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);
 
AAPL
$100.86
Apple Inc.
-0.77
MSFT
$46.76
Microsoft Corpora
+0.52
GOOG
$579.95
Google Inc.
+6.85

MacTech Search:
Community Search:

Software Updates via MacUpdate

Capture One Pro 8.0.0.433 - RAW workflow...
Capture One Pro 8 is a professional RAW converter offering you ultimate image quality with accurate colors and incredible detail from more than 300 high-end cameras -- straight out of the box. It... Read more
Adobe Acrobat Pro 11.0.09 - Powerful PDF...
Adobe Acrobat allows users to communicate and collaborate more effectively and securely. Unify a wide range of content in a single organized PDF Portfolio. Collaborate through electronic document... Read more
Adobe Reader 11.0.09 - View PDF document...
Adobe 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
iFFmpeg 4.6.1 - Convert multimedia files...
iFFmpeg is a graphical front-end for FFmpeg, a command-line tool used to convert multimedia files between formats. The command line instructions can be very hard to master/understand, so iFFmpeg does... Read more
NTFS 11.3.62 - Provides full read and wr...
Paragon NTFS breaks down the barriers between Windows and OS X. Paragon NTFS effectively solves the communication problems between the Mac system and NTFS, providing full read and write access to... Read more
OS X Yosemite 10.10 DP8 - Developer Prev...
Note: This is a Developer Preview. You must be a registered Apple Mac Developer to download this update. You can also sign up for the free OS X Beta Program to download and preview public beta... Read more
FotoMagico 4.5 - Powerful slideshow crea...
FotoMagico lets you create professional slideshows from your photos and music with just a few, simple mouse clicks. It sports a very clean and intuitive yet powerful user interface. High image... Read more
Screenshot Path 1.2.1 - Change the defau...
Screenshot Path lets you change the folder where OS X saves screenshots. Screenshots are saved by default to the user’s desktop. This is handy for the occasional screenshot but those looking to take... Read more
Fantastical 1.3.16 - Create calendar eve...
Fantastical is the Mac calendar you'll actually enjoy using. Creating an event with Fantastical is quick, easy, and fun: Open Fantastical with a single click or keystroke Type in your event details... Read more
GIMP 2.8.14 - Powerful, free image editi...
GIMP is a multi-platform photo manipulation tool. GIMP is an acronym for GNU Image Manipulation Program. The GIMP is suitable for a variety of image manipulation tasks, including photo retouching,... Read more

Latest Forum Discussions

See All

Zynga Unveils First Look at Looney Tunes...
Zynga Unveils First Look at Looney Tunes Dash! Posted by Ellis Spice on September 16th, 2014 [ permalink ] Be vewy, vewy qwiet. I’m hunting wunners featuwing wabbits, and luckily, it’s duck season wunner season. | Read more »
Guardians on the Go – How to Connect to...
At this point I’m relatively certain most of the country is familiar with Destiny. And for those looking to augment their game a bit, Bungie has changed their Bungie Mobile Companion App to Destiny Companion. | Read more »
Kunin Review
Kunin Review By Jordan Minor on September 16th, 2014 Our Rating: :: SHURIKEN DO ITUniversal App - Designed for iPhone and iPad Kunin is a nervous test of ninja reflexes.   Developer: Dodreams Price: FREE Version Reviewed: 1.0.1... | Read more »
4 KEMCO Titles are Just $0.99 for a Limi...
4 KEMCO Titles are Just $0.99 for a Limited Time Posted by Jessica Fisher on September 16th, 2014 [ permalink ] KEMCO RPGs Destiny Fantasia, Infinite Dunamis, Bonds of the Sk | Read more »
Introducing Flash, the Latest Wearable F...
Introducing Flash, the Latest Wearable Fitness Monitor from Misfit Posted by Jessica Fisher on September 16th, 2014 [ permalink ] The Misfit Flash is the newly-released fitness and sleep monitor from | Read more »
Hyper Trip Review
Hyper Trip Review By Jennifer Allen on September 16th, 2014 Our Rating: :: HYPER TWITCHYUniversal App - Designed for iPhone and iPad Tough and unforgiving, Hyper Trip is a bit like Snake – if Snake was really harsh.   | Read more »
Collectible Card Game Earthcore: Shatter...
Collectible Card Game Earthcore: Shattered Elements is Set to Arrive on iOS in 2015 Posted by Ellis Spice on September 16th, 2014 [ permalink ] Polish developers | Read more »
Boogey Boy Review
Boogey Boy Review By Jennifer Allen on September 16th, 2014 Our Rating: :: PRETTY BUT BASICUniversal App - Designed for iPhone and iPad It looks delightful but lack of Game Center support and more variety really affects the fun... | Read more »
Vizzywig 4K (Photography)
Vizzywig 4K 1.0 Device: iOS iPhone Category: Photography Price: $999.99, Version: 1.0 (iTunes) Description: REQUIRES: iOS 7 on iPhone 5S with 32GB or 64GB. (Do not use iOS 8)The world's FIRST mobile 4K video capture, editing and... | Read more »
The Sleeping Prince Review
The Sleeping Prince Review By Jennifer Allen on September 15th, 2014 Our Rating: :: RESTRICTIVE KINGDOM SAVINGUniversal App - Designed for iPhone and iPad The Sleeping Prince looks and feels great to play, but its lack of peril and... | Read more »

Price Scanner via MacPrices.net

Sprint offers 16GB iPad mini for $199.99 with...
Sprint is offering 1st generation 16GB iPad minis for $199.99 with a 2-year service agreement. Standard MSRP for this iPad is $429. Their price is the lowest available for this model. Read more
2.5GHz Mac mini remains on sale for $549, sav...
B&H Photo has the 2.5GHz Mac mini on sale for $549.99 including free shipping. That’s $50 off MSRP, and B&H will also include a free copy of Parallels Desktop software. NY sales tax only. Read more
Apple refurbished iMacs available for up to $...
The Apple Store has Apple Certified Refurbished iMacs available for up to $300 off the cost of new models. Apple’s one-year warranty is standard, and shipping is free. These are the best prices on... Read more
13″ 2.5GHz MacBook Pro offered for $100 off M...
B&H Photo has the 13″ 2.5GHz MacBook Pro on sale for $999.99 including free shipping plus NY sales tax only. Their price is $100 off MSRP. Read more
Free GIMP Professional Grade Graphics App Ver...
The latest 2.8.14 version of the oddly-named GIMP (acronym for: GNU Image Manipulation Program) open source, high-end image editing and creation alternative to Adobe’s Photoshop and refuge from... Read more
Apple Announces Record Pre-orders for iPhone...
Apple has released metrics showing a record number of first day pre-orders of iPhone 6 and iPhone 6 Plus, with over four million sold in the first 24 hours. Demand for the new iPhones exceeds the... Read more
10% off iPhone 6 and 6 Plus Otterbox cases
Get 10% off on popular Otterbox iPhone 6 and iPhone 6 Plus cases at MacMall through September 19th. Use code OTTERBOX10 to see the discount. Read more
15-inch MacBook Pros on sale for up to $125 o...
Amazon has the new 2014 15″ Retina MacBook Pros on sale for up to $125 off MSRP including free shipping: - 15″ 2.2GHz Retina MacBook Pro: $1899.99 save $100 - 15″ 2.5GHz Retina MacBook Pro: $2374... Read more
27-inch 3.2GHz iMac on sale for $1698, $101 o...
Abt has the 27″ 3.2GHz iMac on sale for $1698 including free shipping. Their price is $101 off MSRP. Read more
More To Making A Larger iPad Than Expanded Sc...
CNET’s Ross Rubin has posted a thoughtful analysis of prospects for a larger display iPad Pro, noting that Microsoft and Samsung currently have the large-display touchscreen tablet category to... Read more

Jobs Board

*Apple* Retail - Multiple Positions (US) - A...
Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, you're also the Read more
*Apple* Retail - Multiple Positions (US) - A...
Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, you're also the Read more
*Apple* Retail - Multiple Positions (US) - A...
Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, you're also the Read more
*Apple* Retail - Multiple Positions (US) - A...
Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, you're also the Read more
*Apple* Retail - Multiple Positions (US) - A...
Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, you're also the Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.