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

Google Earth 7.1.7.2602 - View and contr...
Google Earth gives you a wealth of imagery and geographic information. Explore destinations like Maui and Paris, or browse content from Wikipedia, National Geographic, and more. Google Earth combines... Read more
ClamXav 2.10 - Virus checker based on Cl...
ClamXav is a popular virus checker for OS X. I have been working on ClamXav for more than 10 years now, and over those years, I have invested a huge amount of my own time and energy into bringing... Read more
Tweetbot 2.4.4 - Popular Twitter client.
Tweetbot is a full-featured OS X Twitter client with a lot of personality. Whether it's the meticulously-crafted interface, sounds and animation, or features like multiple timelines and column views... Read more
Sierra Cache Cleaner 11.0.1 - Clear cach...
Sierra Cache Cleaner is an award-winning general purpose tool for macOS X. SCC makes system maintenance simple with an easy point-and-click interface to many macOS X functions. Novice and expert... Read more
Things 2.8.8 - Elegant personal task man...
Things is a task management solution that helps to organize your tasks in an elegant and intuitive way. Things combines powerful features with simplicity through the use of tags and its intelligent... Read more
Remotix 4.1 - Access all your computers...
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
Airfoil 5.1.2 - Send audio from any app...
Airfoil allows you to send any audio to AirPort Express units, Apple TVs, and even other Macs and PCs, all in sync! It's your audio - everywhere. With Airfoil you can take audio from any... Read more
Firefox 49.0.1 - Fast, safe Web browser.
Firefox offers a fast, safe Web browsing experience. Browse quickly, securely, and effortlessly. With its industry-leading features, Firefox is the choice of Web development professionals and casual... Read more
Default Folder X 5.0.7 - Enhances Open a...
Default Folder X attaches a toolbar to the right side of the Open and Save dialogs in any OS X-native application. The toolbar gives you fast access to various folders and commands. You just click on... Read more
Safari Technology Preview 10.1 - The new...
Safari Technology Preview contains the most recent additions and improvements to WebKit and the latest advances in Safari web technologies. And once installed, you will receive notifications of... Read more

4 games like Burly Men at Sea to inspire...
Burly Men at Sea is out today and it looks a treat. It tells the tale of three Scandinavian fishermen who leave the humdrum of their daily lives to go exploring. It’s a beautiful folksy story that unfurls as you interact with the environment... | Read more »
3 reasons you need to play Kingdom: New...
Developed by a tag team of indie developers - Thomas "Noio" van den Berg and Marco "Licorice" Bancale - Kingdom is a vibrant medieval fantasy adventure that casts players as a king or queen who must expand their empire by exploring the vasts lands... | Read more »
JoyCity have launched a brand new King o...
Great news for all of you Game of Dice fans out there - JoyCity have just released a brand new limited edition pack with a really cool twist. The premise of Game of Dice is fairly straightforward, asking you to roll dice to navigate your way around... | Read more »
Burly Men at Sea (Games)
Burly Men at Sea 1.0 Device: iOS Universal Category: Games Price: $4.99, Version: 1.0 (iTunes) Description: Burly Men at Sea is a folktale about a trio of large, bearded fishermen who step away from the ordinary to seek adventure. | Read more »
3 tips for catching the gnarliest waves...
Like a wave breaking on the shore, Tidal Rider swept its way onto the App Store charts this week settling firmly in the top 10. It’s a one-touch high score-chaser in which you pull surfing stunts while dodging seagulls and collecting coins. The... | Read more »
The beginner's guide to destroying...
Age of Heroes: Conquest is 5th Planet Games’ all new turn-based multiplayer RPG, full of fantasy exploration, guild building, and treasure hunting. It’s pretty user-friendly as far as these games go, but when you really get down to it, you’ll find... | Read more »
Infinite Tanks (Games)
Infinite Tanks 1.0.0 Device: iOS Universal Category: Games Price: $4.99, Version: 1.0.0 (iTunes) Description: | Read more »
Agatha Christie - The ABC Murders (FULL)...
Agatha Christie - The ABC Murders (FULL) 1.0 Device: iOS Universal Category: Games Price: $6.99, Version: 1.0 (iTunes) Description: Agatha Christie: The ABC Murders Your weapon is your knowledge. Your wits will be put to the ultimate... | Read more »
HeadlessD (Games)
HeadlessD 1.0 Device: iOS Universal Category: Games Price: $.99, Version: 1.0 (iTunes) Description: HeadlessD is hand-painted dungeon crawler with intuitive touch controls and NO in-app purchases. | Read more »
Leaf for Twitter (Social Networking)
Leaf for Twitter 1.0.1 Device: iOS iPhone Category: Social Networking Price: $4.99, Version: 1.0.1 (iTunes) Description: | Read more »

Price Scanner via MacPrices.net

Apple refurbished 2015 13-inch MacBook Airs a...
Apple has Certified Refurbished 2015 13″ MacBook Airs available starting at $759. An Apple one-year warranty is included with each MacBook, and shipping is free: - 2015 13″ 1.6GHz/4GB/128GB MacBook... Read more
MacBook Airs on sale for up to $100 off MSRP
B&H Photo has 13″ and 11″ MacBook Airs on sale for up to $100 off MSRP. Shipping is free, and B&H charges NY sales tax only: - 11″ 1.6GHz/128GB MacBook Air: $799 $100 MSRP - 11″ 1.6GHz/256GB... Read more
Apple refurbished 12-inch 128GB iPad Pros ava...
Apple has Certified Refurbished 12″ Apple iPad Pros available for up to $160 off the cost of new iPads. An Apple one-year warranty is included with each model, and shipping is free: - 32GB 12″ iPad... Read more
Phone2Action Unveils New Voter Turnout Techno...
Phone2Action, a leading digital advocacy platform, today launched its Tech to Vote Civic Action Center digital advocacy and communications platform on National Voter Registration Day September 27.... Read more
Apple & Deloitte Team Up to Help Business...
Apple and international professional services firm Deloitte have announced a partnership to help companies quickly and easily transform their workflow dynamics by maximizing the power, ease-of-use,... Read more
Chop Commute – See Traffic and Drive Times on...
Shrewsbury, Massachusetts based Indie developer, InchWest has released Chop Commute 1.61, a Mac app that takes the guesswork out of daily commute by showing real-time traffic and drive times right on... Read more
12-inch 32GB WiFi iPad Pros on sale for $50 o...
B&H Photo has 12″ 32GB WiFi Apple iPad Pros on sale for $50 off MSRP, each including free shipping. B&H charges sales tax in NY only: - 12″ Space Gray 32GB WiFi iPad Pro: $749 $50 off MSRP -... Read more
Recent price drops on refurbished iPad minis...
Apple recently dropped prices on several Certified Refurbished iPad mini 4s and 2s as well as iPad Air 2s. An Apple one-year warranty is included with each model, and shipping is free: - 16GB iPad... Read more
Apple refurbished Mac minis available startin...
Apple has Certified Refurbished Mac minis available starting at $419. Apple’s one-year warranty is included with each mini, and shipping is free: - 1.4GHz Mac mini: $419 $80 off MSRP - 2.6GHz Mac... Read more
13-inch 2.5GHz MacBook Pro available for $928...
Overstock has the 13″ 2.5GHz MacBook Pro available for $927.99 including free shipping. Their price is $171 off MSRP. 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
Restaurant Manager (Neighborhood Captain) - A...
…in every aspect of daily operation. WHY YOU'LL LIKE IT: You'll be the Big Apple . You'll solve problems. You'll get to show your ability to handle the stress and Read more
*Apple* Wireless Lead - T-ROC - The Retail O...
…wealth of knowledge in wireless sales and activations to the Beautiful and NEW APPLE Experience store within MACYS.. THIS role, APPLE Wireless Lead, isbrandnewas Read more
Lead *Apple* Advocate - T-ROC - The Retail...
…Company, is proud of its unprecedented relationship with our partner and client, APPLE ,in bringing amazing" APPLE ADVOCATES"to "non" Apple store locations. Read more
*Apple* Advocate - T-ROC - The Retail Outsou...
…Company, is proud of its unprecedented relationship with our partner and client, APPLE ,in bringing amazing" APPLE ADVOCATES"to "non" Apple store locations. Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.