Applications Data Structures and Algorithms (60-254)

ApplicationsData Structures and Algorithms (60-254)

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Encoding and Decoding Messages

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Fixed-length coding

Code for each character has the same number of bits.

QuestionHow many bits do we need to encode uniquely each character in a message made up of characters from an n-letter alphabet? Answer log n bits at least.

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Variable-length coding

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Another encoding scheme

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Prefix codes

Code word of a character is NOT prefix of the code for a character , where .Also, called prefix-free because the term “prefix” is misleading… Meaning of prefixExamples:

00 is prefix of 001110 is prefix of 110111111101 is not prefix of 1111001

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What is interesting in a prefix code?

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Lossless Data Compression

M = a d d d

Encoding using scheme 1

C1= 000111Encoding using scheme 2

C2=00111111

Length(C1)=6

Length(C2)=8 Lossless data compression is possible using prefix codes

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Huffman coding

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Huffman coding

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Huffman coding

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Huffman coding

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Huffman coding

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Huffman coding

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Huffman coding

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Huffman coding

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Huffman coding

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Huffman coding

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Graphs

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Graphs

We denote a graph by G=(V, E) where V = Set of vertices andE = Set of edges

|V| = Number of vertices and|E| = Number of edges

G is directed if the edges are, otherwise G is undirected.

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Graph representation

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Graph representation

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Graph Representation

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Breadth-first search

Also called: level order search

What does it do? Given: G=(V, E), a directed or undirected graph and a “source” vertex s V It systematically explores vertices reachable from s, to construct a breadth-first search tree

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Shortest Path

Length of the path from s to each reachable vertex is the shortest path from s to this vertex Example graph

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Expand node s

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Expand node w

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Expand node r

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Expand node t

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Expand node x

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Expand node v

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Expand node u

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Breath-first Search Tree

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Formal Algorithm

Colour terminology:Gray nodes make up the “frontier” (red line) White nodes (vertices) are the unexplored ones Black nodes are the completely explored ones, lying on the other side of the “frontier”

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Breadth-first search algorithm

BFS(G, s) For each vertex uV –{s} { // Setup loop O(|V|)

u.color WHITE;u.distance ;u.parent NULL; // parent of u

}s.color GRAY;s.distance 0;s.parent NULL;Q Q {s};

(Cont’d.)

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Breadth-first search algorithm

While (Q ) { // Processing loop O(|E|)u head (Q);for each v adjacent to u {

if (v.color = WHITE) {v.color GRAY;v.distance u.distance+1;v.parent u;Q Q {v};

}}Delete u from Q;u.color BLACK;

}

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Analysis

Thm. Running time of BFS is O (|V|+|E|) Pf Queuing operations take O (|V|) time.Each edge in the graph is looked at most once. Hence O (|E|) time and the theorem follows.

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We can prove the following factsFact1:Breath-first-search (BFS) discovers every

vertex reachable from s Fact2:The length of the path from s to a reachable

vertex v, is a shortest path from s to v.

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Weighted Graph

Each edge has a weight (a number).

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Minimum Spanning Tree (MST)

Green edges form a minimum spanning tree of the example graph

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Minimum Spanning Tree

Weight of the MST is: W(T) = 8 + 2 + 4 + 7 + 4 + 2 + 9 + 1 = 37

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Formally

Given a connected, weighed, undirected graph G=(V, E), Find an acyclic subset TE that connects all the vertices such that

is minimum, where weight(u,v) is the weight of the edges connecting u to v.

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Growing an MST, greedily

Let A be a subset of edges of some MST

Def. An edge e=(u,v) of G is safe for A if A {(u,v)} is also a subset of an MST.

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A generic MST algorithm

GENERIC_MST(G,w) A ;

while (A does not form an MST)

find a safe edge (u,v) for A; A A {(u,v)};

return A;

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Cuts and safe edges

Definition: A cut of G is a partition (S, V-S) of V

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A few definitions

Definition: An edge (u,v) E crosses the cut (S,V-S) if one of its end points is in S and the other in V-S Definition: A cut respects the set A if no edge in A crosses the cut Definition: An edge is a light edge crossing a cut if its weight is the minimum of any edge crossing the cut.

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Safe Edges

The following theorem characterizes a safe edge for A TheoremLet A be a subset of E that is included in some MST for G. Let (S,V-S) be any cut of G that respects A, and Let (u,v) be a light edge crossing (S,V-S). Then, edge (u,v) is safe for A.

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Prim’s algorithm

Select vertex aS={a}

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(a, b) is a light edge for (S, V-S)

Select vertex bnew S={a, b}

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(b,c) is a light edge for (S,V-S)

Select vertex cnew S={a, b, c}

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(c, i) is a light edge for (S,V-S) where S={a, b, c}Select vertex inew S={a, b, c, i}

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(c, f) is a light edge for (S,V-S)

Select vertex fnew S={a, b, c, f, i}

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(f, g) is a light edge for (S,V-S)

Select vertex gnew S={a, b, c, g, f, i}

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(g, h) is a light edge for (S,V-S)

Select vertex hnew V-S={d, e}

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(c, d) is a light edge for (S,V-S)

Select vertex dnew V-S={e}

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(d, e) is a light edge for (S,V-S)

Select vertex enew V-S=

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Formal Algorithm

MST_Prim(G, W, r) QV;for (each u Q)

u.key ;

r.key 0;r.parent NULL;while ( Q )

u extract_min(Q);for (each v adjacent to u) if v Q and weight(u,v) < v.keyv.parent u;v.key weight(u,v)

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Important Note on Shortest Path: Dijkstra’s algorithmThe algorithm incrementally constructs a set S of vertices to which the shortest path is known.

It adds a vertex v (from the remaining vertices) to S,whose distance from S (source) is

the shortest of the remaining vertices. The assumption of non-negative weights guarantees that the shortest path to v passes only through vertices in S.

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Priority Queues

Let S be a set of keys, a priority queue supports the following operations on S: 1. Insert (S, x)2. Minimum (S) -- returns min key3. Extract_Min (S) -- returns and removes min key

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Heap as Priority Queue

A priority queue can be implemented using a heap A heap is a complete binary tree, satisfying the heap property:

For a node v,v.key v.left.keyv.key v.right.key

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Single-source shortest paths

Let G be a weighted, directed graph.

A path in the graph from u to v is a sequence of edges that connects u to v. The weight of this path is the sum of the weights of its constituent edges. Let (u, v) denote a minimum weight path from u to v Let s be a source vertex of G

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Single-source shortest path problemCompute a shortest path from s to each vertex v. Remark: A shortest path is a path of minimum weight. Dijkstra’s algorithm:

We assume that the weights of the edges are non-negative.

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Relaxation

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Relax

Relax(u,v,w) {If (v.distance > u.distance +w(u,v)) {

v.distance u.distance +w(u,v);v.parent u;

}}

Initialize_Simple_Source (G, v) {For each vertex v V [G];v.distance ;v.parent NULL;}

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Dijkstra’s algorithm by example

The algorithm maintains a set X of vertices whose final shortest-path weights from s have already been determined

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Add s to X

Relax vertices adjacent to s

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Add x to X

Relax vertices adjacent to x

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Add y to X

Relax vertices adjacent to y

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Add u to X

Relax vertices adjacent to u

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Add v to X

Done

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Formal Algorithm

Dijkstra(G,w,s) Initialize_single_source(G,s)

XQV [G]while (Q )

uextract_min(Q)XX{u}For each vertex v adjacent to u

Relax(u,v,w)