Data Structures

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Data Structures Algorithms Recursion and Trees
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Recursion
Fundamental concept in math and CS Recursive
definition Defined in terms of itself aN
aaN-1, a0 1 Recursive function Calls
itself int exp(int base, int pow) return (pow
0 ? 1 baseexp(base,
pow-1))
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Recursion
Recursive definition (and function) must 1. have
a base case termination condition 2. always
call a case smaller than itself All practical
computations can be couched in a recursive
framework! (see theory of computation)
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Recursion
Recursively defined structures e.g., binary
tree Base case Empty tree has no
nodes Recursion None-empty tree has a root node
with two children, each the root of a binary
tree
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Recursion
Widely used in CS and with trees... Mathematical
recurrences Recursive programs Divide and
Conquer Dynamic Programming Tree traversal DFS
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Recursive Algorithms

• Recursive algorithm solves problem by solving
  one or more smaller instances of same problem
• Recurrence relation factorial
• N! N(N-1)!, for N gt 0, with 0! 1.
• In C, use recursive functions

Int factorial(int N) if (N 0) return 1
return Nfactorial(N-1)
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Recursive Algorithms

• BTW, can often also be expressed as iteration
• E.g., can also write N! computation as a loop

int factorial(int N) for (int t 1, i 1
i lt N i) t i return t
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Euclids Algorithm
Euclid's Algorithm is one of the oldest known
algorithms Recursive method for finding the GCD
of two integers
Base case
int gcd(int m, int n) // expect m gt n if (n
0) return m return gcd(n, m n)
Recursive call to smaller instance
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Divide Conquer
Recursive scheme that divides input into two (or
some fixed number) of (roughly) equal parts Then
makes a recursive call on each part Widely used
approach Many important algorithms Depending on
expense of dividing and combining, can be very
efficient
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Divide Conquer
Example find the maximum element in an array
aN (Easy to do iteratively...) Base case Only
one element return it Divide Split array into
upper and lower halves Recursion Find maximum of
each half Combine results Return larger of two
maxima
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Divide Conquer
Property 5.1 A recursive function that divides
a problem of size N into two independent
(non-empty) parts that it solves, recursively
calls itself less than N times. Prf T(1)
0 T(N) T(k) T(N-k) 1 for recursive call on
size N divided into one part of size k and the
other of size N-k Induct!
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Towers of Hanoi
3 pegs N disks, all on one peg Disks arranged
from largest on bottom to smallest on top Must
move all disks to target peg Can only move one
disk at a time Must place disk on another peg Can
never place larger disk on a smaller one Legend
has it that the world will end when a certain
group of monks finishes the task in a temple
with 40 golden disks on 3 diamond pegs
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Towers of Hanoi
Target peg
Which peg should top disk go on first?
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Towers of Hanoi
How many moves does this take?
How many moves does this take?
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Towers of Hanoi
Property 5.2 The recursive dc algorithm for
the Towers of Hanoi problem produces a
solution that has 2N 1 moves. Prf T(1)
1 T(N) T(N-1) 1 T(N-1) 2 T(N-1) 1
2N 1 by induction
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Divide Conquer
Two other important DC algorithms Binary
search MergeSort
Algorithm Metric Recurrence Approx. Soln.
Binary Search comparisons C(N) C(N/2)1 lg N
MergeSort recursive calls A(N) 2 A(N/2) 1 N
MergeSort comparisons C(N) 2 C(N/2) N N lg N
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Dynamic Programming
In Divide Conquer, it is essential that the
subproblems be independent (partition the
input) When this is not the case, life gets
complicated! Sometimes, we can essentially fill
up a table with values we compute once,
rather than recompute every time they are
needed. This is Dynamic Programming Issue table
may be too big!
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Dynamic Programming

• Fibonacci Numbers
• F0 0
• F1 1
• FN FN-1 FN-2
• Horribly inefficient implementation

int F(int N) if (N lt 1) return 0 if (N
1) return 1 return F(N-1) F(N-2)
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Dynamic Programming

• How bad is this code?
• How many calls does it make to itself?
• F(N) makes F(N1) calls!
• Exponential!!!!

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Dynamic Programming

• Can we do better?
• How?
• Make a table compute once (yellow shapes)
• Fill up table

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Dynamic Programming
Property 5.3 Dynamic Programming reduces the
running time of a recursive function to be at
most the time it takes to evaluate the functions
for all arguments less than or equal to the
given argument, treating the cost of a recursive
call as a constant.
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Trees
A mathematical abstraction Central to many
algorithms Describe dynamic properties of
algorithms Build and use explicit tree data
structures Examples Family tree of
descendants Sports tournaments (Who's
In?) Organization Charts (Army) Parse tree of
natural language sentence File systems
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Types of Trees

• Trees
• Rooted trees
• Ordered trees
• M-ary trees and binary trees
• Defn A tree is a nonempty collection of vertices
  
• and edges such that there is exactly one path
• between each pair of vertices.
• Defn A path is a list of distinct vertices such
  that
• successive vertices have an edge between them
• Defn A graph in which there is at most one path
• between each pair of vertices is a forest.

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Types of Trees
root
internal node
leaf
external node
Binary Tree
Ternary Tree
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Types of Trees
root
parent
node
sibling
child
Rooted Tree
Free Tree
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Tree Representation
Binary Tree
Representation
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Tree Representation
Ordered Tree
Representation
Use linked list for siblings at each
level, Pointer to left child
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Properties of Trees

• A binary tree with N internal nodes has
• N1 external nodes
• A binary tree with N internal nodes has 2N links
• N-1 to internal nodes and N1 to external nodes
• The level of a node is one higher than the level
  of
• its parent, with the root at level 0.
• The path length of a tree is the sum of the
  levels
• of all the trees nodes
• The internal path length is the sum of levels of
• internal nodes external path length is sum of
• levels of external nodes.

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Properties of Trees

• The external path length of any binary tree with
  N
• nodes is 2N greater than the internal path
  length
• The height of a binary tree with N internal nodes
  
• is at least lg N and at most N-1.
• The internal path length of a binary tree with N
• internal nodes is at least N lg(N/4) and
• at most N(N-1)/2.

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Tree Traversal

• Given pointer to a tree, visit every node in the
• tree systematically
• Inorder Visit the left subtree, visit the root,
  then visit
• the right subtree
• Preorder Visit the root, visit the left subtree,
  visit the
• right subtree.
• Postorder Visit the left subtree, visit the
  right subtree,
• visit the root.

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Tree Traversal

• Generic recursive traversal code
• Preorder? Inorder? Postorder?

void traverse(link h, void visit(link)) if
(h NULL) return visit(h)
traverse(h-gtleft, visit) traverse(h-gtright,
visit)
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Tree Traversal

• Generic iterative traversal code
• Preorder? Inorder? Postorder?

void traverse(link h, void visit(link))
STACKltlinkgt s(max) s.push(h) while
(!s.empty()) visit(h s.pop()) if
(h-gtright ! 0) s.push(h-gtright) if (h-gtleft
! 0) s.push(h-gtleft)
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Tree Traversal

• Generic iterative traversal code
• Level order top to bottom, left to right

void traverse(link h, void visit(link))
QUEUEltlinkgt q(max) q.put(h) while
(!q.empty()) visit(h q.get()) if
(h-gtleft ! 0) q.put(h-gtleft) if (h-gtright !
0) q.put(h-gtright)
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Basic Tree Algorithms

• Count number of nodes
• Compute height
• Compute internal path length
• Display tree

int count(link h) if (h NULL) return
0 return 1 count(h-gtleft)
count(h-gtright)
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Basic Tree Algorithms

• Count number of nodes
• Compute height
• Compute internal path length
• Display tree

int height(link h) if (h NULL) return
-1 int u height(h-gtleft) int v
height(h-gtright) return 1 (u gt v ? u v)
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Basic Tree Algorithms

• Display tree (order?)

void printnode(Item x, int h) for (int i0
ilth i) cout ltlt cout ltlt x ltlt endl void
show(link t, int h) if (t NULL)
printnode(,h) return show(h-gtleft,
h1) printnode(t-gtitem, h) show(h-gtright,
h1)
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Tree Algorithms

• Tournament construction
• Start with array (list of competitors)
• Develop into tree with matches
• Divide and conquer
• Split in half
• Make tourney with left (first) half
• Make a tourney with right (last) half
• Make a new node with links to the two tourneys
• Single item tourney leaf with that item
• Item in interior nodes? Winner of tourney!

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Graph Traversal

• Recursive Graph Traversal
• DFS Depth-First Search
• Generalization of tree traversal methods
• Basis for many algorithms for processing graphs
• Code
• Starting at any node v
• Visit v
• Recursively visit each unvisited neighbor of v
• If graph is connected, will visit every node
• Need to be able to mark nodes as visited
• Dont need to do this for trees (why not?)
• Set of edges on which calls are made forms a
• spanning tree

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Graph Traversal

• Recursive Graph Traversal DFS
• Property 5.10 DFS requires time O(VE) in
• a graph with V vertices and E edges using
• the adjacency lists representation
• Adjacency list representation one list node
• corresponding to each edge in the graph, and
• one list head pointer corresponding to each
• vertex in the graph.

void traverse(int k, void visit(int))
visit(k) visitedk TRUE for (link
tadjk k!0 t t-gtnext) if
(!visitedt-gtv) traverse(t-gtv, visit)
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Graph Traversal

• BFS in graph

void traverse(int k, void visit(int))
QUEUEltlinkgt q(VV) q.put(k) while
(!q.empty()) if (!visitedk q.get() )
visit(k) visitedk 1 for (link
tadjk t!0 tt-gtnext) if
(!visitedt-gtv) q.put(t-gtv)