TweetFollow Us on Twitter

Apr 99 Challenge

Volume Number: 15 (1999)
Issue Number: 4
Column Tag: Programmer's Challenge

Apr 99 Challenge

by Bob Boonstra, Westford, MA

Shortest Network

This month's problem was suggested by Michael Kennedy, who wins two Challenge points for making the suggestion. The problem is to find the shortest network of line segments interconnecting a specified set of points. Shortest network algorithms have obvious practical application in constructing transportation and communications networks. In a January 1989, Scientific American article, Marshall Bern and Ronald Graham discussed the shortest network "Steiner" problem as one of a class of NP-hard problems. While no polynomial-time algorithm is known, the article (which, unfortunately, I have not been able to find online) discusses practical algorithms that produce networks slightly longer than the optimal one. Your Challenge for this month is to produce a near-optimal network in minimum time. Fortunately, we have been granted unlimited power of eminent domain, so there are no restrictions on where intermediate nodes may be placed or where connections may be routed.

The prototype for the code you should write is:

#if defined(__cplusplus)
extern "C" {
#endif

typedef struct Node {   /* node coordinates */
   double x;
   double y;
} Node;

typedef struct Connection {
         /* connection between Node[index1] and Node[index2] */
   long index1;
   long index2;
} Connection;

long /* numConnections */ ShortestNetwork(
   long numInitialNodes,         /* number of nodes to connect */
   long *numIntermediateNodes,   /* number of nodes added by ShortestNetwork */
   Node nodes[], 
      /* Nodes 0..numInitialNodes-1 are initialized on entry. */
      /* Nodes numInitialNodes..numInitialNodes+*numIntermediateNodes 
                  are added by ShortestNetwork */
   Connection connections[],   /* connections between nodes */
   long maxNodes,        /* number of entries allocated for nodes */
   long maxConnections   /* number of entries allocated for connections */
);

#if defined(__cplusplus)
}
#endif

Your ShortestNetwork routine will be given a list of numInitialNodes nodes to connect. You may add intermediate nodes to help you form a shorter network, and must produce as output a list of connections between pairs of nodes. The connections must provide a path between any pair of the initial nodes.

Your solution must return the number of intermediate nodes added to the network in *numIntermediateNodes, while storing the location of those nodes in nodes[numInitialNodes+k], k=0..*numIntermediateNodes-1. A connection is specified by storing the indices of the two nodes being connected into the connection array. Your ShortestNetwork routine should return the number of connections created.

The maxNodes and maxConnections parameters indicate how much storage has been allocated for nodes and connections. It is my intention to allocate enough storage for all the nodes and connections your solution might create, but if it turns out that there is not enough storage, your solution should return a value of -1 to indicate that storage was exhausted.

The winner will be the solution that generates the shortest network in the minimum amount of time. Specifically, your solution will be assigned a cost equal to the sum of the distances between nodes in your list of connections, plus a penalty of 10% for each second of execution time. Solutions that do not connect all of the initial nodes will be penalized with a very large cost. The solution with the lowest total cost over a series of networking problems will be the winner.

This will be a native PowerPC Challenge, using the latest CodeWarrior environment. Solutions may be coded in C, C++, or Pascal. Thanks to Michael for suggesting this Challenge.

Three Months Ago Winner

Congratulations to Tom Saxton for submitting the winning solution to the January Sphere Packing Challenge. You may recall that this Challenge was to pack a set of spheres of varying size into a box with minimum volume, and to do so in the shortest amount of time possible. Tom submitted one of only two solutions received for this Challenge, and his was the only one that performed correctly.

Tom's approach is to decide on a footprint for the box to contain the spheres, "drop" the spheres individually into the box until they hit another sphere or the bottom of the box, while attempting to move the dropped sphere around the obstacle without going outside the box footprint. The solution then iterates with random movements to try to converge to a better solution. Tom observed in his submission that the time penalty for this problem (1% per millisecond of execution time) was very severe, making it unproductive to let his algorithm iterate very long. Every tenth of a second of execution time requires a factor of 2 reduction in volume to be productive, a rate of improvement smaller than what Tom was able to achieve.

I evaluated the solutions using six test cases with between 200 and 2000 spheres per test case. As one might expect, execution time grew exponentially with the number of spheres. A test case with 1000 spheres took about 20 times as long to solve as a 200-sphere case, and a 2000-sphere case took about 4 times longer than the 1000-sphere case. Tom's solution generated solutions that, in aggregate, occupied between 1.3 and 3.9 times the volume of individual cubes containing the individual spheres, which suggests that better solutions could be achieved with a more relaxed time penalty.

The table below lists, for each of the solutions submitted, the total volume of the boxes containing the spheres, the total execution time, and the total score including the time penalty, as well as the code and data sizes for each entry. As usual, the number in parentheses after the entrant's name is the total number of Challenge points earned in all Challenges prior to this one.

Name Volume (x1.0E12) Time (secs) Score (x1.0e12) Code Size Data Size
Tom Saxton (79)65.3142.310107.25796372
A. D.***820104

Top Contestants

Listed here are the Top Contestants for the Programmer's Challenge, including everyone who has accumulated 20 or more points during the past two years. The numbers below include points awarded over the 24 most recent contests, including points earned by this month's entrants.

  1. Munter, Ernst 200
  2. Saxton, Tom 99
  3. Boring, Randy 56
  4. Mallett, Jeff 50
  5. Rieken, Willeke 47
  6. Maurer, Sebastian 40
  7. Heithcock, JG 37
  8. Cooper, Greg 34
  9. Murphy, ACC 34
  10. Lewis, Peter 31
  11. Nicolle, Ludovic 27
  12. Brown, Pat 20
  13. Day, Mark 20
  14. Higgins, Charles 20
  15. Hostetter, Mat 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 place20 points
2nd place10 points
3rd place7 points
4th place4 points
5th place2 points
finding bug2 points
suggesting Challenge2 points

Here is Tom's winning Sphere Packing solution:

Spheres.cpp
Copyright © 1999 Tom Saxton

#include "Spheres.h"
#include "VecUtil.h"

#include <math.h>
#include <stdlib.h>

enum {
   fFalse = 0,
   fTrue = 1
};

typedef unsigned long ulong;

// disable asserts
#define Assert(f)

// hard iteration limit
#define cIterLim   10000

// scoring an accepting solutions
#define _FAccept(volNew, volBest) ((volNew) < (volBest))
#define _Score(vol, dtick)      ((vol) * (1.0 + (dtick)*10.0/60.0))

// define this to ignore the time penalty
// #define KEEP_GOING

// time checking parameters
#define dtickSec         60
#define dtickCheckScore      (dtickSec/30)
#define dtickFirstCheck      (dtickSec/30)
#define dtickLastCheck      (10*dtickSec)

static const Position s_normalX = { 1.0, 0.0, 0.0 };
static const Position s_normalY = { 0.0, 1.0, 0.0 };
static const Position s_normalZ = { 0.0, 0.0, 1.0 };
static const Position s_normalXNeg = { -1.0, 0.0, 0.0 };
static const Position s_normalYNeg = { 0.0, -1.0, 0.0 };
static const Position s_normalZNeg = { 0.0, 0.0, -1.0 };

static void _InitStartingPos(
   const long csphere,
   long aisphere[],
   const double aradius[],
   double baseMin,
   double baseBest,
   double baseMax,
   double *pbase,
   Position aposStart[]);
static void _TweakStartingPos(
   const long csphere,
   long aisphere[],
   const double aradius[],
   double baseMin,
   double baseBest,
   double baseMax,
   double *pbase,
   Position aposStart[]);
static void _DropSpheres(
   long csphere,
   const long *paisphere,
   const double aradius[],
   const Position *paposStart,
   Position apos[],
   double base,
   double *pvolume);
static void _DropOneSphere(
   const Position &posStart,
   double radius,
   int csphere,
   const long *paisphere,
   const double aradius[],
   const Position apos[],
   Position * pposResult,
   long * pisphereHit);
static int _FFindObstruction(
   const Position normalMove,
   int fNear,
   const Position &posStart,
   double radius,
   int csphere,
   const long *paisphere,
   const double aradius[],
   const Position apos[],
   Position * pposResult,
   long * pisphereHit);

PackSpheres
void PackSpheres(
  long csphere,        /* input: number of spheres to pack */
  double aradius[],    /* input: radius of each of numSpheres spheres */
  Position aposBest[]  /* output: location of center of each sphere */
)
{
   int isphere;
   double volGuess, vol, volBest;
   double base, baseMin, baseMax, baseBest;
   double radiusLarge, radiusSum;
   ulong tickStart, tickCur;
   
   tickStart = LMGetTicks();
   radiusLarge = radiusSum = 0.0;
   for (isphere = 0, volGuess = 0.0; isphere < csphere; ++isphere)
   {
      double radius = aradius[isphere];
      volGuess += 8.0 * radius * radius * radius;
      
      if (radius > radiusLarge)
         radiusLarge = radius;
      radiusSum += radius;
   }
   
   baseMin = 2.0 * radiusLarge;
   baseMax = 2.0 * radiusSum;
   Assert(baseMin <= baseMax);
   
   baseBest = baseMin;
   _DropSpheres(csphere, NULL, aradius, NULL, aposBest, 
            baseBest, &volBest);
   
   base = baseMax;
   _DropSpheres(csphere, NULL, aradius, NULL, aposBest, 
            base, &vol);
   if (vol < volBest)
   {
      volBest = vol;
      baseBest = base;
   }
   
   base = sqrt(baseMin * baseMax);
   _DropSpheres(csphere, NULL, aradius, NULL, aposBest, 
            base, &vol);
   if (vol < volBest)
   {
      volBest = vol;
      baseBest = base;
   }
   
   char * pbBlock = NewPtr(csphere * sizeof(Position) + 
                  csphere * sizeof(Position) + csphere * sizeof(long));
   
   if (pbBlock != NULL)
   {
      long iIter;

      Position * aposStart = (Position *)pbBlock;
      Position * aposEnd = &aposStart[csphere];
      long * aisphere = (long *)&aposEnd[csphere];
      long tickNext = tickStart + dtickCheckScore;
   double scorePrev = _Score(volBest, LMGetTicks() - tickStart);
#ifdef KEEP_GOING
      double scoreBest = scorePrev;
      int iIterBest = 0;
#endif   
      
      for (iIter = 0; iIter < cIterLim; ++iIter)
      {
         tickCur = LMGetTicks();
         if (tickCur >= tickNext)
         {
            ulong dtickCur = tickCur - tickStart;
            if (dtickCur >= dtickFirstCheck)
            {
               if (dtickCur >= dtickLastCheck)
                  break;
                  
               double score = _Score(volBest, dtickCur);
#ifdef KEEP_GOING
               if (score < scoreBest)
               {
                  scoreBest = score;
                  iIterBest = iIter;
               }
#else
               if (score > scorePrev)
                  break;
#endif
               scorePrev = score;
            }
            while (tickNext < tickCur)
               tickNext += dtickCheckScore;
         }
         
         // pick a new scenario
         if (iIter == 0)
            _InitStartingPos(csphere, aisphere, aradius, 
                  baseMin, baseBest, baseMax, &base, aposStart);
         else
            _TweakStartingPos(csphere, aisphere, aradius, 
               baseMin, baseBest, baseMax, &base, aposStart);
         
         // try the new scenario
         _DropSpheres(csphere, aisphere, aradius, aposStart, 
               aposEnd, base, &vol);
         if (_FAccept(vol, volBest))
         {
            volBest = vol;
            baseBest = base;
         BlockMove(aposEnd, aposBest, csphere * sizeof(Position));
         }
         
         // if the largest sphere determined the height, then reduce baseMax
      if (vol <= 2.0 * (radiusLarge + epsilon) * base * base)
         {
            Assert(base <= baseMax);
            baseMax = base;
         }
      }
   }
   
   if (pbBlock != NULL)
      DisposePtr((Ptr) pbBlock);
}

_InitStartingPos
static void _InitStartingPos(
   const long csphere,
   long aisphere[],
   const double aradius[],
   double baseMin,
   double baseBest,
   double baseMax,
   double *pbase,
   Position aposStart[])
{
   long isphereCur;
   
   *pbase = baseBest;
   for (isphereCur = 0; isphereCur < csphere; ++isphereCur)
   {
      Position *ppos = &aposStart[isphereCur];
      double radiusCur = aradius[isphereCur];
      
      aisphere[isphereCur] = isphereCur;
      ppos->coordinate[0] = 
            GRandInRange(radiusCur, *pbase - radiusCur);
      ppos->coordinate[1] = 
            GRandInRange(radiusCur, *pbase - radiusCur);
      ppos->coordinate[2] = csphere * *pbase;
   }
}

_TweakStartingPos
static void _TweakStartingPos(
   const long csphere,
   long aisphere[],
   const double aradius[],
   double baseMin,
   double baseBest,
   double baseMax,
   double *pbase,
   Position aposStart[])
{
   long isphereCur;
   double dbase;
   
   // change the base size?
   if (GRandInRange(0.0, 1.0) < 0.1)
   {
      dbase = GRandInRange(-1.0, 1.0);
      dbase *= fabs(dbase);
      dbase *= 0.25 * (baseMax - baseMin);
      *pbase = baseBest + dbase;
      *pbase = fmax(baseMin, *pbase);
      *pbase = fmin(baseMax, *pbase);
   }
   
   // rearrange the drop order?
   if (GRandInRange(0.0, 1.0) < 4.0)
   {
      for (long index = csphere; - index > 0; )
      {
         long indexSwap;
         long isphereSav;
         
         indexSwap = ((unsigned long)LRand()) % index;
         Assert(0 <= indexSwap && indexSwap < index);
         isphereSav = aisphere[index];
         aisphere[index] = aisphere[indexSwap];
         aisphere[indexSwap] = isphereSav;
      }
   }
   
   // change the starting positions
   for (isphereCur = 0; isphereCur < csphere; ++isphereCur)
   {
      Position *ppos = &aposStart[isphereCur];
      double radiusCur = aradius[isphereCur];
      
      ppos->coordinate[0] = 
            GRandInRange(radiusCur, *pbase - radiusCur);
      ppos->coordinate[1] = 
            GRandInRange(radiusCur, *pbase - radiusCur);
      ppos->coordinate[2] = csphere * *pbase;
   }
}

_DropSpheres
static void _DropSpheres(
   const long csphere,
   const long *paisphere,
   const double aradius[],
   const Position *paposStart,
   Position aposEnd[],
   double base,
   double *pvol)
{
   long csphereDone;
   
   for (csphereDone = 0; csphereDone < csphere; ++csphereDone)
   {
      Position posStart, posLand;
      double radiusCur;
      long isphereHit;
      long isphereCur;

      isphereCur = paisphere == NULL ? csphereDone : 
            paisphere[csphereDone];

      radiusCur = aradius[isphereCur];
      
      // pick a starting point for the current sphere
      Assert(base >= radiusCur*2.0);
      if (paposStart == NULL)
      {
         posStart.coordinate[0] = 
            GRandInRange(radiusCur, base - radiusCur);
         posStart.coordinate[1] = 
            GRandInRange(radiusCur, base - radiusCur);
      }
      else
      {
         posStart.coordinate[0] = 
               paposStart[isphereCur].coordinate[0];
         posStart.coordinate[1] = 
               paposStart[isphereCur].coordinate[1];
      }
      
      // drop it
   _DropOneSphere(posStart, radiusCur, csphereDone, paisphere, 
         aradius, aposEnd, &aposEnd[isphereCur], &isphereHit);
      
      // try to move it around the sphere it hit
   for (int cIter = 0; isphereHit != -1 && cIter < isphereCur; 
            ++cIter)
      {
         Position vecMove, vecMoveXY, normalMove;
         Position posHit;
         double distH, distMove;
         int icoord;
         
         posHit = aposEnd[isphereHit];
         SubVec(aposEnd[isphereCur], posHit, &vecMove);
         vecMoveXY = vecMove;
         vecMoveXY.coordinate[2] = 0;
         distH = LengthVec(vecMoveXY);
         
         if (distH < epsilon)
            break;
            
         ScaleVec(1.0/distH, vecMoveXY, &normalMove);
         distMove = radiusCur + aradius[isphereHit];
         Assert(distMove > distH - epsilon);
         
         // don't move out of the box
         for (icoord = 0; icoord <= 1; ++icoord)
         {
            if (normalMove.coordinate[icoord] < -epsilon)
            {
               if (posHit.coordinate[icoord] + 
         distMove * normalMove.coordinate[icoord] < radiusCur)
            distMove = (radiusCur - posHit.coordinate[icoord]) / 
                     normalMove.coordinate[icoord];
            }
            else if (normalMove.coordinate[icoord] > epsilon)
            {
               if (posHit.coordinate[icoord] + distMove * 
               normalMove.coordinate[icoord] > base - radiusCur)
                  distMove = (base - radiusCur - 
                                             posHit.coordinate[icoord]) / 
                                          normalMove.coordinate[icoord];
            }
         }
         
         Assert(distMove >= distH - epsilon);
         if (distMove < distH + epsilon)
            break;
            
         AddScaleVec(posHit, distMove, normalMove, &posStart);
         
   _DropOneSphere(posStart, radiusCur, csphereDone, paisphere, 
                  aradius, aposEnd, &posLand, &isphereHit);
         
         if (posLand.coordinate[2] > 
                  aposEnd[isphereCur].coordinate[2] - epsilon)
            break;
         
         aposEnd[isphereCur] = posLand;
      
      }

      // try move it toward the edges
      int fImproved, cIter;
      for (fImproved = fTrue, cIter = 1; fImproved; ++cIter)
      {
         Assert(cIter < 15);
         fImproved = fFalse;
         for (int dir = 0; dir < 4; ++dir)
         {
            Position normalMove;
            int fHit;
            double sEdge;
            Position aposStart[2];
            int cposStart;
            
            switch (dir)
            {
            case 0:
               normalMove = s_normalX;
               sEdge = base - radiusCur;
               break;
            case 1:
               normalMove = s_normalY;
               sEdge = base - radiusCur;
               break;
            case 2:
               normalMove = s_normalXNeg;
               sEdge = -radiusCur;
               break;
            case 3:
               normalMove = s_normalYNeg;
               sEdge = -radiusCur;
               break;
            }
            
            fHit = _FFindObstruction(
                     normalMove,
                     fTrue/*fNear*/,
                     aposEnd[isphereCur],
                     radiusCur,
                     csphereDone,
                     paisphere,
                     aradius,
                     aposEnd,
                     &posLand,
                     &isphereHit);
            
            cposStart = 0;
            if (!fHit || DotVec(posLand, normalMove) > sEdge)
            {
               posLand = aposEnd[isphereCur];
               AddScaleVec(posLand, sEdge - 
                        DotVec(posLand, normalMove), normalMove, 
                                    &aposStart[cposStart++]);
               cposStart = 1;
            }
            else
            {
         LinearComboVec(0.5, posLand, 0.5, aposEnd[isphereCur], 
                                 &aposStart[cposStart++]);
               aposStart[cposStart++] = posLand;
            }
            
for (int iposStart = 0; iposStart < cposStart; ++iposStart)
            {
               _DropOneSphere(aposStart[iposStart], radiusCur, 
            csphereDone, paisphere, aradius, aposEnd, &posLand, 
                     &isphereHit);
               
               if (posLand.coordinate[2] < 
                     aposEnd[isphereCur].coordinate[2] + epsilon)
               {
                  if (aposEnd[isphereCur].coordinate[2] - 
                           posLand.coordinate[2] > radiusCur * 0.05)
                     fImproved = fTrue;
                  aposEnd[isphereCur] = posLand;
            
               }
            }
         }
      }
   }
   
   ComputeVol(csphere, NULL, aradius, aposEnd, base, pvol);
}

_DropOneSphere
static void _DropOneSphere(
   const Position &posStart,
   double radius,
   int csphere,
   const long *paisphere,
   const double aradius[],
   const Position apos[],
   Position * pposResult,
   long * pisphereHit)
{
   Position posBase;
   int fHit;
   
   posBase = posStart;
   posBase.coordinate[2] = 0.0;
   
   *pposResult = posBase;

   fHit = _FFindObstruction(
            s_normalZ,
            fFalse, /* fNear */
            posBase,
            radius,
            csphere,
            paisphere,
            aradius,
            apos,
            pposResult,
            pisphereHit
            );
   
   if (!fHit || pposResult->coordinate[2] < radius)
   {
      *pisphereHit = -1;
      pposResult->coordinate[2] = radius;
   }

   // add some fudge
   pposResult->coordinate[2] += epsilon;
   
#ifdef DEBUG
   for (long csphereChecked = 0; csphereChecked < csphere; 
               ++csphereChecked)
   {
      Position vecT;
      double dist, distGap;
      int isphere;
      
      isphere = paisphere == NULL ? csphereChecked : 
                                                paisphere[csphereChecked];
      
      SubVec(apos[isphere], *pposResult, &vecT);
      dist = LengthVec(vecT);
      distGap = dist - (radius + aradius[isphere]);
      Assert(distGap >= 0.0);
   }
#endif
}

_FFindObstruction
// moving a sphere with specifed radius from posStart in the direction normalMove,
// find the nearest or farthest obstruction
// If there is an obstruction, return the index to the obstructing sphere
// and the position to which the object can move.
static int _FFindObstruction(
   const Position normalMove,
   int fNear,
   const Position &posStart,
   double radius,
   int csphere,
   const long *paisphere,
   const double aradius[],
   const Position apos[],
   Position * pposResult,
   long * pisphereHit)
{
   double zBest;
   
   *pisphereHit = -1;

   for (int csphereChecked = 0; csphereChecked < csphere; 
               ++csphereChecked)
   {
      Position vecToOther, vecPerp, vecParallel;
      double distPerpSq, distSep, distSepSq;
      double z, dz;
      int isphere;
      
      isphere = paisphere == NULL ? csphereChecked : 
                                                paisphere[csphereChecked];
      SubVec(apos[isphere], posStart, &vecToOther);
      ProjectVec(vecToOther, normalMove, &vecParallel);
      SubVec(vecToOther, vecParallel, &vecPerp);
      
      distPerpSq = DotVec(vecPerp, vecPerp);
      distSep = radius + aradius[isphere];
      distSepSq = distSep * distSep;
      
      if (distPerpSq > distSepSq)
         continue;
      
      dz = sqrt(distSepSq - distPerpSq);
      if (fNear)
         dz = -dz;
      z = DotVec(vecParallel, normalMove) + dz;
      
      if (z >= 0.0 && (*pisphereHit == -1 || 
                                    (fNear ? z < zBest : z > zBest)))
      {
         zBest = z;
         *pisphereHit = isphere;
      }
   }
   
   if (*pisphereHit == -1)
      return fFalse;
      
   *pposResult = posStart;
   AddScaleVec(posStart, zBest, normalMove, pposResult);
   
   return fTrue;
}

VecUtil.cpp
#include "Spheres.h"
#include "VecUtil.h"

#include <math.h>
#include <stdlib.h>

enum {
   fFalse = 0,
   fTrue = 1
};

// disable asserts
#define Assert(f)

// math utilities

double GRandInRange(double gLow, double gHigh)
{
   double g;
   
   g = gLow + rand() * (gHigh - gLow) / RAND_MAX;
   Assert(gLow <= g && g <= gHigh);
   return g;
}

// return a long's worth of randomness
long LRand()
{
   long lw;
   
   Assert(RAND_MAX > 256);
   
   lw = 0;
   for (int ib = 0; ib < sizeof(long); ++ib)
      lw = (lw << 8) + (rand() & 0xFF);
   return lw;
}

// vector utilities

void SubVec(const Position &pos1, const Position &pos2, 
      Position *pposResult)
{
   for (int i = 0; i < 3; ++i)
      pposResult->coordinate[i] = pos1.coordinate[i] - 
                                                         pos2.coordinate[i];
}

double DotVec(const Position &pos1, const Position &pos2)
{
   double g = 0.0;
   for (int i = 0; i < 3; ++i)
      g += pos1.coordinate[i] * pos2.coordinate[i];
   return g;
}

double LengthVec(const Position &pos)
{
   return sqrt(DotVec(pos, pos));
}

void ScaleVec(double g, const Position &pos, 
   Position *pposResult)
{
   for (int i = 0; i < 3; ++i)
      pposResult->coordinate[i] = g * pos.coordinate[i];
}

void AddScaleVec(const Position &posBase, double g, 
   const Position &posAdd, Position *pposResult)
{
   for (int i = 0; i < 3; ++i)
      pposResult->coordinate[i] = posBase.coordinate[i] + 
                                                g * posAdd.coordinate[i];
}

void LinearComboVec(double g1, const Position &pos1, double g2, 
      const Position &pos2, Position *pposResult)
{
   for (int i = 0; i < 3; ++i)
      pposResult->coordinate[i] = g1 * pos1.coordinate[i] + 
                                                g2 * pos2.coordinate[i];
}

// project "vec" onto a "normal" vector
void ProjectVec(const Position &vec, const Position &normal, 
      Position *pvecResult)
{
   ScaleVec(DotVec(vec, normal), normal, pvecResult);
}

// sphere stuff

void ComputeVol(
   const long csphere,
   const long *paisphere,
   const double aradius[],
   const Position apos[],
   double base,
   double *pvol)
{
   Position posMin, posMax;
   long index;
   int icoord;
   double radius;
   const Position * ppos;

   posMin = posMax = apos[0];

   for (index = 0; index < csphere; ++index)
   {
      long isphere;
      
      isphere = paisphere == NULL ? index : paisphere[index];
      ppos = &apos[isphere];
      radius = aradius[isphere];
      for (icoord = 0; icoord < 3; ++icoord)
      {
         if (ppos->coordinate[icoord] - radius < 
                  posMin.coordinate[icoord])
            posMin.coordinate[icoord] = ppos->coordinate[icoord] - 
                                                            radius;

         if (ppos->coordinate[icoord] + radius > 
                  posMax.coordinate[icoord])
            posMax.coordinate[icoord] = ppos->coordinate[icoord] + 
                                                            radius;
      }
   }

   *pvol = 1.0;
   
   for (icoord = 0; icoord < 3; ++icoord)
   {
      Assert(posMin.coordinate[icoord] >= -epsilon);
   Assert(base == 0 || posMax.coordinate[icoord] <= base+epsilon 
                                       || icoord == 2);
      *pvol *= posMax.coordinate[icoord] - 
                        posMin.coordinate[icoord];
   }
}

Spheres.h

#if defined(__cplusplus)
extern "C" {
#endif

typedef struct Position {
  double coordinate[3];  /* coordinate[0]==X position, [1]==Y, [2]==Z */
} Position;

void PackSpheres(
  long numSpheres,       /* input: number of spheres to pack */
  double radius[],       /* input: radius of each of numSpheres spheres */
  Position location[]    /* output: location of center of each sphere */
);

#if defined (__cplusplus)
}
#endif

VecUtil.h
// error tolerance

const double epsilon (1.0e-10);

// math utilities

double GRandInRange(double gLow, double gHigh);
long LRand();

// vector utilities

void SubVec(const Position &pos1, const Position &pos2, 
   Position *pposResult);
double DotVec(const Position &pos1, const Position &pos2);
double LengthVec(const Position &pos);
void ScaleVec(double g, const Position &pos, 
   Position *pposResult);
void AddScaleVec(const Position &posBase, double g, 
   const Position &posAdd, Position *pposResult);
void ProjectVec(const Position &vec, const Position &normal, 
   Position *pvecResult);
void LinearComboVec(double g1, const Position &pos1, 
   double g2, const Position &pos2, Position *pposResult);

// sphere stuff

void ComputeVol(
   const long csphere,
   const long *paisphere,
   const double aradius[],
   const Position apos[],
   double base,
   double *pvol);
 

Community Search:
MacTech Search:

Software Updates via MacUpdate

Remotix 3.1.4 - Access all your computer...
Remotix is a fast and powerful application to easily access multiple Macs (and PCs) from your own Mac. Features Complete Apple Screen Sharing support - including Mac OS X login, clipboard... Read more
DesktopLyrics 2.6.6 - Displays current i...
DesktopLyrics is an application that displays the lyrics of the song currently playing in "iTunes" right on your desktop. The lyrics for the song have to be set in iTunes; DesktopLyrics does nothing... Read more
VOX 2.5.1 - Music player that supports m...
VOX is a beautiful music player that supports many filetypes. The beauty is in its simplicity, yet behind the minimal exterior lies a powerful music player with a ton of features and support for all... Read more
NetNewsWire 4.0.0 - RSS and Atom news re...
NetNewsWire is the best way to keep up with the sites and authors you read most regularly. Let NetNewsWire pull down the latest articles, and read them in a distraction-free and Mac-like way. Native... Read more
MacUpdate Desktop 6.0.6 - Search and ins...
MacUpdate Desktop 6 brings seamless 1-click installs and version updates to your Mac. With a free MacUpdate account and MacUpdate Desktop 6, Mac users can now install almost any Mac app on macupdate.... Read more
ForkLift 2.6.5 - Powerful file manager:...
ForkLift is a powerful file manager and ferociously fast FTP client clothed in a clean and versatile UI that offers the combination of absolute simplicity and raw power expected from a well-executed... Read more
Drive Genius 4.1.0 - Powerful system uti...
Drive Genius 4 gives you faster performance from your Mac while also protecting it. The award-winning and improved DrivePulse feature alerts you to hard drive issues before they become major problems... Read more
OnyX 2.9.7 - Maintenance and optimizatio...
OnyX is a multifunctional utility for OS X. It allows you to verify the startup disk and the structure of its System files, to run miscellaneous tasks of system maintenance, to configure the hidden... Read more
DEVONthink Pro 2.8.5 - Knowledge base, i...
DEVONthink Pro is your essential assistant for today's world, where almost everything is digital. From shopping receipts to important research papers, your life often fills your hard drive in the... Read more
Backblaze 4.0.1.878 - Online backup serv...
Backblaze is an online backup service designed from the ground-up for the Mac.With unlimited storage available for $5 per month, as well as a free 15-day trial, peace of mind is within reach with... Read more

Carbo - Handwriting in the Digital Age...
Carbo - Handwriting in the Digital Age 1.0 Device: iOS Universal Category: Productivity Price: $3.99, Version: 1.0 (iTunes) Description: | Read more »
The Apple Watch isn't Great as a Fi...
| Read more »
Show the World What You See With Stre.am...
Live broadcasting is getting popular on mobile devices, which is why you can now get Stre.am, by Infinite Takes. [Read more] | Read more »
PhotoTime's 2.1 Update Adds Apple W...
The latest PhotoTime update is adding even more functionality to the handy photo organizing app. Yep, including Apple Watch support. [Read more] | Read more »
Oh My Glob! Adventure Time Puzzle Quest...
Finn and Jake are taking over D3 Go!'s popular puzzle game series in the upcoming Adventure Time Puzzle Quest. [Read more] | Read more »
Earthcore: Shattered Elements - Tips, Tr...
At first glance, Earthcore: Shattered Elements seems like a rather simple card-battling game. Once you’re introduced to skills that will change quite a bit. Even more so once you start to acquire hero cards. But it’s not so complicated that we... | Read more »
Dungeon999F (Games)
Dungeon999F 1.33 Device: iOS Universal Category: Games Price: $.99, Version: 1.33 (iTunes) Description: "The game you must play at least once in your life!" "The game with potential of million downloads globally!" ...is what the... | Read more »
Mixels Rush - Use Mixes, Maxes and Murps...
Mixels Rush - Use Mixes, Maxes and Murps to Outrun the Nixels 1.0.0 Device: iOS Universal Category: Games Price: $2.99, Version: 1.0.0 (iTunes) Description: Use awesome new Mixels and make crazy combinations to beat the annoying... | Read more »
Battles of the Ancient World II (Games)
Battles of the Ancient World II 1.0 Device: iOS Universal Category: Games Price: $2.99, Version: 1.0 (iTunes) Description: | Read more »
Sword Of Xolan (Games)
Sword Of Xolan 1.0.4 Device: iOS Universal Category: Games Price: $.99, Version: 1.0.4 (iTunes) Description: Sword of Xolan is an action platformer game that includes the juice of pixel art style. Xolan is a young and brave man who... | Read more »

Price Scanner via MacPrices.net

Sale! 16GB iPad mini 3 for $349, save $50
B&H Photo has the 16GB iPad mini 3 WiFi on sale for $349 including free shipping plus NY sales tax only. Their price is $50 off MSRP, and it’s the lowest price available for this model. Read more
Price drop on 2014 15-inch Retina MacBook Pro...
B&H Photo has dropped prices on 2014 15″ Retina MacBook Pros by $200. Shipping is free, and B&H charges NY sales tax only: - 15″ 2.2GHz Retina MacBook Pro: $1799.99 save $200 - 15″ 2.5GHz... Read more
Will iOS 9 Finally Bring Productivity Friendl...
Ah, the irony. From its original announcement in 2010, Apple has doggedly insisted that the iPad remain “simple,” thus arbitrarily limiting its considerable potential as a content creation and... Read more
13-inch 2.5GHz MacBook Pro (refurbished) avai...
The Apple Store has Apple Certified Refurbished 13″ 2.5GHz MacBook Pros available for $829, or $270 off the cost of new models. Apple’s one-year warranty is standard, and shipping is free: - 13″ 2.... Read more
Mac Pros on sale for up to $260 off MSRP
B&H Photo has Mac Pros on sale for up to $260 off MSRP. Shipping is free, and B&H charges sales tax in NY only: - 3.7GHz 4-core Mac Pro: $2799, $200 off MSRP - 3.5GHz 6-core Mac Pro: $3719.99... Read more
Save up to $600 with Apple refurbished Mac Pr...
The Apple Store has Apple Certified Refurbished Mac Pros available for up to $600 off the cost of new models. An Apple one-year warranty is included with each Mac Pro, and shipping is free. The... Read more
Apple drops price on refurbished 15-inch 2.2G...
The Apple Store has dropped their price on the Apple Certified Refurbished 2014 15″ 2.2GHz Retina MacBook Pro to $1609, $390 off original MSRP. Apple’s one-year warranty is included, and shipping is... Read more
Apple restocks refurbished Mac minis for up t...
The Apple Store has restocked Apple Certified Refurbished 2014 Mac minis, with models available starting at $419. Apple’s one-year warranty is included with each mini, and shipping is free: - 1.4GHz... Read more
Cover For iPhone Helps You Create Beautiful P...
Taking photos, editing and sharing them is one of the things most of people do on their iPhones for fun, especially if they’re users of social media. Unfortunately, many iOS photo-editing solutions... Read more
Password Keyboard 1.0 For iOS Handles Logins...
Pfaeffikon, Switzerland based Power APP has introduced Password Keyboard 1.0.0, their new password utility developed for iPhone, iPad and iPod touch devices. Password Keyboard is designed to simplify... Read more

Jobs Board

Project Manager / Business Analyst, WW *Appl...
…a senior project manager / business analyst to work within our Worldwide Apple Fulfillment Operations and the Business Process Re-engineering team. This role will work Read more
Visual Merchandise Manager %u2013 Accessories...
…to be part of an incredible team. Imagine what you could do here. At Apple , great ideas have a way of becoming reality very quickly. Our Accessories assortment makes Read more
*Apple* Solutions Consultant - Retail Sales...
**Job Summary** As an Apple Solutions Consultant (ASC) you are the link between our customers and our products. Your role is to drive the Apple business in a retail Read more
*Apple* Solutions Consultant - Retail Sales...
**Job Summary** The ASC is an Apple employee who serves as the Apple business manager and influencer in a hyper-business critical Reseller's store which delivers Read more
Technical Project Manager - *Apple* Pay - A...
**Job Summary** Apple Pay is seeking an experienced technical PM…manage the rollout of features to merchants for the Apple Pay platform in the US Within this role Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.