Dijkstra’s AlgorithmDijkstra’s Algorithm

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Last TimeLast TimeBreadth First Search, Depth First

Search◦Different algorithms for searching all

nodes in a graph.Topological Sort

◦Produces an ordering of vertices in a graph, given certain constrains (A before B, B before C, etc.)

Dijkstra’s Algorithm◦Finds the shortest paths from a source

vertex to all other vertices.

TodayTodayFinish up Dijkstra’s

◦Path reconstruction

MSTs – Minimum Spanning Tree algorithms◦Prim’s◦Kruskal’s

Dijkstra’s AlgorithmDijkstra’s AlgorithmThis algorithm finds the shortest

path from a source vertex to all other vertices in a weighted directed graph without negative edge weights.

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Dijkstra’s Path Dijkstra’s Path ReconstructionReconstructionExample shown

on the Board

Minimum Spanning TreesMinimum Spanning TreesIn a weighted,

undirected graph, it is a tree formed by connecting all of the vertices with minimal cost.◦ The MST is a tree because

it’s acyclic.◦ It’s spanning because it

covers every vertex.◦ And it’s minimum because

it has minimum cost.

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MSTs – proof that greedy MSTs – proof that greedy worksworks Let G be a graph with

vertices in the set V partitioned into two sets V1 and V2. Then the minimum weight edge, e, that connects a vertex from V1 to V2 is part of a minimum spanning tree of G.

Proof: Consider a MST T of G that does NOT contain the minimum weight edge e. ◦ This MUST have at least

one edge in between a vertex from V1 to V2. (Otherwise, no vertices between those two sets would be connected.)

◦ Let G contain edge f that connects V1 to V2.

◦ Now, add in edge e to T. ◦ This creates a cycle. In

particular, there was already one path from every vertex in V1 to V2 and with the addition of e, there are two.

◦ Thus, we can form a cycle involving both e and f. Now, imagine removing f from this cycle.

◦ This new graph, T' is also a spanning tree, but it's total weight is less than or equal to T because we replaced e with f, and e was the minimum weight edge.

MSTs – proof that greedy MSTs – proof that greedy worksworks

As a spanning tree is created◦ If the edge that is added is the one of

minimum cost that avoids creation of a cycle.

◦ Then the cost of the resulting spanning tree cannot be improved Because any replacement edge would have cost at

least as much as an edge already in the spanning tree.

This is why greedy works!

Kruskal’s AlgorithmKruskal’s AlgorithmLet V = For i=1 to n-1, (where there are n vertices in a graph)

V = V e, where e is the edge with the minimum edge

weight not already in V, and that does NOT

form a cycle when added to V.

Return V

Basically, you build the MST of the graph by continually adding in the smallest weighted edge into the MST that doesn't form a cycle.

◦ When you are done, you'll have an MST. ◦ You HAVE to make sure you never add an edge the

forms a cycle and that you always add the minimum of ALL the edges left that don't.

Kruskal’sKruskal’sv1

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CYCLE!

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All Vertices, we’re done!!

Determine the MST:

Edges in sorted order:Given Graph G:

Kruskal’sKruskal’sThe reason this works:

◦ is that each added edge is connecting between two sets of vertices,

◦ and since we select the edges in order by weight,

◦ we are always selecting the minimum edge weight that connects the two sets of vertices.

Cycle detection:◦ Keep track of

disjoint sets.◦ Initially, each vertex

is in its own disjoint set.

◦ When you add an edge you are unioning two sets.

◦ A union cannot happen if the two vertices are already in the same set. This would create a

cycle.

Prim’s AlgorithmPrim’s AlgorithmThis is quite similar

to Kruskal's with one big difference:◦ The tree that you are

"growing" ALWAYS stays connected. Whereas in Kruskal's

you could add an edge to your growing tree that wasn't connected to the rest of it, here you can NOT do it.

Here is the algorithm:

1) Set S = .

1) Pick any vertex in the graph.

2) Add the minimum edge incident to that vertex to S.

3) Continue to add edges into S (n-2 more times) using the

following rule:

 

Add the minimum edge weight to S that is incident to S

but that doesn't form a cycle when added to S.

Prim’sPrim’sv1

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Edges in sorted order:

Given Graph G:

Determine the MST, using Prim’s starting with vertexV1:

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CYCLE!

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All Vertices, we’re done!!

Example on the boardExample on the board

Huffman EncodingHuffman EncodingCompress the

storage of data using variable length codes. ◦ For example, each

character in a text file is stored using 8 bits.

◦ Nice and easy because we always read in 8 bits for a single character.

Not the most efficient…◦ What if ‘e’ is used

10 times more frequently than ‘q’.

◦ It would be more advantageous for us to use a 7 bit code for e and a 9 bit code for q.

Huffman CodingHuffman CodingFinds the optimal

way to take advantage of varying character frequencies in a particular file. ◦ On average,

standard files can shrink them anywhere from 10% to 30% depending to the character distribution.

The idea behind the coding is to give less frequent characters and groups of characters longer codes.

Also, the coding is constructed in such a way that no two constructed codes are prefixes of each other.

This property about the code is crucial with respect to easily deciphering the code.

Building a Huffman TreeBuilding a Huffman TreeExample on the

board

ReferencesReferencesSlides adapted from Arup Guha’s

Computer Science II Lecture notes: http://www.cs.ucf.edu/~dmarino/ucf/cop3503/lectures/

Additional material from the textbook:Data Structures and Algorithm Analysis in

Java (Second Edition) by Mark Allen WeissAdditional images:

www.wikipedia.comxkcd.com