SLIDE 1
A Heap Is Efficiently Represented As An Array
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1 2 3 4 5 6 7 8 9 10 9 8 6 7 2 6 5 1 7
Moving Up And Down A Heap
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A Heap Is Efficiently Represented As An Array 9 8 7 6 7 2 6 5 - - PowerPoint PPT Presentation
A Heap Is Efficiently Represented As An Array 9 8 7 6 7 2 6 5 - - PowerPoint PPT Presentation
A Heap Is Efficiently Represented As An Array 9 8 7 6 7 2 6 5 1 9 8 7 6 7 2 6 5 1 0 1 2 3 4 5 6 7 8 9 10 Moving Up And Down A Heap 1 9 2 3 8 7 4 7 5 6 6 7 2 6 5 1 8 9 Putting An Element Into A Max
SLIDE 2
SLIDE 3
Putting An Element Into A Max Heap
Complete binary tree with 10 nodes.
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SLIDE 4
Putting An Element Into A Max Heap
New element is 5.
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SLIDE 5
Putting An Element Into A Max Heap
New element is 20.
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SLIDE 6
Putting An Element Into A Max Heap
New element is 20.
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SLIDE 7
Putting An Element Into A Max Heap
New element is 20.
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SLIDE 8
Putting An Element Into A Max Heap
New element is 20.
9 8 6 7 2 6 5 1 7 7 7 20
SLIDE 9
Putting An Element Into A Max Heap
Complete binary tree with 11 nodes.
9 8 6 7 2 6 5 1 7 7 7 20
SLIDE 10
Putting An Element Into A Max Heap
New element is 15.
9 8 6 7 2 6 5 1 7 7 7 20
SLIDE 11
Putting An Element Into A Max Heap
New element is 15.
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SLIDE 12
Putting An Element Into A Max Heap
New element is 15.
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SLIDE 13
Complexity Of Put
Complexity is O(log n), where n is heap size.
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SLIDE 14
Removing The Max Element
Max element is in the root.
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SLIDE 15
Removing The Max Element
After max element is removed.
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SLIDE 16
Removing The Max Element
Heap with 10 nodes.
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Reinsert 8 into the heap.
SLIDE 17
Removing The Max Element
Reinsert 8 into the heap.
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SLIDE 18
Removing The Max Element
Reinsert 8 into the heap.
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SLIDE 19
Removing The Max Element
Reinsert 8 into the heap.
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SLIDE 20
Removing The Max Element
Max element is 15.
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SLIDE 21
Removing The Max Element
After max element is removed.
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SLIDE 22
Removing The Max Element
Heap with 9 nodes.
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SLIDE 23
Removing The Max Element
Reinsert 7.
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SLIDE 24
Removing The Max Element
Reinsert 7.
6 2 6 5 1 7 9 8
SLIDE 25
Removing The Max Element
Reinsert 7.
6 2 6 5 1 7 9 8 7
SLIDE 26
Complexity Of Remove Max Element
Complexity is O(log n).
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SLIDE 27
Note
Heap is not a search data-structure
You are interested in minimum or maximum
depending on the heap
There are efficient ways to convert an array
into a heap.
Requires O(n) time to convert an array into a
heap instead of O(n log n) as suggested by n insertions.
SLIDE 28
Administrivia
TA Office Hours
Wednesdays 3 to 5 pm in the PC lab
Homework 2
Updated the PDF document
Made a few things clear; Corrected a few things
Updated the .zip file
Makefiles in both directories Changes to hashT
emplate.c file
SLIDE 29
Thank You
SLIDE 30
EE717: Advanced Computing foR ELECTRICAL Engineers Lecture 8
Graphs
Shankar Balachandran, IIT Bombay (shankarb@ee.iitb.ac.in)
24 Aug 2014
SLIDE 31
Graphs
G = (V,E) V is the vertex set. Vertices are also called nodes and points. E is the edge set.
Each edge connects two different vertices. Captures relationships between vertices
Directed edge has an orientation (u,v).
u v
SLIDE 32
Graph Representation
Adjacency Matrix Adjacency Lists
- Linked Adjacency Lists
- Array Adjacency Lists
SLIDE 33
Adjacency Matrix
0/1 n x n matrix, where n = # of vertices A(i,j) = 1 iff (i,j) is an edge
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1 2 3 4 5 1 2 3 4 5 1 1 1 1 1 1 1 1 1 1
SLIDE 34
Adjacency Matrix Properties
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1 2 3 4 5 1 2 3 4 5 1 1 1 1 1 1 1 1 1 1
- Diagonal entries are zero.
- Adjacency matrix of an undirected graph is
symmetric.
- A(i,j) = A(j,i) for all i and j.
SLIDE 35
Adjacency Matrix (Digraph)
2 3 1 4 5
1 2 3 4 5 1 2 3 4 5 1 1 1 1 1 1
- Diagonal entries are zero.
- Adjacency matrix of a digraph need not be
symmetric.
SLIDE 36
Adjacency Matrix
n2 bits of space For an undirected graph, may store only lower
- r upper triangle (exclude diagonal).
- (n-1)n/2 bits
O(n) time to find vertex degree and/or
vertices adjacent to a given vertex.
SLIDE 37
Adjacency Lists
Adjacency list for vertex i is a linear list of vertices adjacent from vertex i. An array of n adjacency lists.
2 3 1 4 5
aList[1] = (2,4) aList[2] = (1,5) aList[3] = (5) aList[4] = (5,1) aList[5] = (2,4,3)
SLIDE 38
Linked Adjacency Lists
Each adjacency list is a chain.
2 3 1 4 5
aList[1] aList[5] [2] [3] [4] 2 4 1 5 5 5 1 2 4 3
Array Length = n # of chain nodes = 2e (undirected graph) # of chain nodes = e (digraph)
SLIDE 39
Array Adjacency Lists
Each adjacency list is an array list.
2 3 1 4 5
aList[1] aList[5] [2] [3] [4] 2 4 1 5 5 5 1 2 4 3
Array Length = n # of list elements = 2e (undirected graph) # of list elements = e (digraph)
SLIDE 40
Weighted Graphs
Cost adjacency matrix.
- C(i,j) = cost of edge (i,j)
Adjacency lists => each list element is a pair
(adjacent vertex, edge weight)
SLIDE 41
Graph Search Methods
- A vertex u is reachable from vertex v iff there
is a path from v to u.
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SLIDE 42
Graph Search Methods
- A search method starts at a given vertex v and
visits/labels/marks every vertex that is reachable from v.
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SLIDE 43
Graph Search Methods
- Many graph problems solved using a search method.
- Path from one vertex to another.
- Is the graph connected?
- Find a spanning tree.
- Etc.
- Commonly used search methods:
- Breadth-first search.
- Depth-first search.