Graph Algorithms and Huffman Coding

The requirements are feasibility, local optimality, and irrevocability.

Prim's algorithm builds the MST from any starting vertex, while Kruskal’s algorithm starts from the least weighted edge and adds edges without forming a cycle.

The initial step involves selecting an arbitrary vertex from the graph's vertices to start the subtree.

<p>The primary objective is to form an acyclic subgraph with |V| - 1 edges, where the sum of the edge weights is minimized.</p> Signup and view all the answers

<p>Dijkstra’s algorithm systematically explores the nearest vertices, updating their distances based on the current shortest path until the destination is reached.</p> Signup and view all the answers

<p>Irrevocability means that once a choice is made, it cannot be undone or changed in subsequent steps.</p> Signup and view all the answers

<p>Sorting the edges helps to ensure that edges are considered in order of their weights, which is fundamental for constructing the MST efficiently.</p> Signup and view all the answers

<p>Prim’s algorithm prefers list data structures.</p> Signup and view all the answers

<p>The initial distance for the source vertex is set to 0.</p> Signup and view all the answers

<p>The time complexity of Kruskal’s algorithm is O(|E| log |E|).</p> Signup and view all the answers

<p>A spanning tree is a connected acyclic subgraph containing all vertices, while a minimum spanning tree is the spanning tree with the smallest total weight.</p> Signup and view all the answers

<p>Dynamic Huffman encoding is a variant of Huffman coding that updates the coding tree with each new character read from the source text.</p> Signup and view all the answers

<p>Fixed length encoding uses the same number of bits for each symbol, while variable length encoding assigns different lengths based on symbol frequency.</p> Signup and view all the answers

<p>A Huffman tree is used to create variable-length codes where each leaf represents a symbol and the path to it corresponds to its binary code.</p> Signup and view all the answers

<p>Edge weights determine the total weight of the spanning tree, and the minimum spanning tree has the smallest sum of these weights.</p> Signup and view all the answers

<p>Prim’s algorithm updates distances to neighbors of the selected vertex, aiming to find the minimum edge connecting the unvisited vertices.</p> Signup and view all the answers

<p>A variable-length code is assigned based on the frequencies of corresponding characters, where more frequent characters get shorter codes.</p> Signup and view all the answers

<p>Lower bound arguments are mathematical proofs that show the minimum time or space required by any algorithm for a problem.</p> Signup and view all the answers

<p>A P problem is a decision problem that can be solved in polynomial time by deterministic algorithms; for example, basic math operations.</p> Signup and view all the answers

<p>NP problems are decision problems that can be verified in polynomial time but not necessarily solved in polynomial time; an example is integer factorization.</p> Signup and view all the answers

<p>NP-complete problems are the most difficult in NP, where any NP problem can be reduced to them polynomially; an example is the traveling salesman problem.</p> Signup and view all the answers

<p>A decision tree is a graphical representation of decisions and their possible consequences, with nodes for decisions and branches for outcomes.</p> Signup and view all the answers

<p>A decision tree for finding the maximum of two numbers compares the values and branches based on whether one is greater than the other.</p> Signup and view all the answers

<p>NP problems allow solutions to be verified in polynomial time, while P problems can be solved in polynomial time.</p> Signup and view all the answers

<p>The first step is to select an arbitrary vertex from the graph's vertices to start the initial subtree.</p> Signup and view all the answers

<p>Prim's algorithm selects the nearest vertex with the smallest edge weight that is not already in the tree.</p> Signup and view all the answers

<p>Avoiding cycles is important to maintain the properties of a tree, specifically ensuring that it remains acyclic.</p> Signup and view all the answers

<p>The first edge added is a-b with weight 2.</p> Signup and view all the answers

<p>The next vertex added is 'c' with a weight of 3 from the edge b-c.</p> Signup and view all the answers

<p>You stop when all vertices in the graph have been included in the Minimum Spanning Tree.</p> Signup and view all the answers

<p>The total weight of the Minimum Spanning Tree is 11.</p> Signup and view all the answers

<p>The criterion is to select edges based on the smallest weight connecting a visited vertex to an unvisited vertex.</p> Signup and view all the answers

<p>Class P includes problems that can be solved in polynomial time, while class NP includes problems whose solutions can be verified in polynomial time.</p> Signup and view all the answers

<p>NP-complete problems are those in NP that are as difficult as any other problem in NP, meaning every NP problem can be reduced to them in polynomial time.</p> Signup and view all the answers

<p>One major challenge is approximation, as numerical algorithms often involve rounding and can produce errors that accumulate and affect solution accuracy.</p> Signup and view all the answers

<p>Stability issues can cause small changes in the input to lead to disproportionately large changes in the output, making the algorithms unreliable.</p> Signup and view all the answers

<p>An ill-conditioned problem is one where small changes to the input result in large changes to the output, complicating accurate solution finding.</p> Signup and view all the answers

<p>Prim's algorithm builds the minimum spanning tree by continually adding the smallest edge that connects a vertex in the tree to a vertex outside the tree, ensuring no cycles are formed.</p> Signup and view all the answers

<p>First, sort all the edges of the graph by their weights. Then, initialize an empty forest and add edges from the sorted list while avoiding cycles until a spanning tree is formed.</p> Signup and view all the answers

<p>Dijkstra's algorithm finds the shortest path from a source vertex to all other vertices in a weighted graph with non-negative edge weights.</p> Signup and view all the answers

<p>Occurrence probabilities determine the weights of the characters, allowing Huffman's algorithm to produce a more efficient encoding by minimizing the average length of the codewords.</p> Signup and view all the answers

<p>The expected number of bits per character indicates the average length of the encoded output, reflecting the efficiency of the encoding scheme.</p> Signup and view all the answers

<p>Information-theoretic arguments utilize the amount of information that must be produced as a basis to demonstrate that any algorithm must perform at least a certain amount of work.</p> Signup and view all the answers

<p>A trivial lower bound is the simplest estimate of the minimum work required based on input size and output produced, such as for multiplication of two n x n matrices.</p> Signup and view all the answers

<p>An irrevocable choice in greedy algorithms means that once a decision is made, it cannot be changed or undone in subsequent steps.</p> Signup and view all the answers