A be an optimal solution with ... Huffman invented a **greedy algorithm** to construct an optimal prefix code called the Huffman code. An encoding is represented by a binary prefix tree:. • Let k be the number of rooms picked by the **greedy algorithm**. Then, at some point t, |B(t)| ≥ k (i.e., there are at least k events happening at time t). • **Proof** –Let t be the starting time of the first event to be scheduled in room k –Then, by the **greedy** choice, room k was the least number room available at that time. . . class so far, take it! See Figure . for a visualization of the resulting **greedy** schedule. We can write the **greedy algorithm** somewhat more formally as shown in in Figure .. (Hopefully the ﬁrst line is understandable.) After the initial sort, the **algorithm** is a simple linear-time loop, so the entire **algorithm** runs in O(nlogn) time. Coin-Changing: Analysis of **Greedy Algorithm** Theorem. Greed is optimal for U.S. coinage: 1, 5, 10, 25, 100. Pf. (**by induction** on x)! Consider optimal way to change c k ! x < c k+1: **greedy** takes coin k.! We claim that any optimal solution must also take coin k. –if not, it needs enough coins of type c 1, , c k-1to add up to x. 4th gen 4runner transmission fluid change. Your **proof** **by induction** must show that there cannot exist a solution that is better than the one found by the **greedy** **algorithm**.Here is an example of a complete write up of a **proof** **by induction** that the earliest finishing time **algorithm** finds the best solution to the activity selection problem: gas.pdf. **Greedy** Trick or Treat.. After designing the **greedy algorithm**, it is important to analyze it, as it often fails if we cannot nd a **proof** for it. We usually prove the correctnesst of a **greedy algorithm** by contradiction: assuming there is a better solution, show that it is actually no better than the **greedy algorithm**. 8.1 Fractional Knapsack.Using this lemma, we can prove that the **greedy algorithm** is correct. Illustrate the **algorithm** . **Proof** of correctness. Typically the **greedy algorithms** are easy to write. Proving that they construct the optimal solution can be difficult. We prove Prim's **algorithm** is correct **by induction** on the growing tree constructed by the **algorithm**.

Then, the **greedy** will take a coin of k = 1 and will set x ← x − 1. That the **greedy** solves here optimally is more or less trivial. **Induction** hypothesis: k. The **greedy** solves optimally for any value of x such that c k − 1 ≤ x < c k. **Induction** step: k + 1. Show that the **greedy** solves optimally for any value of x such that c k ≤ x < c k + 1. **Proof** of Optimality Theorem 1 The solution generated by **Greedy**-Activity-Selector is opti-mum. **Proof**. Let A= (x 1;:::;x k) be the solution generated by the **greedy algorithm**, where x 1 <x 2 < <x k. It sufﬁces to show the following two claims. (1) Ais feasible. (2) No more interval can be added to Awithout violating the “mu-tually disjoint. At this point I got curious and started doing **greedy** easys and still can't manage to do a single one after looking at 5. Is it normal to suck this bad at **greedy**. In this blog post, I am going to cover 2 fundamental **algorithm** design principles: **greedy algorithms** and dynamic programming. **Greedy Algorithm**. A **greedy algorithm**, as the name suggests,.

Observation. **Greedy algorithm** never schedules two incompatible lectures in the same classroom. Theorem. **Greedy algorithm** is optimal. Pf. Let d = number of classrooms that the **greedy algorithm** allocates. Classroom d is opened because we needed to schedule a job, say j, that is incompatible with all d-1 other classrooms. I'll write copies x [coinval$] notationally. O* will convert 5x[1$] → 1x[5$] , because it's better to use less Recall: G (the **greedy** solution) takes as many 100, 10, 5, 1 as possible, starting from 100, working its way down to 1. Contribute to aneksamun/ **greedy** - coin - change development by creating an account on GitHub.

coin change **greedy algorithm proof**. mississippi resale certificate. Coin Changing Goal. Given currency denominations: 1, 5, 10, 25, 100, ... Pf. (**by induction** on x). n Consider optimal way to change ck ≤ x < ck+1 : **greedy** takes coin k. n We claim that any optimal solution must also take coin k. - if not, it needs enough. 4. TWO BASIC **GREEDY** CORRECTNESS **PROOF** METHODS 4 4 Two basic **greedy** correctness **proof** methods The material in this section is mainly based on the chapter from **Algorithm** Design [4]. 4.1 Staying ahead Summary of method If one measures the **greedy algorithm's** progress in a step-by-step fashioin, one sees that it does better than any other **algorithm** at.**GREEDY**. Topological Sorting **Algorithm** Analysis (Correctness). **Proof by induction** on number of vertices : •, no edges, the vertex itself forms topological ordering • Suppose our **algorithm** is correct for any graph with less than vertices • Consider an arbitrary DAG on vertices • Must contain a vertex with in-degree (we proved it) • Deleting that vertex and all outgoing edges gives us a. **Algorithm** 加权任务调度问题贪婪解的证明,**algorithm**,dynamic-programming,**greedy**,**proof**-of-correctness,**Algorithm**,Dynamic Programming,**Greedy**,**Proof** Of Correctness,我试图证明完全正确（部分正确+终止），但我似乎只能证明任意示例输入（而不是一般输入） 例如，我创建了一个包含作业及其相关属性（截止日期和利润）的表： 从. 7e8 engine code ford fusion. **Greedy Algorithms Proofs** of correctness •It can sometimes feel like more of an art than a science 1.**Proof by induction** on the **greedy** decision 2.**Proof by induction** on an exchange argument 1. Either by contraction 2. Or by exchanging. **Algorithms** AppendixI:ProofbyInduction[Sp’16] **Proof by induction**: Let n be an arbitrary integer greater than 1. 3 **Greedy Algorithm** Principles 30 ... We now examine a second example of **proof by induction** . Proposition 4. Fix c>−1 to be a constant. For each n∈N, we have that (1 + c)n ≥1 + nc. **Proof** . The **proof** is **by induction** on n∈N. Base Case. Consider the base case of n= 0. One of the simplest methods for showing that a **greedy algorithm** is correct is to use a “**greedy** stays ahead” argument. This style of **proof** works by showing that, according to some measure, the **greedy algorithm** always is at least as far ahead as the optimal solution during each iteration of the **algorithm**.

Prereq: CNMT 110 Start studying C949- Data Structures and **Algorithms** I Start studying C949- Data Structures and **Algorithms** I. You will have 2 hours time to answer the questions Use this representation to Midterm Exams: Midterm exams often come at the midpoint in the semester QUESTION 1 What are the three characteristics of Big Data, and what are the main. **GREEDY ALGORITHMS** The **proof** of the correctness of a **greedy algorithm** is based on three main steps: 1: The **algorithm** terminates, i.e. the while loop performs a ﬁnite number of iterations. 2: The partial solution produced at every iteration of the **algorithm** is a subset of an optimal solution, i.e. for each. The **Greedy Algorithm** Stays Ahead **Proof by induction**: Base case(s):Verify that. Observation. **Greedy algorithm** never schedules two incompatible lectures in the same classroom. Theorem. **Greedy algorithm** is optimal. Pf. Let d = number of classrooms that the **greedy algorithm** allocates. Classroom d is opened because we needed to schedule a job, say j, that is incompatible with all d-1 other classrooms. These d jobs each end. • Let k be the number of rooms picked by the **greedy algorithm**. Then, at some point t, |B(t)| ≥ k (i.e., there are at least k events happening at time t). • **Proof** –Let t be the starting time of the first event to be scheduled in room k –Then, by the **greedy** choice, room k was the least number room available at that time. The **algorithm** proceeds by successive subtractions in two loops: IF the test B ≥ A yields "yes" or "true" (more accurately, the number b in location B is greater than or equal to the number a in location A) THEN, the **algorithm** specifies. **Induction** **Proof** of **Algorithm** [**Greedy** Graph Coloring] Having a G = ( V, E) with each vertex having a range. **Algorithm** 加权任务调度问题贪婪解的证明,**algorithm**,dynamic-programming,**greedy**,**proof**-of-correctness,**Algorithm**,Dynamic Programming,**Greedy**,**Proof** Of Correctness,我试图证明完全正确（部分正确+终止），但我似乎只能证明任意示例输入（而不是一般输入） 例如，我创建了一个包含作业及其相关属性（截止日期和利润）的表： 从.

•**Greedy algorithm** for coin changing –Order coins in decreasing order –Select coins one at a time (divide x by denomination) –Solution: contains a = 3, b = 1,. The **Greedy Algorithm** Stays Ahead **Proof by induction**: Base case(s):Verify that the claim holds for a set of initial instances. Inductive step:Prove that, if the claim holds for the. class so far, take it! See Figure . for a visualization of the resulting **greedy** schedule. We can write the **greedy algorithm** somewhat more formally as shown in in Figure .. (Hopefully the ﬁrst line is understandable.) After the initial sort, the **algorithm** is a simple linear-time loop, so the entire **algorithm** runs in O(nlogn) time. 7e8 engine code ford fusion. **Greedy Algorithms Proofs** of correctness •It can sometimes feel like more of an art than a science 1.**Proof by induction** on the **greedy** decision 2.**Proof by induction** on an exchange argument 1. Either by contraction 2. Or by exchanging. **Algorithms** AppendixI:ProofbyInduction[Sp’16] **Proof by induction**: Let n be an arbitrary integer greater than 1.

• Let k be the number of rooms picked by the **greedy algorithm** . Then, at some point t, |B(t)| ≥ k (i.e., there are at least k events happening at time t). • **Proof** -Let t be the starting time of the first event to be scheduled in room k -Then, by the **greedy** choice, room k was the least number room available at that time. We can write the **greedy algorithm** somewhat more formally as shown in in Figure .. (Hopefully the ﬁrst line is understandable.) After the initial sort, the ... The **proof** might be easier to understand if we unroll the **induction** slightly. **Proof**: Let hg 1,g 2,...,g ki be the sequence of classes chosen by the **greedy**. { **Proof** by counterexample: x = 1;y = 3;xy = 3; 3 6 1 **Greedy Algorithms** De nition 11.2 (**Greedy Algorithm**) An **algorithm** that selects the best choice at each step, instead of considering all sequences of steps that may lead to an optimal solution. It’s usually straight-forward to nd a **greedy algorithm** that is feasible, but hard to nd a **greedy**. Here is an example of a complete write up of a **proof by induction** that the earliest finishing time **algorithm** finds the best solution to the activity selection problem: gas.pdf. **Greedy** Trick or Treat. **GREEDY ALGORITHMS** The **proof** of the correctness of a **greedy algorithm** is based on three main steps: 1: The **algorithm** terminates, i.e. the while. of compatible jobs selected by the **greedy** and optimal **algorithm** respectively, ordered by increasing ﬁnish time. Lemma 1. For all , we have: . **Proof**. (**By induction**) Base case: is true, why? • Assume holds for : • For th job, note that (why?) • Using inductive hypothesis: • **Greedy** picks earliest ﬁnish time among compatible jobs (which. apache ramada weight. 4. TWO BASIC **GREEDY** CORRECTNESS **PROOF** METHODS 4 4 Two basic **greedy** correctness **proof** methods The material in this section is mainly based on the chapter from **Algorithm** Design [4]. 4.1 Staying ahead Summary of method If one measures the **greedy** **algorithm's** progress in a step-**by**-step fashioin, one sees that it does better than any other **algorithm** at.

Some of them are: Brute Force. Divide and Conquer. **Greedy** Programming. Dynamic Programming to name a few. In this article, you will learn about what a **greedy algorithm** is and how you can use this technique to solve a lot of programming problems that otherwise do not seem trivial. Imagine you are going for hiking and your goal is to reach the. An Empirical Study for Inversions-Sensitive Sorting **Algorithms**. by Amr Elmasry. Download Free PDF Download PDF Download Free PDF View PDF. Integrating coordinated checkpointing and recovery mechanisms into DSM synchronization barriers.. Observation. **Greedy algorithm** never schedules two incompatible lectures in the same classroom. Theorem. **Greedy algorithm** is optimal. Pf. Let d = number of classrooms that the **greedy algorithm** allocates. Classroom d is opened because we needed to schedule a job, say j, that is incompatible with all d-1 other classrooms. 4. TWO BASIC **GREEDY** CORRECTNESS **PROOF** METHODS 4 4 Two basic **greedy** correctness **proof** methods The material in this section is mainly based on the chapter from **Algorithm** Design [4]. 4.1 Staying ahead Summary of method If one measures the **greedy algorithm**’s progress in a step-by-step fashioin, one sees that it does better than any other **algorithm** at. Let d(v) be the label found by the **algorithm** and let (v) be the shortest path distance from s-to-v. We want to show that d(v) = (v) for every vertex vat the end of the **algorithm**, showing that the **algorithm** correctly computes the distances. We prove this **by induction** on jRjvia the following lemma: Lemma: For each x2R, d(x) = (x).

Jun 18, 2022 · **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 the best fit for** Greedy.** For example consider the Fractional Knapsack Problem.. The crypto fear & greed index of alternative.me provides an easy overview of the current sentiment of the Bitcoin / crypto market at a glance. This is a plot of the Fear & Greed Index over time, where a value of 0 means "Extreme Fear" while a value of 100 represents "Extreme Greed". After designing the **greedy algorithm**, it is important to analyze it, as it often fails if we cannot nd a **proof** for it. We usually prove the correctnesst of a **greedy algorithm** by contradiction: assuming there is a better solution, show that it is actually no better than the **greedy algorithm**. 8.1 Fractional Knapsack.Using this lemma, we can prove that the **greedy algorithm** is correct. **Proof by induction** BASE: There is an optimal solution that contains **greedy** activity 1 as first activity. Let A be an optimal solution with ... Huffman invented a **greedy algorithm** to construct an optimal prefix code called the Huffman code. An encoding is represented by a binary prefix tree:. **Greedy** stays ahead usually use **induction** Exchange start with the first difference between **greedy** and optimal. TU/e **Algorithms** (2IL15) - Lecture 2 11 A = {a 1,, a n}: set of n activities Lemma: Let a i be an activity in A that ends first.Then there is an optimal solution to the Activity-Selection Problem for A that includes a i.

Greedy Algorithms Proofs of correctness •It** can sometimes feel like more of an art than a science 1.Proof by induction on the greedy decision 2.Proof by induction on an exchange argument 1. Either by contraction** 2. Or** by exchanging with the** optimal solution 3.Whatever works. class so far, take it!. **GREEDY ALGORITHMS** The **proof** of the correctness of a **greedy algorithm** is based on three main steps: 1: The **algorithm** terminates, i.e. the while loop performs a ﬁnite number of iterations. 2: The partial solution produced at every iteration of the **algorithm** is a subset of an optimal solution, i.e. for each. The **Greedy Algorithm** Stays Ahead **Proof by induction**: Base case(s):Verify that. Dijkstra’s **Algorithm**: Correctness Invariant. For each , is length of a shortest path **Proof**. [**By induction** on ]. Base case: , and . Assume holds for some . Let be next node added to • Suppose some other path in is shorter • Let be the ﬁrst edge along that leaves • Let be the subpath from to. • Let k be the number of rooms picked by the **greedy algorithm** . Then, at some point t, |B(t)| ≥ k (i.e., there are at least k events happening at time t). • **Proof** -Let t be the starting time of the first event to be scheduled in room k -Then, by the **greedy** choice, room k was the least number room available at that time. 2 / 4 Theorem (Feasibility): Prim's **algorithm** returns a spanning tree. **Proof**: We prove by **induction** that after k edges are added to T, that T forms a spanning tree of S.As a base case, after 0 edges are added, T is empty and S is the single node {v}. Also, the set S is connected by the edges in T because v is connected to itself by any set of edges. Therefore, T connects S and satisfies |T. The MST problem can be solved by a **greedy algorithm** because the the locally optimal solution is also the globally optimal solution. This fact is described by the **Greedy**-Choice Property for MSTs, and its **proof** of correctness is given via a "cut and paste" argument common for **greedy proofs**.Lemma 2 (**Greedy**-Choice Property for MST).For any cut. 4. TWO BASIC **GREEDY**. . The coin of the highest value, less than the remaining change owed, is the local optimum. (In general, the change-making problem requires dynamic programming to find an optimal solution; however, most currency systems are special cases where the **greedy** strategy does find an optimal solution.). "/>. The MST problem can be solved by a **greedy algorithm** because the the locally optimal solution is also the globally optimal solution. This fact is described by the **Greedy**-Choice Property for MSTs, and its **proof** of correctness is given via a "cut and paste" argument common for **greedy proofs**.Lemma 2 (**Greedy**-Choice Property for MST).For any cut. 4. TWO BASIC **GREEDY**. The MST problem can be solved by a **greedy algorithm** because the the locally optimal solution is also the globally optimal solution. This fact is described by the **Greedy**-Choice Property for MSTs, and its **proof** of correctness is given via a "cut and paste" argument common for **greedy proofs**.Lemma 2 (**Greedy**-Choice Property for MST).For any cut. 4. TWO BASIC **GREEDY**.

{ **Proof** by counterexample: x = 1;y = 3;xy = 3; 3 6 1 **Greedy Algorithms** De nition 11.2 (**Greedy Algorithm**) An **algorithm** that selects the best choice at each step, instead of considering all sequences of steps that may lead to an optimal solution. It’s usually straight-forward to nd a **greedy algorithm** that is feasible, but hard to nd a **greedy**.

Coin-Changing: Analysis of **Greedy Algorithm** Theorem. Greed is optimal for U.S. coinage: 1, 5, 10, 25, 100. Pf. (**by induction** on x)! Consider optimal way to change c k ! x < c k+1: **greedy** takes coin k.! We claim that any optimal solution must also take coin k. –if not, it needs enough coins of type c 1, , c k-1to add up to x. Theorem A Greedy-Activity-Selector solves the activity-selection problem. Proof The proof is by induction on n. For the base case, let** n **=1. The statement trivially holds. For the induction step, let n 2, and assume that the claim holds for all values of n less than the current one. We may assume that the activities are already sorted according to. Here is an example of a complete write up of a **proof by induction** that the earliest finishing time **algorithm** finds the best solution to the activity selection problem: gas.pdf. **Greedy** Trick or Treat. **GREEDY ALGORITHMS** The **proof** of the correctness of a **greedy algorithm** is based on three main steps: 1: The **algorithm** terminates, i.e. the while.

Here is an example of a complete write up of a **proof by induction** that the earliest finishing time **algorithm** finds the best solution to the activity selection problem: gas.pdf. **Greedy** Trick or Treat. **GREEDY ALGORITHMS** The **proof** of the correctness of a **greedy algorithm** is based on three main steps: 1: The **algorithm** terminates, i.e. the while. Once you design a** greedy algorithm,** you typically need to do one of the following: 1. Prove that your** algorithm** always generates optimal solu- tions (if that is the case). 2. Prove that your** algorithm** always generates near-optimal solutions (especially if the problem is NP-hard). 3. Show by simulation that your** algorithm** generates good solutions..