day04 - CS1 Introduction to Computation Day 4 October 8...

Caltech CS 1 -- Fall 2008
CS1:
Introduction to Computation
Day 4: October 8, 2008
Dynamic Evolution of Computations

Caltech CS 1 -- Fall 2008
Before we begin...
• Let's select house ombudspeople!
• Meet at noon on Thursday at the
Athenaeum (olive walk entrace)
• Starting tomorrow!

Caltech CS 1 -- Fall 2008
Today, let’s think about:
• How do our computations unfold?
– in space
– in time
• This will lead us to consider:
• efficiency of program execution
(first look at a huge subject)

Caltech CS 1 -- Fall 2008
Design and analysis (again)
• Last time saw how to design recursive
procedures
• There are many ways to design a
procedure that solves the same problem
• Analysis will help us to distinguish "good"
solutions from "bad" ones
– based on some criterion for "good" and "bad"
• Design and analysis always interact!

Caltech CS 1 -- Fall 2008
Example
• Recall from last time:
(define (sum-integers n)
(if (= n 0)
0
(+ n (sum-integers (- n 1)))))

Caltech CS 1 -- Fall 2008
Revisited (summarizing steps)
• Evaluate: (sum-integers 3)
• (if (= 3 0) 0 (+ 3 (sum-integers (- 3 1))))
• (if #f 0 (+ 3 (sum-integers (- 3 1))))
• (+ 3 (sum-integers (- 3 1)))
• (+ 3 (sum-integers 2))
• (+ 3 (if (= 2 0) 0 (+ 2 (sum-integers (- 2 1)))))
• (+ 3 (+ 2 (sum-integers 1)))
• (+ 3 (+ 2 (if (= 1 0) 0 (+ 1 (sum-integer (- 1 1))))))

Caltech CS 1 -- Fall 2008
Revisited (summarizing steps)
• (+ 3 (+ 2 (+ 1 (sum-integers 0))))
• (+ 3 (+ 2 (+ 1 (if (= 0 0) 0 …))))
• (+ 3 (+ 2 (+ 1 (if #t 0 …))))
• (+ 3 (+ 2 ( + 1 0)))
• …
• 6

Caltech CS 1 -- Fall 2008
Evolution of computation
• (sum-integers 3)
• (+ 3 (sum-integers 2))
• (+ 3 (+ 2 (sum-integers 1)))
• (+ 3 (+ 2 (+ 1 (sum-integers 0))))
• (+ 3 (+ 2 (+ 1 0)))

Caltech CS 1 -- Fall 2008
What can we say about the
computation?
On input N :
• How many calls to
sum-integers ?
N+1
• How much work per
call?
(sum-integers 3)
(+ 3 (sum-integers 2))
(+ 3 (+ 2 (sum-integers 1)))
(+ 3 (+ 2 (+ 1 (sum-integers 0))))
(+ 3 (+ 2 (+ 1 0)))

Caltech CS 1 -- Fall 2008
Work per call?
• (+ 3 (sum-integers 2))
• (+ 3 (if (= 2 0) 0 (+ 2 (sum-integers (- 2 1)))))
• (+ 3 (+ 2 (sum-integers 1)))
compare if add * add call
*(subtract costs same as add)

Caltech CS 1 -- Fall 2008
What can we say about the
computation?
on input N :
• How many calls to
sum-integers ?
N+1
• How much work per
call?
constant
• one comparison, one
if , two additions, one
function call
(at most)
• How many deferred
operations at end?
(sum-integers 3)
(+ 3 (sum-integers 2))
(+ 3 (+ 2 (sum-integers 1)))
(+ 3 (+ 2 (+ 1 (sum-integers 0))))
(+ 3 (+ 2 (+ 1 0)))

Caltech CS 1 -- Fall 2008
Deferred operations?
• (+ 3 (+ 2 (+ 1 0)))
• Last call to sum-integers gives the result 0
– base case
• but have to use this result to "finish up"
computations that were previously started
to get final result
• We call those incomplete computations
deferred operations or pending operations

Caltech CS 1 -- Fall 2008
What can we say about the
computation?
on input N :
• How many calls to
sum-integers ?
N+1
• How much work per
call?
constant
• one comparison, one
if , two additions, one
function call
(at most)
• How many deferred
operations at end?