**Greedy** **algorithms** , divide and conquer, dynamic programming. ... **Proof** idea 1: **algorithm** makes the best choice at each step, so it must choose the largest number of mutually compatible jobs. ... Prove by **induction** on r. Claim: m = k. Claim: The **greedy** **algorithm** returns an optimal set A.

# Greedy algorithm proof by induction

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**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. Coin-Changing: Analysis of **Greedy Algorithm**. Observation. **Greedy algorithm** is sub-optimal without nickels. Counterexample. 30¢. n **Greedy**: 25, 1, 1, 1, 1, 1. n Optimal: 10, 10, 10. n Lemma: For any optimal solution, Oi <= Gi, for 1≤i≤k (k = # breakpoints in O) (**proof by induction**). n Base case (first job). The coin of the highest value, less than the remaining change owed, is the local.

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**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.

. **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. 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).

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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. { **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**. **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. 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.

**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.

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