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
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) 65.3 142.3 10107.2 5796 372 A. D. * * * 820 104

### 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 place 20 points 2nd place 10 points 3rd place 7 points 4th place 4 points 5th place 2 points finding bug 2 points suggesting Challenge 2 points

Here is Tom's winning Sphere Packing solution:

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

PackSpheres
void PackSpheres(
long csphere,        /* input: number of spheres to pack */
Position aposBest[]  /* output: location of center of each sphere */
)
{
int isphere;
double volGuess, vol, volBest;
double base, baseMin, baseMax, baseBest;
ulong tickStart, tickCur;

tickStart = LMGetTicks();
for (isphere = 0, volGuess = 0.0; isphere < csphere; ++isphere)
{

}

Assert(baseMin <= baseMax);

baseBest = baseMin;
baseBest, &volBest);

base = baseMax;
base, &vol);
if (vol < volBest)
{
volBest = vol;
baseBest = base;
}

base = sqrt(baseMin * baseMax);
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)
baseMin, baseBest, baseMax, &base, aposStart);
else
baseMin, baseBest, baseMax, &base, aposStart);

// try the new scenario
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[],
double baseMin,
double baseBest,
double baseMax,
double *pbase,
Position aposStart[])
{
long isphereCur;

*pbase = baseBest;
for (isphereCur = 0; isphereCur < csphere; ++isphereCur)
{
Position *ppos = &aposStart[isphereCur];

aisphere[isphereCur] = isphereCur;
ppos->coordinate[0] =
ppos->coordinate[1] =
ppos->coordinate[2] = csphere * *pbase;
}
}

_TweakStartingPos
static void _TweakStartingPos(
const long csphere,
long aisphere[],
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];

ppos->coordinate[0] =
ppos->coordinate[1] =
ppos->coordinate[2] = csphere * *pbase;
}
}

_DropSpheres
static void _DropSpheres(
const long csphere,
const long *paisphere,
const Position *paposStart,
Position aposEnd[],
double base,
double *pvol)
{
long csphereDone;

for (csphereDone = 0; csphereDone < csphere; ++csphereDone)
{
Position posStart, posLand;
long isphereHit;
long isphereCur;

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

// pick a starting point for the current sphere
if (paposStart == NULL)
{
posStart.coordinate[0] =
posStart.coordinate[1] =
}
else
{
posStart.coordinate[0] =
paposStart[isphereCur].coordinate[0];
posStart.coordinate[1] =
paposStart[isphereCur].coordinate[1];
}

// drop it

// 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);
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 = (radiusCur - posHit.coordinate[icoord]) /
normalMove.coordinate[icoord];
}
else if (normalMove.coordinate[icoord] > epsilon)
{
if (posHit.coordinate[icoord] + distMove *
distMove = (base - radiusCur -
posHit.coordinate[icoord]) /
normalMove.coordinate[icoord];
}
}

Assert(distMove >= distH - epsilon);
if (distMove < distH + epsilon)
break;

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;
break;
case 1:
normalMove = s_normalY;
break;
case 2:
normalMove = s_normalXNeg;
break;
case 3:
normalMove = s_normalYNeg;
break;
}

fHit = _FFindObstruction(
normalMove,
fTrue/*fNear*/,
aposEnd[isphereCur],
csphereDone,
paisphere,
aposEnd,
&posLand,
&isphereHit);

cposStart = 0;
if (!fHit || DotVec(posLand, normalMove) > sEdge)
{
posLand = aposEnd[isphereCur];
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)
{
&isphereHit);

if (posLand.coordinate[2] <
aposEnd[isphereCur].coordinate[2] + epsilon)
{
if (aposEnd[isphereCur].coordinate[2] -
fImproved = fTrue;
aposEnd[isphereCur] = posLand;

}
}
}
}
}

ComputeVol(csphere, NULL, aradius, aposEnd, base, pvol);
}

_DropOneSphere
static void _DropOneSphere(
const Position &posStart,
int csphere,
const long *paisphere,
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,
csphere,
paisphere,
apos,
pposResult,
pisphereHit
);

if (!fHit || pposResult->coordinate[2] < radius)
{
*pisphereHit = -1;
}

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);
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,
int csphere,
const long *paisphere,
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);
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;

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,
{
for (int i = 0; i < 3; ++i)
pposResult->coordinate[i] = posBase.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 Position apos[],
double base,
double *pvol)
{
Position posMin, posMax;
long index;
int icoord;
const Position * ppos;

posMin = posMax = apos[0];

for (index = 0; index < csphere; ++index)
{
long isphere;

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

posMax.coordinate[icoord])
posMax.coordinate[icoord] = ppos->coordinate[icoord] +
}
}

*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 */
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,
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 Position apos[],
double base,
double *pvol);
```

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The addition of multiplayer to The Battle of Polytopia has catapulted the game from a fun enough time waster to a fully-fledged 4X experience on your phone. We've been playing quite a few matches over the past week or so, and we've put together a... | Read more »
All the best games on sale for iPhone an...
Hi there, and welcome to our round up of all the best games that are on sale for iOS at the moment. It's not a vintage week in terms of numbers, but I'm pretty sure that every single one of these is worth picking up if you haven't already played... | Read more »
Disc Drivin' 2 Guide - Tips for the...
We're all still playing quite a bit of Disc Drivin' 2 over here at 148Apps, and we've gotten pretty good at it. Now that we've spent some more time with the game and unlocked more powerups, check out some of these more advanced tips: | Read more »
Alto's Odyssey Guide - How to Tackl...
Alto’s Odyssey is a completely stunning and serene runner, but it can also be a bit tricky. Check out these to try and keep your cool while playing this endless runner: Don’t focus too much on tasks [Read more] | Read more »
Here's everything you need to know...
Alto's Odyssey is a really, really good game. If you don't believe me, you should definitely check out our review by clicking this link right here. It takes the ideas from the original Alto's Adventure, then subtly builds on them, creating... | Read more »
Alto's Odyssey (Games)
Alto's Odyssey 1.0.1 Device: iOS Universal Category: Games Price: \$4.99, Version: 1.0.1 (iTunes) Description: Just beyond the horizon sits a majestic desert, vast and unexplored. Join Alto and his friends and set off on an endless... | Read more »
Vainglory 5v5: Everything you need to kn...

## Price Scanner via MacPrices.net

Use your Apple Education discount and save up...
Purchase a new Mac using Apple’s Education discount, and take up to \$400 off MSRP. All teachers, students, and staff of any educational institution with a .edu email address qualify for the discount... Read more
Apple Canada offers 2017 21″ and 27″ iMacs fo...
Canadian shoppers can save up to \$470 on the purchase of a 2017 current-generation 21″ or 27″ iMac with Certified Refurbished models at Apple Canada. Apple’s refurbished prices are the lowest... Read more
9″ iPads available online at Walmart for \$50...
Walmart has 9.7″ Apple iPads on sale for \$50 off MSRP for a limited time. Sale prices are for online orders only, in-store prices may vary: – 9″ 32GB iPad: \$279.99 \$50 off – 9″ 128GB iPad: \$379.99 \$... Read more
15″ Apple MacBook Pros, Certified Refurbished...
Save \$360-\$420 on the purchase of a 2017 15″ MacBook Pro with Certified Refurbished models at Apple. Apple’s refurbished prices are the lowest available for each model from any reseller. An standard... Read more
Amazon restocks MacBook Pros with models avai...
Amazon has restocked 15″ and 13″ Apple MacBook Pros with models on sale for up to \$251 off MSRP. Shipping is free. Note that stock of some Macs may come and go (and some sell out quickly), so check... Read more
Lowest price of the year: 15″ 2.8GHz Apple Ma...
Amazon has the 2017 Space Gray 15″ 2.8GHz MacBook Pro on sale today for \$251 off MSRP. Shipping is free: – 15″ 2.8GHz Touch Bar MacBook Pro Space Gray (MPTR2LL/A): \$2148, \$251 off MSRP Their price is... Read more
Apple restocks full line of Certified Refurbi...
Apple has restocked a full line of Apple Certified Refurbished 2017 13″ MacBook Pros for \$200-\$300 off MSRP. A standard Apple one-year warranty is included with each MacBook, and shipping is free.... Read more
Lowest sale price available for 13″ 1.8GHz Ma...
Focus Camera has the 2017 13″ 1.8GHz/128GB Apple MacBook Air on sale today for \$829 including free shipping. Their price is \$170 off MSRP, and it’s the lowest price available for a current 13″... Read more
21-inch 2.3GHz iMac on sale for \$999, \$100 of...
B&H Photo has the 2017 21″ 2.3GHz iMac (MMQA2LL/A) in stock and on sale for \$999 including free shipping plus NY & NJ tax only. Their price is \$100 off MSRP. Read more
Apple refurbished Mac minis in stock again st...
Apple has restocked Certified Refurbished Mac minis 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

## Jobs Board

*Apple* Retail - Multiple Positions - Apple,...
Job Description: Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more
*Apple* Retail - Multiple Positions - Apple,...
Job Description:SalesSpecialist - Retail Customer Service and SalesTransform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more
*Apple* Retail - Multiple Positions - Apple,...
Job Description: Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more
*Apple* Solutions Consultant - Apple (United...
# Apple Solutions Consultant Job Number: 113523441 Orange, CA, California, United States Posted: 21-Feb-2018 Weekly Hours: 40.00 **Job Summary** Are you passionate Read more
*Apple* Retail - Multiple Positions - Apple,...
Job Description:SalesSpecialist - Retail Customer Service and SalesTransform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more