Andrew Cooke | Contents | Latest | RSS | Previous | Next

C[omp]ute

Welcome to my blog, which was once a mailing list of the same name and is still generated by mail. Please reply via the "comment" links.

Always interested in offers/projects/new ideas. Eclectic experience in fields like: numerical computing; Python web; Java enterprise; functional languages; GPGPU; SQL databases; etc. Based in Santiago, Chile; telecommute worldwide. CV; email.

Personal Projects

Choochoo Training Diary

Last 100 entries

Surprise Paradox; [Books] Good Author List; [Computing] Efficient queries with grouping in Postgres; [Computing] Automatic Wake (Linux); [Computing] AWS CDK Aspects in Go; [Bike] Adidas Gravel Shoes; [Computing, Horror] Biological Chips; [Books] Weird Lit Recs; [Covid] Extended SIR Models; [Art] York-based Printmaker; [Physics] Quantum Transitions are not Instantaneous; [Computing] AI and Drum Machines; [Computing] Probabilities, Stopping Times, Martingales; bpftrace Intro Article; [Computing] Starlab Systems - Linux Laptops; [Computing] Extended Berkeley Packet Filter; [Green] Mainspring Linear Generator; Better Approach; Rummikub Solver; Chilean Poetry; Felicitations - Empowerment Grant; [Bike] Fixing Spyre Brakes (That Need Constant Adjustment); [Computing, Music] Raspberry Pi Media (Audio) Streamer; [Computing] Amazing Hack To Embed DSL In Python; [Bike] Ruta Del Condor (El Alfalfal); [Bike] Estimating Power On Climbs; [Computing] Applying Azure B2C Authentication To Function Apps; [Bike] Gearing On The Back Of An Envelope; [Computing] Okular and Postscript in OpenSuse; There's a fix!; [Computing] Fail2Ban on OpenSuse Leap 15.3 (NFTables); [Cycling, Computing] Power Calculation and Brakes; [Hardware, Computing] Amazing Pockit Computer; Bullying; How I Am - 3 Years Post Accident, 8+ Years With MS; [USA Politics] In America's Uncivil War Republicans Are The Aggressors; [Programming] Selenium and Python; Better Walking Data; [Bike] How Fast Before Walking More Efficient Than Cycling?; [COVID] Coronavirus And Cycling; [Programming] Docker on OpenSuse; Cadence v Speed; [Bike] Gearing For Real Cyclists; [Programming] React plotting - visx; [Programming] React Leaflet; AliExpress Independent Sellers; Applebaum - Twilight of Democracy; [Politics] Back + US Elections; [Programming,Exercise] Simple Timer Script; [News] 2019: The year revolt went global; [Politics] The world's most-surveilled cities; [Bike] Hope Freehub; [Restaurant] Mama Chau's (Chinese, Providencia); [Politics] Brexit Podcast; [Diary] Pneumonia; [Politics] Britain's Reichstag Fire moment; install cairo; [Programming] GCC Sanitizer Flags; [GPU, Programming] Per-Thread Program Counters; My Bike Accident - Looking Back One Year; [Python] Geographic heights are incredibly easy!; [Cooking] Cookie Recipe; Efficient, Simple, Directed Maximisation of Noisy Function; And for argparse; Bash Completion in Python; [Computing] Configuring Github Jekyll Locally; [Maths, Link] The Napkin Project; You can Masquerade in Firewalld; [Bike] Servicing Budget (Spring) Forks; [Crypto] CIA Internet Comms Failure; [Python] Cute Rate Limiting API; [Causality] Judea Pearl Lecture; [Security, Computing] Chinese Hardware Hack Of Supermicro Boards; SQLAlchemy Joined Table Inheritance and Delete Cascade; [Translation] The Club; [Computing] Super Potato Bruh; [Computing] Extending Jupyter; Further HRM Details; [Computing, Bike] Activities in ch2; [Books, Link] Modern Japanese Lit; What ended up there; [Link, Book] Logic Book; Update - Garmin Express / Connect; Garmin Forerunner 35 v 230; [Link, Politics, Internet] Government Trolls; [Link, Politics] Why identity politics benefits the right more than the left; SSH Forwarding; A Specification For Repeating Events; A Fight for the Soul of Science; [Science, Book, Link] Lost In Math; OpenSuse Leap 15 Network Fixes; Update; [Book] Galileo's Middle Finger; [Bike] Chinese Carbon Rims; [Bike] Servicing Shimano XT Front Hub HB-M8010; [Bike] Aliexpress Cycling Tops; [Computing] Change to ssh handling of multiple identities?; [Bike] Endura Hummvee Lite II; [Computing] Marble Based Logic; [Link, Politics] Sanity Check For Nuclear Launch; [Link, Science] Entropy and Life

© 2006-2017 Andrew Cooke (site) / post authors (content).

O(n) and O(n^2) in a Dynamic Programming Problem

From: andrew cooke <andrew@...>

Date: Sat, 6 Aug 2011 14:39:51 -0400

I posted an answer at
http://stackoverflow.com/questions/6967853/dynamic-programing-can-interval-of-even-1s-and-0s-be-found-in-linear-time
that I want to copy here, beause I am worried that it will be deleted by the
moderators.

The question asks for a linear solution to a problem which generates O(N^2)
results.  That seems impossible, but if you look in more detail there's a cute
argument that hinges on the difference between "finding" and "printing" the
results:


A linear solution is possible (sorry, earlier I argued that this had to be
n^2) if you're careful to not actually print the results!

First, let's define a "score" for any set of zeros and ones as the number of
ones minus the number of zeroes.  So (0,1) has a score of 0, while (0) is -1
and (1,1) is 2.

Now, start from the right.  If the right-most digit is a 0 then it can be
combined with any group to the left that has a score of 1.  So we need to know
what groups are available to the left, indexed by score.  This suggests a
recursive procedure that accumulates groups with scores.  The sweep process is
O(n) and at each step the process has to check whether it has created a new
group and extend the table of known groups.  Checking for a new group is
constant time (lookup in a hash table).  Extending the table of known groups
is also constant time (at first I thought it wasn't, but you can maintain a
separate offset that avoids updating each entry in the table).

So we have a peculiar situation: each step of the process identifies a set of
results of size O(n), but the calculation necessary to do this is constant
time (within that step).  So the process itself is still O(n) (proportional to
the number of steps).  Of course, actually printing the results is O(n^2).

I'll write some Python code to test/demonstrate.

Here we go:

    SCORE = [-1,1]
    
    class Accumulator:
    
        def __init__(self):
            self.offset = 0
            self.groups_to_right = {} # map from score to start index
            self.even_groups = []
            self.index = 0
    
        def append(self, digit):
            score = SCORE[digit]
            # want existing groups at -score, to sum to zero
            # but there's an offset to correct for, so we really want
            # groups at -(score+offset)
            corrected = -(score + self.offset)
            if corrected in self.groups_to_right:
                self.even_groups.append(
                    (self.index, self.groups_to_right[corrected]))
            # this updates all the known groups
            self.offset += score
            # this adds the new one, which should be at the index so that
            # index + offset = score (so index = score - offset)
            groups = self.groups_to_right.get(score-self.offset, [])
            groups.append(self.index) 
            self.groups_to_right[score-self.offset] = groups
            # and move on
            self.index += 1
            #print self.offset
            #print self.groups_to_right
            #print self.even_groups
            #print self.index
    
        def dump(self):
            # printing the results does take longer, of course...
            for (end, starts) in self.even_groups:
                for start in starts:
                    if start < end:
                        print (start, end)
    
        @staticmethod
        def run(input):
            accumulator = Accumulator()
            print input
            for digit in input:
                accumulator.append(digit)
            accumulator.dump()
            print
    
    Accumulator.run([0,1,0,0,1,1,1,1,0])

And the output:

    dynamic: python dynamic.py 
    [0, 1, 0, 0, 1, 1, 1, 1, 0]
    (0, 1)
    (1, 2)
    (1, 4)
    (3, 4)
    (0, 5)
    (2, 5)
    (7, 8)

You might be worried that some additional processing (the filtering for `start
< end`) is done in the dump routine that displays the results.  But that's
because I am working around Python's lack of linked lists (I want to both
extend a list and save the previous value in constant time).

It may seem surprising that the result is of *size* O(n^2) while the process
of *finding* the results is O(n), but it's easy to see how that is possible:
at one "step" the process identifies a number of groups (of size O(n)) by
associating the current point (`end` in `dump()`) with a list of end points
(`ends`).


Andrew

Comment on this post