Data Structures

1

• Data Structures Algorithms
• Recursion and Trees

• Richard Newman

2

• 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))

3

• 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)

4

• Trees

• 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 a tree

5

• Roadmap to Recursion

• Widely used in CS and with trees...
• Mathematical recurrences
• Recursive programs
• Divide and Conquer
• Dynamic Programming
• Tree traversal
• DFS

6

• 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)
7

• 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
8

• Euclid's 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
9

• 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

10

• 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

11

• 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!

12

• Tower 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

13

• Tower of Hanoi

Target peg
Which peg should top disk go on first?
14

• Tower of Hanoi

How many moves does this take?
How many moves does this take?
15

• Tower 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
16

• 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
17

• 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.

18

• 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)
19

• Dynamic Programming

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

13
8
5
2
5
3
3
1
1
1
1
2
2
2
3
1
1
1
1
1
1
0
1
1
1
0
1
0
2
1
1
1
0
1
0
1
0
1
0
1
0
20

• Dynamic Programming

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

13
8
5
2
5
3
3
1
1
1
1
2
2
2
3
1
1
1
1
1
1
0
1
1
1
0
1
0
2
1
1
1
0
1
0
1
0
1
0
1
0
21

• 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.

22

• 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

23

• 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.