TweetFollow Us on Twitter

Nov 95 Challenge
Volume Number:11
Issue Number:11
Column Tag:Programmer’s Challenge

Programmer’s Challenge

By Bob Boonstra, Westford, Massachusetts

Note: Source code files accompanying article are located on MacTech CD-ROM or source code disks.

Enclosing Bounds

The Challenge this month is based on a suggestion by Mike Scanlin, who remains a fan of the column. (We’re still waiting for Mike’s first Challenge entry, however.) The problem is to write a routine that will return a rectangle enclosing all non-white pixels in a selected area of an image. This code might be useful in a drawing or painting program, where the user would be allowed to select a subset of the image by clicking and dragging, and the software would select all of the elements of the image contained within that selection. The prototype of the code you will write is:

void EnclosingBounds(
 PixMapHandle pm,  /* handle to PixMap containing image */
 Rect selection, /* subset of image to enclose */
 Rect *enclosingRect /* enclosing rect return value */

Your code should examine all of the pixels within the selection rectangle of the PixMap and return the smallest rectangle containing all of the non-white pixels. Pixels outside the selection rectangle should be ignored. The bounds rectangle of the PixMap will be no larger than 2048 pixels in each dimension, the baseAddr pointer will be longword aligned, and rowBytes will be a multiple of 4. You should deal with pixelSize values of 1, 8, or 32, with values of 8 and 32 being weighted most heavily in measuring performance. For PixMaps with indexed pixels (cmpCount==1), the color table will contain white as the first table entry (as all good color tables are supposed to). For PixMaps with direct pixels, the unused (alpha) bits of each pixel will be zero.

You may use either the Metrowerks or the Symantec compilers for this native PowerPC Challenge. If you have any questions, or would like some test data for your code, please send me e-mail at one of the Programmer’s Challenge addresses, or directly to

Two Months Ago Winner

Congratulations to Eric Lengyel (Blacksburg, VA) for submitting the fastest and smallest entry to the Reversible Scrambling Algorithm Challenge. Despite an unfortunate delay in publication of the magazine that left participants with less time than usual to complete the Challenge, three of the four entries I received by the extended deadline worked correctly, at least in part.

You might recall that the Challenge was to write code that would raise a large integer message to a power and compute the remainder modulo another large integer. The name of the Challenge comes from the fact that this technique is reversible, given properly chosen integers. Eric is a graduate student in Mathematics at Virginia Tech, and he took advantage of a highly optimized multiple precision integer arithmetic library that he had written as part of a number theory project involving the factorization of very large numbers.

Each of the working entries converted the BigNum representation provided in the problem into one that right-justified numbers into a fixed-length data structure. While this imposes a restriction on the maximum size integer that the code can handle, this assumption was permitted by the problem statement. In Eric’s code, the restriction is controlled by a single #define statement.

Eric uses a binary exponentiation algorithm to raise the message to the specified power, and takes advantage of facts from number theory that allow the remainder to be computed at each step of the exponentiation. The time to perform the exponentiation is therefore proportional to the logarithm of the exponent. Eric’s multiplication and division routines use the 68020’s capability to compute the 64-bit product of two longwords and to divide a 64-bit dividend by a longword. The multiplication, division, exponentiation, and compare routines in Eric’s code are general purpose and could be used in any 68K application that needs large integers.

Honorable mention goes to Ernst Munter, who submitted an entry in pure C that was actually the fastest code for the short modulus test cases. Unfortunately, his entry did not produce correct results for the longer moduli.

Here are the times and code sizes for the entries that worked correctly (or partially correctly). Execution time is presented for two specific test cases, with modulus lengths of 22 and 88 bytes, respectively, as well as the total time for all of the test cases I ran. Cases that produced incorrect results are indicated with an asterisk. Numbers in parens after a person’s name indicate that person’s cumulative point total for all previous Challenges, not including this one.

Name time1 time2 Total time code data

(22) (88)
bytes bytes

Eric Lengyel 47 463 2083 1190 0

Xan Gregg (51) 35 967 3175 1558 0

Ernst Munter (C entry) (90) 17 * * 4266 11788

Top 20 Contestants of All Time

Here are the Top 20 Contestants for the Programmer’s Challenges to date. The numbers below include points awarded for this month’s entrants. (Note: ties are listed alphabetically by last name - there are more than 20 people listed this month because of ties.)

Rank Name Points

1. [Name deleted] 176

2. Munter, Ernst 90

3. Karsh, Bill 78

4. Stenger, Allen 65

5. Gregg, Xan 61

6. Larsson, Gustav 60

7. Riha, Stepan 51

8. Goebel, James 49

9. Nepsund, Ronald 47

10. Cutts, Kevin 46

11. Mallett, Jeff 44

12. Kasparian, Raffi 42

13. Vineyard, Jeremy 42

14. Darrah, Dave 31

15. Landry, Larry 29

16. Elwertowski, Tom 24

17. Lee, Johnny 22

18. Noll, Robert 22

19. Anderson, Troy 20

20. Beith, Gary 20

21. Burgoyne, Nick 20

22. Galway, Will 20

23. Israelson, Steve 20

24. Landweber, Greg 20

25. Lengyel, Eric 20

26. Pinkerton, Tom 20

There are three ways to earn points: (1) scoring in the top 5 of any Challenge, (2) being the first person to find a bug in a published winning solution or, (3) being the first person to suggest a Challenge that I use. The points you can win are:

1st place 20 points

2nd place 10 points

3rd place 7 points

4th place 4 points

5th place 2 points

finding bug 2 points

suggesting Challenge 2 points

Here is Eric’s winning solution:


Copyright © 1995 Eric Lengyel
I call my fixed length numbers “BigFixed” and translate from BigNum’s to BigFixed’s in the PowerAndRemainder 
routine.  These are the assembly language routines which are the guts of my program:

(1) PowerMod - raises a number to a power and reduces it by a modulus.  It uses a fast binary exponentiation 
algorithm, reducing by the modulus at each step.
(2) Multiply - multiplies 2 BigNum’s together.
(3) MultQ - mutliplies a BigNum by a long int.
(4) Divide - divides one BigNum by another and supplies the quotient and remainder.
(5) Compare - determines the ordering of 2 BigNum’s.

Some of the loops have been expanded to make more efficient use of the instruction cache.

#define NumSize 72

typedef struct BigNum
   short           numDig;
   unsigned char   *dig;
} BigNum;

typedef struct BigFixed
   unsigned char   dig[NumSize*4];
} BigFixed;

/* We need 72 longs because the division routine needs the most significant longword to be zero and the 
speed optimization requires that NumSize be a multiple of four. */

void PowerAndRemainder(BigNum *msg, BigNum *exp, BigNum *n,
   BigNum *res);
void PowerMod(BigFixed *msg, BigFixed *exp, BigFixed *n,
   BigFixed *res);
void Multiply(BigFixed *src1, BigFixed *src2, BigFixed *dst);
void MultQ(BigFixed *src1, long src2, BigFixed *dst);
void Divide(BigFixed *end, BigFixed *sor, BigFixed *dst);
short Compare(BigFixed *src1, BigFixed *src2);


void PowerAndRemainder(BigNum *msg, BigNum *exp, BigNum *n,
   BigNum *res)
   short      a, b, numDigits;
   BigFixed   msg0, exp0, n0, res0;
   for (a = 0; a < NumSize*4; a++)
      b = NumSize*4 - msg->numDig;
      if (a < b) msg0.dig[a] = 0;
      else msg0.dig[a] = msg->dig[a - b];
      b = NumSize*4 - exp->numDig;
      if (a < b) exp0.dig[a] = 0;
      else exp0.dig[a] = exp->dig[a - b];
      b = NumSize*4 - n->numDig;
      if (a < b) n0.dig[a] = 0;
      else n0.dig[a] = n->dig[a - b];
   PowerMod(&msg0, &exp0, &n0, &res0);
   a = 0;
   while (res0.dig[a] == 0) a++;
   numDigits = res->numDig = NumSize*4 - a;
   for (b = 0; b < numDigits; b++)
      res->dig[b] = res0.dig[a++];


void PowerMod(BigFixed *msg, BigFixed *exp, BigFixed *n,
   BigFixed *res)
   BigFixed   acc, scrap;

   LEA      acc, A0            ;   Start with one in
   MOVEQ    #NumSize/4-2, D0   ;   accumulator
1} ; Test a bit in current
                               ;   longword of exponent
   BEQ      @1                 ;   If zero, skip multiply
   PEA      acc                ;   Multiply accumulator
   PEA      acc                ;   by base
   MOVE.L   msg, -(A7)
   JSR      Multiply
   ADDA.W   #12, A7
   MOVE.L   n, -(A7)           ;   Compare accumulator
   PEA      acc                ;   to modulus
   JSR      Compare
   ADDQ.W   #8, A7
   TST.B    D0
   BMI      @1                 ;   If it’s less, skip
                               ;   reduction
   PEA      scrap              ;   Reduce modulo “n”
   MOVE.L   n, -(A7)
   PEA      acc
   JSR      Divide
   ADDA.W   #12, A7

/* Multiply src1 by src2 and put product in dst */

void Multiply(BigFixed *src1, BigFixed *src2, BigFixed *dst)
   short      topStop, botStop;
   BigFixed   acc, line;
   MOVEM.L  D0-D7/A0-A4, -(A7)
   LEA      acc, A0            ;   Clear accumulator
   MOVEQ    #NumSize/4-1, D0
D5       ;   Do 64-bit multiply
   ADD.L    D2, D5             ;   Add carry to low
                               ;   longword of product
   CLR.L    D2                 ;   Use D2 as dummy to
                               ;   extend carry
   ADDX.L   D2, D6             ;   Add zero to high
                               ;   longword with carry
   MOVE.L   D6, D2             ;   Anything in high
                               ;   longword gets carried
   MOVE.L   D5, 00(A2, D3.W*4) ;   Store low longword in
                               ;   partial product
   SUBQ.W   #1, D3             ;   Loop through all
   CMP.W    topStop, D3        ;   longwords in top number
   BGE      @1
   MOVEA.L  A2, A0             ;   Now add partial product
                               ;   to accumulator
   MOVE.L   D4, D0             ;   Calculate correct
                               ;   position in product
   LEA      acc, A1            ;   Get accumulator’s addr
   ADDQ.W   #1, D0
   ADDA.W   #NumSize * 4, A0
   LSL.W    #2, D0
   ADDA.W   D0, A1
   MOVE.W   D4, D1
   MOVE.L   -(A1), D0          ;   Get longword of product
   SUBQ     #1, D1
   ADD.L    -(A0), D0          ;   Add longword of
   MOVE.L   D0, (A1)           ;   partial product
   TST.W    D1                 ;   If no more longwords,
   BMI      @2                 ;   then branch
/* Multiply src1 by src2 and put product in dst */

void MultQ(BigFixed *src1, long src2, BigFixed *dst)
   BigFixed   pro;
   MOVEM.L  D0-D7/A0/A1, -(A7)
   LEA      pro, A0            ;   Clear product
   MOVEQ    #NumSize/4-1, D0
D4          ;   Do 64-bit multiply
                               ;   by bottom number
   ADD.L    D2, D4             ;   Add carry
   CLR.L    D2                 ;   Use D2 as dummy to
                               ;   extend carry
   ADDX.L   D2, D5             ;   Add zero with carry
   MOVE.L   D5, D2             ;   High longword
                               ;   becomes carry
   MOVE.L   D4, 00(A1, D0.W*4) ;   Put partial product
                               ;   into result
   SUBQ.W   #1, D0             ;   Loop through all
   CMP.W    D1, D0             ;   longwords in top #
   BGE      @1

/* Divide end (dividend) by sor (divisor) and put quotient in dst.  Remainder will end
    up in end */

void Divide(BigFixed *end, BigFixed *sor, BigFixed *dst)
   long      pq;
   BigFixed  quo, line;
   MOVEM.L  D0-D7/A0-A4, -(A7)
   LEA      quo, A0            ;   Clear quotient
   MOVEQ    #NumSize/4-1, D0
D4          ;   Do 64-bit division

/* Compare src1 and src2.  Returns 1 if src1 > src2, 0 if they’re equal, and -1 if src1 < 
    rc2. */

short Compare(BigFixed *src1, BigFixed *src2)
   MOVEM.L  D1/D2/A0/A1, -(A7)
   MOVEA.L  src1, A0           ;   Get src1’s address
   MOVEA.L  src2, A1           ;   Get src2’s address
   MOVEQ    #1, D0             ;   Start with +1
   MOVE.L   (A0)+, D2
   CMP.L    (A1)+, D2          ;   Compare 1st longwords
   BLT      @1                 ;   If src1 less, branch
   BNE      @2                 ;   If !=, src1 must
   MOVE.L   (A0)+, D2          ;   be greater
   CMP.L    (A1)+, D2          ;   Cmp 3 more longwords
   BCS      @1                 ;   (Unsigned)
   BNE      @2
   MOVE.L   (A0)+, D2
   CMP.L    (A1)+, D2
   BCS      @1
   BNE      @2
   MOVE.L   (A0)+, D2
   CMP.L    (A1)+, D2
   BCS      @1
   BNE      @2
   MOVEQ    #NumSize/4-2, D1   ;   Number of longwords
                               ;   remaining / 4


Community Search:
MacTech Search:

Software Updates via MacUpdate

Parallels Desktop 18.0.0 - Run Windows a...
Parallels allows you to run Windows and Mac applications side by side. Choose your view to make Windows invisible while still using its applications, or keep the familiar Windows background and... Read more
TechTool Pro 16.0.1 - Hard drive and sys...
TechTool Pro has long been one of the foremost utilities for keeping your Mac running smoothly and efficiently. With the release of this version, it has become more proficient than ever. Main... Read more
Adobe Acrobat Reader 22.002.20191 - View...
Adobe Acrobat Reader allows users to view PDF documents. You may not know what a PDF file is, but you've probably come across one at some point. PDF files are used by companies and even the IRS to... Read more
Adobe Acrobat DC 22.002.20191 - Powerful...
Acrobat DC is available only as a part of Adobe Creative Cloud, and can only be installed and/or updated through Adobe's Creative Cloud app. Adobe Acrobat DC with Adobe Document Cloud services is... Read more
Skype - Voice-over-internet p...
Skype is a telecommunications app that provides HD video calls, instant messaging, calling to any phone number or landline, and Skype for Business for productive cooperation on the projects. This... Read more
Vitamin-R 4.14 - Personal productivity t...
Vitamin-R creates the optimal conditions for your brain to work at its best by structuring your work into short bursts of distraction-free, highly focused activity alternating with opportunities for... Read more
Bookends 14.1.1 - Reference management a...
Bookends is a full-featured bibliography/reference and information-management system for students and professionals. Bookends uses the cloud to sync reference libraries on all the Macs you use.... Read more
Mactracker 7.11.5 - Database of all Mac...
Mactracker provides detailed information on every Mac computer ever made, including items such as processor speed, memory, optical drives, graphic cards, supported OS X versions, and expansion... Read more
Visual Studio Code 1.70.0 - Cross-platfo...
Visual Studio Code provides developers with a new choice of developer tool that combines the simplicity and streamlined experience of a code editor with the best of what developers need for their... Read more
RapidWeaver 8.10.1 - Create template-bas...
RapidWeaver is a next-generation Web design application to help you easily create professional-looking Web sites in minutes. No knowledge of complex code is required, RapidWeaver will take care of... Read more

Latest Forum Discussions

See All

Keep Your Smatphone’s Case On When Using...
The original Gamevice was born as a sort of offshoot of the weird Wikipad gaming tablet/controller/hybrid thing way back in 2014. Interestingly, the first Gamevice controller for iOS only supported the iPad mini and launched in 2015, with versions... | Read more »
SwitchArcade Round-Up: A ‘Splatoon 3’ Ni...
Hello gentle readers, and welcome to the SwitchArcade Round-Up for August 9th, 2022. In today’s article, we’ve got some news about a Splatoon 3 Nintendo Direct, a review of QUByte’s Thunderbolt Collection, a single new release summary, and the usual... | Read more »
Orangepixel’s Pacifist Survival Game ‘Re...
Back in June we learned that long-time mobile developer Orangepixel, who also makes games for PC and consoles (including the Atari VCS!), would be bringing the unique survival game Residual to mobile devices sometime this year. Originally launched... | Read more »
Football Club Management 23 launches tod...
Go Play Games has announced that its popular Club Soccer Director game has undergone a rebrand, correcting the sport in its title and becoming Football Club Management. Launching August 4th, Football Club Management will give players full reign... | Read more »
Kingdom Hearts Dark Road Final Chapter R...
We’ve been waiting quite a while for the finale to KINGDOM HEARTS Union X Dark Road (Free) that was set to arrive in winter last year. | Read more »
Point-and-Click Space Adventure ‘Dexter...
Following its launch on Steam and Nintendo Switch this past March, indie point-and-click adventure Dexter Stardust is making its way to iOS devices this week. | Read more »
SwitchArcade Round-Up: Reviews Featuring...
Hello gentle readers, and welcome to the SwitchArcade Round-Up for August 8th, 2022. In today’s article, we’ve got a bit of news to dig into, a pair of reviews to read, some dubious new releases to summarize, and the usual lists of new and outgoing... | Read more »
‘The House of Da Vinci 3’ is the Third I...
The House of Da Vinci 3 is out now on iOS, with the Android version to follow in the next few weeks. Players can also expect to dive into the third instalment of the popular mystery franchise on Mac, PC and Switch in the coming months, as they... | Read more »
Into the Breach Mobile Interview: Subset...
Subset Games finally brought Into the Breach (Free) to mobile platforms through Netflix Gaming, and it is a fantastic conversion. Read my review of it here. | Read more »
XD Entertainment release details for Sea...
XD Entertainment has revealed the launch of Season 2 for their hit team-based shooter T3 Arena. Entitled Lights On, Season 2 brings a new character, multiple new maps and tweaks to some older mechanics. [Read more] | Read more »

Price Scanner via

In stock today: Mac Studio models for up to $...
Apple retailer Expercom has Mac Studio models in stock today and on sale for up to $400 off Apple’s MSRP, depending on configuration. Their prices are the lowest price available for a Mac Studio from... Read more
Mac mini with M1 CPU and 512GB of storage on...
Amazon has the M1 Mac mini with a 512GB SSD in stock today on sale for $749.99 including free shipping. Their price is $150 off Apple’s MSRP, and it’s the lowest price available for this... Read more
Need a Mac or iPad for school? Get a free App...
Apple’s Back to School promotion for 2022 continues to run through September 26, 2022. As part of this promotion, Apple will include a free $150 Apple Gift Card with the purchase of any MacBook Air,... Read more
Apple Watch SE on sale for $50 off MSRP
Amazon has Apple Watch SE GPS models on sale for $50 off MSRP for a limited time, each including free shipping. Their prices are the lowest currently available for SE Watches: – 40mm Apple Watch SE... Read more
Save $310 on a 14″ 24-core GPU M1 Max MacBook...
Save $310 on 14″ MacBook Pros with 24-core M1 Max processors at Apple (32GB RAM/1TB SSD) with these Certified Refurbished models in stock today for $2789 in Space Gray or Silver colors. Regular price... Read more
14″ M1 Pro MacBook Pros available today at Ap...
Apple has Certified Refurbished standard-configuration 14″ MacBook Pros with M1 Pro CPUs available today for up to $250 off original MSRP, starting at $1799. Each model features a new outer case,... Read more
13″ MacBook Air with M2 CPU, in Starlight, on...
Apple retailer Expercom has the new Starlight 13″ MacBook Air with an M2 CPU (8GB RAM/256GB SSD) on sale for $1135.05, shipped, through August 12, 2022. Their price is $64 off Apple’s MSRP, and it’s... Read more
14″ M1 Pro MacBook Pro with 1TB SSD on sale f...
Expercom is offering a $200 instant discount on the 14″ M1 Pro MacBook Pro with a 1TB SSD through August 12, 2022. Their discount reduces the price of this configuration to $1999 shipped — the lowest... Read more
10″ 256GB WiFi iPads are on sale for $399, sa...
Amazon has new 10.2″ 9th generation 256GB WiFi iPads in stock and on sale today for $80 off Apple’s MSRP, only $399. Shipping is free. Note that stock (and prices) at Amazon can change quickly. These... Read more
13-inch M2 MacBook Pro with 512GB SSD on sale...
Amazon has the new 13″ Silver MacBook Pro with an M2 CPU and 512GB SSD in stock and on sale today for $1349 shipped. Their price is $150 off Apple’s MSRP. Be sure to purchase from Amazon rather than... Read more

Jobs Board

Solutions Engineering Manager - *Apple* - S...
…in our Hardware and Advanced Solutions group leading and developing our Apple technical practice to increase revenue and profitability. The ideal candidate would Read more
Operations Associate - *Apple* Blossom Mall...
Operations Associate - Apple Blossom Mall Location:Winchester, VA, United States ( - Apple Read more
Cashier - *Apple* Blossom Mall - JCPenney (...
Cashier - Apple Blossom Mall Location:Winchester, VA, United States ( - Apple Blossom Mall Read more
Omnichannel Associate - *Apple* Blossom Mal...
Omnichannel Associate - Apple Blossom Mall Location:Winchester, VA, United States ( - Apple Read more
Sephora Beauty Advisor - *Apple* Blossom Ma...
Sephora Beauty Advisor - Apple Blossom Mall Location:Winchester, VA, United States ( - Apple Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.