CSE233 Course Review

1
CSE233 Course Review

• CSE, POSTECH

2
What we studied in the 1st half

• Week 1 Overview of C, Program performance
• Week 2 Array-based and linked representations
  of Lists
• Week 3 Arrays and Matrices
• Week 4 Performance Measurement
• Week 5 Stacks
• Week 6 Queues
• Week 7 Skip lists and Hashing
• Week 8 Review and Midterm Exam

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What we studied in the 2nd half

• Week 9 Binary and other trees
• Week 10 Priority queues, Heaps, Leftist trees
• Week 11 Tournament trees and Bin packing
• Week 12 Binary search trees
• Week 13 AVL trees
• Week 14 Graphs
• Week 15 Graph Search Methods
• Week 16 Review and Final exam

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• Trees

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Tree Terminology

• A tree t is a finite nonempty set of elements
• The element at the top is called the root
• The remaining elements, if any, are partitioned
  into trees, which are called the subtrees of t.
• Elements next in the hierarchy are the children
  of the root.
• Elements next in the hierarchy are the
  grandchildren of the root, and so on.
• Elements at the lowest level of the hierarchy are
  the leaves.

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Leaves, Parent, Grandparent, Siblings, Ancestors,
Descendents
Leaves Mike,AI,Sue,Chris
Parent(Mary) Joe
Grandparent(Sue) Mary
Siblings(Mary) Ann,John
Ancestors(Mike) Ann,Joe
Descendents(Mary)Mark,Sue
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Levels and Height

• Root is at level 1 and its children are at level
  2.
• Height depth number of levels

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Node Degree

• Node degree is the number of children it has

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Tree Degree

• Tree degree is the maximum of node degrees

tree degree 3
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• Binary Trees

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Binary Tree

• A finite (possibly empty) collection of elements
• A nonempty binary tree has a root element and the
  remaining elements (if any) are partitioned into
  two binary trees
• They are called the left and right subtrees of
  the binary tree

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Difference Between a Tree a Binary Tree

• A binary tree may be empty a tree cannot be
  empty.
• No node in a binary tree may have a degree more
  than 2, whereas there is no limit on the degree
  of a node in a tree.
• The subtrees of a binary tree are ordered those
  of a tree are not ordered.

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Binary Tree for Mathematical Expressions
Figure 11.5 Expression Trees
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Full Binary Tree

• A full binary tree of height h has exactly 2h-1
  nodes.
• Numbering the nodes in a full binary tree
• Number the nodes 1 through 2h-1
• Number by levels from top to bottom
• Within a level, number from left to right

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Complete Binary Tree with N Nodes

• Start with a full binary tree that has at least n
  nodes
• Number the nodes as described earlier.
• The binary tree defined by the nodes numbered 1
  through n is the n-node complete binary tree.
• A full binary tree is a special case of a
  complete binary tree

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Example of Complete Binary Tree

• Complete binary tree with 10 nodes.

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Incomplete Binary Trees

• Complete binary tree with some missing elements

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Binary Tree Representation

• Array representation
• Linked representation

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Array Representation of Binary Tree

• The binary tree is represented in an array by
  storing each element at the array position
  corresponding to the number assigned to it.

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Right-Skewed Binary Tree

• An n node binary tree needs an array whose length
  is between n1 and 2n.
• Right-skewed binary tree wastes the most space
• What about left-skewed binary tree?

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Linked Representation of Binary Tree

• The most popular way to present a binary tree
• Each element is represented by a node that has
  two link fields (leftChild and rightChild) and an
  element field

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• Priority Queues

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Priority Queues

• A priority queue is a collection of zero or more
  elements ? each element has a priority or value
• Unlike the FIFO queues, the order of deletion
  from a priority queue (e.g., who gets served
  next) is determined by the element priority
• Elements are deleted by increasing or decreasing
  order of priority rather than by the order in
  which they arrived in the queue

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Priority Queues

• Operations performed on priority queues
• 1) Find an element, 2) insert a new element, 3)
  delete an element, etc.
• In a Min priority queue, find/delete operation
  finds/deletes the element with minimum priority
• In a Max priority queue, find/delete operation
  finds/deletes the element with maximum priority
• Two or more elements can have the same priority

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Implementation of Priority Queues

• Implemented using heaps and leftist trees
• Heap is a complete binary tree that is
  efficiently stored using the array-based
  representation
• Leftist tree is a linked data structure suitable
  for the implementation of a priority queue

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• Min (Max) Trees

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Max (Min) Tree

• A max tree (min tree) is a tree in which the
  value in each node is greater (less) than or
  equal to those in its children (if any)
• Nodes of a max or min tree may have more than two
  children (i.e., may not be binary tree)

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Max Tree Example
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Min Tree Example
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• Heaps

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Heaps - Definitions

• A max heap (min heap) is a max (min) tree that is
  also a complete binary tree

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Max Heap with 9 Nodes
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Min Heap with 9 Nodes
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Array Representation of Heap

• A heap is efficiently represented as an array.

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Insertion into a Max Heap

• New element is 20
• Are we finished?

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Insertion into a Max Heap

• Exchange the positions with 7
• Are we finished?

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Insertion into a Max Heap

• Exchange the positions with 8
• Are we finished?

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Insertion into a Max Heap

• Exchange the positions with 9
• Are we finished?

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Deletion from a Max Heap

• Max element is in the root
• What happens when we
• delete an element?

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Deletion from a Max Heap

• After the max element is removed.
• Are we finished?

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Deletion from a Max Heap

• Heap with 10 nodes.
• Reinsert 8 into the heap.

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Deletion from a Max Heap

• Reinsert 8 into the heap.
• Are we finished?

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Deletion from a Max Heap

• Exchange the position with 15
• Are we finished?

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Deletion from a Max Heap

• Exchange the position with 9
• Are we finished?

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Max Heap Initialization

• Heap initialization means to construct a heap by
  adjusting the tree if necessary
• Example input array -,1,2,3,4,5,6,7,8,9,10,11
  

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Max Heap Initialization
- Start at rightmost array position that has a
child.
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Max Heap Initialization
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Max Heap Initialization
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Max Heap Initialization
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Max Heap Initialization
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Max Heap Initialization

• Are we finished?
• Done!

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Exercise 12.7

• Do Exercise 12.7
• theHeap -, 10, 2, 7, 6, 5, 9, 12, 35, 22, 15,
  1, 3, 4

• 12.7 (a) complete binary tree

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Exercise 12.7 (b)

• 12.7 (b) The heapified tree

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Exercise 12.7 (c)

• 12.7 (c) The heap after 15 is inserted is

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Exercise 12.7 (c)

• 12.7 (c) The heap after 20 is inserted is

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Exercise 12.7 (c)

• 12.7 (c) The heap after 45 is inserted is

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Exercise 12.7 (d)

• 12.7 (d) The heap after the first remove max
  operation is

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Exercise 12.7 (d)

• 12.7 (d) The heap after the second remove max
  operation is

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Exercise 12.7 (d)

• 12.7 (d) The heap after the third remove max
  operation is

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• Leftist Trees

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Leftist Trees

• Leftist tree is a tree which tends to lean to the
  left
• Leftist tree structures are useful for
  applications
• to meld (i.e., combine) pairs of priority queues
• using multiple queues of varying size
• External node a special node that replaces each
  empty subtree
• Internal node a node with non-empty subtrees
• Extended binary tree a binary tree with
  external nodes added

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Height-Biased Leftist Tree (HBLT)

• Let s(x) be the length (height) of a shortest
  path from node x to an external node in its
  subtree
• If x is an external node, s(x) 0
• If x is an internal node, s(x) min s(L), s(R)
  1, where L and R are left and right children of
  x
• A binary tree is a height-biased leftist tree
  (HBLT) iff at every internal node, the s value of
  the left child is greater than or equal to the s
  value of the right child
• A max HBLT is an HBLT that is also a max tree
• A min HBLT is an HBLT that is also a min tree

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Weight-Biased Leftist Tree (WBLT)

• Let the weight, w(x), of node x to be the number
  of internal nodes in the subtree with root x
• If x is an external node, w(x) 0
• If x is an internal node, its weight is one more
  than the sum of the weights of its children
• A binary tree is a weight-biased leftist tree
  (WBLT) iff at every internal node, the w value of
  the left child is greater than or equal to the w
  value of the right child
• A max (min) WBLT is a max (min) tree that is also
  a WBLT

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Extended Binary Tree
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Melding max HBLTs
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Applications of Heaps

• Sort (heap sort)
• Machine scheduling
• Huffman codes

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Heap Sort

• use element key as priority
• Algorithm
• put elements to be sorted into a priority queue
  (i.e., initialize a heap)
• extract (delete) elements from the priority queue
• if a min priority queue is used, elements are
  extracted in non-decreasing order of priority
• if a max priority queue is used, elements are
  extracted in non-increasing order of priority

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Machine Scheduling Problem

• m identical machines
• n jobs/tasks to be performed
• The machine scheduling problem is to assign jobs
  to machines so that the time at which the last
  job completes is minimum

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NP-hard Problems

• The class of problems for which no one has
  developed a polynomial time algorithm.
• No algorithm whose complexity is O(nk ml) is
  known for any NP-hard problem (for any constants
  k and l)
• NP stands for Nondeterministic Polynomial
• NP-hard problems are often solved by heuristics
  (or approximation algorithms), which do not
  guarantee optimal solutions
• Longest Processing Time (LPT) rule is a good
  heuristic for minimum finish time scheduling.

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Huffman Codes

• For text compression, the LZW method relies on
  the recurrence of substrings in a text
• Huffman codes is another text compression method,
  which relies on the relative frequency (i.e., the
  number of occurrences of a symbol) with which
  different symbols appear in a text
• Uses extended binary trees
• Variable-length codes that satisfy the property,
  where no code is a prefix of another
• Huffman tree is a binary tree with minimum
  weighted external path length for a given set of
  frequencies (weights)

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• Tournament Trees

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Tournament Trees

• Like the heap, a tournament tree is a complete
  binary tree that is most efficiently stored using
  array-based binary tree
• Used to obtain efficient implementations of two
  approximation algorithms for the bin packing
  problem (another NP-hard problem)
• Types of tournament trees winner loser trees
• The tournament is played in the sudden-death mode
• Tournament trees are also called selection trees

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Winner Trees

• A winner tree for n players is a complete binary
  tree with n external nodes and n-1 internal
  nodes. Each internal node records the winner of
  the match.
• To determine the winner of a match, we assume
  that each player has a value
• In a min (max) winner tree, the player with the
  smaller (larger) value wins

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Loser Trees

• A loser tree for n players is also a complete
  binary tree with n external nodes and n-1
  internal nodes. Each internal node records the
  loser of the match.
• The overall winner is recorded in tree0

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• Bin Packing Problems

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Bin Packing Problem

• We have bins that have a capacity binCapacity and
  n objects that need to be packed into these bins
• Object i requires objSizei, where 0 lt
  objSizei ? binCapacity, units of capacity
• Feasible packing - an assignment of objects to
  bins so that no bins capacity is exceeded
• Optimal packing - a feasible packing that uses
  the fewest number of bins
• Goal pack objects with the minimum number of
  bins
• The bin packing problem is an NP-hard problem

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Truck Loading Problem

• Have parcels to pack into trucks
• Each parcel has a weight
• Each truck has a load limit
• Goal Minimize the number of trucks needed
• Equivalent to the bin packing problem

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Bin Packing Approximation Algorithms

• First Fit (FF)
• First Fit Decreasing (FFD)
• Best Fit (BF)
• Best Fit Decreasing (BFD)

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First Fit (FF) Bin Packing

• Bins are arranged in left to right order.
• Objects are packed one at a time in a given
  order.
• Current object is packed into the leftmost
  bininto which it fits.
• If there is no bin into which current object
  fits,start a new bin.

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Best Fit (BF) Bin Packing

• Let binj.unusedCapacity denote the capacity
  available in bin j
• Initially, the available capacity is binCapacity
  for all bins
• Object i is packed into the bin with the least
  unusedCapacity that is at least objSizei
• If there is no bin into which current object
  fits,start a new bin.

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First Fit Decreasing (FFD) Bin Packing

• This method is the same as FF except that the
  objects reordered in a decreasing size so that
  objSizei ? objSizei1, 1 ? i lt n

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Best Fit Decreasing (BFD) Bin Packing

• This method is the same as BF except that the
  objects are reordered as for FFD

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• Binary Search Trees

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Binary Search Tree

• A binary tree that may be empty. A nonempty
  binary search tree satisfies the following
  properties
• Each node has a key (or value), and no two nodes
  have the same key (i.e., all keys are distinct).
• For every node x, all keys in the left subtree of
  x are smaller than that in x.
• For every node x, all keys in the right subtree
  of x are larger than that in x.
• The left and right subtrees of the root are also
  binary search trees

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Examples of Binary Trees

• Which of the above trees are binary search trees?
• ? (b) and (c)
• Why isnt (a) a binary search tree?
• ? It violates the property 3

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Indexed Binary Search Trees

• Binary search tree.
• Each node has an additional field LeftSize.
• Support search and delete operations by rankas
  well as all the binary search tree operations.
• LeftSize
• the number of elements in its left subtree
• the rank of an element with respect to the
  elements in its subtree (e.g., the fourth element
  in sorted order)

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Indexed Binary Search Tree Example

• LeftSize values are in red.

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The Operation Ascend()

• How can we output all elements in ascending
  order of keys?
• Do an inorder traversal (left, root, right).
• What would be the output?
• 2, 6, 8, 10, 15, 20, 25, 30, 40

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The Operation Search(key, e)

• Search begins at the root
• If the root is NULL, the search tree is empty and
  the search fails.
• If key is less than the root, then left subtree
  is searched
• If key is greater than the root, then right
  subtree is searched
• If key equals the root, then the search
  terminates successfully
• The time complexity for search is O(height)
• See Program 11.4 for the search operation code

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The Operation Insert(key, e)

• To insert a new element e into a binary search
  tree, we must first verify that its key does not
  already exist by performing a search in the tree
• If the search is successful, we do not insert
• If the search is unsuccessful, then the element
  is inserted at the point the search terminated
• Why insert it at that point?
• The time complexity for insert is O(height)
• See Figure 14.3 for examples
• See Program 14.5 for the insert operation code

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Insert Example
We wish to insert an element with the key
35. Where should it be inserted?
92
Insert Example
Insert an element with the key 7.
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Insert Example
Insert an element with the key 18.
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The Operation Delete(key, e)

• For deletion, there are three cases for the
  element to be deleted
• Element is in a leaf.
• Element is in a degree 1 node (i.e., has exactly
  one nonempty subtree).
• Element is in a degree 2 node (i.e., has exactly
  two nonempty subtrees).

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Case 1 Delete from a Leaf

• For case 1, we can simply discard the leaf node.
• Example, delete a leaf element. key7

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Case 2 Delete from a Degree 1 Node

• Example Delete key40

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Case 3 Delete from a Degree 2 Node

• Example Delete key10

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Case 3 Delete from a Degree 2 Node

• Replace with the largest key in the left
  subtree(or the smallest in the right subtree)
• Which node is the largest key in the left
  subtree?

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Case 3 Delete from a Degree 2 Node
The largest key must be in a leaf or degree 1
node.
100

• AVL Trees

101
Balanced Binary Search Trees

• Trees with a worst-case height of O(log n) are
  called balanced trees
• An example of a balanced tree is AVL
  (Adelson-Velsky and Landis) tree

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AVL Tree

• Definition
• Binary tree.
• If T is a nonempty binary tree with TL and TR as
  its left and right subtrees, then T is an AVL
  tree iff
• TL and TR are AVL trees, and
• hL hR ? 1 where hL and hR are the heights of
  TL and TR, respectively

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AVL Search Trees

• An AVL search tree is a binary search tree that
  is also an AVL tree

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Indexed AVL Search Trees

• An indexed AVL search tree is an indexed binary
  search tree that is also an AVL tree

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Balance Factor

• To facilitate insertion and deletion, a balance
  factor (bf) is associated with each node
• The balance factor bf(x) of a node x is defined
  as height(x?leftChild) height(x?rightChild)
• Balance factor of each node in an AVL tree must
  be 1, 0, or 1

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AVL Tree with Balance Factors

• Is this an AVL tree?
• What is the balance factor for each node in this
  AVL tree?
• Is this an AVL search tree?

107
Inserting into an AVL Search Trees

• If we use the strategy of Program 14.5 to insert
  an element into an AVL search tree, the result
  may not be an AVL tree
• That is, the tree may become unbalanced
• If the tree becomes unbalanced, we must adjust
  the tree to restore balance - this adjustment is
  called rotation

108
Inserting into an AVL Search Tree
Insert(9)

• Where is 9 going to be inserted into?
• After the insertion, is the tree still an AVL
  search tree? (i.e., still balanced?)

109
Imbalance Types

• After an insertion, when the balance factor of
  node A is 2 or 2, the node A is one of the
  following four imbalance types
• LL new node is in the left subtree of the left
  subtree of A
• LR new node is in the right subtree of the left
  subtree of A
• RR new node is in the right subtree of the right
  subtree of A
• RL new node is in the left subtree of the right
  subtree of A

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Rotation

• Definition
• To switch children and parents among two or three
  adjacent nodes to restore balance of a tree.
• A rotation may change the depth of some nodes,
  but does not change their relative ordering.

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Left Rotation

• Definition
• In a binary search tree, pushing a node A down
  and to the left to balance the tree.
• A's right child replaces A, and the right child's
  left child becomes A's right child.

112
Right Rotation

• Definition
• In a binary search tree, pushing a node A down
  and to the right to balance the tree.
• A's left child replaces A, and the left child's
  right child becomes A's left child.

113
AVL Rotations

• To balance an unbalanced AVL tree (after an
  insertion), we may need to perform one of the
  following rotations LL, RR, LR, RL

Figure 15.3 Inserting into an AVL search tree
114
An LL Rotation
115
An LR Rotation
116
Single and Double Rotations

• Single rotations the transformations done to
  correct LL and RR imbalances
• Double rotations the transformations done to
  correct LR and RL imbalances
• The transformation to correct LR imbalance can be
  achieved by an RR rotation followed by an LL
  rotation
• The transformation to correct RL imbalance can be
  achieved by an LL rotation followed by an RR
  rotation (do Exercise 15.13)
• See Figure 15.6 for the AVL-search-tree-insertion
  algorithm

117
Deletion from an AVL Search Tree

• To delete an element from an AVL search tree, we
  can use Program 14.6
• Deletion of a node may also produce an imbalance
• Imbalance incurred by deletion is classified
  intothe types R0, R1, R-1, L0, L1, and L-1
• Rotation is also needed for rebalancing

118
An R0 Rotation
Figure 15.7 An R0 rotation (single rotation)
119
An R1 Rotation
Figure 15.8 An R1 rotation (single rotation)
120
An R-1 Rotation
121

• Graphs

122
Topics related to Graphs

• Graph terminology vertex, edge, adjacent,
  incident, degree, cycle, path, connected
  component, spanning tree
• Types of graphs undirected, directed, weighted
• Graph representations adjacency matrix, array
  adjacency lists, linked adjacency lists
• Graph search methods breath-first, depth-first
  search
• Algorithms
• to find a path in a graph
• to find the connected components of an undirected
  graph
• to find a spanning tree of a connected undirected
  graph

123
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 vertices.
• Edges are also called arcs and lines.
• Vertices i and j are adjacent vertices iff (i, j)
  is an edge in the graph
• The edge (i, j) is incident on the vertices i and
  j

124
Graphs

• Undirected edge has no orientation (no arrow
  head)
• Directed edge has an orientation (has an arrow
  head)
• Undirected graph all edges are undirected
• Directed graph all edges are directed

125
Path and Simple Path

• A sequence of vertices P i1, i2, , ik is an i1
  to ik path in the graph G(V, E) iff the edge
  (ij, ij1) is in E for every j, 1 j lt k
• A simple path is a path in which all vertices,
  except possibly in the first and last, are
  different

126
Length (Cost) of a Path

• Each edge in a graph may have an associated
  length (or cost). The length of a path is the sum
  of the lengths of the edges on the path
• What is the length of the path 5, 9, 11, 10?

127
Subgraph Cycle

• Let G (V, E) be an undirected graph
• A graph H is a subgraph of graph G iff its vertex
  and edge sets are subsets of those of G
• A cycle is a simple path with the same start and
  end vertex

128
Spanning Tree

• Let G (V, E) be an undirected graph
• A connected undirected graph that contains no
  cycles is a tree
• A subgraph of G that contains all the vertices of
  G and is a tree is a spanning tree
• A spanning tree has n vertices and n-1 edges
• The spanning tree that costs the least is called
  the minimum-cost spanning tree

129
Bipartite Graph

• A bipartite graph is a special graph where the
  set of vertices can be divided into two disjoint
  sets U and V such that no edge has both
  end-points in the same set.
• A simple undirected graph G (V, E) is called
  bipartite if there exists a partition of the
  vertex set V V1 U V2 so that both V1 and V2 are
  independent sets.

130
Graph Properties

• The degree of vertex i is the no. of edges
  incident on vertex i.
• The sum of vertex degrees 2e (where e is the
  number of edges)
• In-degree of vertex i is the number of edges
  incident to i
• Out-degree of vertex i is the number of edges
  incident from i

131
Complete Undirected/Directed Graphs

• A complete undirected graph has n(n-1)/2 edges
  (i.e., all possible edges) and is denoted by Kn

• A complete directed graph (also denoted by Kn) on
  n vertices contains exactly n(n-1) edges

132
Path Finding

• Path between 1 and 8

• What is a possible path its length?
• A path is 1, 2, 5, 9, 8 and its length is 20.

133
Connected Graph

• Let G (V, E) be an undirected graph
• G is connected iff there is a path between every
  pair of vertices in G

134
Representation of Unweighted Graphs

• The most frequently used representations for
  unweighted graphs are
• Adjacency Matrix
• Linked adjacency lists
• Array adjacency lists

135
Adjacency Matrix

• 0/1 n x n matrix, where n of vertices
• A(i, j) 1 iff (i, j) is an edge.

136
Adjacency Matrix Properties

• Diagonal entries are zero.
• Adjacency matrix of an undirected graph is
  symmetric (A(i,j) A(j,i) for all i and j).

137
Adjacency Matrix for Digraph

• Diagonal entries are zero.
• Adjacency matrix of a digraph need not be
  symmetric.
• See Figure 16.9 for more adjacency matrices

138
Adjacency Lists

• Adjacency list for vertex i is a linear list of
  vertices adjacent from vertex i.
• An array of n adjacency lists for each vertex of
  the graph.

139
Linked Adjacency Lists

• Each adjacency list is a chain.Array length
  n. of chain nodes 2e (undirected graph) of
  chain nodes e (digraph)

• See Figure 16.11 for more linked adjacency lists

140
Array Adjacency Lists

• Each adjacency list is an array list.Array
  length n. of chain nodes 2e (undirected
  graph) of chain nodes e (digraph)

• See Figure 16.12 for more array adjacency lists

141
Representation of Weighted Graphs

• Weighted graphs are represented with simple
  extensions of those used for unweighted graphs
• The cost-adjacency-matrix representation uses a
  matrix C just like the adjacency-matrix
  representation does
• Cost-adjacency matrix C(i, j) cost of edge (i,
  j)
• Adjacency lists each list element is a
  pair(adjacent vertex, edge weight)

142

• Graph Search Methods

143
Graph Search Methods

• Many graph problems solved by a search method
• Finding a path from one vertex to another.
• Determining whether a graph is connected
• Find a spanning tree
• Finding a minimum-cost path/spanning tree
• Breadth-first search (BFS)
• Method of starting at a vertex and identifying
  all vertices reachable from it
• Similar to the level-order traversal of a binary
  tree
• Depth-first search (DFS)
• an alternative to BFS
• Similar to the pre-order traversal of a binary
  tree

144
Tips on the Final Exam

• Make sure you do all the exercises mentioned in
  the lecture notes
• Make sure you understand on how to solve the
  problems in the assignments
• Good luck!

145
Tips on your LIFE

• Have a Life Plan
• Yearly life plan at the beginning of each year
• Plan for 5, 10, 20, 30, 40, 50 years ahead
• Have a Role Model
• Bill Gates, Steve Jobs, Larry Page and Sergey
  Brin
• Alan Turing, Vincent Cerf and Leonard Kleinrock
• ???, ???, ???
• ????
• Have a Wonderful Life!