CSE 326: Data Structures Lecture #4

CSE 326: Data StructuresLecture #4 Alon Halevy Spring Quarter 2001

Agenda • Today: • Finish complexity issues. • Linked links (Read Ch 3; skip “radix sort”)

Direct Proof of Recursive Fibonacci • Recursive Fibonacci: int Fib(n) if (n == 0 or n == 1) return 1 else return Fib(n - 1) + Fib(n - 2) • Lower bound analysis • T(0), T(1) >= b T(n) >= T(n - 1) + T(n - 2) + c ifn > 1 • Analysis let  be (1 + 5)/2 which satisfies 2 =  + 1 show by induction on n that T(n) >= bn - 1

Direct Proof Continued • Basis: T(0)  b > b-1 and T(1)  b = b0 • Inductive step: Assume T(m)  bm - 1 for all m < n T(n)  T(n - 1) + T(n - 2) + c  bn-2 + bn-3 + c  bn-3( + 1) + c = bn-32 + c  bn-1

Fibonacci Call Tree 5 3 4 3 2 2 1 1 2 0 1 1 0 1 0

Learning from Analysis • To avoid recursive calls • store all basis values in a table • each time you calculate an answer, store it in the table • before performing any calculation for a value n • check if a valid answer for n is in the table • if so, return it • Memoization • a form of dynamic programming • How much time does memoized version take?

Kinds of Analysis • So far we have considered worst case analysis • We may want to know how an algorithm performs “on average” • Several distinct senses of “on average” • amortized • average time per operation over a sequence of operations • average case • average time over a random distribution of inputs • expected case • average time for a randomized algorithm over different random seeds for any input

Amortized Analysis • Consider any sequence of operations applied to a data structure • your worst enemy could choose the sequence! • Some operations may be fast, others slow • Goal: show that the average time per operation is still good

E D C B A A F B C D E F Stack ADT • Stack operations • push • pop • is_empty • Stack property: if x is on the stack before y is pushed, then x will be popped after y is popped What is biggest problem with an array implementation?

Stretchy Stack Implementation int data[]; int maxsize; int top; Push(e){ if (top == maxsize){ temp = new int[2*maxsize]; copy data into temp; deallocate data; data = temp; } else { data[++top] = e; } Best case Push = O( ) Worst case Push = O( )

Stretchy Stack Amortized Analysis • Consider sequence of n operations push(3); push(19); push(2); … • What is the max number of stretches? • What is the total time? • let’s say a regular push takes time a, and stretching an array containing k elements takes time kb, for some constants a and b. • Amortized =(an+b(2n-1))/n = a+2b-(1/n)= O(1) log n

Average Case Analysis • Attempt to capture the notion of “typical” performance • Imagine inputs are drawn from some random distribution • Ideally this distribution is a mathematical model of the real world • In practice usually is much more simple – e.g., a uniform random distribution

Example: Find a Red Card • Input: a deck of n cards, half red and half black • Algorithm: turn over cards (from top of deck) one at a time until a red card is found. How many cards will be turned over? • Best case = • Worst case = • Average case: over all possible inputs (ways of shuffling deck)

Summary • Asymptotic Analysis – scaling with size of input • Upper bound O, Lower bound  • O(1) or O(log n) great • O(2n) almost never okay • Worst case most important – strong guarantee • Other kinds of analysis sometimes useful: • amortized • average case

List ADT ( A1 A2 … An-1 An ) length = n • List properties • Ai precedes Ai+1 for 1  i < n • Ai succeeds Ai-1 for 1 < i  n • Size 0 list is defined to be the empty list • Key operations • Find(item) = position • Find_Kth(integer) = item • Insert(item, position) • Delete(position) • Next(position) = position • What are some possible data structures?

Implementations of Linked Lists Array: 1 7 9 2 3 4 5 6 8 10 H W 1 I S E A S Y Can we apply binary search to an array representation? Linked list: (optional header) (a b c) a b c  L

Linked List vs. Array linked list array sorted array Find(item) = position Find_Kth(integer)=item Find_Kth(1)=item Insert(item, position) Insert(item) Delete(position) Next(position) = position

Tradeoffs • For what kinds of applications is a linked list best? • Examples for an unsorted array? • Examples for a sorted array?

Implementing in C++ (optional header) (a b c) a b c  Create separate classes for • Node • List (contains a pointer to the first node) • List Iterator (specifies a position in a list; basically, just a pointer to a node) Pro: syntactically distinguishes uses of node pointers Con: a lot of verbage! Also, is a position in a list really distinct from a list? L

CSE 326: Data Structures Lecture #4