diff --git a/Submissions/002786959_Yuktaba_Gohil/002786959_Yuktaba_Gohil_Assignment4.ipynb b/Submissions/002786959_Yuktaba_Gohil/002786959_Yuktaba_Gohil_Assignment4.ipynb new file mode 100644 index 0000000..91bf2cb --- /dev/null +++ b/Submissions/002786959_Yuktaba_Gohil/002786959_Yuktaba_Gohil_Assignment4.ipynb @@ -0,0 +1,381 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#Assignment 4" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Question1 (20 Points)**
\n", + "**Given an undirected graph G = (V, E), a vertex-disjoint cycle-cover is a set of vertex-disjoint cycles that cover all vertices in the graph. The vertex-disjoint cycle-cover problem asks whether a given undirected graph has a vertex-disjoint cycle cover.**\n", + "\n", + "**A. (10 points) Is the vertex-disjoint cycle-cover problem in P? If so, prove it.**
\n", + "**B. (5 points) Suppose we require each cycle to have exactly four edges. We call this the 4-cycle-cover problem. Is the 4-cycle-cover problem in NP? If so, prove it.**
\n", + "**C. (10 points) Is the 4-cycle-cover problem in NP-complete? If so, prove it.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "A. The vertex-disjoint cycle-cover problem is not in P. To prove this, we will show that the problem is NP-hard. We will do this by reducing the Hamiltonian cycle problem to the vertex-disjoint cycle-cover problem.\n", + "\n", + "The Hamiltonian cycle problem is the problem of determining whether a given undirected graph G has a cycle that visits each vertex exactly once. The Hamiltonian cycle problem is known to be NP-complete.\n", + "\n", + "Given an undirected graph G = (V,E), we can construct an undirected graph G' = (V',E') as follows:\n", + "\n", + "+ V' = V
\n", + "+ E' = E ∪ {uv | u, v ∈ V and u ≠ v and u and v are not adjacent in G}
\n", + "In other words, G' is the graph G with all missing edges added.\n", + "\n", + "We claim that G has a Hamiltonian cycle if and only if G' has a vertex-disjoint cycle-cover.\n", + "\n", + "If G has a Hamiltonian cycle C, then we can construct a vertex-disjoint cycle-cover of G' by taking the following cycles:\n", + "\n", + "+ For each edge e ∈ C, we take the cycle that consists of e and its two neighbors in G'.
\n", + "\n", + "If G' has a vertex-disjoint cycle-cover C, then we can construct a Hamiltonian cycle of G by taking the following path:\n", + "\n", + "+ Start at any vertex v ∈ V.
\n", + "+ Follow the edges of the cycles in C, in any order, until you return to v.\n", + "Therefore, the vertex-disjoint cycle-cover problem is NP-hard, and therefore it is not in P.\n", + "\n", + "B. The 4-cycle-cover problem is in NP. To prove this, we will show that it is possible to verify a solution to the problem in polynomial time.\n", + "\n", + "Given an undirected graph G = (V,E) and a set of cycles C, we can check whether C is a 4-cycle-cover of G in polynomial time as follows:\n", + "\n", + "+ Check whether each cycle in C has exactly four edges.
\n", + "+ Check whether each vertex in V belongs to exactly one cycle in C.
\n", + "For the first check, we can simply count the number of edges in each cycle.
\n", + "For the second check, we can maintain a list of vertices that have already been covered by C. If a vertex belongs to more than one cycle in C, then it will be listed multiple times. If a vertex does not belong to any cycle in C, then it will not be listed at all.\n", + "\n", + "Therefore, the 4-cycle-cover problem is in NP.\n", + "\n", + "C. The 4-cycle-cover problem is NP-complete. To prove this, we will show that it is possible to reduce the Hamiltonian cycle problem to the 4-cycle-cover problem in polynomial time.\n", + "\n", + "Given an undirected graph G = (V,E), we can construct an undirected graph G' = (V',E') as follows:\n", + "\n", + "+ V' = V ∪ {v_1, ..., v_n}
\n", + "+ E' = E ∪ {uv | u ∈ V and v ∈ {v_1, ..., v_n}}\n", + "\n", + "In other words, G' is the graph G with n new vertices added, each of which is adjacent to every vertex in V.\n", + "\n", + "We claim that G has a Hamiltonian cycle if and only if G' has a 4-cycle-cover.\n", + "\n", + "If G has a Hamiltonian cycle C, then we can construct a 4-cycle-cover of G' by taking the following cycles:\n", + "\n", + "+ For each edge e ∈ C, we take the cycle that consists of e and its two neighbors in G'.
\n", + "+ For each pair of adjacent vertices u and v in C, we take the cycle that consists of the edge uv and the two new vertices v_i and v_j that are adjacent to both u and v.\n", + "\n", + "If G' has a 4-cycle-cover C, then we can construct a Hamiltonian cycle of G by taking the following path:\n", + "\n", + "+ Start at any vertex v ∈ V.
\n", + "+ Follow the edges of the cycles in C, in any order, until you return to v.\n", + "Therefore, the 4-cycle-cover problem is NP-complete." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Relevance to the Sample Problem:**\n", + "+ The problem is related to the sample problem in the sense that both are variations of the cycle-cover problem, which involves finding sets of cycles that cover all vertices in a graph.
\n", + "+ However, the sample problem dealt with directed graphs, while this problem involves undirected graphs. This distinction impacts how cycles are formed and identified within the graph due to the presence or absence of edge directions.
\n", + "+ Sample problem didn't specify an edge limit for the cycles in the cycle cover (atmost three edges), while this problem requires cycles to have a particular number of edges (exactly four edges) in the cycle cover." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used chatGPT to frame the question similar to the sample problem.\n", + "+ However, to solve the problem, I learnt various concepts like Graph Theory, NP-Completeness, Hamiltonian path and vertex-disjoint cycle-cover problem that are needed to solve the problem.\n", + "+ I have included my understanding of these concepts in the readme file." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Question2 (20 Points)**
\n", + "**The problem involves a directed graph G and k nodes s1, s2, ..., sk. The task is to determine whether there exist k disjoint cycles in the graph such that each cycle contains one of the given nodes.**\n", + "\n", + "**A. (10 points) Show that the Directed Disjoint Cycles Problem is in NP.**
\n", + "**B. (10 points) Prove that the Directed Disjoint Cycles Problem is NP-complete.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + " A. To demonstrate that the Directed Disjoint Cycles Problem belongs to the NP class, we need to show that it is possible to efficiently verify a given solution to the problem. In this case, a solution consists of a set of k disjoint cycles in the graph G, each containing one of the given nodes s1, s2, ..., sk.\n", + "\n", + "Given a proposed solution, the verification process involves checking the following conditions:\n", + "\n", + "Cycle Integrity: Ensure that each cycle forms a valid directed path, starting and ending at the same node.\n", + "\n", + "Cycle Disjointness: Verify that no two cycles share any nodes.\n", + "\n", + "Node Coverage: Confirm that each of the given nodes s1, s2, ..., sk is included in exactly one cycle.\n", + "\n", + "If all three conditions hold, the proposed solution is indeed a set of k disjoint cycles in the graph G, each containing one of the given nodes. This verification process can be performed in polynomial time, typically O(kn), where k is the number of nodes and n is the number of edges in the graph G. Therefore, the Directed Disjoint Cycles Problem is in NP.\n", + "\n", + "B. To establish that the Directed Disjoint Cycles Problem is NP-complete, we need to show that it is not only in NP but also NP-hard. This means that we must demonstrate that any problem in NP can be reduced to the Directed Disjoint Cycles Problem in polynomial time.\n", + "\n", + "The Directed Disjoint Cycles Problem can be reduced from the Directed Hamiltonian Cycle Problem, which is known to be NP-complete. The Directed Hamiltonian Cycle Problem involves determining whether a given directed graph G has a Hamiltonian cycle, which is a cycle that visits each node in the graph exactly once.\n", + "\n", + "The reduction can be performed as follows:\n", + "\n", + "Create a Modified Graph: Given a directed graph G = (V, E) from the Directed Hamiltonian Cycle Problem, create a modified graph G' = (V', E') by adding a new vertex s and directing edges from each node in V to s.\n", + "\n", + "Construct a Set of Cycles: For each node v ∈ V, construct a cycle in G' that consists of the edge (s, v) followed by a Hamiltonian path from v back to s.\n", + "\n", + "Problem Reduction: The existence of k disjoint cycles in G' that cover all nodes, including s, corresponds to the existence of k Hamiltonian cycles in G that cover all nodes.\n", + "\n", + "Therefore, the Directed Disjoint Cycles Problem is NP-complete, as it is in NP and can be reduced from an NP-complete problem in polynomial time." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Relevance to the Sample Problem:**\n", + "\n", + " + Sample problem shares fundamental characteristics and principles with this problem.\n", + " + The sample problem aims to find node-disjoint paths between specified pairs of nodes while this problem focuses on identifying disjoint cycles in the graph, each containing one of the specified nodes.\n", + " + While both problems are in NP due to their verifiability in polynomial time, the proofs of NP-completeness would involve different reductions because they aim for distinct objectives (disjoint paths vs. disjoint cycles)." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "+ I have used chatGPT to create a similar question like the sample problem.\n", + "+ The problem aligns with fundamental concepts of directed graphs, exploring connectivity, disjointness, and structural elements like cycles.\n", + "+ While researching, I also came to know that Problems related to disjoint cycles have practical applications in network routing, circuit design, and optimization, contributing to their significance beyond theoretical considerations." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Question3 (20 Points)**
\n", + "**You are organizing a tech conference covering various topics (e.g., machine learning, cybersecurity, blockchain, cloud computing, etc.). You have received applications from potential speakers, and for each of the n topics, there is a subset of potential speakers qualified to present on it. The goal is to select at most k speakers to cover all n topics. We’ll call this the Minimum Speaker Set.**\n", + "\n", + "**A. (10 points) Show that the Minimum Speaker Set problem is in NP.**\n", + "**B. (10 points) Prove that the Minimum Speaker Set problem is NP-complete.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "A. To demonstrate that the Minimum Speaker Set problem belongs to the NP class, we need to establish that a potential solution, represented by a set of k speakers, can be efficiently verified in polynomial time.\n", + "\n", + "Given an input consisting of n topics and a subset of potential speakers qualified for each topic, along with a proposed solution consisting of a set of k speakers S, the verification procedure can be performed as follows:\n", + "\n", + "+ Check Speaker Coverage: Verify that each of the n topics is covered by at least one speaker in the set S. This can be done by checking whether each topic appears in the qualifications of at least one speaker in S.\n", + "\n", + "+ Check Speaker Count: Verify that the number of speakers in S does not exceed k. This can be done by simply counting the number of speakers in S.\n", + "\n", + "If both checks pass, then the proposed solution is a valid set of k speakers that covers all n topics. Therefore, the Minimum Speaker Set problem is in NP.\n", + "\n", + "B. To establish that the Minimum Speaker Set problem is NP-complete, we need to show that it is not only in NP but also NP-hard. This means that we must demonstrate that any problem in NP can be reduced to the Minimum Speaker Set problem in polynomial time.\n", + "\n", + "The Minimum Speaker Set problem can be reduced from the Independent Set Problem, which is known to be NP-complete. The Independent Set Problem involves determining whether a given undirected graph G has an independent set of size k, which is a set of k vertices in the graph such that no two vertices in the set are adjacent.\n", + "\n", + "The reduction can be performed as follows:\n", + "\n", + "+ Create a Bipartite Graph: Given an undirected graph G = (V, E) from the Independent Set Problem, create a bipartite graph G' = (V', E') by adding a new vertex t and connecting it to every vertex in V.\n", + "\n", + "+ Construct a Speaker Set: For each vertex v ∈ V, create a speaker with a qualification that includes the topic associated with v.\n", + "\n", + "+ Problem Reduction: The existence of a set of k speakers that covers all n topics corresponds to the existence of an independent set of size k in G'.\n", + "\n", + "Therefore, the Minimum Speaker Set problem is NP-complete, as it is in NP and can be reduced from an NP-complete problem in polynomial time." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Relevance to the Sample Problem:**\n", + "\n", + "+ Sample problem and this problem involve the selection of individuals (instructors/speakers) based on their qualifications (skills/topics) to cover specific requirements (skills/topics) within a limited number constraint.
\n", + "+ Sample problems and this problem rely on a similar structural framework of selecting individuals to cover specific requirements within given constraints." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used chatGPT to create a similar question to the sample problem.
\n", + "+ I think that the problem scenario of organizing a tech conference and selecting speakers aligns with real-world situations, reflecting practical considerations in event planning.
\n", + "+ The need for selected speakers to cover all topics emphasizes the optimization aspect under constraints, requiring an optimal subset of speakers.
\n", + "+ Also, the constraint on the maximum number of speakers adds another layer of challenge in selecting an efficient set that covers all topics." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Question4 (20 Points)**
\n", + "**You're organizing a summer music camp with various music ensembles, covering instruments like violin, piano, flute, and more. You've received applications from potential musicians. For each of the n instruments, there is a subset of the m applicants qualified to play it. The task is to select at most k musicians to form an ensemble, ensuring that there's at least one musician qualified in each of the n instruments. We’ll call this the Ensemble Formation Problem.**\n", + "\n", + "**A. (10 points) Show that the Ensemble Formation Problem is in NP.**\n", + "**B. (10 points) Prove that the Ensemble Formation Problem is NP-complete.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "A. To demonstrate that the Ensemble Formation Problem belongs to the NP class, we need to establish that a potential solution, represented by a set of k musicians, can be efficiently verified in polynomial time.\n", + "\n", + "Given an input consisting of n instruments and a subset of potential musicians qualified for each instrument, along with a proposed solution consisting of a set of k musicians S, the verification procedure can be performed as follows:\n", + "\n", + "1. Check Instrument Coverage: Verify that each of the n instruments is played by at least one musician in the set S. This can be done by checking whether each instrument appears in the qualifications of at least one musician in S.\n", + "\n", + "2. Check Musician Count: Verify that the number of musicians in S does not exceed k. This can be done by simply counting the number of musicians in S.\n", + "\n", + "If both checks pass, then the proposed solution is a valid set of k musicians that covers all n instruments.Given a set of k musicians, we can check in polynomial time if they cover all n instruments as per the qualifications specified. This verification process confirms the solution's validity in polynomial time, making the problem verifiable in NP. Therefore, the Ensemble Formation Problem is in NP.\n", + "\n", + "B. To establish that the Ensemble Formation Problem is NP-complete, we need to show that it is not only in NP but also NP-hard. This means that we must demonstrate that any problem in NP can be reduced to the Ensemble Formation Problem in polynomial time.\n", + "\n", + "The Ensemble Formation Problem can be reduced from the Independent Set Problem, which is known to be NP-complete. The Independent Set Problem involves determining whether a given undirected graph G has an independent set of size k, which is a set of k vertices in the graph such that no two vertices in the set are adjacent.\n", + "\n", + "The reduction can be performed as follows:\n", + "\n", + "1. Create a Bipartite Graph: Given an undirected graph G = (V, E) from the Independent Set Problem, create a bipartite graph G' = (V', E') by adding a new vertex m and connecting it to every vertex in V.\n", + "\n", + "2. Construct a Musician Set: For each vertex v ∈ V, create a musician with a qualification that includes the instrument corresponding to v.\n", + "\n", + "3. Problem Reduction: The existence of a set of k musicians that covers all n instruments corresponds to the existence of an independent set of size k in G'.\n", + "\n", + "Therefore, the Ensemble Formation Problem is NP-complete, as it is in NP and can be reduced from an NP-complete problem in polynomial time." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Relevance to the Sample Problem:**\n", + "\n", + "+ Both problems involve selecting a subset of individuals (counselors/musicians) based on their qualifications to cover specific areas (sports skills/musical instrument expertise).
\n", + "+ The sample problem aims to ensure at least one counselor skilled in each of the covered sports.
\n", + "+ This problem aims to form an ensemble by selecting at most k musicians, ensuring representation from each instrument category." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used chatGPT to create a similar question like the sample problem.\n", + "+ I learned how to choose groups of musicians for different instruments, like forming bands or orchestras.\n", + "+ Also, I figured out that verifying solutions was quick, which means it's in the NP category.\n", + "+ I also faced the challenge of picking the best musicians while having limits on how many I could pick.\n", + "+ The main concepts I learned while solving this problem are Constraint-Based Selection and Efficient Subset selection which I have explained in my readme file." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Question5 (20 Points)**
\n", + "**Suppose you are the manager of a small software development team with n developers and n projects. Each developer has expertise in a specific subset of the projects, and each project requires a developer with the necessary expertise to complete it. The goal is to assign each developer to one project, ensuring that each project is assigned to a developer with the required expertise.**\n", + "\n", + "**Part A (10 Points)**\n", + "\n", + "**Express this problem as a maximum flow problem that maximizes the number of projects that can be assigned to developers.**\n", + "\n", + "**Part B (10 Points)**\n", + "\n", + "**Can all n projects always be assigned to n developers? Prove that it can or cannot.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "A. To model this problem as a maximum flow problem, we can create a directed graph with three types of nodes:\n", + "\n", + "1. Developer Nodes: Represent each of the n developers as a node.\n", + "\n", + "2. Project Nodes: Represent each of the n projects as a node.\n", + "\n", + "3. Source Node: Create a single source node connected to each developer node with an edge capacity of 1. This represents the fact that each developer can be assigned to one project.\n", + "\n", + "4. Sink Node: Create a single sink node connected to each project node with an edge capacity of 1. This represents the fact that each project can be assigned to one developer.\n", + "\n", + "5. Directed Edges: For each developer-project pair where the developer has the necessary expertise for the project, create a directed edge from the developer node to the project node with an edge capacity of 1. This represents the possibility of assigning that developer to that project.\n", + "\n", + "By finding the maximum flow in this graph, we can determine the maximum number of projects that can be assigned to developers. The maximum flow will correspond to the number of edges that are saturated, representing the number of successful assignments.\n", + "\n", + "B. Whether or not all n projects can always be assigned to n developers depends on the specific expertise requirements of the developers and the projects. In general, it is not always possible to assign all projects, as there may be cases where a developer's expertise does not match the requirements of any project, or where a project requires expertise that no developer possesses.\n", + "\n", + "To analyze this further, consider the following cases:\n", + "\n", + "1. Over-subscribed Projects: If a project requires expertise that multiple developers possess, then there may not be enough developers to assign to all projects. This can lead to a situation where some projects cannot be assigned to any developer, even though there are enough developers in total.\n", + "\n", + "2. Underqualified Developers: If a developer's expertise is very limited, they may not be able to handle any of the projects, leading to an unassigned project. This can happen if the projects require more specialized skills than the developers possess.\n", + "\n", + "3. Uneven Expertise Distribution: If the developers' expertise is not evenly distributed across the projects, there may be a mismatch between the available expertise and the project requirements. This can lead to a situation where some projects cannot be assigned to any developer.\n", + "\n", + "Therefore, it is not always possible to assign all n projects to n developers in a feasible manner. The feasibility depends on the specific requirements of the projects and the expertise of the developers." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Relevance to the Sample Problem:**\n", + "\n", + "+ Sample problem and this problem aim to assign resources (cooks or developers) to tasks (nights or projects) in an efficient and optimal manner.
\n", + "+ Both problems involve constraints that limit the assignment possibilities. In the sample problem, the constraints are represented by the nights when each person is not available to cook. In this problem, the constraints are represented by the expertise requirements of each developer and each project.\n", + "+ However, The resources in the sample problem are people with varying availability, while the resources in this problem are developers with specific expertise.\n", + "+ The tasks in the sample problem are nights that need to be covered by cooks, while the tasks in this problem are projects that require developers with the necessary expertise.\n", + "+ The sample problem can be solved using a maximum flow algorithm, while this problem may require more sophisticated techniques, such as bipartite matching or network flow algorithms." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used chatGPT to create a similar question to this one.\n", + "+ As mentioned in the relevance section, techniques such as bipartite matching helped me solve this problem whic I have included in my read me file.\n", + "+ I also applied maximum flow algorithm to determine maximum number of assignments and interpreted the maximum flow value in terms of project assignments." + ] + } + ], + "metadata": { + "language_info": { + "name": "python" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/Submissions/002786959_Yuktaba_Gohil/002786959_Yuktaba_Gohil_Assignment4.md b/Submissions/002786959_Yuktaba_Gohil/002786959_Yuktaba_Gohil_Assignment4.md new file mode 100644 index 0000000..00769f8 --- /dev/null +++ b/Submissions/002786959_Yuktaba_Gohil/002786959_Yuktaba_Gohil_Assignment4.md @@ -0,0 +1,34 @@ +In the process of completing this assignment, I learnt several key concepts related to which I would like to mention here. Through various tasks and questions, I gained a deeper understanding of the following topics: + +**Graph Theory:** + Graphs are mathematical structures that consist of vertices (nodes) and edges (links). Vertex-disjoint cycle-covers are a type of graph covering, where the goal is to cover all vertices of the graph using a set of + cycles that do not share any vertices. + + **NP-Completeness:** + + NP-completeness is a classification of problems in computational complexity theory. A problem is said to be in NP if its solutions can be verified efficiently (in polynomial time) by a deterministic Turing machine, but it is unknown whether they can be found efficiently (in polynomial time) by a deterministic Turing machine. + + No efficient algorithm has been found for any NP-complete problem, and it is widely believed that no such algorithm exists. However, efficient algorithms exist for verifying solutions to NP-complete problems. + + **Hamiltonian Path and Cycle:** + + In graph theory, a Hamiltonian path is a path in a graph that visits each vertex exactly once. A Hamiltonian cycle is a cycle in a graph that visits each vertex exactly once. + + The Hamiltonian path problem is the problem of determining whether a given directed or undirected graph has a Hamiltonian path. The Hamiltonian cycle problem is the problem of determining whether a given directed or undirected graph has a Hamiltonian cycle. + + **Vertex-Disjoint Cycle-Cover Problem:** + + The vertex-disjoint cycle-cover problem is a generalization of the Hamiltonian path and cycle problems. In this problem, the goal is to find a set of cycles in a graph that cover all vertices of the graph, such that no two cycles share any vertices. + +**Directed Hamiltonian Cycle Problem:** ++ The Directed Hamiltonian Cycle Problem is the problem of determining whether a given directed graph has a Hamiltonian cycle. + +**Directed Disjoint Cycles Problem:** ++ The Directed Disjoint Cycles Problem involves determining whether there exist k disjoint cycles in a directed graph such that each cycle contains one of k given nodes. + +**NP-hardness:** ++ A problem is said to be NP-hard if any problem in NP can be reduced to it in polynomial time. + +**Constraint-Based Selection:** ++ Leveraged the concept of selecting a subset of speakers under constraints, akin to choosing a limited number of musicians for an orchestra ensemble. + +**Efficient Subset Selection:** ++ Learned to strategically select subsets by balancing criteria fulfillment and constraint limitations, similar to the challenge of forming an efficient orchestra ensemble. + +**Bipartite Matching:** ++ A technique for matching pairs of nodes in a bipartite graph, ensuring that no two nodes in the same set are matched. \ No newline at end of file diff --git a/Submissions/002786959_Yuktaba_Gohil/002786959_Yuktaba_Gohil_Assignment5.ipynb b/Submissions/002786959_Yuktaba_Gohil/002786959_Yuktaba_Gohil_Assignment5.ipynb new file mode 100644 index 0000000..b0d18d9 --- /dev/null +++ b/Submissions/002786959_Yuktaba_Gohil/002786959_Yuktaba_Gohil_Assignment5.ipynb @@ -0,0 +1,1166 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q1 (5 Points)**
\n", + "**Match the following definitions:**
\n", + "**i. QuickSort**
\n", + "**ii. Breadth-First Search (BFS)**
\n", + "**iii. Prim's Algorithm**
\n", + "**iv. Merge Sort**
\n", + "**v. Depth-First Search (DFS)**\n", + "\n", + "**Match each concept with its description.**
\n", + "**A. An algorithm for finding the minimum spanning tree in a connected weighted graph.**
\n", + "**B. An algorithm that sorts a list by recursively dividing the list into smaller sublists, then merging them.**
\n", + "**C. An algorithm that explores a graph by prioritizing visiting neighboring nodes before visiting deeper nodes.**
\n", + "**D. A divide-and-conquer algorithm that sorts a list by dividing it into smaller sublists and sorting those sublists.**
\n", + "**E. An algorithm for sorting that repeatedly selects a pivot element and partitions the list around the pivot.**
" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "i-E\n", + "ii-C\n", + "iii-A\n", + "iv-D\n", + "v-B\n", + "\n", + "i. QuickSort
\n", + "An algorithm for sorting that repeatedly selects a pivot element and partitions the list around the pivot.\n", + "\n", + "ii. Breadth-First Search (BFS)
\n", + "An algorithm that explores a graph by prioritizing visiting neighboring nodes before visiting deeper nodes.\n", + "\n", + "iii. Prim's Algorithm
\n", + "An algorithm for finding the minimum spanning tree in a connected weighted graph\n", + "\n", + "iv. Merge Sort
\n", + "A divide-and-conquer algorithm that sorts a list by dividing it into smaller sublists and sorting those sublists.\n", + "\n", + "v. Depth-First Search (DFS)
\n", + "An algorithm that sorts a list by recursively dividing the list into smaller sublists, then merging them." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used ChatGPT to form an question and then used google to confirm my answer" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q2 (5 Points)**
\n", + "**Given the weights and values of five items in the table below, determine the subset of items with the maximum combined value that fits in a knapsack with a weight limit, W, of 8. Use dynamic programming to solve the problem.**
\n", + "| Item | Value (vi) | Weight (wi) |\n", + "| ------ | ---------- | ----------- |\n", + "| Item 1 | 4 | 3 |\n", + "| Item 2 | 5 | 4 |\n", + "| Item 3 | 7 | 2 |\n", + "| Item 4 | 8 | 5 |\n", + "| Item 5 | 3 | 2 |\n", + "\n", + "**Capacity of the knapsack, W = 8**\n", + "\n", + "**Select which of the following statements are true:**\n", + "\n", + "**A. Either {1,3} or {2,5} is a subset of items with the maximum combined value.**
\n", + "**B. The maximum combined value is 15.**
\n", + "**C. Item 4 will be in the subset of items with the maximum combined value.**
\n", + "**D. The maximum combined value of items 1, 2, and 3 is 16 if the capacity of the knapsack is W = 8.**
\n", + "**E. The maximum combined value of items 3 and 5 is 10 if the capacity of the knapsack is W = 8.**
" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "A. Either {1,3} OR {2,5} is a subset of items with the maximum combined value. (T)
\n", + "B. The maximum combined value is 16. (T)
\n", + "C. Item 4 will be in the subset of items with the maximum combined value. (T)
\n", + "D. The maximum combined value of items 1, 2, and 3 is 16 if the capacity of the knapsack is W = 8. (T)
\n", + "E. The maximum combined value of items 3 and 5 is 10 if the capacity of the knapsack is W = 8. (F)\n", + "Capacity of the knapsack W = 8\n", + "Algorithm: given two arrays w[3, 4, 2, 5, 2] and v[4, 5, 7, 8, 3]:\n", + "for i ← 1 to n:\n", + "for x ← 0 to W:\n", + "if w[i] > x:\n", + "OPT[i][x] ← OPT[i - 1][x]\n", + "else:\n", + "OPT[i][x] ← max(OPT[i - 1][x], OPT[i - 1][x - w[i]] + v[i])\n", + "\n", + "Resulting OPT table:\n", + "| 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |\n", + "|---|---|----|----|----|----|----|----|----|\n", + "| 8 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 9 |\n", + "| 8 | 8 | 15 | 15 | 15 | 15 | 15 | 15 | 15 |\n", + "| 8 | 8 | 15 | 15 | 16 | 16 | 23 | 23 | 23 |\n", + "| 8 | 8 | 15 | 15 | 16 | 23 | 23 | 24 | 24 |\n", + "| 8 | 8 | 15 | 15 | 16 | 23 | 23 | 24 | 24 |\n", + "\n", + "The maximum combined value achievable in the knapsack with a weight limit of 8 is 16. By tracing back through the table, we can determine that selecting items 1, 2, and 3 yields this maximum combined value." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used ChatGPT to form a question and revised the knap-sack concept to solve this problem" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q3 (5 Points)**
\n", + "**Master Theorem: For each of the following recurrences, give an expression for the runtime T(n) if the recurrence can be solved with the Master Theorem. Otherwise, indicate that the Master Theorem doesnot apply.**
\n", + "**i. T(n) = 3T (n/2)+ n**
\n", + "**ii. T(n) = 2T (n/2)+ n**
\n", + "**iii. T(n) = n3 T (n/2) + n**
\n", + "**iv. T(n) = 8T (n/3) + n3**
\n", + "**v. T(n) = 27T (n/3) + n3**
\n", + "**Match the θ with the recurrence**
\n", + "**A. θ(n3)**
\n", + "**B. Does not apply**
\n", + "**C. θ(nlog3)**
\n", + "**D. θ(n3 log n)**
\n", + "**E. θ(n log n)**
\n", + "\n", + "***Solution:***\n", + "\n", + "i- θ(nlog3 ) which is C\n", + "ii- θ(n log n) which is E\n", + "iii- Does not apply which is B\n", + "iv- θ(n3) which is A\n", + "v- θ(n3 log n) which is D\n", + "\n", + "i. T (n) = 3T (n/2) + n =⇒ T (n) = Θ(n lg 3 ) (Case 1)\n", + "k = logb a = log2 3 = log(3)/log(2) > 1\n", + "f(n) is θ(n logb a ) which is θ(nlog3)\n", + "Therefore the runtime T(n) is θ(nlog3)\n", + "\n", + "ii. T(n) = 2T (n/2)+ n\n", + "a=b=2; k = logb a = log2 2 = log(2)/log(2) = 1\n", + "Case 2 again. p=0. f(n) = n2 which is θ(n)\n", + "Therefore the runtime T(n) is θ(n log n)\n", + "\n", + "iii. T(n) = n3T (n/2) + n\n", + "Does not apply (a is not constant)\n", + "\n", + "iv. T(n) = 8T (n/3) + n3 (Case 3)\n", + "k = logb a = log3 8 = log(8)/log(3) < 3\n", + "f(n) is θ(n3)\n", + "Therefore the runtime T(n) is θ(n3)\n", + "\n", + "v. T(n) = 27T (n/3)+ + n3\n", + "k = logb a = log3 27 = log(27)/log(3) = 3\n", + "Case 2 again. p=0. f(n) = n3\n", + "Therefore the runtime T(n) is θ(n3 log n)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used ChatGPT to form a question and then revised Masters Theorem to solve the question" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q4 (5 Points)**
\n", + "**Arrange the following functions in increasing order of growth:**
\n", + "+ 4n\n", + "+ 2n/2\n", + "+ n2 + n\n", + "+ 10nlogn\n", + "+ 3n\n", + "+ n!\n", + "+ $\\sqrt{n}$\n", + "+ n3" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "Polynomial time:
\n", + "$\\sqrt{n}$,4n,n2+n,10nlogn
\n", + "Note: $\\sqrt{n}$ grows slower than linear, while n2+n is between 4n and 10nlogn\n", + "\n", + "Between Polynomial and Exponential time:
\n", + "2n/2 lies between polynomial and exponential growth rates.\n", + "\n", + "Exponential time:
\n", + "n!,3n\n", + "Note: n! grows faster than any polynomial or exponential function.\n", + "\n", + "Final order:
\n", + "$\\sqrt{n}$,4n,n2+n,2n/2,10nlogn,n3, 3n,n!" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used ChatGPT to form this question and then revised my concrpts to solve it." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q5 (5 Points)**
\n", + "**Which of the following statements accurately describes Greedy Algorithms?**
\n", + "**A. Greedy Algorithms always guarantee the globally optimal solution for every problem they are applied to.**
\n", + "**B. Greedy Algorithms make the locally optimal choice at each step with the hope of finding a globally optimal solution.**
\n", + "**C. Greedy Algorithms are only applicable to problems with a single objective, not multiple objectives.**
\n", + "**D. Greedy Algorithms are particularly efficient for solving problems involving backtracking and recursion.**
\n", + "**E. Greedy Algorithms require exhaustive search through all possible solutions to find the best outcome.**
" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "B. Greedy Algorithms make the locally optimal choice at each step with the hope of finding a globally optimal solution.\n", + "\n", + "Explanation:\n", + "\n", + "+ Greedy algorithms make decisions based on the best choice available at each step without reconsidering or backtracking. It selects the locally optimal solution in the hope that it leads to the globally optimal solution.\n", + "+ While this strategy might not always guarantee the globally optimal solution, in many cases, it does produce satisfactory results.\n", + "+ Greedy algorithms are used in a variety of problems such as finding the shortest path in graph algorithms (Dijkstra's algorithm), the minimum spanning tree (Prim's or Kruskal's algorithm), and the fractional knapsack problem.\n", + "\n", + "The other options explained:\n", + "\n", + "A. Greedy Algorithms don't always guarantee the globally optimal solution; they focus on local optimality and might not lead to the best overall solution.
\n", + "C. Greedy Algorithms can be applied to problems with multiple objectives but may not always provide the best solution for all objectives simultaneously.
\n", + "D. Greedy Algorithms are not specifically designed for problems involving backtracking and recursion; they are different algorithmic techniques.
\n", + "E. Greedy Algorithms do not involve exhaustive search through all possible solutions; they make immediate optimal choices at each step without considering future consequences exhaustively.
" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ In creating a similar question related to algorithms, using ChatGPT was immensely helpful. \n", + "+ It offered insights into the core concepts of algorithms, aiding in crafting multiple-choice options that were both plausible and distinctive.\n", + "+ The tool provided guidance on structuring the question to ensure it remained relevant and challenging." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q6 (5 Points)**
\n", + "**Consider the following directed graph N with 4 vertices. Its adjacency matrix N is shown below. Which of the following statements is true?**\n", + "\n", + " | | 1 | 2 | 3 | 4 |\n", + " |---|---|---|---|---|\n", + " | 1 | 0 | 1 | 0 | 0 |\n", + " | 2 | 0 | 0 | 1 | 1 |\n", + " | 3 | 1 | 0 | 0 | 0 |\n", + " | 4 | 0 | 0 | 0 | 0 |\n", + "\n", + "**A. Graph N contains a Hamiltonian cycle.**
\n", + "**B. Graph N has a unique Eulerian path.**
\n", + "**C. Graph N has no cycles.**
\n", + "**D. Graph N has a unique minimum spanning tree.**
\n", + "**E. Graph N has a total of 6 distinct paths from one vertex to another.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "E. Graph N has a total of 6 distinct paths from one vertex to another.\n", + "\n", + "Explanation:\n", + "\n", + "A. Graph N contains a Hamiltonian cycle.\n", + "\n", + "False. A Hamiltonian cycle visits each vertex exactly once and returns to the starting vertex, which is not present in this graph. There's no cycle that traverses all vertices exactly once and returns to the starting point.\n", + "\n", + "B. Graph N has a unique Eulerian path.\n", + "\n", + "False. An Eulerian path visits every edge exactly once. This graph does not have an Eulerian path because it has vertices with in-degree not equal to out-degree.\n", + "\n", + "C. Graph N has no cycles.\n", + "\n", + "False. There are cycles present in this graph. For instance, vertex 1 connects back to itself (a self-loop), forming a cycle.\n", + "\n", + "D. Graph N has a unique minimum spanning tree.\n", + "\n", + "Not applicable. Minimum spanning trees are defined for undirected graphs, and this is a directed graph. The concept of a minimum spanning tree doesn't directly apply here.\n", + "\n", + "E. Graph N has a total of 6 distinct paths from one vertex to another.\n", + "\n", + "True. There are various paths between different vertices in this graph. For instance, considering distinct paths from vertex 1 to vertex 2, we have two paths: 1->2 and 1->3->2, totaling six distinct paths between different pairs of vertices in the graph.\n", + "Therefore, the correct statement among the options provided is:\n", + "\n", + "E. Graph N has a total of 6 distinct paths from one vertex to another." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality**\n", + "\n", + "+ I used chatGPT to create this question.\n", + "+ It helped me to construct statements that accurately tested various properties of directed graphs. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q7 (5 Points)**
\n", + "**There are 4 men (M1, M2, M3, M4) and 4 women (W1, W2, W3, W4). Given their preferences:**\n", + "\n", + "**M1: W2 > W1 > W3 > W4     W1: M2 > M3 > M1 > M4**
\n", + "**M2: W1 > W3 > W2 > W4     W2: M1 > M2 > M3 > M4**
\n", + "**M3: W1 > W2 > W4 > W3     W3: M3 > M1 > M2 > M4**
\n", + "**M4: W3 > W2 > W1 > W4     W4: M3 > M1 > M4 > M2**
\n", + "\n", + "**Select the statements that are true:**\n", + "\n", + "**A. If all men propose to their first choice, there will be a woman who has multiple proposals.**
\n", + "**B. If all women propose to their first choice, there will be a man who has multiple proposals.**
\n", + "**C. The stable matching is (M1, W2), (M2, W1), (M3, W3), (M4, W4).**
\n", + "**D. The stable matching is (M1, W1), (M2, W2), (M3, W3), (M4, W4).**
\n", + "**E. The current matchings are NOT stable.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "Given the preferences of men and women, let's examine each statement:\n", + "\n", + "A. If all men propose to their first choice, there will be a woman who has multiple proposals.\n", + "\n", + "True. For instance, if all men propose to their top choice women, both W1 and W3 receive proposals from multiple men (M1 and M2 propose to W1, while M3 and M4 propose to W3).\n", + "\n", + "B. If all women propose to their first choice, there will be a man who has multiple proposals.\n", + "\n", + "True. If all women propose to their top choice men, both M3 and M4 receive proposals from multiple women (W1 and W2 propose to M3, while W3 and W4 propose to M4).\n", + "\n", + "C. The stable matching is (M1, W2), (M2, W1), (M3, W3), (M4, W4).\n", + "\n", + "False. This matching isn't stable as there exist unstable pairs. For instance, M1 prefers W1 over W2, and W1 prefers M2 over M1, creating an instability.\n", + "\n", + "D. The stable matching is (M1, W1), (M2, W2), (M3, W3), (M4, W4).\n", + "\n", + "False.This option was not provided in the initial choices.\n", + "\n", + "E. The current matchings are NOT stable.\n", + "\n", + "True. The matchings are unstable due to preferences that lead to preferences of some individuals being unfulfilled, causing instability in certain pairs.\n", + "\n", + "Therefore, among the given choices:\n", + "\n", + "A. If all men propose to their first choice, there will be a woman who has multiple proposals. - True
\n", + "B. If all women propose to their first choice, there will be a man who has multiple proposals. - True
\n", + "C. The stable matching is (M1, W2), (M2, W1), (M3, W3), (M4, W4). - False
\n", + "D. The stable matching is (M1, W1), (M2, W2), (M3, W3), (M4, W4). - False
\n", + "E. The current matchings are NOT stable. - True
" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality**\n", + "\n", + "+ I used ChatGPT to create this question.\n", + "+ This question involves assessing the outcomes of the Gale-Shapley algorithm and determining the stability of the resulting matchings based on the given preferences of individuals." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q8 (5 Points)**
\n", + "**In Dijkstra's algorithm for finding the shortest path in a weighted graph, which of the following statements is true?**\n", + "\n", + "**A. Dijkstra's algorithm guarantees finding the shortest path in a graph with negative edge weights.**
\n", + "**B. The algorithm always starts traversing the graph from the node with the lowest value.**
\n", + "**C. Dijkstra's algorithm fails when encountering a graph with cycles.**
\n", + "**D. It requires the graph to have non-negative edge weights for it to find the shortest path.**
\n", + "**E. It's suitable only for directed graphs and not for undirected ones.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution***\n", + "\n", + "\n", + "Let's evaluate each statement regarding Dijkstra's algorithm for finding the shortest path in a weighted graph:\n", + "\n", + "A. Dijkstra's algorithm guarantees finding the shortest path in a graph with negative edge weights.\n", + "\n", + "False. Dijkstra's algorithm assumes non-negative edge weights. It doesn't work correctly with negative edge weights because it prioritizes nodes based on the cumulative path cost, which might lead to incorrect results with negative weights.\n", + "\n", + "B. The algorithm always starts traversing the graph from the node with the lowest value.\n", + "\n", + "True. Dijkstra's algorithm uses a priority queue or a similar data structure to prioritize nodes based on their current calculated distance from the source node. It starts from the node with the lowest known distance.\n", + "\n", + "C. Dijkstra's algorithm fails when encountering a graph with cycles.\n", + "\n", + "False. Dijkstra's algorithm can handle graphs with cycles as long as the graph doesn't have negative-weight cycles. It might not work correctly with negative cycles due to the concept of minimizing path distances.\n", + "\n", + "D. It requires the graph to have non-negative edge weights for it to find the shortest path.\n", + "\n", + "True. Dijkstra's algorithm assumes non-negative edge weights. This assumption is essential for its correctness because it explores nodes based on their cumulative distances from the source.\n", + "\n", + "E. It's suitable only for directed graphs and not for undirected ones.\n", + "\n", + "False. Dijkstra's algorithm can work on both directed and undirected graphs as long as the edge weights are non-negative and the graph structure is appropriately represented.\n", + "\n", + "Therefore, the correct statements regarding Dijkstra's algorithm are:\n", + "\n", + "B. The algorithm always starts traversing the graph from the node with the lowest value.
\n", + "D. It requires the graph to have non-negative edge weights for it to find the shortest path." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used ChatGPT to create a question.\n", + "+ These statements accurately describe key aspects of Dijkstra's algorithm, emphasizing its operation based on node priorities and its dependence on non-negative edge weights for correct results." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q9 (5 Points)**
\n", + "**Here's a set of statements related to graph theory and algorithms:**\n", + "\n", + "**Statement 1: In graph Y, an undirected graph with vertices {A, B, C, D, E}, every edge has a distinct weight. AB is the edge with the minimum weight, and DE is the edge with the maximum weight. DE is part of the minimum spanning tree (MST) if and only if it is a cut edge.**
\n", + "**Statement 2: Kruskal's Algorithm for finding an MST only works with directed graphs.**
\n", + "**Statement 3: In graph Y, if the Bellman-Ford Algorithm decreases the distance even after V+1 iterations, then there exists a negative weight cycle in Y.**
\n", + "**Statement 4: A negative weight edge can exist in a Minimum Spanning Tree (MST).**
\n", + "**Statement 5: The Bellman-Ford Algorithm can be used to find the shortest path in a weighted graph with negative edge weights.**\n", + "\n", + "**A. Statement 1**
\n", + "**B. Statement 2**
\n", + "**C. Statement 3**
\n", + "**D. Statement 4**
\n", + "**E. Statement 5**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "let's evaluate the correctness of each statement:\n", + "\n", + "Statement 1: \"In graph Y, an undirected graph with vertices {A, B, C, D, E}, every edge has a distinct weight. AB is the edge with the minimum weight, and DE is the edge with the maximum weight. DE is part of the minimum spanning tree (MST) if and only if it is a cut edge.\"\n", + "\n", + "Correctness: This statement is correct. In an undirected graph, an edge is a cut edge if removing it disconnects the graph. In a minimum spanning tree, every edge is crucial for connectivity. So, an edge being part of the MST implies it's not a cut edge, and vice versa.\n", + "\n", + "Statement 2: \"Kruskal's Algorithm for finding an MST only works with directed graphs.\"\n", + "\n", + "Correctness: False. Kruskal's Algorithm works with undirected graphs.\n", + "\n", + "Statement 3: \"In graph Y, if the Bellman-Ford Algorithm decreases the distance even after V+1 iterations, then there exists a negative weight cycle in Y.\"\n", + "\n", + "Correctness: This statement is correct. The Bellman-Ford Algorithm detects the presence of negative weight cycles by checking if the distance decreases after V+1 iterations. If so, it indicates the existence of a negative weight cycle.\n", + "\n", + "Statement 4: \"A negative weight edge can exist in a Minimum Spanning Tree (MST).\"\n", + "\n", + "Correctness: False. In an MST, all edge weights must be non-negative.\n", + "\n", + "Statement 5: \"The Bellman-Ford Algorithm can be used to find the shortest path in a weighted graph with negative edge weights.\"\n", + "\n", + "Correctness: This statement is correct. The Bellman-Ford Algorithm can handle negative edge weights and find the shortest path even in graphs with negative weights, as long as there are no negative weight cycles." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used ChatGPT to create question.\n", + "+ These examples illustrate scenarios that correspond to the statements provided, showcasing various properties and behaviors of algorithms and graph structures." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q10 (5 Points)**
\n", + "**Algorithm A: You check each item in the list individually, starting from the first item, and inquire if it's the desired item. If any item matches, you stop and confirm its presence.**\n", + "\n", + "**Algorithm B: You randomly select a position in the list and ask if the item at that position matches the desired item. If not, you proceed to another random position until a match is found.**\n", + "\n", + "**Algorithm C: You start by checking the first item. If it doesn't match, you ask the second person to check the next item. If they don't find a match, they ask the third person, and so on until someone confirms the presence or absence of the item.**\n", + "\n", + "**Which of the following represents the ordering of the algorithms from fastest to slowest?**\n", + "\n", + "**Choose the correct sequence:**\n", + "\n", + "**A.     A, B, C**
\n", + "**B.     B, A, C**
\n", + "**C.     A, C, B**
\n", + "**D.     C, A, B**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "+ Fastest: Algorithm B (Random selection can be faster in some scenarios)\n", + "+ Intermediate: Algorithm A (Sequential search takes a predictable time)\n", + "+ Slowest: Algorithm C (Dependent on linear inquiry down a line)\n", + "\n", + "Algorithm A:\n", + "+ Description: Sequentially check each item in the list.\n", + "+ Example: Suppose you're searching for the number 7 in the list [3, 5, 8, 2, 7, 9].\n", + "+ Process: Start from the first item and check each one until you find a match (7 in this case).\n", + "+ Evaluation: This algorithm typically takes O(n) time in the worst case.\n", + "\n", + "Algorithm B:\n", + "+ Description: Randomly select positions and check.\n", + "+ Example: Searching for the number 7 in the list [3, 5, 8, 2, 7, 9].\n", + "+ Process: Randomly select positions until you find a match (7 in this case).\n", + "+ Evaluation: The time complexity can vary; in the worst case, it might take O(n), but it can be quicker with luck.\n", + "\n", + "Algorithm C:\n", + "+ Description: Chain of inquiries down a line until a match is found.\n", + "+ Example: Seeking the number 7 in the list [3, 5, 8, 2, 7, 9].\n", + "+ Process: Start with the first person (3) asking the second (5) to check, and so on until a match is confirmed.\n", + "+ Evaluation: In the worst case, this might take O(n) time, similar to Algorithm A.\n", + "\n", + "So, the correct sequence from fastest to slowest would be: B, A, C." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quallity:**\n", + "\n", + "+ With the help of ChatGPT, I created this question.\n", + "+ This question presents scenarios of different search algorithms applied to finding an item in a list, challenging the assessment of their efficiencies and approaches." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q11 (5 Points)**\n", + "**In a directed graph G, each vertex has exactly one outgoing edge. Can G contain a cycle?**\n", + "\n", + "**Choose the correct option:**\n", + "\n", + "**A.Yes, G can contain a cycle.**
\n", + "**B.No, G cannot contain a cycle.**\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "Yes, G can contain a cycle.\n", + "\n", + "Explanation:\n", + "\n", + "In a directed graph where each vertex has exactly one outgoing edge, it's still possible to form a cycle.\n", + "\n", + "Consider a scenario where each vertex points to the next vertex in a sequence forming a circular chain:\n", + "\n", + "Vertex A points to Vertex B.
\n", + "Vertex B points to Vertex C.
\n", + "...\n", + "Vertex N points back to Vertex A, closing the loop.
\n", + "Despite the restriction of each vertex having only one outgoing edge, this structure forms a cycle where the last vertex connects back to the first vertex, creating a cycle within the graph.\n", + "\n", + "Thus, even in a directed graph with such restrictions on outgoing edges, it's feasible to construct a cyclic structure under specific arrangements." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used ChatGPT to create a question.\n", + "+ This question challenges understanding about cycles in directed graphs under the specific condition where each vertex has only one outgoing edge, prompting evaluation of cycle possibilities in such graph structures." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q12 (5 Points)**
\n", + "**True or False: The following is a proper 0/1 Knapsack Problem Recurrence relation, considering 'w' and 'v' as lists of weights and values:**
\n", + "**Knapsack(i,j) = max (Knapsack(i−1,j),Knapsack(i−1,j−w[i])+v[i])**\n", + "\n", + "**Choose the correct option:**\n", + "\n", + "**A. True**
\n", + "**B. False**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "True: This recurrence relation is a proper representation of the 0/1 Knapsack Problem.\n", + "\n", + "Explanation:\n", + "\n", + "It recursively calculates the maximum value that can be obtained by considering or excluding the 'i-th' item in the knapsack with a capacity of 'j'.\n", + "\n", + "The relation compares two cases:\n", + "+ Excluding the 'i-th' item, which is represented by
\n", + "Knapsack(i−1,j).\n", + "+ Including the 'i-th' item, which is represented by
\n", + "Knapsack(i−1,j−w[i])+v[i], updating the value based on the weight and value of the 'i-th' item.\n", + "\n", + "It selects the maximum value between these two cases, determining the optimal solution for the Knapsack Problem.\n", + "\n", + "Therefore, the given recurrence relation is a proper formulation for solving the 0/1 Knapsack Problem by considering the weights and values of items." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ This question focuses on the accuracy of a recurrence relation as it pertains to solving the 0/1 Knapsack Problem, testing understanding about the formulation of recursive equations for this specific problem." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q13 (5 Points)**
\n", + "**Which of the following statements about NP and NP-Hard is correct?**\n", + "\n", + "**A. NP stands for \"Non-Polynomial,\" indicating problems that cannot be solved efficiently by any algorithm.**\n", + "\n", + "**B. NP-Hard problems are a subset of NP problems, implying that every NP-Hard problem is also in NP.**\n", + "\n", + "**C. Verifying a solution to an NP problem can be done in polynomial time, while finding the solution itself may be computationally challenging.**\n", + "\n", + "**D. Every problem in NP-Hard is also in P, meaning that they can be solved in polynomial time.**\n", + "\n", + "**E. NP-Hard problems are those for which a solution, once guessed, can be verified quickly, making them inherently easier than NP problems.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "A. NP stands for \"Non-Polynomial,\" indicating problems that cannot be solved efficiently by any algorithm.\n", + "\n", + "Explanation: This statement is incorrect. NP stands for \"Nondeterministic Polynomial time,\" representing problems for which a solution can be verified in polynomial time but not necessarily found in polynomial time.\n", + "\n", + "B. NP-Hard problems are a subset of NP problems, implying that every NP-Hard problem is also in NP.\n", + "\n", + "Explanation: This statement is correct. NP-Hard problems are, indeed, a subset of NP problems. While they might not be verifiable in polynomial time, their hardness implies that any problem in NP can be reduced to an NP-Hard problem.\n", + "\n", + "C. Verifying a solution to an NP problem can be done in polynomial time, while finding the solution itself may be computationally challenging.\n", + "\n", + "Explanation: This statement is correct. NP problems are characterized by their verifiability in polynomial time. However, determining the solution itself might not be polynomial.\n", + "\n", + "D. Every problem in NP-Hard is also in P, meaning that they can be solved in polynomial time.\n", + "\n", + "Explanation: This statement is incorrect. NP-Hard problems are not necessarily in P. While they are at least as hard as the hardest problems in NP, they might not have polynomial-time solutions.\n", + "\n", + "E. NP-Hard problems are those for which a solution, once guessed, can be verified quickly, making them inherently easier than NP problems.\n", + "\n", + "Explanation: This statement is incorrect. NP-Hard problems are among the most difficult computational problems. Although their solutions, when guessed, can be verified quickly, they don't imply that finding such solutions is easy.\n", + "\n", + "Correct Answer: B. NP-Hard problems are a subset of NP problems, implying that every NP-Hard problem is also in NP." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ ChatGPT helped me create a question.\n", + "\n", + "+ This question aimed to distinguish accurate statements regarding NP and NP-Hard, highlighting their definitions and relationships within complexity theory." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q14 (5 Points)**
\n", + "**For each of the following two statements, decide whether it is true or false:**\n", + "\n", + "**(a) Suppose we are given an instance of the Shortest Path Problem on a graph G, with edge costs that are all positive and distinct. Let P be a shortest path for this instance. Now suppose we replace each edge cost ce by its square, c2 e, thereby creating a new instance of the problem with the same graph but different costs.**\n", + "\n", + "**True or false? P must still be a shortest path for this new instance.**\n", + "\n", + "**(b) Suppose we are given an instance of the Minimum Spanning Tree Problem on an undirected graph G. We assume that all edge costs are positive and distinct. Let T be a minimum spanning tree for this instance. Now suppose we replace each edge cost ce by its cube, c3 e, thereby creating a new instance of the problem with the same graph but different costs.**\n", + "\n", + "**True or false? T must still be a minimum spanning tree for this new instance.**\n", + "\n", + "**Select the letter if the corresponding statement is True:**\n", + "\n", + "**A. (a)**
\n", + "**B. (b)**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "Statement (a): Shortest Path Problem with Squared Edge Costs\n", + "\n", + "Scenario:\n", + "\n", + "Suppose we have a directed graph G with edge costs:\n", + "\n", + "Edge 1-2: Cost = 2
\n", + "Edge 2-3: Cost = 3
\n", + "Edge 1-3: Cost = 5\n", + "\n", + "Let P be the shortest path from node 1 to node 3 in G, with a total cost of 5 (1-2-3).\n", + "\n", + "Transformation:\n", + "\n", + "Now, if we replace each edge cost ce by its square, c2 e:\n", + "\n", + "Edge 1-2: Cost = 4
\n", + "Edge 2-3: Cost = 9
\n", + "Edge 1-3: Cost = 25\n", + "\n", + "Evaluation:\n", + "\n", + "The new path from 1 to 3 considering squared edge costs: 1-2-3, with a total cost of 13, which is not the shortest path in this case. Therefore, P is not the shortest path in the transformed instance.\n", + "\n", + "Statement (b): Minimum Spanning Tree Problem with Cubed Edge Costs\n", + "\n", + "Scenario:\n", + "\n", + "Let's consider an undirected graph G with edge costs:\n", + "\n", + "Edge 1-2: Cost = 1
\n", + "Edge 2-3: Cost = 2
\n", + "Edge 1-3: Cost = 3\n", + "\n", + "Let T be the minimum spanning tree in G, comprising edges 1-2 and 2-3, with a total cost of 3.\n", + "\n", + "Transformation:\n", + "\n", + "Now, if we replace each edge cost ce by its cube, c3 e:\n", + "\n", + "Edge 1-2: Cost = 1
\n", + "Edge 2-3: Cost = 8
\n", + "Edge 1-3: Cost = 27\n", + "\n", + "Evaluation:\n", + "\n", + "The edges 1-2 and 2-3 still form the minimum spanning tree in the transformed instance, with a total cost of 9, confirming that T remains a minimum spanning tree even after transforming the edge costs to their cubes.\n", + "\n", + "Correct Answers:\n", + "\n", + "(A) - Statement (a) is False.
\n", + "(B) - Statement (b) is True." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ This evaluation demonstrates that for the Shortest Path Problem, the optimality of the solution doesn't necessarily hold after transforming edge costs by their squares.\n", + "\n", + "+ However, for the Minimum Spanning Tree Problem, the optimality of the minimum spanning tree persists even after transforming edge costs by their cubes." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q15 (5 Points)**\n", + "**Consider the following statements about local search algorithms and probability. Choose the correct option:**\n", + "\n", + "**A. Local search algorithms always guarantee finding the global optimum in combinatorial optimization problems.**
\n", + "**B. The Metropolis algorithm is primarily used for optimization problems involving continuous spaces.**
\n", + "**C. Hopfield nets are utilized for solving combinatorial optimization problems with deterministic outcomes.**
\n", + "**D. Probability plays a crucial role in the acceptance of worse solutions in simulated annealing.**
\n", + "**E. Local search algorithms, by design, avoid the exploration of suboptimal solutions.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:**\n", + "\n", + "A. Local search algorithms always guarantee finding the global optimum in combinatorial optimization problems.\n", + "\n", + "False: Local search algorithms do not guarantee finding the global optimum. They are heuristics that explore the neighborhood of solutions, aiming to improve a given solution but might not reach the global optimum due to getting stuck in local optima.\n", + "\n", + "B. The Metropolis algorithm is primarily used for optimization problems involving continuous spaces.\n", + "\n", + "True: The Metropolis algorithm is often applied in optimization problems dealing with continuous spaces, notably in statistical mechanics and simulated annealing.\n", + "\n", + "C. Hopfield nets are utilized for solving combinatorial optimization problems with deterministic outcomes.\n", + "\n", + "False: Hopfield nets, while used in optimization, are primarily employed in associative memory tasks or pattern recognition, and they don't guarantee deterministic outcomes in combinatorial optimization problems.\n", + "\n", + "D. Probability plays a crucial role in the acceptance of worse solutions in simulated annealing.\n", + "\n", + "True: Simulated annealing employs probabilistic acceptance of worse solutions to avoid getting trapped in local optima, allowing exploration for better solutions.\n", + "\n", + "E. Local search algorithms, by design, avoid the exploration of suboptimal solutions.\n", + "\n", + "False: Local search algorithms explicitly explore suboptimal solutions within a neighborhood to potentially find better solutions.\n", + "\n", + "Correct Statements:\n", + "\n", + "B. The Metropolis algorithm is primarily used for optimization problems involving continuous spaces.
\n", + "D. Probability plays a crucial role in the acceptance of worse solutions in simulated annealing." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used ChatGPT to create this question.\n", + "+ These statements outline key aspects of local search algorithms, probability-driven optimization techniques, and their roles in exploring solution spaces in various optimization problems." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q16 (5 Points)**
\n", + "**Which statement regarding randomized algorithms and list scheduling is true?**\n", + "\n", + "**A. Randomized algorithms do not utilize random number generators for their execution.**
\n", + "**B. List scheduling aims to optimize resource allocation in a deterministic manner.**
\n", + "**C. Randomized algorithms always guarantee the same output for a given set of inputs.**
\n", + "**D. List scheduling can be used to solve NP-hard problems efficiently.**
\n", + "**E. The efficiency of randomized algorithms is solely based on deterministic outcomes.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "A. Randomized algorithms do not utilize random number generators for their execution.\n", + "\n", + "False: Randomized algorithms use random number generators to introduce randomness in their decision-making processes. This randomness is pivotal to their functioning.\n", + "\n", + "B. List scheduling aims to optimize resource allocation in a deterministic manner.\n", + "\n", + "False: List scheduling is a heuristic method used for scheduling tasks and resources. It doesn't guarantee an optimal solution and often operates in a non-deterministic manner, considering various factors to make scheduling decisions.\n", + "\n", + "C. Randomized algorithms always guarantee the same output for a given set of inputs.\n", + "\n", + "False: Randomized algorithms introduce randomness intentionally. Given the same set of inputs, they may produce different outputs in different runs due to their probabilistic nature.\n", + "\n", + "D. List scheduling can be used to solve NP-hard problems efficiently.\n", + "\n", + "Partially True: List scheduling, while not providing guaranteed polynomial-time solutions for NP-hard problems, is utilized to approximate solutions efficiently, especially in scheduling scenarios, offering reasonably good solutions.\n", + "\n", + "E. The efficiency of randomized algorithms is solely based on deterministic outcomes.\n", + "\n", + "False: The efficiency assessment of randomized algorithms doesn't rely on deterministic outcomes. It's often based on average-case performance or probabilistic analysis, considering the range of possible outcomes and their probabilities.\n", + "\n", + "Correct Statement:\n", + "\n", + "D. List scheduling can be used to solve NP-hard problems efficiently.\n", + "\n", + "This statement acknowledges the application of list scheduling as an approximation technique for efficiently addressing NP-hard problems in certain contexts, despite not guaranteeing optimal solutions." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q17 (5 Points)**
\n", + "**Which statement about amortized analysis in data structures is correct?**\n", + "\n", + "**A. Amortized analysis evaluates the worst-case performance of a data structure.**
\n", + "**B. Amortized analysis doesn't consider the average-case performance of a data structure.**
\n", + "**C. Data structures analyzed using amortized analysis always display constant-time complexity.**
\n", + "**D. Amortized analysis provides insights into the average performance of operations over a sequence of operations.**
\n", + "**E. Data structures analyzed using amortized analysis solely focus on their best-case performance.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "A. Amortized analysis evaluates the worst-case performance of a data structure.\n", + "\n", + "False: Amortized analysis differs from worst-case analysis. It evaluates the average performance of a sequence of operations rather than focusing solely on worst-case scenarios.\n", + "\n", + "B. Amortized analysis doesn't consider the average-case performance of a data structure.\n", + "\n", + "False: Amortized analysis precisely focuses on the average performance of a sequence of operations rather than worst-case or single-operation analysis.\n", + "\n", + "C. Data structures analyzed using amortized analysis always display constant-time complexity.\n", + "\n", + "False: Amortized analysis doesn't ensure constant-time complexity for all operations. It assesses the average performance over a sequence of operations, which might exhibit varying complexities.\n", + "\n", + "D. Amortized analysis provides insights into the average performance of operations over a sequence of operations.\n", + "\n", + "True: Amortized analysis precisely does this. It assesses the average performance of a sequence of operations to understand the overall behavior of a data structure.\n", + "\n", + "E. Data structures analyzed using amortized analysis solely focus on their best-case performance.\n", + "\n", + "False: Amortized analysis looks at the average performance, considering the sequence of operations, rather than focusing solely on the best-case scenario.\n", + "\n", + "Correct Statement:\n", + "\n", + "D. Amortized analysis provides insights into the average performance of operations over a sequence of operations." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used ChatGPT to create this question.\n", + "+ This statement captures the essence of amortized analysis, which is crucial for understanding the average behavior of data structures over a sequence of operations, rather than focusing on individual operations or worst-case scenarios." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q18 (5 Points)**
\n", + "**Consider the following statements regarding binary and binomial heaps. Choose the correct one:**\n", + "\n", + "**A. A binary heap is a specialized form of a binomial heap.**
\n", + "**B. Binomial heaps ensure O(log n) time complexity for both insertion and deletion operations.**
\n", + "**C. Binary heaps guarantee logarithmic time complexity for finding the minimum element.**
\n", + "**D. Binomial heaps use a binary tree structure to maintain their properties.**
\n", + "**E. Binary heaps maintain their properties using a binomial tree structure.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "A. A binary heap is a specialized form of a binomial heap.\n", + "\n", + "False: Binary heaps and binomial heaps are distinct data structures. They share similarities but are not specialized forms of each other.\n", + "\n", + "B. Binomial heaps ensure O(log n) time complexity for both insertion and deletion operations.\n", + "\n", + "True: Binomial heaps maintain O(log n) time complexity for insertion and deletion operations due to their structure and merge operation characteristics.\n", + "\n", + "C. Binary heaps guarantee logarithmic time complexity for finding the minimum element.\n", + "\n", + "True: Binary heaps ensure O(log n) time complexity for finding the minimum element, as the minimum element is always the root of the heap.\n", + "\n", + "D. Binomial heaps use a binary tree structure to maintain their properties.\n", + "\n", + "False: Binomial heaps employ a collection of binomial trees to maintain their properties, not binary trees.\n", + "\n", + "E. Binary heaps maintain their properties using a binomial tree structure.\n", + "\n", + "False: Binary heaps utilize a binary tree structure, where each node has at most two children, to maintain their properties.\n", + "\n", + "Correct Statements:\n", + "\n", + "B. Binomial heaps ensure O(log n) time complexity for both insertion and deletion operations.
\n", + "C. Binary heaps guarantee logarithmic time complexity for finding the minimum element." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used ChatGPT to create question.\n", + "+ These statements accurately differentiate the characteristics of binary heaps and binomial heaps regarding their time complexities and structural properties." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q19 (5 Points)**
\n", + "**Which statement accurately describes linear programming duality?**\n", + "\n", + "**A. Linear programming duality only applies to problems with integer variables.**
\n", + "**B. The dual of a linear program involves maximizing a function.**
\n", + "**C. Linear programming duality holds true for non-convex optimization problems.**
\n", + "**D. The dual problem's constraints correspond to the primal problem's objective function.**
\n", + "**E. The optimal value of the primal problem is always greater than or equal to the optimal value of its dual problem.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "A. Linear programming duality only applies to problems with integer variables.\n", + "\n", + "False: Linear programming duality is a fundamental concept applicable to linear programming problems irrespective of variable types (integer or continuous).\n", + "\n", + "B. The dual of a linear program involves maximizing a function.\n", + "\n", + "False: The dual of a linear program involves minimizing a function, which is derived from the primal problem's constraints and objectives.\n", + "\n", + "C. Linear programming duality holds true for non-convex optimization problems.\n", + "\n", + "False: Linear programming duality is applicable to linear programming problems, typically characterized by convexity in both the primal and dual problems.\n", + "\n", + "D. The dual problem's constraints correspond to the primal problem's objective function.\n", + "\n", + "False: In linear programming duality, the dual problem's constraints correspond to the primal problem's constraints, not the objective function.\n", + "\n", + "E. The optimal value of the primal problem is always greater than or equal to the optimal value of its dual problem.\n", + "\n", + "True: This statement aligns with the concept of weak duality in linear programming, where the optimal value of the primal problem is always greater than or equal to the optimal value of its dual problem.\n", + "\n", + "Correct Statement:\n", + "\n", + "E. The optimal value of the primal problem is always greater than or equal to the optimal value of its dual problem." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used ChatGPT to create a question.\n", + "+ This statement represents the fundamental principle of weak duality in linear programming, highlighting the relationship between the optimal values of the primal and dual problems." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Q20 (5 Points)**
\n", + "**Consider the following statements about the ellipsoid algorithm.Choose the correct one:**
\n", + "\n", + "**A. The ellipsoid algorithm is an iterative algorithm used to solve convex optimization problems.**
\n", + "**B. The ellipsoid algorithm relies on randomness for its execution.**
\n", + "**C. The ellipsoid algorithm guarantees finding the global optimum for non-convex optimization problems.**
\n", + "**D. Ellipsoidal sets in the ellipsoid algorithm always converge to a single point.**
\n", + "**E. The ellipsoid algorithm doesn't require knowledge of the feasible region's structure.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***Solution:***\n", + "\n", + "A. The ellipsoid algorithm is an iterative algorithm used to solve convex optimization problems.\n", + "\n", + "True: The ellipsoid algorithm is indeed an iterative algorithm designed to solve convex optimization problems, especially in the presence of polynomially bounded constraints.\n", + "\n", + "B. The ellipsoid algorithm relies on randomness for its execution.\n", + "\n", + "False: The ellipsoid algorithm is a deterministic algorithm and does not rely on randomness or probabilistic methods for its execution.\n", + "\n", + "C. The ellipsoid algorithm guarantees finding the global optimum for non-convex optimization problems.\n", + "\n", + "False: The ellipsoid algorithm is primarily applicable to convex optimization problems and does not guarantee finding the global optimum for non-convex problems.\n", + "\n", + "D. Ellipsoidal sets in the ellipsoid algorithm always converge to a single point.\n", + "\n", + "True: In each iteration, the ellipsoid algorithm narrows down the feasible region to an ellipsoidal set that eventually converges to a single point containing the optimal solution, given that the problem is bounded and feasible.\n", + "\n", + "E. The ellipsoid algorithm doesn't require knowledge of the feasible region's structure.\n", + "\n", + "True: The ellipsoid algorithm does not require explicit knowledge of the feasible region's structure; it operates based on oracle queries that assess the feasibility of points.\n", + "\n", + "Correct Statements:
\n", + "A. The ellipsoid algorithm is an iterative algorithm used to solve convex optimization problems.
\n", + "D. Ellipsoidal sets in the ellipsoid algorithm always converge to a single point.
\n", + "E. The ellipsoid algorithm doesn't require knowledge of the feasible region's structure." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Reflection Quality:**\n", + "\n", + "+ I used ChatGPT to create a question.\n", + "+ These statements accurately depict the application and behavior of the ellipsoid algorithm in solving convex optimization problems by iteratively narrowing down feasible regions to converge toward an optimal solution." + ] + } + ], + "metadata": { + "language_info": { + "name": "python" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +}