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
$102.23
Apple Inc.
+2.47
MSFT
$44.61
Microsoft Corpora
+0.53
GOOG
$520.91
Google Inc.
+0.07

MacTech Search:
Community Search:

Software Updates via MacUpdate

NTFS 12.0.39 - 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
RestoreMeNot 2.0.3 - Disable window rest...
RestoreMeNot provides a simple way to disable the window restoration for individual applications so that you can fine-tune this behavior to suit your needs. Please note that RestoreMeNot is designed... Read more
Macgo Blu-ray Player 2.10.9.1750 - Blu-r...
Macgo Mac Blu-ray Player can bring you the most unforgettable Blu-ray experience on your Mac. Overview Macgo Mac Blu-ray Player can satisfy just about every need you could possibly have in a Blu-ray... Read more
Apple iOS 8.1 - The latest version of Ap...
The latest version of iOS can be downloaded through iTunes. Apple iOS 8 comes with big updates to apps you use every day, like Messages and Photos. A whole new way to share content with your family.... Read more
TechTool Pro 7.0.5 - Hard drive and syst...
TechTool Pro is now 7, and this is the most advanced version of the acclaimed Macintosh troubleshooting utility created in its 20-year history. Micromat has redeveloped TechTool Pro 7 to be fully 64... Read more
PDFKey Pro 4.0.2 - Edit and print passwo...
PDFKey Pro can unlock PDF documents protected for printing and copying when you've forgotten your password. It can now also protect your PDF files with a password to prevent unauthorized access and/... Read more
Yasu 2.9.1 - System maintenance app; per...
Yasu was originally created with System Administrators who service large groups of workstations in mind, Yasu (Yet Another System Utility) was made to do a specific group of maintenance tasks... Read more
Hazel 3.3 - Create rules for organizing...
Hazel is your personal housekeeper, organizing and cleaning folders based on rules you define. Hazel can also manage your trash and uninstall your applications. Organize your files using a... Read more
Autopano Giga 3.7 - Stitch multiple imag...
Autopano Giga allows you to stitch 2, 20, or 2,000 images. Version 3.0 integrates impressive new features that will definitely make you adopt Autopano Pro or Autopano Giga: Choose between 9... Read more
MenuMeters 1.8 - CPU, memory, disk, and...
MenuMeters is a set of CPU, memory, disk, and network monitoring tools for Mac OS X. Although there are numerous other programs which do the same thing, none had quite the feature set I was looking... Read more

Latest Forum Discussions

See All

Clips Review
Clips Review By Jennifer Allen on October 21st, 2014 Our Rating: :: CONVENIENT PASTINGUniversal App - Designed for iPhone and iPad Making copying and pasting more powerful than usual, Clips is a great way to move stuff around.   | Read more »
MonSense Review
MonSense Review By Jennifer Allen on October 21st, 2014 Our Rating: :: ORGANIZED FINANCESiPhone App - Designed for the iPhone, compatible with the iPad Organize your finances with the quick and easy to use, MonSense.   | Read more »
This Week at 148Apps: October 13-17, 201...
Expert App Reviewers   So little time and so very many apps. What’s a poor iPhone/iPad lover to do? Fortunately, 148Apps is here to give you the rundown on the latest and greatest releases. And we even have a tremendous back catalog of reviews; just... | Read more »
Angry Birds Transformers Review
Angry Birds Transformers Review By Jennifer Allen on October 20th, 2014 Our Rating: :: TRANSFORMED BIRDSUniversal App - Designed for iPhone and iPad Transformed in a way you wouldn’t expect, Angry Birds Transformers is a quite... | Read more »
GAMEVIL Announces the Upcoming Launch of...
GAMEVIL Announces the Upcoming Launch of Mark of the Dragon Posted by Jessica Fisher on October 20th, 2014 [ permalink ] Mark of the Dragon, by GAMEVIL, put | Read more »
Interview With the Angry Birds Transform...
Angry Birds Transformers recently transformed and rolled out worldwide. This run-and-gun title is a hit with young Transformers fans, but the ample references to classic Transformers fandom has also earned it a place in the hearts of long-time... | Read more »
Hail to the King: Deathbat Review
Hail to the King: Deathbat Review By Rob Thomas on October 20th, 2014 Our Rating: :: SO FAR AWAYUniversal App - Designed for iPhone and iPad Hail to the King: Deathbat may feel like “Coming Home” for Avenged Sevenfold’s faithful,... | Read more »
Find Free Food on Campus with Ypay
Find Free Food on Campus with Ypay Posted by Jessica Fisher on October 20th, 2014 [ permalink ] iPhone App - Designed for the iPhone, compatible with the iPad | Read more »
Strung Along Review
Strung Along Review By Jordan Minor on October 20th, 2014 Our Rating: :: GOT NO STRINGSUniversal App - Designed for iPhone and iPad A cool gimmick and a great art style keep Strung Along from completely falling apart.   | Read more »
P2P file transferring app Send Anywhere...
File sharing services like Dropbox have security issues. Email attachments can be problematic when it comes to sharing large files. USB dongles don’t fit into your phone. Send Anywhere, a peer-to-peer file transferring application, solves all of... | Read more »

Price Scanner via MacPrices.net

Strong iPhone, Mac And App Store Sales Drive...
Apple on Monday announced financial results for its fiscal 2014 fourth quarter ended September 27, 2014. The Company posted quarterly revenue of $42.1 billion and quarterly net profit of $8.5 billion... Read more
Apple Posts How-To For OS X Recovery
OS X 10.7 Lion and later include OS X Recovery. This feature includes all of the tools you need to reinstall OS X, repair your disk, and even restore from a Time Machine backup. OS X Recovery... Read more
Mac OS X Versions (Builds) Supported By Vario...
Apple Support has posted a handy resource explaining which Mac OS X versions (builds) originally shipped with or are available for your computer via retail discs, downloads, or Software Update. Apple... Read more
Deals on 2011 13-inch MacBook Airs, from $649
Daily Steals has the Mid-2011 13″ 1.7GHz i5 MacBook Air (4GB/128GB) available for $699 with a 90 day warranty. The Mid-2011 13″ 1.7GHz i5 MacBook Air (4GB/128GB SSD) is available for $649 at Other... Read more
2013 15-inch 2.0GHz Retina MacBook Pro availa...
B&H Photo has leftover previous-generation 15″ 2.0GHz Retina MacBook Pros now available for $1599 including free shipping plus NY sales tax only. Their price is $400 off original MSRP. B&H... Read more
Updated iPad Prices
We’ve updated our iPad Air Price Tracker and our iPad mini Price Tracker with the latest information on prices and availability from Apple and other resellers, including the new iPad Air 2 and the... Read more
Apple Pay Available to Millions of Visa Cardh...
Visa Inc. brings secure, convenient payments to iPad Air 2 and iPad mini 3as well as iPhone 6 and 6 Plus. Starting October 20th, eligible Visa cardholders in the U.S. will be able to use Apple Pay,... Read more
Textkraft Pocket – the missing TextEdit for i...
infovole GmbH has announced the release and immediate availability of Textkraft Pocket 1.0, a professional text editor and note taking app for Apple’s iPhone. In March 2014 rumors were all about... Read more
C Spire to offer iPad Air 2 and iPad mini 3,...
C Spire on Friday announced that it will offer iPad Air 2 and iPad mini 3, both with Wi-Fi + Cellular, on its 4G+ LTE network in the coming weeks. C Spire will offer the new iPads with a range of... Read more
Belkin Announces Full Line of Keyboards and C...
Belkin International has unveiled a new lineup of keyboard cases and accessories for Apple’s newest iPads, featuring three QODE keyboards and a collection of thin, lightweight folios for both the... Read more

Jobs Board

*Apple* Solutions Consultant (ASC) - Apple (...
**Job Summary** The ASC is an Apple employee who serves as an Apple brand ambassador and influencer in a Reseller's store. The ASC's role is to grow Apple Read more
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
*Apple* Retail - Multiple Positions (US) - A...
Job Description: Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more
Position Opening at *Apple* - Apple (United...
…customers purchase our products, you're the one who helps them get more out of their new Apple technology. Your day in the Apple Store is filled with a range of Read more
Position Opening at *Apple* - Apple (United...
**Job Summary** At the Apple Store, you connect business professionals and entrepreneurs with the tools they need in order to put Apple solutions to work in their Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.